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Lancaster University
Department of Environmental Science
JOINT TWO-DIMENSIONAL INVERSION OF GEOELECTROMAGNETIC AND SEISMIC
REFRACTION DATA WITH CROSS-GRADIENTS CONSTRAINT
by Luis Alonso Gallardo
A thesis submitted to LANCASTER UNIVERSITY for the degree of Doctor of Philosophy
Lancaster, England, October 2004
Declaration
This thesis has been composed by me and except where acknowledgment
is made, the work is original. No part of the work presented in this thesis
has been submitted in support of an application for another degree or
qualification of this or any other academic institute.
Luis Alonso Gallardo
Dedication
This work is dedicated to… my wife Angelica who has walked every inch of this journey along side me, sharing dreams and giving me strength during the toughest moments. Erik and Alina trusting they will always remember all those times in that English land where they left part of a joyful childhood while living memorable “big adventures”. my mother Esther Delgado and my late father Francisco Gallardo that fed my curiosity in my childhood projects that shaped my mind to the search of knowledge. the Romero-Pacheco family that always encouraged me to achieve my goals. my friends from CICESE that have always shown me their trust, they are remembered with gratitude.
Acknowledgment
My deepest thank to Dr. Enrique Gómez who introduced me to the basics in electromagnetic theory and whose unswerving support made possible the fulfilment of my aspirations I acknowledge Dr. Max Meju for providing me with this enterprising opportunity and for being supportive of my research. His teaching in the art of scientific writing is most valuable for my future. I am grateful to Dr. Marco A. Perez for his advice, support and for looking after me during my stay away from CICESE. My sincere appreciation to Dr José Frez who sowed my passion for inverse theory and provided me with invaluable skills and tools which have been vital for the physical materialization of my inversion algorithms. I am indebted to Renata Romanowicz for being my counsellor in those crucial stages and for her sincere friendship, and to my dearest friend Adel K. Mohamed for introducing me to the university life while in Leicester. To all those friends in different places, rooms LG506 and B531 in Lancaster and in Leicester University, I thank them for their friendship during my “sporadic” visits to my desk. Special thanks go to Richie Brown, Graham Andrews, Gareth Morris and Lee Burbery for their advice, especially in language matters. I acknowledge SUPERA-ANUIES the financial support for my studies via contract 5167 and CICESE for the permission granted for my stay in England.
Joint two-dimensional inversion of geoelectromagnetic and seismic refraction data with cross-gradients constraint
Luis Alonso Gallardo
thesis submitted for degree of Doctor of Philosophy
October 2004
Abstract
While the integration of multiple geophysical models is a necessary practice, the
procedures adopted have either lacked quantitative support or have a very rigid
framework that suits only specific geological environments and demands a detailed
knowledge of the studied site. In this thesis I took up the challenge of finding and
implementing a generalized methodology for the joint inversion of disparate data sets
based strictly on quantitative and objective features of the geophysical models and
data as an aid for model integration.
To achieve this aim I developed the concept of cross-gradients that evaluates
quantitatively the structural similarity between any pair of multidimensional images
and implemented it as a link for the purpose of joint inversion. I incorporated this
novel structural constraint in several objective functions involving geoelectromagnetic
and seismic refraction data and two-dimensional (2D) environments. The objective
functions adopt the least squares formalism and seek geometrically similar models
that satisfy the disparate geophysical data sets to the misfit given by the data errors. I
also propose a methodology of solution that involves a gradual search of the optimal
models that overcomes the problems of non-linearity of the objective function and
avoids local minima.
The algorithms of joint inversion have been applied to synthetic test and field data.
The resulting models have been proven successful because they satisfy the data and
are geometrically similar. The models also go beyond this success and suggest a
possible zonation within the models into plausible geophysical domains at a field site,
each with characteristic resistivity-velocity behaviour.
The properties of the cross-gradients constraint and the results obtained in joint
inversion testify to the contribution of my research towards a more objective
integration of multidimensional models and seem to offer multiple opportunities for
future research.
Table of Contents
1. Introduction…………………………………………………………..1
1.1. The conventional inversion of single geophysical fields and its
limitations………………………………………………………..1
1.2. Combining geophysical methods…………..………………………...3
1.3. Joint inversion of uncorrelated data………………………………….6
1.3.1. The petrophysical approach for joint inversion………………....7
1.3.2. The structural approach for joint inversion…………………….9
1.4. Problem definition and research aims……………………………….11
1.5. The pilot geophysical data and models………...……………………14
1.6. Thesis outline and achievements…………..……………………….15
2. Forward modelling in two-dimensional media……………………16
2.1. DC resistivity modelling in two-dimensional environments……..…….16
2.1.1. DC resistivity forward modelling theory….……………..……17
2.1.2. Adopted DC resistivity forward modelling and derivative
computation approach………………………………..……20
2.2. Magnetotelluric (MT) modelling in two-dimensional environments……26
2.2.1. MT forward modelling theory……………………………….28
2.2.2. Adopted MT forward modelling and derivative computation
approach…………………………………………………..32
2.3. Seismic modelling of first arrivals and raytracing……………….……37
i
2.3.1. Theory of forward modelling for first arrivals………………...39
2.3.2. Adopted seismic forward modelling and derivative computation
approach………………………………….………………..42
3. Joint inversion using cross-gradients……………………………...47
3.1. Combination of geoelectromagnetic and seismic refraction
field data (Paper 1)……….…..…….…………………………...47
3.2. Cross-gradients joint inversion for improved characterisation
(Paper 2)……….………………………………………………49
3.3. Cross-gradients joint inversion formulation for DC resistivity
and seismic refraction data (Paper 3)……………………………...50
3.4. Cross-gradients joint inversion of magnetotelluric and seismic
refraction data (Paper 4)……………………..…………………..52
3.5. Cross-gradients approach for joint image reconstruction (Paper 5)….…53
4. Summary of results and discussions……………………………….54
4.1. The complementary nature of multiple geophysical data……….….….54
4.2. The cross-gradients concept for quantification of structural
similarity………………………….……………………………55
4.3. The target objective function for joint inversion…………….………..57
4.4. The minimisation of the objective function for joint inversion…….…..60
4.5. Results for synthetic and field examples………...…………………..65
4.6. The implications of jointly inverted models for subsurface
characterisation………………..………………………………..66
4.7 Overall assessment of jointly inverted models…………...…………...67
ii
5. Conclusions and suggestions for further research………………..70
5.1. Conclusions……………………………………….……………..70
5.1.1. The concept of cross-gradients………………………...……70
5.1.2. The joint inversion algorithm……………………………….71
5.1.3. The features of models resulting from joint inversion………….72
5.2. Suggestions for further research……………………………………73
6. References…………………………………………………………..77
Paper 1…………………………………………………………………84
Paper 2…………………………………………………………………89
Paper 3…………………………………………………………………94
Paper 4…………………………………………….………………….106
Paper 5…………………………………………….………………….122
iii
List of papers
This thesis is based on the following papers which are reprinted in the corresponding
section with permission from the copyright holder, where appropriate.
Paper 1. Meju, M. A., Gallardo, L. A., Mohamed, A. K. (2003) Evidence for
correlation of electrical resistivity and seismic velocity in heterogeneous
near-surface materials. Geophys. Res. Lett., 30(7), 1373,
doi:10.1029/2002GL016048.
Paper 2. Gallardo, L. A., Meju, M. A. (2003) Characterization of heterogeneous near-
surface materials by joint 2D inversion of dc resistivity and seismic data.
Geophys. Res. Lett., 30(13), 1658, doi:10.1029/2003GL017370.
Paper 3. Gallardo, L. A., Meju, M. A. (2004) Joint two-dimensional dc resistivity and
seismic traveltime inversion with cross-gradients constraints. J. Geophys.
Res. 109, B03311, doi:10.1029/2003JB002716.
Paper 4. Gallardo, L. A., Meju, M. A. (2005) Joint 2D cross-gradients inversion of
magnetotelluric and seismic refraction data: Implication for structural and
lithological classification (manuscript submitted to Geophys. Res. Lett.)
Paper 5. Gallardo, L. A., Meju, M. A., Pérez-Flores, M. A. (2005) A Quadratic
Programming Approach for Joint Image Reconstruction: Mathematical and
Geophysical Examples. Inverse Problems 21, 435-452.
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List of figures
Figure 1-1: Illustration of the process of integration of geophysical models generated
by the separate inversion of their respective data. This process of integration
can be complicated by the mutual incompatibilities between the geophysical
models produced from the inaccurate data…………………………………..3
Figure 1-2: Illustration of the method of sequential inversion. In this method the
processing of the second model (II) is constrained by the other geophysical
model generated first (I) and “absorbs” some of its features………………..5
Figure 1-3: Concept of joint inversion using the petrophysical approach. The
petrophysical parameter (φ) is used to correlate both geophysical parameters
(ρ and V) and its distribution may be obtained straight from the joint
inversion or in later model transformations………………………………….8
Figure 1-4: Concept of joint inversion using structural constraints. The structure of the
model is parameterized by common features (A, B, C) which provide the
link for the joint inversion of both geophysical data………………………..9
Figure 2-1: Schematic diagram of the homogeneous earth and the potential (V)
associated with a point source of current (I) at position A………………..18
Figure 2-2: Four-electrode survey system commonly used for two-dimensional
electrical resistivity. A, B refer to the position of electrodes for inducing
current and M, N to the position of electrodes for measuring electric
potential…………………………………………………………………….19
Figure 2-3: Coordinate and reference system used for the formulation of the DC
resistivity two-dimensional forward problem. The prime coordinates denote
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the position of conductive elements in the subsurface while the natural
coordinates refer to position of field measurements………………………..21
Figure 2-4: Discretization of the cells of the model and their re-sampling in sub-cells
to compute forward and derivative computations. The sub-cells are simulated
using lines of constant resistivity…………………………………………...25
Figure 2-5: Magnetotelluric field setup (after Vozoff, 1972)…………….…………..27
Figure 2-6: Illustration of the two-dimensional resistivity model, orientation of
surveying modes and sub-gridding used for the magnetotelluric forward and
derivative computation. xi represents the horizontal position of an MT site
and zi-1, zi the depths to the top and bottom respectively of a specific cell
below the MT site as used in the derivative formulation………………..….30
Figure 2-7: Typical seismic survey profile for refraction experiments. The receivers
register the ground movement propagated from the source through the
subsurface…………………………………………………………………..38
Figure 2-8: Illustration of the propagation of a square wavefront as used in Vidale’s
(1988) progressive finite difference scheme. The crosses denote the nodes of
the seismic grid (timed or to be timed)……………………………………..43
Figure 2-9: Stencils implemented for the forward finite difference scheme of the
arrival times. a) stencil type I to follow headwaves (Hole and Zelt, 1995), b)
stencil Type II (Vidale, 1988), c) stencil type III (Vidale, 1988). The stencils
are modified from the original schemes to work with discrete velocity for the
cells rather than in the nodes……………………………………………….45
Figure 2-10: Example of the timefield (dark contours in seconds) and raypaths (red
lines) for a source located at profile position 200 km. In this figure some
causality violations occur on nodes at surface position A because the rays
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that first reach them should come from trajectories refracted on the sharp
vertical contact rather than on the deepest layer……………………………46
Figure 3-1: Map of the Quorn pilot field site. Collocated DC resistivity,
audiomagnetotelluric and seismic refraction data were measured along
profile 20 W. The schematic geological section illustrates the heterogeneous
geological units expected and their stratigraphic position as described in
Paper 1. The inset shows the location of the site…………………………..48
Figure 4-1: Illustration of examples of model boundaries that are accepted in the
cross-gradients framework as structurally similar (a and b) and dissimilar
(c)…………………………………………………………………………...58
Figure 4-2: Illustration of the sequence of the two-stage minimization process
(reproduced from Paper 5, Figure 2)……………………………………….62
Figure 4-3: Evolution of the joint inversion process. Shown are the resultant resistivity
and velocity models for each iteration. Note the gradual development of
common structural features in both sets of models during the process
(Reproduced from Paper 3, Figure 10)…………………………………….64
vii
1. Introduction
Studies of the inaccessible regions within the Earth are of immense economic and
scientific interests. However, they also pose a special challenge to the geophysicist.
To access these hidden regions, a geophysicist uses or induces physical fields into the
ground with the expectation that slight perturbations of these fields will give some
insight on the subsurface physical property distribution. Measuring and understanding
these fields form the cornerstone of applied geophysics.
A vital step for achieving this understanding is the ability to quantitatively predict the
physical fields for a geophysical model of the earth (geophysical forward problem).
However, there is a second step, probably that with the most practical interest, which
allows the estimation of the particular distribution of a physical property in the
subsurface from geophysical measurements. This mapping will then aid the
construction of a complete model or image of the subsurface for economic or
scientific applications (geological model). This second step is known as the inverse
problem and the process of reconstructing the image of the subsurface using field
observations is aptly termed geophysical inversion. My research is devoted to this
aspect of geophysics.
1.1. The conventional inversion of single geophysical fields and its limitations
The results of geophysical inversion are usually summarized in geophysical models
that map the distribution of the physical property sensed by the particular field
induced in the subsurface. However these geophysical models require a further
1
interpretative step to transform them into complete and meaningful geological models
from which practical or scientific deductions can be made. To make reliable
geological or petrophysical inferences, it is necessary to use accurate geophysical
models and assess their accuracy. A key issue, however, is how to generate such
accurate models from geophysical inversion.
Theoretically, abundant geophysical data reduce the uncertainty on the geophysical
models. However, the non-linear behaviour of some geophysical fields and the non-
uniqueness of the associated model limit the benefits of data redundancy. Examples of
this are the problem of detection of hidden low-velocity and blind (or shadow) zones
in seismic refraction experiments (see for example Dobrin and Savit, 1988, p. 460;
Kearey and Brooks, 1991, p. 107) and the non-uniqueness in gravity and magnetic
models (e.g. Blakely, 1995, p. 216). Instead, the non-uniqueness can be alleviated by
the use of extraneous data in the form of a priori information (Jackson, 1979) or
complementary geophysical data, commonly those sensing the same physical
parameters but using different physical measurements combined in a least squares
framework (e.g. Vozoff and Jupp, 1975; Sasaki, 1989; Yang et al., 1999; Benech et
al., 2002; de la Vega et al., 2003).
To improve the accuracy of a particular geophysical model it is necessary to increase
the amount of complementary data and constraints. However, this is not a guarantee
for its geological correctness. For instance, no matter how accurate a particular
geophysical model is, it will not be sufficient to discern between two materials with
the same physical property. To overcome this difficulty, multiple geophysical data
sensitive to different physical properties must be included. Examples of geological
2
models derived from integrating several geophysical models are abundant in the
literature (e.g. Eberhart-Philips et al., 1995; Nelson et al., 1996; Anderson et al.,
2004). Furthermore, detailed discussions in Berge et al. (2000) and Kozlovskaya
(2001) illustrate the necessity of combining several geophysical properties in order to
derive petrophysically meaningful parameters.
1.2. Combining geophysical methods
The interpretation of multiple geophysical data sensitive to different physical
properties can be based on independently processed models (e.g. van Overmeeren,
1981; Nelson et al., 1996), which are then compared and fused into a conceptual
geological model as illustrated in Figure 1-1. Despite generating useful geological
models, this procedure of data integration is subjective and often suffers from the lack
of a robust quantitative framework.
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Separate inversionData I
Separate inversionData II
Geological constraints
Geological model
Geophysical model IGeophysical data I
Geophysical data IIGeophysical model II
Model integration
φ
Figure 1-1: Illustration of the process of integration of geophysical models generated by the
separate inversion of their respective data. This process of integration can be complicated by the
mutual incompatibilities between the geophysical models produced from the inaccurate data.
3
Some insight into quantitative integration in analogous problems is given in the fields
of remote sensing and medical imaging where multiple images can be fused or
corrected to produce better images of high-resolution (e.g. Brown, 1992; Zhang,
2004). However, in geophysical applications, fusing directly two inaccurate
geophysical images generated separately also fuses their differences and this could
reduce the reliability of eventual geological inferences. Rather than treat the
inaccurate models in this questionable manner, one may ask whether the limited and
inaccurate data could also produce alternative models that show more similarities
between them.
Solving this challenging problem would not only permit an easier geophysical
integration but it would also increase the reliability of any geological or petrophysical
deduction made from multiple geophysical data. However, realistically defining what
similarities mean and ultimately, how two geophysical models can be assessed as
more or less similar are non-trivial challenges.
While a generalised and quantitative criterion to define these similarities is not
reported in the literature, some subjective methodologies that attempt to produce
geophysical models that are somewhat concordant have been proposed, one of which
is the method of sequential inversion (e.g. Lines et al., 1988). In this method the
geophysical data considered to have the best resolution capability are processed first
and some of the features (e.g depths of layers, physical properties of contained bodies,
etc.) of the geophysical model generated are selected and used to constrain the
processing of the second set of geophysical data (e.g. Lines et al., 1988; Nath et al.,
2000). This approach is illustrated in Figure 1-2 and is frequently used for the
4
generation of seismic and gravity models in tectonic studies or in oil industry
applications (e.g. Lees and Vandecar, 1991; Anderson et al., 2004) and occasionally in
combination with electrical resistivity or electromagnetic data (e.g. Scott et al., 2000).
Apart from the subjectivity in the selection of the features that can be regarded as
common among the models, the main disadvantage of this approach is that it seems to
disregard the data set deemed to have less resolution and limit its contribution to the
final geological model and it might “double-validate” an incorrect hypothesis derived
from the first data set.
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Geological constraints
Geophysical model IGeophysical data I
Geophysical data II
Geophysical model II
A
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φ
Figure 1-2: Illustration of the method of sequential inversion. In this method the processing of the
second model (II) is constrained by the other geophysical model generated first (I) and “absorbs”
some of its features.
Another methodology is termed cooperative inversion, which Lines et al. (1988)
define as the estimation of a subsurface model that is consistent with various
independent geophysical data sets. This definition however is not restrictive in the
criteria to integrate such a subsurface model or the methodology for its estimation. As
5
such, it is broad enough to include the possibility of sequential or joint inversion of
correlated or uncorrelated data following subjective criteria.
1.3. Joint inversion of uncorrelated data
Central to the interpretation of multiple disparate geophysical data is the concept of
joint inversion. This concept has been recurrently used in the literature (e.g. Vozzof
and Jupp, 1975; Lines et al., 1988; Zhang and Morgan, 1996; Haber and Oldenburg,
1997; Berge et al., 2000; Gallardo-Delgado et al., 2003) and it has been equally
applied to the inversion of correlated and uncorrelated data assuming a range of
common features between the geophysical models. In this thesis, joint inversion refers
strictly to the simultaneous objective estimation of similar (or integrated) models
from disparate data. In this process, several sets of geophysical data sensitive to
different geophysical parameters must interact according to their individual precision
and model resolution to produce better geophysical models with common attributes.
In the core of this definition of joint inversion is the concept of similarity or
commonality between the models. While there is as yet no generalized measure for
these similarities that is applicable to any pair of geophysical models, several
approaches assume that, for some particular models, one or more features can be
regarded as common and used to perform joint inversion. These can be classified into
(i) petrophysical approach and (ii) structural (or geometrical) approach.
6
1.3.1. The petrophysical approach for joint inversion
The joint inversion methodologies that follow the petrophysical approach are based on
the fact that for some specific geological environments, multiple geophysical
parameters can be correlated via physical or empirical relationships. Examples of such
inversion involving direct relationships between parameters are the joint inversion of
arrival times of shear and compressional seismic waves using the value of the VS/VP
ratio (e.g. de Natale et al., 2004) and joint inversion of seismic and gravity data (e.g.
Roecker et al., 2004) based on empirical relationships between seismic velocity and
density like those derived by Birch (1961), Gardner et al. (1974) or Christensen and
Mooney (1995).
In other cases where direct relationships between geophysical parameters have not
been established, valid petrophysical relationships that involve different geophysical
parameters can be combined using one or more petrophysical attributes as the
common factor. For instance the porosity factor implicit in the Archie and Wyllie
equations (Archie, 1942; Wyllie et al., 1956) can be used to find a relationship
between electrical resistivity and seismic velocities (e.g. Rudman et al., 1975; Marquis
and Hyndman, 1992). More examples can be found in relationships between VP and VS
for porosity estimations (e.g. Berryman et al., 2002) and other empirical approaches
like those developed by Tillman and Stöcker (2000), Kozlovskaya (2001) and
Kazatchenko et al. (2004).
The relationships can be used for either fusing multiple geophysical models into one
integrated petrophysical model (for example, a porosity map extracted directly from
7
the combination of the geophysical parameters as in Berryman et al., 2002) or to
produce multiple geophysical models with common features (e.g. Roecker et al.,
2004). This procedure is illustrated in Figure 1-3. Unfortunately, the advantage of
joining the geophysical data to produce integrated models or petrophysical images
also becomes its main drawback as this methodology may neglect the uncertainty of
the petrophysical relationship itself (e.g. Roecker et al, 2004). For example, if a
specific relationship developed for a particular geological environment is wrongly
assumed as a link for joint inversion of data from other environments, it will lead to
erroneous results that may only be used to test the validity of the petrophysical
assumption afterwards.
Even if a petrophysical property is sampled, there might also be others that influence
the relationship and make it variable along the studied site. This implies that there is a
need for the careful involvement of all the petrophysical attributes known to influence
the geophysical properties.
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= ( )= ( )
fgφφV
V= ( )w
Petrophysicalconstraints
Geophysical model II
Geophysical data I
Geophysical data II
Geophysical model I
V
φ
Figure 1-3: Concept of joint inversion using the petrophysical approach. The petrophysical
parameter (φ) is used to correlate both geophysical parameters (ρ and V) and its distribution may
be obtained straight from the joint inversion or in later model transformations.
8
1.3.2. The structural approach for joint inversion
For the structural approach, the correlating factor is given by the subsurface
distribution of the physical properties, which is somehow defined as being common to
all the geophysical models (see Figure 1-4). The main difficulty in this methodology
is how to define mathematically the concepts of structure and structural commonality,
and how to express them in an objective formulation.
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= (A,B,C)= (A,B,C)
fgV
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Geophysical model I
V
A
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Figure 1-4: Concept of joint inversion using structural constraints. The structure of the model is
parameterized by common features (A, B, C) which provide the link for the joint inversion of
both geophysical data.
A straightforward idea is to make up the geophysical models using a group of finite
bodies with clearly defined borders and assume the full coincidence of their borders.
Some examples of this approach of joint inversion can be seen in gravity, magnetic
and seismic multidimensional applications (e.g. Afnimar et al., 2002; Ditmar, 2002;
Gallardo-Delgado et al., 2003; Gallardo et al., 2004) and geoelectromagnetic-seismic
applications (Hering et al., 1995; Manglik and Verma, 1998; Kis, 2002). Despite the
9
popularity of this approach, it has the disadvantage of requiring an accurate a priori
knowledge of the number of bodies involved and their possible a priori shapes and
distribution (see discussion in Gallardo et al., 2004).
Another more flexible way to construct geophysical models is through an ensemble of
homogeneous elements of fixed geometries but variable geophysical parameters.
However, a major issue for these models is how to define the concept of structure
itself. Examples of structural joint inversion using this type of models are rare and so
far there are only two published schemes that follow this principle: one proposed by
Zhang and Morgan (1996) and other by Haber and Oldenburg (1997). These two
methodologies were developed for two-dimensional models using the position of the
largest property changes (Laplacian measures) as mathematical indicators of the
structural boundaries and then aim for their full coincidence in the geophysical
images. The underlying principle of looking for boundaries is similar to that of the
finite bodies described above but it does not seem to need any a priori knowledge of
these bodies. However, the difficulty here is how to “normalize” (Zhang and Morgan,
1996) or “select” (Haber and Oldenburg, 1997) how big the changes need to be for
them to be accurately recognized as a real boundary to emphasize in the jointly
inverted models.
Similar to the petrophysical approach, models resulting from joint inversion using the
structural approach have proven to be appropriate for some applications where the
models lend themselves to the hypothesis of full commonality and when acceptable
assumptions (e.g. number of bodies) are made. However, some questions regarding
the assumptions in this approach are: Are the boundaries or changes in all the
10
geophysical models really spatially coincident? Are all the contrasts between bodies
or structures significant or large enough to be detected in all the geophysical models?
1.4. Problem Definition and Research Aims
It is clear that the quest for a generalized methodology to produce somewhat similar
models of different geophysical parameters that satisfy their respective disparate
measured data is as yet unfulfilled. So far, the published methodologies for joint
inversion have shown severe limitations. On one hand, methodologies based on
common petrophysical attributes require an accurate previous knowledge of every
specific geological environment to guarantee the validity of each petrophysical
relationship perhaps through direct testing, and thereby losing the generality of the
joint inversion criteria and the non-invasive nature of the geophysical deductions. The
fact that two geophysical parameters are assumed to be correlated functionally via a
petrophysical function collapses two useful pieces of information into one, while the
other is sacrificed (subordinated) in the search for the commonality of the models (for
example to generate a porosity map from two geophysical parameters). On the other
hand, methodologies based on common structural attributes are rigid in the concept of
structure itself and require further assumptions (e.g. the shape of included bodies or
magnitude of property changes) to generate geometrically common models that, in the
absence of reliable previous knowledge, may be based on subjective criteria.
11
Some relevant questions raised by the above discussions are:
q1: What features can be used to define the resemblance between any two
models and which makes them “perceptibly” similar without making them
interdependent (i.e. lose their information)?
q2: Can these features be generalized and therefore valid for any geological
environment or geophysical data?
q3: Can these features be quantitatively evaluated and compared to assess
how similar any two models are and can they be implemented in an
automated process for their simultaneous estimation?
q4: Are there “similar” geophysical models that can justify their respective
noisy data?
q5: Are they unique?
q6: Assuming that two similar geophysical models that satisfy their respective
data exist, will that guarantee improvements in the geological deductions
derived from them?
The main aim of my research is to develop an effective multidimensional joint
inversion approach that will address the issues highlighted in the foregoing
discussions.
12
To achieve this aim, my study is built upon specific objectives with clear deliverables,
viz:
1. Finding an objective and quantifiable criterion to measure the similarities between
any two geophysical methods that are based on fundamentally different phenomena.
This criterion should preferably be valid regardless of the dimensionality of the
problem (1D, 2D or 3D), the type of geophysical data and the geological medium.
2. Incorporate such a similarity measure in a robust joint inversion procedure to
produce better-integrated models (similar models) that exploit all the disparate data
available. The procedure should avoid subjective criteria to subordinate any type of
data. It should be robust enough to account for data errors and also flexible to accept
models with different discretization or even continuous models.
3. Demonstrate the generality of its application and the improvement in image
resolution or model parameter estimation through experimental and theoretical tests
on a diverse range of geophysical situations.
4. Explore possible implications of the resulting joint inversion models in
characterisation.
13
1.5. The pilot geophysical data and models
For the purposes of this work I focused on collocated geoelectromagnetic (DC
resistivity and magnetotelluric) and seismic refraction experiments. The properties
and wide use of these geophysical techniques make them appropriate for this research
on joint inversion of disparate data. First, geoelectromagnetic and seismic techniques
obey different physical phenomena (diffusion and wave propagation) and their fields
are sensitive to different physical properties, namely electrical resistivity (or its
inverse, conductivity) and seismic velocity. The use of active sources in these methods
allows a detailed data sampling that reduces the uncertainty in the geophysical
models. These techniques are popular and equally applied in tomographic experiments
for shallow and deep geophysical targets (e.g. Riley, 1993). Their popularity and high
resolution have also motivated the development and application of abundant
petrophysical transformations (e.g. Berge et al., 2000; Kozlovskaya, 2001).
To explore the multidimensional aspect of joint inversion I use two-dimensional (2D)
models, which should suffice for the objectives of this research. First, two-
dimensional models allow both structural and petrophysical attributes to be studied
without needing the more computationally expensive rigorous 3D cases. The common
execution of geophysical surveys in 2D profiles ensures more data availability for
testing purposes.
14
1.6. Thesis outline and achievements
For the development of this research I followed closely the solution of the set
objectives and most of the partial results were written and submitted to scientific
journals. These publications now constitute the main bulk of this thesis, which I
organize in the following format:
Chapter 2 provides a short review of the theory of the electrical, electromagnetic and
seismic refraction techniques and describes briefly the adopted forward
modelling and derivative formulations.
Chapter 3 introduces the concept of cross-gradients and the methodology of joint
inversion as the main developments of my research by referring to the relevant
constitutive papers that are appended at the end of this thesis and by describing
their role in the context of my research.
Chapter 4 summarizes and discusses the developments and results in joint inversion in
the framework of the specific set objectives and the issues highlighted in this
introductory chapter.
Chapter 5 concludes this thesis highlighting the achievements and their importance
and suggesting the future research opportunities offered by my theory and
methodology of joint inversion.
Chapter 6 lists the references cited.
Appended Papers 1 to 5 contain the complete published or submitted papers that
support this thesis.
15
2. Forward modelling in two-dimensional media
As previously described, the physical fields induced in the Earth carry information
about what lies in the subsurface. Understanding these fields and their relationship
with physical models form the basis for the geophysical inverse problem. In this
chapter I review the principles of the geophysical techniques used for the development
of joint inversion. I briefly describe the fundamental equations that relate the physical
fields to the respective geophysical properties, the methodologies of solution adopted
for the forward problems and for the derivatives required for the formulation of the
inverse problem.
2.1. DC resistivity modelling in two-dimensional environments
Geophysical methods based on the propagation of a steady electric current have a
wide range of applications and are currently highly popular for groundwater and
engineering studies (see Riley, 1993). In DC resistivity techniques an electric current
is injected into the ground using electrodes. This electric current travels into the
ground and polarizes the subsurface materials according to their electric properties (in
this case resistivity). The resulting electric field is measured on the surface in the form
of electric potential between pairs of grounded electrodes.
Conventionally, the electric potential measured on the surface is compared with that
expected for a homogeneous underground (of unit resistivity) and the ratio provides a
first insight of the average resistivity of the ground. The value of this weighted
average of resistivity is known as apparent resistivity and it depends on the relative
16
location of the electrode array and the actual subsurface resistivity distribution.
Several apparent resistivity data measured using different deployments and sites
provide different views and complement each other for a “de-averaging” that will lead
to the reconstruction of a subsurface model that reflects the resistivity distribution
more accurately.
2.1.1. DC resistivity forward modelling theory
The behaviour of a steady electrical current in the subsoil is described by the steady
state equation of continuity given by:
tq
J v
∂∂
−=⋅∇v
(2-1)
where is the current density and qJv
v is the volume charge density. This Poisson-type
equation describes the behaviour of the current for every point in the subsurface
coincident or not with any source of current. To relate the distribution of the electric
current with the physical property of interest, i.e. electric resistivity (ρ), it is necessary
to use a relationship known as a constitutive equation. In this case the “microscopic
form” of Ohm’s law for linear isotropic media given by
EJwv
σ= (2-2)
17
provides the required link. In Equation 2-2 σ refers to the electrical conductivity and
Ev
to the electric field. This electric field is, in the absence of non-stationary magnetic
fields, conservative and can be expressed in terms of the electric potential V using
V−∇=Ev
. (2-3)
Equations 2-1, 2-2 and 2-3 describe the electric phenomenon for steady current in
terms of the properties sought and the electric potential and they are preferably
combined into the general equation (e.g. Hohmann, 1988):
tqv
∂∂
−=∇⋅∇+∇ VV2 σσ . (2-4)
The forward problem for DC resistivity modelling solves Equation 2-4 in terms of the
electric potential due to a particular resistivity distribution. The simplest example is
the solution to Equation 2-4 for a homogeneous half-space, which is given by (see
Figure 2-1)
r2IVπρ
= . (2-5)
I
r
AirA
V= I2 rπ
Figure 2-1: Schematic diagram of the homogeneous earth and the potential (V) associated with a
point source of current (I) at position A.
18
Although Equation 2-5 is exact only for a constant-resistivity medium, it is used on
heterogeneous media to normalize the potential (V) and the electric current (I)
measured. This ratio is known as apparent resistivity (ρapp) and constitutes a weighted
average of the true resistivity. Specific expressions for the apparent resistivity depend
on the distribution of the electric and potential electrodes of the electrode array.
Among the multiple arrays that can be used, the most popular for two-dimensional
surveys consists of a system of four collinear electrodes as illustrated in Figure 2-2.
I VAirA B M N
Figure 2-2: Four-electrode survey system commonly used for two-dimensional electrical
resistivity. A, B refer to the position of electrodes for inducing current and M, N to the position of
electrodes for measuring electric potential.
For this type of array, a general formula to compute the apparent resistivity is:
IV
IV11112
1
krrrr NBMBNAMA
app =⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=
−
−−−−
πρ , (2-6)
where k is the geometrical factor. rA-M, rA-N, rB-M and rB-N denote the distance to the
corresponding electrodes as illustrated in Figure 2-2.
19
Analytical solutions to Equation 2-4 for two- or three-dimensional models are scarce
in the literature and are only available for simple geometrical models that under an
appropriate coordinate system (e.g. cylindrical system) simplify the incorporation of
the boundary conditions (e.g. Van Nostrand, 1953; Ward and Hohmann, 1988). For
more complex two- or three-dimensional media, Equation 2-4 is solved by numerical
procedures involving finite difference (e.g. Mufti, 1976; Dey and Morrison, 1979;
Spitzer, 1995), finite element (e.g. Coggon, 1971; Fox et al., 1980; Pridmore et al.,
1981) and integral equation (e.g. Lee, 1975; Snyder, 1976; Pérez-Flores et al., 2001;
Ma, 2002) techniques. Although the advantages and disadvantages on speed and
accuracy seem to balance out in each technique, the speed in both forward and
derivative computations becomes particularly important for the development and
testing of joint inversion. I therefore selected an approach based on the linear integral
equation of Pérez-Flores et al. (2001) that, despite being less accurate and time
consuming in its numerical solution, is efficient because it does not require re-
computation of forward and derivative responses in iterative procedures.
2.1.2. Adopted DC resistivity forward modelling and derivative computation approach
The adopted methodology for DC resistivity forward modelling is based on the non-
linear integral equations developed by Gomez-Treviño (1987). This approach uses the
scaling properties of Maxwell’s equations to produce non-linear integral equations for
electromagnetic problems that directly relate the electromagnetic fields and
conductivity rather than their perturbations. The corresponding equation for static
electric fields ( Ev
) can be expressed as (Pérez-Flores et al., 2001):
20
')'(),'(),',(),('
dvrrErrGrEV
E σσσσ vvvvvvvv⋅−= ∫ (2-7)
where rv and 'rv denote the position of the receiver and any conductor in the
subsurface respectively (see Figure 2-3) , EGvv
is the dyadic Green’s function (see
Hohmann, 1988; Pérez-Flores et al., 2001), V’ denotes the volume in the subsurface
and dv’ the corresponding differential element.
AirE( )r
r
r
r
x
z
Figure 2-3: Coordinate and reference system used for the formulation of the DC resistivity two-
dimensional forward problem. The prime coordinates denote the position of conductive elements
in the subsurface while the natural coordinates refer to position of field measurements.
Equation 2-7 is non linear as the dyadic Green’s function and the electric field depend
on the conductivity heterogeneities within the model. To overcome this difficulty and
keep the complete kernel as a Frêchet derivative as defined by Gómez-Treviño (1987),
Pérez-Flores (1995) approximates the Green’s function and the electric field with
those of a half-space model with homogeneous conductivity (σh) thus making
),',(),',( hEE rrGrrG σσ vvvvvvvv≈ and ),'(),'( hrErE σσ vvvv
≈ . The expressions for the tensor
21
Green’s function and electric field in the homogeneous half-space can be found in
Pérez-Flores et al. (2001).
Because Ev
is a conservative field the electric potential V at a position can be
conveniently integrated in the x-direction (see Figure 2-3) and Equation 2-7, for a
specific source of current at
Mrv
Arv
, becomes:
∫ ⋅−='
')'(),,'(),',(),,(VV
hAhMMA dvrrrErrUrr σσσσ vvvvvvvv. (2-8)
),',( hM rrU σvvv being the line integration along the x-axis of the first row of
),',( hE rrG σvvvv. Considering that for a half-space
3')'(
2I),,'(
A
A
hhA
rrrr
rrE vv
vvvvv
−
−=
πσσ (2-9)
and
3')'(
21),',(
rrrr
rrUM
M
hhM vv
vvvvv
−
−−=
πσσ , (2-10)
Equation 2-8 can be written in terms of resistivity as
∫ −
−⋅
−
−=
'
2
2
332 ')'()'(')'(
')'(
4I),,(V
V hM
M
A
AMA dvrr
rrrr
rrrr
rr vv
vv
vv
vv
vvvv ρ
σσ
πρ . (2-11)
22
Pérez-Flores et al. (2001) also assume the occurrence of models with low contrasts in
resistivity approximating 1)'(2
2
≈h
rσ
σ v
. Equation 2-11 can then be used to compute the
apparent resistivity for a collinear four-electrode array (see Figure 2-2) as:
[ ]),,(V),,(V),,(V),,(V4 2 ρρρρπ
ρ NBMBNAMAapp rrrrrrrrk vvvvvvvv+−−= . (2-12)
A further step to transform Equation 2-12 to use directly the logarithmic values of
resistivity (and apparent resistivity) is shown in Pérez-Flores (1995) and Pérez-Flores
et al. (2001).
For any four-electrode array, Equation 2-12 corresponds to the sum of four integral
expressions of the type of Equation 2-11 and can be summarized as:
∫ ∫ ∫ ⎥⎦
⎤⎢⎣
⎡=
∞
∞−x z
a dxdzdyzyxwzxk ''')',','()','(4 2 ρπ
ρ (2-13)
or
∫ ∫=x z
a dxdzzxWzxk '')','()','(4 2 ρπ
ρ . (2-14)
In these expression W(x′, z′) depend on the positions between sources and receivers
relative to the subsurface structure according to Equation 2-11. Pérez-Flores et al.
(2001) deduced the analytical expressions for W(x′,z′) for any collinear four-electrode
array. Paper 5 (Figure 9) shows an example of the values for the function W(x′,z′).
23
Note that Equation 2-14 is linear and therefore can be used in linear inverse problem
formulations.
For implementation in joint inversion, it was convenient to subdivide the subsurface
model into two-dimensional rectangular cells coincident in space and position in all
the geophysical models to be jointly inverted. For development purposes, the size of
the cells is constrained by the available computing capability; however, the cells are
finely re-sampled as appropriate to improve the accuracy of the forward modelling
algorithm. To solve Equation 2-14 I followed the procedure of Pérez-Flores (1995)
and subdivide each cell in multiple rectangular sub-cells as illustrated in Figure 2-4.
The number of sub-cells is selected so that the evaluation of the W-coefficients in
Equation 2-14 is efficient without under-sampling the subsurface model. The spread
of the subsurface model is set long enough to allow the natural decay of the electric
potential. An appropriate discretization can be evaluated using the identity (Pérez-
Flores, 1995)
1'')','(4 2 =∫ ∫
x z
dxdzzxWkπ . (2-15)
One obvious advantage of this methodology of solution is that it linearizes the
problem and, as shown in the following chapters, has no time cost for iterative
procedures required by, for example, the seismic inversion or, as shown in Chapter 3,
the joint inversion procedure itself. However, this methodology has the disadvantage
of being inaccurate for the forward modelling computation when the model differs
significantly from the assumption of low contrast in resistivity. Nevertheless, it has
already been shown to have adequate precision for models with complex structures
24
(e.g. Pérez-Flores and Gómez-Treviño, 1997). In Paper 3, I use the finite-difference
algorithm of Dey and Morrison (1979) to generate more accurate test data. Note that
other forward modelling techniques could also be used for joint inversion procedures
but special attention must be paid to the efficiency and computational implementation
to ensure adequate speed in the procedure of joint inversion as discussed in Papers 3,
4 and 5.
Airx
z
Figure 2-4: Discretization of the cells of the model and their re-sampling in sub-cells to compute
forward and derivative computations. The sub-cells are simulated using lines of constant
resistivity.
25
2.2. Magnetotelluric (MT) modelling in two-dimensional environments
The magnetotelluric technique uses natural electromagnetic fields generated by
thunderstorms or current systems in the magnetosphere (Vozoff, 1991). These
electromagnetic fields impinge on the Earth’s surface almost in the form of plane
waves and propagate towards its interior. These electromagnetic fields travel
following a combination of diffusive and wave behaviours and are modified according
to the distribution of the earth resistivity (or conductivity) and the frequency of the
fields.
The surveying principle is to measure horizontal electric and magnetic fields on the
earth’s surface because the changes between their relative amplitudes and phases will
depend on the physical property sought. The classical MT field set up is shown in
Figure 2-5. The system consists of grounded electric dipoles that measure the
horizontal components of the electric field in two perpendicular directions (Ex and Ey)
and three magnetic coils that measure the three mutually orthogonal components of
the magnetic field (Hx, Hy and Hz). The method uses continuous recording of the
electromagnetic fields over a period of time using a narrow sampling rate. These
recorded time series are conventionally analysed in the frequency domain (Fourier
domain) to yield two sounding curves in two orthogonal directions. Further details of
the surveying technique can be found in the literature (e.g. Vozoff, 1972; Vozoff,
1991).
26
Figure 2-5: Magnetotelluric field setup (after Vozoff, 1972).
The main advantage of this method is that it is broadband and can utilise natural or
artificial signals of sufficient frequencies (e.g. from 0.01 to 100,000 Hz) to probe
different depths in the subsurface. It can provide information from relatively shallow
(10 m) to very deep targets (500000 m) that will be inaccessible to other
electromagnetic techniques. Unlike potential fields such as gravity and magnetism,
which are also used for deep targets, the multi-frequency content of the
electromagnetic source provides information from different depths. This technique can
extend the applications of joint inversion to deep targets using electrical conductivity
when combined with deep refraction profiles associated with wide-angle reflection
surveys or seismological networks.
27
2.2.1. MT forward modelling theory
The propagation of the magnetotelluric waves can be described mathematically using
Maxwell’s equations. In the frequency domain, for isotropic non-permanently
magnetized or polarized materials, these equations are given by (e.g. Ward and
Hohmann, 1988):
HiEvv
ωµ−=×∇ (2-16)
EiHvv
)( εωσ +=×∇ (2-17)
vqE −=⋅∇v
ε (2-18)
0=⋅∇ Hv
. (2-19)
In these expressions Ev
is the electric field, Hv
the magnetic field, ω the frequency, µ
the magnetic permeability, ε the electric permitivity, σ the conductivity, qv is the
volume charge density and i = 1− . In the range of frequencies commonly used for
MT surveys (~ 100 to 0.0001 Hz) the effects of displacement currents in Equation 2-
17 are negligible and the effects due to variations of µ in Equations 2-16 and 2-17 can
be neglected for most applications, thus reducing Equations 2-16 and 2-17 to
HiEvv
0ωµ−=×∇ (2-20)
EHvv
σ=×∇ (2-21)
28
where µ0 = 4πx10-7 H/m is the magnetic permeability of free space. The solution to
these equations provides the electromagnetic responses of the subsurface model that
are required in forward modelling.
The solution of Equations 2-20 and 2-21 for a homogeneous half-space can be
expressed in terms of the medium resistivity as:
2
0
1⎟⎠⎞
⎜⎝⎛=
HE
ωµρ . (2-22)
In this case, the electromagnetic fields Ev
and Hv
are always perpendicular and
shifted by 45º regardless of their orientation. However, such properties of the
electromagnetic fields change for real 1D, 2D and 3D applications. To simplify the
analysis of the observed fields in terms of a more intuitive parameter, Cagniard (1953)
proposed a useful transformation and introduced the concept of apparent resistivity
and phase differences for MT measurements. Following conventional nomenclature
used for two-dimensional media oriented as illustrated in Figure 2-6, the conventional
definitions of apparent resistivity and phase are given by:
2
0
1
y
xaTM H
Eωµ
ρ = , (2-23)
2
0
1
x
yaTE H
Eωµ
ρ = , (2-24)
29
⎟⎟⎠
⎞⎜⎜⎝
⎛=
y
xTM H
Eargφ (2-25)
and ⎟⎟⎠
⎞⎜⎜⎝
⎛=
x
yTE H
Eargφ . (2-26)
The subscripts TM (Transverse Magnetic) and TE (Transverse Electric) are
meaningful only for two-dimensional structures such as that depicted in Figure 2-6.
Although in two- and three-dimensional media more observables, that may involve
Hz, can be defined (e.g. “Tipper” as in Vozoff (1972) or “Rotational invariants” as in
Romo et al.(1999)), I will use the more convenient data defined by Equations 2-23, 2-
24, 2-25 and 2-26 to be used in joint inversion.
Airx
zi-1
xsy
z
ExHy
TM Mode
Ey
Hx
TE Mode
zi
Figure 2-6: Illustration of the two-dimensional resistivity model, orientation of surveying modes
and sub-gridding used for the magnetotelluric forward and derivative computation. xi represents
the horizontal position of an MT site and zi-1, zi the depths to the top and bottom respectively of a
specific cell below the MT site as used in the derivative formulation.
30
For two-dimensional models, Maxwell’s equations 2-20 and 2-21 decouple into
transverse electric (TE) and transverse magnetic (TM) sets of equations (e.g. Swift,
1971; Rodi, 1976) given by:
yzx E
xH
zH σ=
∂∂
−∂∂
, (2-27)
xy Hi
zE
0ωµ=∂
∂ (2-28)
and zy Hi
xE
0ωµ−=∂
∂ (2-29)
for the TE mode and
,0 yzx Hi
xE
zE
ωµ−=∂∂
−∂∂
(2-30)
xy E
zH
σ−=∂
∂ (2-31)
and zy E
xH
σ=∂
∂ (2-32)
for the TM mode.
31
These equations, or Equations 2-20 and 2-21 directly, have been solved using several
approaches to reproduce MT responses of two- and three-dimensional models. The
common approaches have been based on integral equations (e.g. Hohmann, 1975;
Wannamaker et al., 1984) or numerical solutions of the differential equations using
finite differences (e.g. Mackie et al., 1988; Madden and Mackie, 1989; Smith and
Booker, 1991) and finite element (e.g. Coggon, 1971; Rodi, 1976) techniques. All of
them have relative advantages and drawbacks in processing time and accuracy.
However for the purpose of the development of joint inversion, computational speed
is a necessary consideration.
2.2.2. Adopted MT forward modelling and derivative computation approach
Based on speed considerations, I adopted the methodology of forward and derivative
computation developed by Smith and Booker (1991). This approach consists of two
finite difference schemes based on nodes (one for the TE mode and one for the TM
mode).
The TE forward modelling algorithm of Smith and Booker (1991) aims to solve the
equation
yyy Ei
xE
zE
σωµ 02
2
2
2
=∂
∂+
∂
∂, (2-33)
which results from the substitution of Equations 2-28 and 2-29 in 2-27. The complex
field estimated in a subsurface grid is interpolated right below the MT sites and used
to estimate Hx on the surface using Equation 2-28. These values are then substituted in
32
Equations 2-24 and 2-26 to calculate the TE apparent resistivity and phase responses
of the model.
Similarly, the finite difference scheme of Smith and Booker (1991) for the TM mode
solves the equation
yyy Hi
zH
zxH
x 0ωµρρ =⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
(2-34)
resulting from Equations 2-30, 2-31 and 2-32. The Hy field estimated in a grid of
nodes is also interpolated below the measurement sites and used to estimate Ex on the
surface using Equation 2-31. These values are substituted in Equations 2-23 and 2-25
to calculate the TM apparent resistivity and phase responses of the model.
As illustrated in Figure 2-6, the MT forward grid over-samples the cells of the two-
dimensional model used for joint inversion, so it becomes detailed enough for the
computation of the forward response. Although in principle any forward modelling
technique can be adapted in the scheme of joint inversion, I preferred to keep
consistency and use Smith and Booker (1991) methodologies for both forward and
derivative computation.
Smith and Booker (1991) solve Equations 2-33 and 2-34 using a centred scheme using
five-points for the internal nodes, and add appropriate boundary conditions. For the
TE mode they subject the electrical fields to fit that of the one-dimensional model in
the farthermost lateral limits of the grid as a Dirchlet-type condition. At the bottom of
the model they use an impedance boundary condition that satisfies
33
yy Ei
zE
σωµ0−=∂
∂ (2-35)
while the top of the model is padded with several air node-layers to allow the natural
decay of the fields constraining the horizontal magnetic fields (Hx) to be unity on the
surface. For the TM mode, the boundary conditions are identical but it does not
require the padding of air node-layers above the surface. All these boundary
conditions are similar to those applied also by Mackie and Madden (1993). The matrix
equation result of the finite difference scheme and the boundary conditions is solved
using some conventional procedures of linear algebra as described in Smith and
Booker (1991).
The reciprocity principle could be used to calculate the partial derivatives using the
forward modelling scheme of Smith and Booker (1991). However, it would involve
the solution of additional forward problems (e.g. Mackie and Madden, 1993) and
increase the processing time. Instead, Smith and Booker (1991) developed an
approach that can resemble the solution of a one-dimensional problem for each site.
Smith and Booker (1991) based their pseudo one-dimensional approach on the
premise that the horizontal gradients of the transverse electric and magnetic fields are
smaller than the corresponding vertical gradients and assume that they can use the
horizontal gradients of the previous (or initial) model instead of the fields of the actual
model and simplify the corresponding equation to one-dimensional problems. This
way, for the TE mode, Equation 2-33 simplifies to the first order differential equation:
34
( ) 02 00 =−+∂∂ δσωµδδ iVVVz (2-36)
where y
xy
y EHi
zE
EV 0
1 ωµ=∂∂
= and V0 refer to values computed from a previous model.
The solution to Equation 2-36 is a numerical integral of the form:
∫∞
−=
0
202
0
0 ),(),()0,(
)0,( dzzxzxExE
ixV ssy
sys δσ
µδ (2-37)
where xs refers to the position of the MT sounding along the profile and Ey0 is
computed from the previous model. Naming ρi the resistivity of the cell of the model
used for inversion right below the measurement site on xs between the depths zi-1 and
zi (illustrated in Figure 2-6), the partial derivatives for the MT responses in terms of
logarithm of resistivity and phase are:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡=
∂∂ ∫
−
iz
iz
sysxsy
i
i
a dzzxExHxE
real1
20
00
0
10
10 ),()0,()0,()(log
)(log σρρ
(2-
38)
and
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡=
∂∂ ∫
−
iz
iz
sysxsy
ie
i
dzzxExHxE
imag1
20
00
0
10
),()0,()0,(2
)10(log)(log
σρ
φ. (2-39)
35
For the TM mode, Equation 2-34 can be simplified using the same assumptions as for
the TE mode but with
y
x
HEV = . (2-40)
The resulting expressions for the partial derivatives are identical to those of the TE
mode if Ey is substituted by Ex and Hx by Hy in the corresponding places in Equations
2-36, 2-37, 2-38 and 2-39.
The methodology of Smith and Booker (1991) has proved to be fast but at the high
cost of being inaccurate because the underlying model does not exactly match a two-
dimensional MT model. Nevertheless, with an appropriate interpolating model, the
discrepancy can remain within the order of the data measurement errors expected in
many MT surveys, as shown in the synthetic examples of Smith and Booker (1991).
Although for the purpose of development of joint inversion this factor is not the most
relevant, it could be further reduced by the beneficial interpolating information
provided by the other method (in this case, seismics) that is incorporated in the
inversion process.
36
2.3. Seismic modelling of first arrivals and raytracing
Seismic methods are among the most widely used geophysical methods and have
found many applications in prospecting and monitoring of resources (e.g. Riley, 1993)
as well as in earthquake and global earth studies (e.g. Lillie, 1999). In seismic
methods an elastic wave is naturally or artificially induced into the ground and during
its propagation the subsurface materials modify the direction and amplitude of the
particle displacements in the wave. The displacements that occur on the earth’s
surface or in boreholes are recorded and analysed in the search for information about
the subsurface. This analysis leads to important deductions concerning the location or
description of an earthquake source and the distribution of the elastic and attenuating
factors of the subsurface materials, including local, deep, and global features of the
solid earth.
Although full time series of the three-dimensional ground movement can be recorded,
I focus exclusively on the travel time of the first arrivals to determine the elastic
properties of the subsurface materials in the form of seismic velocities. The technique
based on the analysis of first arrivals is usually referred to as seismic refraction
because most of the elastic waves that first arrive at the surface correspond to head
waves critically refracted on the surface of more rigid materials in the subsoil.
Although this methodology has had vast applications for shallow studies, it is equally
applied to deep crustal targets when it is executed with powerful sources and longer
source-receiver offsets. Examples of this are the wide-angle reflection surveys (e.g.
Zelt et al., 1999; Korenaga et al., 2000) and the earthquake seismograms recorded in
local or global networks. In addition, other surveying methodologies like cross-
37
borehole seismic tomography are also based on the measurement and interpretation of
the travel time of the first-arriving waves and are also covered by the same theory. An
illustration of a typical two-dimensional seismic refraction surveying design used in
the synthetic and field experiments of this thesis is shown in Figure 2-7.
Figure 2-7: Typical seismic survey profile for refraction experiments. The receivers register the
ground movement propagated from the source through the subsurface.
The diverse surveying techniques, applications, depth coverage and resolution make
the seismic refraction a strategic technique to incorporate in the development of joint
inversion of disparate data sets.
38
2.3.1. Theory of forward modelling for first arrivals
The phenomenon of elastic propagation for an isotropic medium can be formulated
from the second Newton’s law, Hooke’s law and the definitions of stress and strain.
This leads to the elastodynamic equation given by (see Aki and Richards, 1980, p.
64):
( ) uuftu vvvv
×∇×∇+⋅∇∇++=∂∂ µµλη )2(2
2
, (2-41)
where η is the density of the material, uv is the particle displacement, fv
is a body
force and λ and µ are the Lamé constants that describe the elastic properties for
isotropic media.
Equation 2-41 can be reformulated in terms of Helmoltz potentials (φ and ψv
) into a
pair of decoupled equations:
ηφφ Φ=∇−
∂∂ 22
2
2
PVt (2-42)
and ηψψ Ψ
=∇−∂∂
vv
v22
2
2
SVt (2-43)
where
ψφ vv×∇+∇=u , (2-44)
39
Ψ×∇+Φ∇=vvv
f , (2-45)
ηµλ 2+
=pV (2-46)
and ηµ
=SV . (2-47)
Equations 2-42 and 2-43 correspond to the P- and S-wave equations respectively with
the phase wave speeds VP and VS given by equations 2-46 and 2-47.
Although 2-42 and 2-43 can be used to model the travel times (and amplitude) of the
time series of the seismic records (e.g. Luo and Schuster, 1991), there are other more
common approaches based on the geometrical concepts of wavefronts and rays. The
wavefront is a travelling discontinuity in the displacement of the particles originated
by a discontinuity in the seismic source (Aki and Richards, 1980, p. 89). Denoting as
, the travel time required by the wavefront generated at point to reach
another point
),( SrrT vvSrv
rv , the wave equation for an inhomogeneous medium can be decoupled
as (Aki and Richards, 1980, p. 90):
0)()(),(),(
)(2)()(),(),(
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛−∇⋅∇⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−∇⋅∇rrrrTrrT
rrrrrTrrT SSSS v
vvvvv
vv
vvvvv
µη
µλη
(2-48)
thus leading to the eikonal equation
40
)(1),(),(rV
rrTrrT SS vvvvv
=∇⋅∇ (2-49)
where )(rV vcan be either P-wave speed or S-wave speed according to the respective
wavefront. A seismic ray is defined by a continuous trajectory normal to the
wavefront as it propagates in the medium. This trajectory can be determined from the
travel times of the wavefront ( ),( SrrT vv) or by directly solving the ray tracing equations
(e.g. Julian and Gubbins, 1977; Virieux et al., 1988). For any specific ray trajectory,
the travel time of the wavefront can be computed by
∫=r
Sr
S rVdlrrT
v
v
vvv
)(),( , (2-50)
which is integrated along the raypath.
Although Equations 2-42 and 2-43 are preferred for modelling full waveforms and to
generate synthetic seismograms (e.g. Tarantola, 1987, p. 438), the alternative
equations are preferred to compute travel times. For instance, Vidale (1988), Vidale
(1990) and Qin et al., (1992) use finite difference solutions of the eikonal equation
(Equation 2-49); Julian and Gubbins (1977), Michelena and Harris (1991) and Hole
(1992) solve the geometrical problem of ray tracing and use Equation 2-50 to compute
the travel times whereas Saito (1990), Vinje et al., (1993) and Zhang and Toksöz
(1998) reconstruct simultaneously wavefronts and raypaths.
41
2.3.2. Adopted seismic forward modelling and derivative computation approach
The selection of a fast procedure to compute travel times and their derivatives is a key
factor in this research on joint multidimensional inversion. Techniques based on ray
tracing have disadvantages in terms of processing time, along with the usual problems
with shadow zones and multiplicity of raypaths (Vidale, 1988; Zhang and Toksöz,
1998). In contrast, alternative methodologies based on wavefronts are attractive
because the travel times are swiftly computed and the wavefronts can penetrate into
shadow zones, find diffracted trajectories and head waves (Vidale, 1988; Qin et al.,
1992; Zhang and Toksöz, 1998). Because of this, tomographic methods based on
wavefronts (or combined wavefronts and raytracing) have become increasingly
popular (e.g. Zhang and Toksöz, 1998; Zelt and Barton, 1998; Zelt et al., 1999;
Korenaga et al., 2000). The method of Vidale (1988) in particular is recognized to be
one of the fastest methods to compute travel times (Matarese, 1993 in Zhang and
Toksöz, 1998), this being particularly favourable for the development of joint
inversion.
This study adopts the procedure of Vidale (1988) and some of the improvements made
by Hole and Zelt (1995) to compute the field of travel times for each seismic source.
The methodology of Vidale (1988) is based on the propagation of the wavefront from
the source to the whole subsurface model producing the field of travel times ( ),( SrrT vv).
The propagation is performed by timing sequential nodes of the subsurface model
using a finite difference scheme for the eikonal equation (Equation 2-49) in expanding
rectangular rings that progress out from the source as illustrated in Figure 2-8.
42
Vidale (1988, 1990) developed and tested the accuracy of his approach for models of
the subsoil composed exclusively of equally spaced nodes. Taking this into account, I
discretize the two-dimensional model of the subsoil into a regular grid that divides the
cells used for inversion (common to the resistivity model as well) into regular sub-
cells with constant seismic velocity.
V(x,z)
Airx
z
Wave front
Nodesalready timed
Seismicsource (t=0)
Figure 2-8: Illustration of the propagation of a square wavefront as used in Vidale’s (1988)
progressive finite difference scheme. The crosses denote the nodes of the seismic grid (timed or to
be timed).
The finite difference scheme of Vidale (1988) uses several stencils according to the
position of the expanding nodes of undetermined travel time and their neighbours with
travel time known. These stencils are applied either individually or simultaneously to
determine the fastest arrivals. The stencils I applied are illustrated in Figure 2-9 and
the corresponding formulae are:
43
-Stencil type I for direct waves (Figure 2-9a)
⎩⎨⎧
+=hshs
tt2
11I min (2-51)
-Stencil type II (Figure 2-9b)
213
222II )(2 ttshtt −−+= (2-52)
-Stencil type III (Figure 2-9c)
)( 134122
2III ttshtt −−+= (2-53)
In first instance, the selection of any of these schemes depends on the node
availability. The stencil type II is preferred for more “circular” wavefronts where no
other front node is available (Vidale, 1988) and the node type III for plane wavefronts
propagating towards the node to be timed. For each linear front in expansion (see
Figure 2-8), the nodes are timed out in order of the arrival of the wavefront (the nodes
with the shortest expected travel times are timed first). In all cases the stencil type I is
applied to follow possible head waves.
The main drawback of the technique is that it is inaccurate for boundaries with large
velocity contrasts (Qin et al., 1992; Hole and Zelt, 1995). This inaccuracy results from
the strategy of front propagation that does not account for wavefronts that could arrive
from outside the already expanded area leading to violations of the causality principle
44
of the technique (see Figure 2-10) and producing negative radicands in Equations 2-
52 and 2-53. However, Hole and Zelt (1995) reduce this problem without sacrificing
the speed of the procedure (c.f. Qin et al., 1992).
s1
t1 tIh
s2
t1t2
t3
s
tIII
t1
t2
t3
tII
s1
s2
s +s1 2
a) b)
Figure 2-9: Stencils implemented for the forward finite difference scheme of the arrival times. a)
stencil type I to follow headwaves (Hole and Zelt, 1995), b) stencil Type II (Vidale, 1988), c)
stencil type III (Vidale, 1988). The stencils are modified from the original schemes to work with
discrete velocity for the cells rather than in the nodes.
For the computation of the required derivatives for inversion I trace the rays from the
field of travel times according to the definition of raypaths and then follow a
conventional procedure used in seismic tomography. Following Vidale (1988) the rays
are traced from the receivers to the source to ensure the arrival of a ray to every
receiver regardless of the existence of shadow zones and diffracted or head waves.
Once the rays are traced, they are used to compute the corresponding Frêchet
derivatives following the definition of the perturbation of the travel times for a given
trajectory as described in Hole (1992). In this procedure the derivatives are computed
assuming a small perturbation in the slowness of Equation 2-50 and a fixed trajectory
provided by the previous field of travel times using:
45
∫=0
),(l
dlzxst δδ (2-54)
where l0 refers to the ray path trajectory generated from the field of travel times
associated to a previous (or initial) velocity model. For a cell with constant velocity,
the derivative is simply the length of the linear segment of the ray in the cell in transit.
Figure 2-10 provides an illustrative example of the field of travel times ( ) and
raypaths on a two-dimensional model. This example shows the geometrical features of
the propagation and some distortions on the travel times that result from causality
violations in the forward modelling scheme of Vidale (1988).
),( SrrT vv
0 20 40 60 80 100 120 140 160 180 200Distance (km)
-50-40-30-20-10
0
Dept
h(k
m)
-50-40-30-20-10
0
2 3 4 5 6 7 8
Seismic Velocity (km/s)
A
Figure 2-10: Example of the timefield (dark contours in seconds) and raypaths (red lines) for a
source located at profile position 200 km. In this figure some causality violations occur on nodes
at surface position A because the rays that first reach them should come from trajectories
refracted on the sharp vertical contact rather than on the deepest layer.
46
3. Joint inversion using cross-gradients
The original contribution of my research for joint inversion includes: the exploration
of the combined effect of conventional geoelectromagnetic and seismic models, the
development of the cross-gradients as a mathematical measure of the structural
similarity, the incorporation of the cross-gradients constraint to produce structurally
based joint inversion methodologies for geoelectromagnetic and seismic refraction
data as well as the comparative evaluation of the results of joint inversion of synthetic
and field data.
The details of the developments and results are published in the constitutive papers
that are appended at the end of this thesis (Papers 1 to 5) and this chapter only
describes their contents in the context of the objectives and deliverables set in the
introductory chapter.
3.1. Combination of geoelectromagnetic and seismic refraction field data (Paper
1)
A basic step of this research is to gauge the potential of the combination of
geoelectromagnetic and seismic refraction data and explore their possibilities for joint
inversion. A geological site in Quorn in England was selected as a pilot field site (see
Figure 3-1). This site resulted convenient because a research group from Leicester
University had already carried out vast geophysical work, which included a profile
with collocated geoelectromagnetic and seismic measurements, data types considered
here for the development of joint inversion.
47
Figure 3-1: Map of the Quorn pilot field site. Collocated DC resistivity, audiomagnetotelluric and
seismic refraction data were measured along profile 20 W. The schematic geological section
illustrates the heterogeneous geological units expected and their stratigraphic position as
described in Paper 1. The inset shows the location of the site.
My pilot study of seismic inversion contributed to Paper 1. This paper presents the
conventional results of separate inversion and gauges their potential for geological
associations and geophysical (resistivity-velocity) correlations as derived for the
Quorn pilot experiment. Paper 1 also provides a comparative framework for assessing
the results of joint inversion on field situations.
The geophysical images presented in Paper 1 (Figures 3, 4 and 5) are obtained from
separate processes and they all show good agreement. While the similarities between
the resistivity images is easily justified by the DC resistivity and MT data themselves,
the concordance with the seismic velocity model may be questionable. Apart from the
geological target, the only linking factor is that the seismic initial model was based on
the MT model. Although this procedure may resemble that of the sequential inversion,
it is not a decisive factor because additional inversion experiments started with flat
models achieved identical results. This demonstrates that the geological target is
48
effectively sensed by the involved data sets and therefore, appropriate for testing joint
inversion.
The correlative cross-plots in Paper 1 compare the resistivity and seismic velocity
values of areas of the subsurface with common geoelectromagnetic and seismic data
coverage. In these cross-plots two characteristic trends are clearly distinguishable.
However, to ensure these trends are produced by the corresponding geophysical data,
the differences in discretization/sampling rate must be properly accounted for besides
the data coverage. For instance, the smooth MT model in Paper 1 (Figure 5) is
interpolated from a discrete model with coarse cells and this may have over-sampled
the trend depicted in Figure 6b in Paper 1.
3.2. Cross-gradients joint inversion for improved characterisation (Paper 2)
The results from Paper 1 encouraged me to focus on the search for a structural link
rather than a generalized geophysical relationship. For this I developed a mathematical
function of cross-gradients defined by:
),,(),,(),,( zyxqzyxpzyxt ∇×∇=v
(3-1)
where p(x,y,z) and q(x,y,z) are the model parameters and is the named cross-
gradients vector. This function is introduced in Paper 2, although its properties are
discussed in more detail in Paper 3 and in Chapter 4.
),,( zyxt→
49
Although alternative functions can be used to measure geometrical similarities (see
Papers 3 and 5), I prefer the cross-gradients function because it has several
mathematical advantages, which are summarized in Chapter 4. However, the non-
linearity of the cross-gradients function also poses some computational difficulties. To
overcome these difficulties and adopt the cross-gradients function as a basis for joint
inversion, I take several numerical approaches. I adopt an asymmetric three-cell
scheme for the cross-gradients function (among other alternative schemes) and expand
this function on first order Taylor series. This facilitates the incorporation of an
iterative inversion procedure such as that introduced in Paper 2.
The Quorn apparent resistivity and seismic refraction data are jointly inverted and the
models compared to those of the separate inversion in Paper 1. The jointly inverted
resistivity and velocity models are geometrically more similar suggesting that a
clearer geophysical characterisation is achievable from models resulting from joint
inversion. This is demonstrated in the resistivity-velocity cross-plot, which shows
trends that can be sub-classified in the search for the maximum possible detail (see
Figure 4 in Paper 2). The example also shows that the analysis of the data coverage
(e.g. seismic ray distribution) and the use of a common model sampling are important
to classify the resistivity-velocity trends and the associated subsurface zoning.
3.3. Cross-gradients joint inversion formulation for DC resistivity and seismic
refraction data (Paper 3)
Paper 3 is the core of my research. It shows the concept of cross-gradients in detail.
This paper explains how the cross-gradients function quantitatively evaluates the
50
structural similarity between two multidimensional models and how it provides an
objective link for structural joint inversion. Paper 3 also shows the numerical and
computational details of the algorithm that I developed for the joint inversion of DC
resistivity and seismic refraction data. This paper also gives proof of the
improvements of the models that resulted from the joint inversion of test and field data
in terms of structural similarity and closeness to the theoretical test models.
There are several considerations for the DC resistivity and seismic data processing. In
particular, the success on the inversion of the synthetic and field data in Paper 3
depended not only on the appropriate selection of the data and model discretization
but also on the balance between the involved covariance matrices (e.g CSS, Crr, C00,
CRR in Paper 3, Equation 10) and the damping factors (αr, αS, β). Whereas the
physical significance of the covariance matrices can be statistically founded, the
physical meaning of the damping factors is darkened by aspects such as the
standardization of the roughness values, the physical units adopted for each particular
example, the amount of constraints/data and the stability of the process. I find
particularly successful results when I select initial large values of β, which are
gradually reduced in logarithmic steps (see the named two-stage procedure in Paper
3). For simplicity, I prefer to hold the α values fixed during the process; however, I
select such value after repeated experiments (namely automatic or simple trial-error
methodologies) as performed in conventional inversion experiments.
51
3.4. Cross-gradients joint inversion of magnetotelluric and seismic refraction
data (Paper 4)
Paper 4 proves the generality of the cross-gradients concept for joint inversion when
the joint inversion methodology is applied to jointly invert magnetotelluric and
seismic refraction data for deeper targets. The models result of joint inversion of
magnetotelluric and seismic refraction data also show improved resemblance when
compared to models derived from conventional separate inversion.
The joint inversion of MT and seismic refraction data is carried out identically to that
of the joint inversion of DC resistivity and seismic refraction data shown in Papers 2
and 3. This involves not only the cross-gradients constraint but also other practical
aspects like the selection of damping factors and covariance matrices. The major
difference is that, due to the limited accuracy of the derivative formulation approach
adopted for MT, the appropriate selection of achievable values for the standard
deviation of the MT data is important to facilitate the convergence of the process.
The results of the joint inversion of Quorn MT and seismic data in Paper 4 confirm
the consistency of the joint inversion methodology as the presented models show
structures and MT resistivity-velocity trends that closely resemble those obtained in
Paper 2.
52
3.5. Cross-gradients approach for joint image reconstruction (Paper 5)
Paper 5 explores the generality of the joint inversion based on cross-gradients using
mathematical test functions and a new methodology of solution based on quadratic
programming for special band-limited images with the objective of pushing the
methodology beyond purely geophysical applications. The results are compared in
terms of model improvement, computational efficiency and memory requirements
with those obtained from separate inversion and from the joint inversion methodology
presented in Paper 3.
The approach presented in this paper is the result of experiments that I performed to
explore diverse formulations of objective functions and numerical solutions of the
cross-gradients constraint. For instance, I formulated an objective function based on
the RMS value of the cross-gradients (standard least squares formulation); however,
the process showed disadvantages in convergence and the results I obtained were not
successful. In contrast, the quadratic programming formulation of Paper 5, despite
some computational drawbacks, provides a convenient procedure to solve multiple
imaging problems apart from geophysics.
53
4. Summary of results and discussions
In general, the development of the concept of cross-gradients and the corresponding
algorithms for joint inversion are the main contributions of my research. The
publications that form the bulk of this thesis explain the cross-gradients concept, the
formulation for joint inversion and also present comparative results of separate and
joint inversion of geoelectromagnetic and seismic refraction data. The results prove
the goodness of the cross-gradients function as a link to the joint inversion of disparate
data sets in multidimensional environments and the computational methodology of
solution.
In this chapter I summarize the relevant results, discuss their implications and
highlight the achievements that contributed towards solving the issues set out in the
introductory chapter.
4.1. The complementary nature of multiple geophysical data
The results of the pilot experiment using separate inversion of geolectromagnetic and
seismic refraction field data from Quorn near Leicester in Paper 1 illustrate the
advantages of the combined use of multiple geophysical methods for studying
complex shallow targets. In this example, the corresponding electrical resistivity and
seismic velocity models show a significant agreement in aspects of the delineated
structures in both geophysical images, even when they are processed separately. The
concordance on the delineated structures and their resistivity and seismic velocity
values facilitate the association of possible geological units. In addition, the
54
characteristic resistivity-velocity cross-plots (Paper 1, Figure 6) show correlative
trends, which yielded some clues for the deduction of interrelationships of both
geophysical parameters for the Quorn site that could have petrophysical significance.
The Quorn pilot experiment described in Paper 1 also furnished an excellent
comparative framework for gauging the features of the models that were improved by
the joint inversion of the same data. The comparison of the results obtained by
separate and joint inversion experiments (Paper 1 and Paper 2) demonstrates the
superiority of joint inversion to produce better matching geophysical images with
clearer geophysical interrelationships.
4.2. The Cross-gradients concept for quantification of structural similarity
The most relevant features of the cross-gradients function that make it convenient for
the evaluation of model similarities, are:
i) It is quantifiable. The cross-gradients function (Paper 2, Equation 1) is defined by
the cross product of collocated property changes that result in a vector entity for each
position. The magnitude of this vector is indicative of the magnitude of the property
changes of two models at corresponding positions that point to mutually perpendicular
directions. Larger magnitudes are related to less similarity because they imply large
changes in the model parameters that are directionally inconsistent thus indicating the
existence of important physical boundaries that cross each other (see Figure 4-1;
Paper 5, Figure 1).
55
ii) It is objective. The magnitude of the cross-gradients function attains a value of zero
only for a state of complete structural similarity between two given models (if the two
properties vary at the same position, they must vary in the same direction). This
furnishes one objective target value for every corresponding position in the two
multidimensional models to achieve structural similarity. Some alternative functions
which do not offer this objective target are discussed in Paper 3.
iii) It is general. The cross-gradients function is defined mathematically and it does
not include any geophysical assumption. Because of this, it can evaluate similarities
between any pair of models or images irrespective of the physical parameters
involved, coordinate system, model parameterization, etc. For instance, Paper 2
(Equation 3) shows a discrete version of the cross-gradients function appropriate to
the parameterization and discretization of the resistivity and seismic velocity models
shown but alternative equations for different type of models can be easily derived.
This can apply to first derivative continuous or discontinuous (e.g. blocky or layered)
models.
iv) It is simple. The cross-gradients function is mathematically simple and shows no
problems of discontinuities or singularities. As a second order function, it is
compatible with conventional least squares techniques and fits in the context of other
quadratic functions.
Based on the properties discussed above, it is remarked that the cross-gradients
function properly address the questions q1 and q2 raised in the introductory chapter.
This novel function defines the geometrically common features that make any pair of
56
images or models to be perceived as similar or not and offer a means to quantify the
similarity.
4.3. The target objective function for joint inversion
Given the convenient features of the cross-gradients function, I combined this function
with other complementary constraints to formulate an objective function that, when
minimised, produces models that satisfy the requirements set in the definition of joint
inversion in the introductory chapter, which are:
i) Achieving structural similarity
Paper 2 introduces the concept of cross-gradients constraint (cross-gradients function
equal to zero for the whole model space) as the requirement for two models to be
structurally identical. This constraint is set out as part of the functional target that will
guarantee the similarity between two models in the form of full colinearity of
collocated property changes.
Because the cross-gradients constraint seeks similarity in terms of direction (i.e.
disregarding the amplitude of the changes), it also gives the constraint the flexibility
of accepting gradients equal to zero as indicative of similarity. This fact is important
because, as illustrated in Figure 4-1, those models with boundaries where one
physical property does not change are also admitted as similar. This situation seems to
be particularly important in some geological targets. For instance, the boundary
between fresh and salt groundwater constitutes a strong electrical resistivity boundary
57
but it represents no change in seismic velocity. The zero-gradient situation also
complements smoothness-flatness constraints to allow simple (flat) solutions for those
areas of the models not well constrained by the observed data. See for example
Figures 6c and 6d in Paper 3 and Figures 6e and 6f in Paper 5, where the property
values surrounding the geophysical targets are reduced to constant values.
p1
p1
p1
q1
q1
q1 q1q2
q2
p2
p2
p2
a) b) c)
p q=0x p q=0x p q=0x
Figure 4-1: Illustration of examples of model boundaries that are accepted in the cross-gradients
framework as structurally similar (a and b) and dissimilar (c).
ii) Achieving simultaneous data fit
The fit of the data is stated in a least squares sense, which is weighted by the standard
deviation of the data errors (Papers 2, 3, 4, and 5). This provides a convenient
statistical framework to balance the contribution of the independent data sets and
facilitates the mathematical and computational solution of the objective function.
The final misfit achieved in the joint inversion of different synthetic data and also
actual field data from DC resistivity (Papers 2, 3 and 5), magnetotelluric (Paper 4)
and seismic refraction methods (Papers 2, 3, 4 and 5), as well as other mathematical
58
functions (Paper 5) was satisfactory, attaining unity values for the normalized misfits
despite the limited accuracy of some of the forward computation methodologies
adopted in this study of joint inversion. This proves the general success of the least
squares formulation for data fit in joint inversion.
iii) Reducing non-uniqueness
Paper 3 (Equations 13 and 14) and Paper 5 (Figure 6) demonstrate that models that
simultaneously satisfy the geophysical data and the cross-gradients constraint are not
necessarily unique. Additional information like that provided by a priori models or a
smoothness assumption is then required to overcome this insufficiency of information.
For mathematical and computational convenience, I incorporated smoothness and a
priori model constraints in a least squares fashion weighting them by damping factors
and covariance matrices of the a priori model parameters, respectively. The results of
joint inversion in Papers 2, 3, 4 and 5 show how these two measures combine
appropriately with the cross-gradients constraint to solve those areas not fully
constrained by the data.
In case any deterministic information is known, it can also be incorporated into the
objective function. An example of this is given by the inequality constraints
incorporated in Paper 5, which reduce the range of model solutions and exclude
unfeasible models (e.g. Paper 5, Figure 6).
59
Papers 2 and 3 also discuss the possibility of incorporation of a priori petrophysical
relationships into the objective function to constrain the models; however, I did not
introduce this information into the objective function to focus on the implications of
the cross-gradients constraints for joint inversion.
4.4. The minimisation of the objective function for joint inversion
The objective functions formulated in Papers 2, 3, 4 and 5 include non-linear
functions for which a simple linear algebraic minimum is not achievable. Instead, I
developed solution methodologies that use exact linear algebraic (Papers 2, 3 and 4)
or a quadratic programming (Paper 5) minimisation of a linearized objective function
in an iterative procedure.
i) The linearized objective function
To formulate a generalized procedure of solution, the cross-gradients and the synthetic
response functions are substituted by their corresponding First order Taylor series
expansions (Papers 2, 3, 4 and 5) to produce a linearized version of the objective
function. Papers 2, 3 and 4 show that the minimisation of this linearized objective
function using Lagrange multipliers leads to a simple set of linear equations.
Similarly, Paper 5 shows that alternative solution methodologies for this objective
function can also be adapted to suit specific model requirements (e.g. inequality
constraints). The use of different solution schemes that overcome other computational
difficulties also seems viable (e.g. use of conjugate gradient schemes for large-scale
problems) but this has not been implemented here.
60
The models that satisfy this linearized objective function are updated iteratively until
they converge to the desired minimum of the original non-linear objective function.
Unfortunately, in this iterative procedure the convergence to a unique optimal model
depends not only on the sufficiency of the data and a priori information, but also on
the behaviour of the non-linear functions, the selected starting point and the trajectory
followed during the iterative process.
ii) The two-stage minimisation process
When exact a priori information about the features of the optimal models sought is not
available, a common choice to start an iterative process is the use of flat models. In
this case it is also appropriate because constant-value models simultaneously satisfy
the cross-gradients and smoothness constraints and help to concentrate the process in
the assimilation of model features that improve the data fit.
Paper 3 (Equations 9 and 14) show that for the prescribed objective function, starting
with an initial flat model when the model response functions are linear leads to the
conventional solutions of separate inversion straight from the first step. This
represents a local minimum and if the geometrical aspects of the models are
significantly different, the models find it difficult to move away from this minimum
following the linearized version of the cross-gradients constraint. As a result, the
models do not achieve geometrical similarity. Analogous problems are also found
whenever a large improvement on data fit is aimed for in a single iterative step. This
fact suggests the necessity of a gradual small-step approximation to models that
satisfy the data fit.
61
I found a satisfactory gradual fit procedure in the “two-stage minimization” process
described in detail in Paper 3 and illustrated in Paper 5 (Figure 2, reproduced here as
Figure 4-2). The target of the main stage is to guarantee that the optimal models
satisfy the final misfit and that the process is gradually carried out in small steps. The
final success of this stage is illustrated by the gradual misfit decrease in the test and
field examples (Paper 3, Figures 7 and 9; Paper 5, Figures 8 and 11) and the
achievement of the final target misfit (Paper 2, Figure 2; Paper 3, Figures 5 and 11;
Paper 4, Figure 3; Paper 5, Figure 8).
Read initial p and q modelsand regularization parameters
0 0
Select initial target misfit .
Start
yes
yes
no
no
Stop
Is the final target misfit =1 achieved?
Compute predicted data:f( ) and ( )p qg
Have the updated models changed significantly (i.e. >>0)?t
Update p and q models (quadratic programming solution)
Set normal equations (c, Q) as well as cross-gradients and inequality constraints
Reduce to new target misfit .
Mai
n st
age
itera
tion
Sub-
stag
e it
erat
ion
(sea
rch
for s
imila
r im
ages
)
(sea
rch
for d
ata
fit)
Figure 4-2: Illustration of the sequence of the two-stage minimization process (reproduced from
Paper 5, Figure 2)
62
A sub-stage process runs inside the main iterative loop and it searches for similar
models that satisfy the target misfit imposed for the main-stage. Paper 3 (Figure 10)
(reproduced here as Figure 4-3) shows pairs of models for several misfit targets that
illustrate the success of this procedure for the Quorn DC resistivity and seismic
refraction field data. Theoretically, exactly similar models can only be obtained when
the non-linear cross-gradients constraint is satisfied exactly. In the sub-stage process
this only occurs when the corresponding models from two consecutive iterations are
equal (i.e. when the model converges exactly). I also found in experimental results
that when two consecutive models tend to converge the RMS value of the cross-
gradients function tends to decrease (e.g. Paper 3, Figure 9). This suggests the
possibility of defining threshold values for the RMS of consecutive model changes as
stated in Paper 3 (Equations 17 and 18).
It is remarked that the discussions made in this and the previous sections answer the
question q3 raised in the introductory chapter and demonstrate the viability of the
simultaneous estimation of similar models that satisfy their respective data. This
includes the postulation of an objective function (discussed in section 4.3) and a
computational solution procedure (discussed in section 4.4).
63
Figure 4-3: Evolution of the joint inversion process. Shown are the resultant resistivity and
velocity models for each iteration. Note the gradual development of common structural features
in both sets of models during the process (Reproduced from Paper 3, Figure 10) .
64
4.5. Results for synthetic and field examples
i) Test models and processing parameters
The synthetic test examples in Papers 3, 4 and 5 are all composed of geometrically
simple but challenging and illustrative models that avoid the use of correlating
parameters. The test data were preferably generated using alternative forward
modelling schemes (Papers 3 and 4) and noise-corrupted to resemble normal
surveying conditions. The level of error is properly accounted for by the data error
variances included in the objective function, which determined the individual
contribution of the data in the joint models. The a priori model variances assumed for
those regions in the model space that are not constrained by the data are kept large to
allow the models enough flexibility to fit the actual data. The selection of appropriate
damping factors (as explained in Paper 3) is based on direct experimentation.
ii) Main features of the resulting models
The models resulting from the joint inversion of synthetic (Papers 3, 4 and 5) and
field data (Paper 2) are always compared to those derived from conventional separate
inversion. When several methods of solution are discussed (Paper 5), then the models
so generated are also compared. These comparisons show that:
- The models resulting from joint inversion are more geometrically similar
than those obtained from separate inversion of the same data. This is
based on the cross-gradients values (Paper 3, Figures 8 and 12; Paper 5,
65
Figure 7), which are at least ten times smaller that those computed from
models obtained from separate inversion.
- The theoretical responses of the jointly inverted models fit the data to the
specified level of error (Papers 2, 3, 4 and 5) and the random nature of
the residuals is presented in some examples (Paper 5, Figure 8).
It is remarked that the experimental results of joint inversion using cross-gradients in
synthetic and true geological targets demonstrate the existence of models that have the
geometrically concordant features expected for joint inversion thus addressing the
question q4 raised in the introduction. The results in Paper 5 (Figure 6) show how the
manipulation of additional a priori constraints can influence the solution and provide
several different pairs of models that are similar between them and fit the data. This
example, apart of addressing q5, demonstrates the importance of a priori information
in joint inversion.
4.6. The implications of jointly inverted models for subsurface characterisation
The comparison between separate and joint inversion results of Quorn field data show
the improvement of the model in relation to the geometrical similarity between the
seismic and electrical images. To explore the implications of this geometrical
similarity one step forward, the direct geophysical values are analysed in an attempt to
demonstrate that joint inversion using cross-gradients (despite being geometrically
founded) could also provide a means to facilitate a better geological association. The
resistivity and seismic values in Paper 2 are thus compared pixel by pixel to find out
66
whether the geophysical correlations occurring in the separately inverted models in
Paper 1 are enhanced or not in the models resulting from joint inversion.
Paper 2 (Figure 4) shows the distribution of the characteristic resistivity-velocity
cross-plot as extracted directly from the resistivity and seismic models resulting from
joint inversion. The plot confirms the existence of characteristic trends that are
correlatable to local zones in the subsurface. This may indicate some consistency of
the geological materials and possibly petrophysical attributes. However such an
association is not tested by invasive means at the studied site and there is also the
possibility that those trends correspond to an artefact induced by the combination of
smoothness and cross-gradients constraints rather than by the geophysical data.
It is remarked that the correlative trends found, whatever their explanation, could be
useful for geological discrimination and zoning and the chances of deducing some
petrophysical features seem better than from results of separate inversion. However,
additional work is required to gauge this potential and properly answer the question q6
set in the introductory chapter.
4.7. Overall assessment of jointly inverted models
Perhaps the most relevant questions to answer are whether the models resulting from
joint inversion of disparate data sets are more precise than those obtained from
separate inversion and whether the geological deductions made from them are more
reliable or not. The answers to these questions would require a detailed analysis
beyond the scope of the present thesis. However, while this thesis has no formal
67
section treating uncertainty analysis, it contains some relevant results that give some
insight into this issue.
The results of joint inversion of Quorn field data shown in Paper 2, demonstrate the
existence of similar geophysical models that satisfy the geophysical data. The test
models in Papers 3, 4 and 5 were designed to be structurally identical (i.e. to satisfy
the cross-gradients constraint). In the later case, the two-stage solution procedure for
joint inversion using cross-gradients offered a means to predict models with good
resemblance that fit noise corrupted data satisfactorily and are closer to the theoretical
test models.
The formulation and solutions to the linearized objective function for joint inversion
using Lagrange multipliers in Papers 2, 3 and 4 make clear that the cross-gradients
constraint does not solve the problem of non-uniqueness per se. The linearized
solution (e.g. Paper 3, Equations 13 and 14) requires the inverse of the normal
equation matrices from the standard least squares formulation, i.e. it requires that least
squares solutions for the separate inversions exist. This suggests that the uniqueness of
model solutions largely depends on the proper data coverage, data precision (data
weighting) and damping factors. The multiple models shown in Paper 5 (Figure 6)
give account of the existence of multiple solutions that achieve satisfactory data fit
and model similarities.
While a detailed statistical sampling of the space of model solutions of joint inversion
seems unaffordable even for small-scale problems, the changes of a model around a
specific optimal point can be evaluated more easily. A common procedure like linear
68
statistical measures may be useful for this purpose, but this issue is left as a topic for
future study.
The problems of non-uniqueness and accuracy of the jointly inverted models should
have an impact in the final geological deductions and therefore must be taken into
account in future research. A subsurface zoning based exclusively on models resulting
from joint inversion (e.g. Paper 2, Figures 4 and 5) still deserves to be more critically
evaluated. For this, conventional statistical tests of the geophysical models and
comparison with ground truth data may be useful.
69
5. Conclusions and suggestions for further research
From the results obtained and the discussion presented earlier, it is notable that the
concept of cross-gradients for structural joint inversion that I developed contributes
towards solving some of the issues raised in the context of joint inversion from a
structural point of view. At the same time it also uncovers new questions that pose
new challenges in the field of joint inversion worthy of future research. In this chapter
I summarize the relevant conclusions driven by my research and stress some key
issues that need to be considered in future studies of joint processing of multiple
multidimensional images. I also suggest particular aspects of joint inversion where the
concept of cross-gradients may play an important role in the near future.
5.1. Conclusions
5.1.1. The concept of cross-gradients
The cross-gradients function defines a geometrical interrelationship between any two
multidimensional images obtained from geophysical or other imaging methodologies
and furnishes a simple, generalized and quantitative means to evaluate structural
(geometrical) similarities between them.
The cross-gradients function, applied as a constraint, offers an objective target in the
search for geometrically similar multidimensional models. This objective target
proved compatible and complementary with some of the most conventional
70
constraints used in geophysical inverse problems (least squares data fitting,
smoothness, ridge regression and inequality constraints).
In the field of geophysical inversion, the structural link provided by the cross-
gradients constraint made possible the joint inversion of the disparate geophysical data
studied here (DC resistivity/magnetotelluric and seismic refraction data). Because of
its generality, this constraint could link geophysical models from heterogeneous
environments regardless of the knowledge of direct relationships between the
geophysical parameters. The concept of cross-gradients also appears to offer good
promise for the simultaneous processing of multiple images in other scientific fields.
The cross-gradients constraint furnishes a link that is not only generalised but also
flexible because it does not necessarily induce models with abrupt collocated property
changes. In fact, it allows models with common boundaries defined by gradual
property changes and it even permits models with boundaries undetected by one data
set but sensed by the other. It therefore enhances the ability to exploit the peculiarities
of different geological environments.
5.1.2. The joint inversion algorithm
The cross-gradients constraint could be easily combined with other conventional
constraints to define appropriate objective functions to jointly process
geoelectromagnetic and seismic refraction data in the pursuit of structurally similar
models that satisfy the data. The inconvenience of being a non-linear equality
71
constraint could be solved by adopting an iterative method of solution based on a
linearized scheme in the form of a two-stage minimization process.
The two-stage minimization process adopted in this study is convenient because it
solves the objective function in steps that allow the gradual incorporation of common
structural features in the models thus facilitating the search for two similar models that
satisfy the target misfit. An important part of this process is how to update the models
for a particular iteration. For this purpose, the Lagrange multiplier method was
adopted and found to be efficient because it led to simple algebraic linear solutions
that optimise the processing time and memory requirements.
5.1.3. The features of models resulting from joint inversion.
The models resulting from joint inversion using cross-gradients were derived
simultaneously following objective criteria that control their structural similarities and
data fit. Because of this, the models found were geometrically similar and fit the data
to their respective statistical target levels.
The structural similarity achieved for the resulting models is defined by the
distribution rather than the actual magnitudes of the geophysical parameters. This not
only facilitates the deduction of possible geological associations but also allows the
possibility of estimating additional characteristic geophysical trends at a post-
processing stage that may help to infer attributes with geological meaning.
72
5.2 Suggestions for further research
The cross-gradients concept developed in my research has so far provided an answer
to the need for a generalized procedure for the joint inversion of disparate data sets.
My research has also shown the advantages inherent in the models derived by joint
inversion. As such, all the limitations in the forward modelling algorithms adopted
and the assumptions made to simplify or accelerate the process of experimentation
taken as a personal choice should be re-visited in future work and new features to
tackle specific scientific/economical problems must be incorporated.
i) The quest for more accuracy:
The schemes adopted for forward and derivative computation in this thesis were
selected mainly on the basis of a computational speed. This limited the resolution
capability and amount of detail that the data could have provided to the two-
dimensional models presented. It is highly desirable that future research in this field
considers the incorporation (or judicious combination) of more accurate forward and
derivative computation schemes.
One aspect that is not solved by the cross-gradients constraints is the use of theoretical
models that are not compatible with the geological reality. Examples of this are the
assumptions of two-dimensional models in highly three-dimensional environments.
This, apart from making the search for similar models more difficult, may erroneously
incorporate “common” but non-existent features into the models. Another example is
the under-sampling of the physical models due to limitations in computing capabilities
73
so that the detailed common features sought by joint inversion may remain
undetected. These two aspects however can only be solved by redesigning the
subsurface models themselves and undertaking the challenges this may pose, such as
incorporation of algorithms for the solution of large-scale problems or restating the
original mathematical formulations of solution of the objective function shown here.
The results shown in this thesis clearly point towards a gained confidence in the
geophysical models. However, there is not a quantitative assessment of such
improvement. At the present time, I cannot state by how much the cross-gradients
technique reduces the non-uniqueness of the individual models and whether all the
similar models derived from joint inversion will always (as in the synthetic examples
presented here) be closer to the true model or not. Similarly, models resulting from
joint inversion can be regarded as “biased” towards better estimates at minimum cost
to the data fit; however, the uncertainty of the models is not yet quantified and its
consequences on the geophysical associations should not be obviated. In analogy with
the individual inversion case, the issue of model assessment is important and deserves
by itself a whole set of research.
ii) the quest for efficiency
The numerical-computational methods I applied to minimise the set objective function
for joint inversion can be considered appropriate for the number of variables involved
in two-dimensional models. However, in problems of larger scale (especially three-
dimensional) models or for “real-time” solutions they might become inefficient. Any
74
research in this direction must find new (or adapt known) faster methodologies for the
forward, derivative or inverse computations.
iii) Towards new integrated fields and applications
The implications of the models resulting from joint inversion in economic or scientific
applications is a very attractive issue as it involves not only a clearer source of
subsurface information but also the derivation of “geophysical signatures” that might
offer the opportunity of new deductions that seem unreachable from conventional
separate techniques.
The concept of cross-gradients is not restricted to geophysical assumptions and as
such it can be incorporated in any scientific field that deals with processing of
multidimensional images. It offers the possibility of incorporation of other
geophysical methodologies or even cross-field images that can provide new integral
views of the studied target.
The use of cross-gradients for the integration of different fields and the exploration of
applications to specific geological environments and other targets will demand the
future attention and involvement of multidisciplinary research teams.
The solution offered by this generalized joint inversion method constitutes not only
one step forward towards the scientific understanding of complex systems like the
earth but also opens vast new fields for future advances in scientific research in areas
75
like image fusion in remote sensing imagery, medical image registration, automated
pattern recognition for signal analysis, etc.
76
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Vozoff, K., Jupp, D. L. B. (1975) Joint inversion of geophysical data. Geophys. J. R.
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Wyllie, M. R. J., Gregory, A. R., Gardner, L. W. (1956) Elastic wave velocities in
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Yang, C. H., Tong, L. T., Huang, C. F. (1999) Combined application of dc and TEM
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83
Paper 1
Evidence for correlation of electrical resistivity and seismic velocity
in heterogeneous near-surface materials
by Max A. Meju, Luis A. Gallardo and Adel K. Mohamed
84
Evidence for correlation of electrical resistivity and seismic velocity in
heterogeneous near-surface materials
Max A. Meju and Luis A. Gallardo1
Department of Environmental Science, Lancaster University, Lancaster, UK
Adel K. Mohamed2
Department of Geology, University of Leicester, Leicester, UK
Received 4 August 2002; revised 24 October 2002; accepted 26 November 2002; published 4 April 2003.
[1] The electrical resistivity and seismic velocitydistributions over a buried hillside have been obtainedusing non-invasive controlled experiments on coincidentprofiles and 2D image reconstructions. The optimal imagesare in structural agreement and allow the deduction of twoopposite resistivity-velocity trends in the near-surfacematerials. For both trends, the resistivity (r) and p-wavevelocity (Vp) are related in the form Log10 r = mLog10Vp + cwith the respective constantsm and c having different signs inunconsolidated and consolidated materials. INDEX
TERMS: 5102 Physical Properties of Rocks: Acoustic properties;
5109 Physical Properties of Rocks: Magnetic and electrical
properties; 5114 Physical Properties of Rocks: Permeability and
porosity. Citation: Meju, M. A., L. A. Gallardo, and A. K.
Mohamed, Evidence for correlation of electrical resistivity and
seismic velocity in heterogeneous near-surface materials,Geophys.
Res. Lett., 30(7), 1373, doi:10.1029/2002GL016048, 2003.
1. Introduction and Problem Definition
[2] Electrical and seismic relationship in the subsurface[Faust, 1953] is a subject of on-going debate as correlationbetween anomalous electrical conductivities and low veloc-ities are increasingly observed in non-invasive deep crustalstudies [see e.g., Marquis and Hyndman, 1992 and refer-ences therein]. In much of the attempts to reconcile elec-trical and seismic observations in deep wells, the commonthread is that resistivity and velocity are both functions ofporosity [see e.g., Rudman et al., 1975] which is also theunifying assumption in non-invasive experiments currentlyfocusing on correlating deep crustal data of variable qualityfrom approximately coincident regional studies [e.g., Mar-quis and Hyndman, 1992]. There is a need to studyheterogeneous near-surface materials for any such relation-ships especially as this may have implications for improvedstructural [cf. Eberhart-Phillips et al., 1995], petrophysicaland environmental characterizations and for the develop-ment of algorithms for effective joint multidimensionalinterpretation of electrical and seismic field data. If porosityis also the connecting factor in the near-surface, it is logicalto expect that additional insight may be gained fromcollocated studies of exposed fractured crystalline and
porous sedimentary materials. Portable audiofrequency mag-netotelluric (AMT), transient electromagnetic (TEM) and dcresistivity (herein collectively dubbed geoelectromagnetic orGEM) as well as seismic refraction methods can be adaptedto near-surface studies. Also, the available sophisticatedmulti-dimensional inverse modelling schemes for interpret-ing traditional GEM and seismic field data [e.g., Mackie etal., 1997; Zelt and Barton, 1998] can be appropriately scaledto handle near-surface imaging problems. It is thus opportuneto collect high quality, spatially dense measurements alongthe same survey lines and invert them to determine anyresistivity-velocity relationships at shallow depths.[3] In this letter, we present the results of coincident GEM
and seismic experiments to investigate near-surface resistiv-ity-velocity relations at a selected area in Quorn in England(Figure 1). The Mountsorrel granodiorite (MG) forms thebedrock in Quorn and surrounding areas. This body wasunroofed, deeply weathered and eroded (resulting in a highlyirregular surface) during Permo-Triassic times and was sub-sequently overlain by the Mercian Mudstone (MM) deposits.Heterogeneous glacial drift deposits form a 1–3 m thicksurficial blanket in the area. MG outcrops in the southernmargin of the study site and is believed to descend northwardsunder sedimentary cover. It is heavily fractured at outcrop andpresumably at depth (based on field observations at the largesthardrock quarry in western Europe located ca. 400 m south of
Figure 1. Location map showing the geophysical surveygrid at Quorn in England. The lines run N-S and are 20 mapart.
GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 7, 1373, doi:10.1029/2002GL016048, 2003
1Also at CICESE, Mexico.2Now at Dept. of Geology, Mansoura University, Mansoura, Egypt.
Copyright 2003 by the American Geophysical Union.0094-8276/03/2002GL016048$05.00
26 -- 1
our survey grid). The Quorn site is thus an excellent naturallaboratory for testing the applicability of the hypothesis ofelectrical-seismic relations in heterogeneous porous and frac-tured materials. The questions we seek to answer using a 2Ddata imaging approach are: (1) Are there any correlatabletrends in the vertical and lateral distribution of resistivity andvelocity within MG, MM and cover materials? (2) Do thenear-surface resistivity-velocity trends follow those predicted[e.g.,Marquis and Hyndman, 1992] for deep crustal systemsand if not, what are the possible causes of the discrepancy?
2. Field Experiments and Model Correlations
2.1. Collocated High-Resolution Profiling
[4] TEM, dc resistivity, seismic refraction and AMTsurveying, in that order, have been conducted at the Quornsite. The site is a relatively flat grazing ground and topo-graphic heights were available from a previous differentialGPS survey using the Magellan 5000 PRO system. The
TEM profiling employed contiguous (20 m-sided) trans-mitter loops along six N-S survey lines (80E to 20W) shownin Figure 1 and served to pinpoint any spatial variability orsignificant fracture-zones in the bedrock and hence the bestlocation for the collocated 2D GEM and seismic experi-ments. Areal maps of the TEM voltage responses forselected time-windows (not presented here) showed spatialvariability with significant differences in amplitude betweennorth and south of position 180S.[5] Line 20W was chosen for detailed 2D profiling
experiments based on TEM information. Bi-directionalSchlumberger dc soundings were made at selected positions
Figure 2. Example of TEM and bi-directional dc andAMT sounding curves from position 45S on line 20W.Shown are the north-south (xy) and east-west (yx) apparentresistivities.
Figure 3. 2D resistivity model for line 20W. Shown are the optimal model (top plot) and the fit of the model responses(ornamented solid line) to field data (round symbols) at six sounding locations (bottom plot).
Figure 4. Optimal 2D velocity model for line 20W. Themodel is shown in the bottom diagram. The fit to the fieldrecordings for different shot points (differentiated bysymbols) is shown in the top plot.
26 - 2 MEJU ET AL.: NEAR-SURFACE RESISTIVITY-VELOCITY CORRELATION
(ca. 40 m apart) with N-S and E-W expanding electrodearrays (AB/2 of 1.5 to 90 m). Seismic travel-time data wererecorded along the line using a multi-channel seismographwith a sledgehammer as energy source and a geophonespacing of 2 m. The source was used at both ends of theprofile and at two intermediate points along the line togenerate continuous forward and reverse profiles of poten-tial refractors. Finally, AMT data were simultaneouslyrecorded in two orthogonal directions in the frequencyrange 10 Hz to 100 KHz using a station spacing (andelectric dipole lengths) of 15 m. Sample dc, TEM andAMT apparent resistivity (ra) data from station 45S on line20W are presented in Figure 2 using a convenient common-scale [Meju, 2002, equations 1 and 2] in which AB/2 (or Lin metres) is converted to the equivalent transient time (t inmsec) using the relation t = (pm0L
2)/2ra where ra is in �mand m0 = 4p � 10�7 H/m; t is then converted to anequivalent MT frequency. Notice the agreement betweenthe various ra sounding curves. The AMT data are relativelypoor in quality.
2.2. Model Comparability: Resistivity-VelocityRelations
[6] The inverse problem was to reconstruct the smoothest2D distribution of the relevant physical parameters in thesubsurface that explained the field observations to within apreset (1 rms) error. Only the in-line (N-S) measurements online 20W have been inverted to yield 2D images requiredfor the comparability analysis. The in-line AMT data weretaken as the TM-mode responses, corrected for static shift
using TEM data (cf. Figure 2) [e.g., Sternberg et al., 1988],and the noisy sections smoothed before inversion. Popular,finite-difference based, conjugate gradient inversionschemes [Mackie et al., 1997; Zelt and Barton, 1998] wereadapted to the task of imaging the Quorn AMT apparentresistivity and seismic travel-time data. A different 2Dinversion algorithm [Perez-Flores et al., 2001] was usedfor imaging the dc resistivity data.[7] The optimal models from dc resistivity (Figure 3),
seismic refraction (Figure 4) and AMT (Figure 5) imagingshow similar subsurface structural features suggesting thatthere may be a geological basis for correlating these models.The configuration of the boundary between the bedrock andits cover materials can be discerned (approximated by the100 �m and 3000 m/s contours) in these models.[8] The dc and AMT resistivities are in good accord
(Figures 3 and 5) and so either model can serve forcorrelation with seismic velocity. An interesting observationis that the resistivity (r in �m) and p-wave velocity (Vp inm/s) distributions (sampled at coincident grid positions orpixels in the 2D models) seem to be related in the form (seeFigures 6a and 6b)
Log10r ¼ mLog10VP þ c ð1Þ
where the constants m and c respectively have values of 3.88and �11 for the consolidated rocks (>3m deep) at this site(see trend B in Figure 6a). An inverse relation appears tohold for the unconsolidated soil/drift deposits (i.e., top 3 m)where m =�3.88 and c = 13 (see trend A in Figure 6a). Notethat Rudman et al. [1975, equation 10] interrelated ra andvelocity logs from 700–1300 m deep wells (see Figure 6b)using an equation derived assuming ra and Vp to be functionsof porosity. If we further assume that the transit time of theelastic wave in the solid grains is very small compared to thatin the pore fluid, their equation simplifies to Log10 ra =(mLog10Vp � mLog10B) where m and B are empiricalconstants. This is identical to our experimentally determinedrelation for the consolidated rocks at Quorn and wouldsuggest that porosity is also a connecting factor for resistivityand velocity in the near-surface.[9] The Quorn AMT-seismic relation is compared in
Figure 6b with the predicted resistivity-velocity trend for
Figure 5. Optimal AMT resistivity model for line 20W.The 13 sounding positions (15m apart) are indicated at thetop.
Figure 6. Relationships between logarithmic resistivity and velocity. Shown are: (a) Dc resistivity and (b) AMT resistivityversus seismic p-wave velocity on line 20W. The depth of sampling (in metres) is shown for selected points (pixels). Note theidentified trends A and B of inverse slope in (a). Trend B was constrained to pass through well estimated points thus givingless emphasis to contributions (e.g. zone C in (a)) from unresolved deep features in our seismic model. In (b), trends D and Eare taken respectively from Marquis and Hyndman [1992, Figure 4] and Rudman et al. [1975, Figure 7] for comparison.
MEJU ET AL.: NEAR-SURFACE RESISTIVITY-VELOCITY CORRELATION 26 - 3
a deep well [Rudman et al., 1975, Figure 7] and a lowporosity deep crustal system (aspect ratio of 0.01 andArchie’s law exponent of 1.2) [Marquis and Hyndman,1992, Figure 4]. Notice that the various curves (representingdifferent crustal depths) show the same basic trend. Furtherwork is required to fully understand the significance ofthese relationships or trends.[10] Laboratory measurements on cores [Mazac et al.,
1988] suggest that resistivity increases with decreasingsaturated permeability in the aerated zone of heterogeneoussoils over weathered granite. It is also known that Vp
increases with degree of grain packing in unconsolidatedmaterials while Vp increases as the natural logarithm ofpermeability in consolidated materials [Marion et al., 1992].It is thus probable that fracture or saturated permeabilitydecreases with depth in MM and MG with a correspondingrise in both resistivity and velocity. In the granular coversediments (top 3m), saturated permeability would appear toincrease (and hence r decreases) with depth. Accordingly,and because of probable air pockets in the shallower vadosezone, Vp appears to increase with depth in these coversediments causing the observed reversed resistivity-velocitytrend (A in Figure 6a).
3. Conclusion
[11] Two-dimensional imaging of data from GEM andseismic profiling over porous sediments and fracturedgranodiorite at Quorn have yielded concordant images ofthe near-surface. Analysis of the 2D images suggests thepresence of correlatable trends in the near-surface resistivityand velocity distributions at this site and is interpreted aslending support to the hypothesis that porosity or fracturepermeability may be a key factor in understanding electri-cal-seismic relations in both consolidated and unconsoli-dated crustal materials. We suggest that joint 2D imaging ofGEM and seismic profile data may be a useful strategy for
improved resistivity-velocity correlations in near-surfacestudies.
[12] Acknowledgments. The authors are grateful to Doug Groom forproviding the STRATAGEM-EH4 system and Peter Fenning for providingthe TEM field system used in this study. We thank Nasir Ahmed for makingthe seismic refraction data available and Colin Zelt and M. Perez-Flores forpermission to use their inversion codes. We thank two anonymousreviewers for their very constructive comments.
ReferencesEberhart-Phillips, D., W. D. Stanley, B. D. Rodriguez, and W. J. Lutter,Surface seismic and electrical methods to detect fluids related to faulting,J. Geophys. Res., 97, 12,919–12,936, 1995.
Faust, L. Y., A velocity function including lithologic variation, Geophysics,18, 271–288, 1953.
Mackie, R., S. Rieven, W. Rodi, User manual and software for two-dimen-sional inversion of magnetotelluric data, Earth Resources Lab., Mass.Inst. of Technol., Cambridge, 1997.
Marion, D., A. Nur, H. Yin, and D. Han, Compressional velocity andporosity in sand-clay mixtures, Geophysics, 57, 554–563, 1992.
Marquis, G., and R. D. Hyndman, Geophysical support for aqueous fluidsin the deep crust: seismic and electrical relationships, Geophys. J. Int.,110, 91–105, 1992.
Mazac, O., M. Cislerova, and T. Vogel, Application of geophysical methodsin describing spatial variability of saturated hydraulic conductivity in thezone of aeration, J. Hydrology, 103, 117–126, 1988.
Meju, M. A., Geoelectromagnetic exploration for natural resources: models,case studies and challenges, Surv. Geophys., 23, 133–205, 2002.
Perez-Flores, M. A., S. Mendez-Delgado, and E. Gomez-Trevino, Imaginglow-frequency and dc electromagnetic fields using a simple linear ap-proximation, Geophysics, 66, 1067–1081, 2001.
Rudman, A. J., J. F. Whaley, R. F. Blake, and M. E. Biggs, Transformationof resistivity to pseudovelocity logs, AAPG Bull., 59, 1151–1165, 1975.
Sternberg, B. K., J. C. Washburne, and L. Pellerin, Correction for the staticshift in magnetotellurics using transient electromagnetic soundings, Geo-physics, 53, 1459–1468, 1988.
Zelt, C. A., and P. J. Barton, Three-dimensional seismic refraction tomo-graphy: a comparison of two methods applied to data from the FaeroeBasin, J. Geophys. Res., 103, 7187–7210, 1998.
�����������������������M. A. Meju and L. A. Gallardo, Department of Environmental Science,
Lancaster University, Lancaster, LA1 4YQ, United Kingdom. ([email protected]; [email protected])A. K. Mohamed, Department of Geology, Mansoura University,
Mansoura, Egypt.
26 - 4 MEJU ET AL.: NEAR-SURFACE RESISTIVITY-VELOCITY CORRELATION
Paper 2
Characterization of heterogeneous near-surface materials by joint 2D
inversion of dc resistivity and seismic data
by Luis A. Gallardo and Max A. Meju
89
Characterization of heterogeneous near-surface materials by joint 2D
inversion of dc resistivity and seismic data
Luis A. Gallardo1 and Max A. MejuDepartment of Environmental Science, Lancaster University, Lancaster, UK
Received 19 March 2003; revised 7 May 2003; accepted 22 May 2003; published 1 July 2003.
[1] We have developed a robust 2D joint inversionscheme incorporating the new concept of cross-gradients ofelectrical resistivity and seismic velocity as constraints soas to investigate more precisely the resistivity-velocityrelationships in complex near-surface environments. Theresults of joint inversion of dc resistivity and seismictraveltime data from collocated experiments suggest thatone can distinguish between different types or facies ofunconsolidated and consolidated materials, refining apreviously proposed resistivity-velocity interrelationshipderived from separate inversions of the respective datasets. A consistent interpretive structural model can beobtained from the joint inversion models. INDEX TERMS:
0935 Exploration Geophysics: Seismic methods (3025); 0902
Exploration Geophysics: Computational methods, seismic; 0925
Exploration Geophysics: Magnetic and electrical methods; 3260
Mathematical Geophysics: Inverse theory. Citation: Gallardo,
L. A., and M. A. Meju, Characterization of heterogeneous near-
surface materials by joint 2D inversion of dc resistivity and
seismic data, Geophys. Res. Lett., 30(13), 1658, doi:10.1029/
2003GL017370, 2003.
1. Introduction
[2] Establishing the precise relationship between electricalresistivity and seismic compressional (P) wave velocity inheterogeneous near-surface materials is a fundamental prob-lem in hydrogeophysics andmay lead to improved petrophys-ical characterization necessary for understanding flow andtransport processes in shallow-depth environments. Correla-tions between these two physical properties are increasinglyobserved at different spatial scales in collocated experiments[e.g., Rudman et al., 1975; Meju et al., 2003 and referencestherein] but the relevant data from multidimensional non-invasive dc resistivity and seismic refraction investigationsof the near-surface have conventionally been inverted sepa-rately leading sometimes to unequivocal models. Joint in-version of such data is a better approach as shown forone-dimensional (1D) structures [e.g., Hering et al., 1995;Manglik and Verma, 1998] and allows an objective testing ofthe current resistivity-velocity interrelationships derivedfrom separate interpretations of experimental data [e.g.Mejuet al., 2003]. However, joint multidimensional resistivity-velocity inversion is a difficult task as there is no establishedanalytical relationship between resistivity and velocity.[3] There are two emerging philosophical approaches to
joint 2D inversion which are underpinned either by the
assumption that resistivity and velocity are both functions ofporosity and water saturation (petrophysical approach) orthat both methods are sensing the same underlying geologywhich in turn structurally controls the distribution of petro-physical properties (structural approach). The petrophysicalapproach is attractive but the validity or efficacy of theexisting correlation models in general geological media is amatter of ongoing debate since there is no simple or singlerelationship for accurately predicting the whole range ofeffects like variations in clay content, shape or interconnec-tions of the pores on the geophysical measurements in truefield situations [see Kozlovskaya, 2001]. Our concern hereis: how can we recover a reliable joint inversion model overporous as well as non-porous heterogeneous natural materi-als for which the predictive model relating resistivity andseismic velocity may not always hold?[4] We prefer the structural approach and posit that
petrophysical information may be derived from the resultantmodels. To our knowledge, the only reported account of two-dimensional joint structural inversion of non-invasive dcresistivity and seismic sounding data in the literature is thatof Zhang and Dale Morgan [1996] where, using syntheticdata, they assume that the laplacian of a dc resistivity imageshould resemble that of the seismic image. Their approachrequired a scaling factor to weight the relevance of each part(i.e., dc or seismic laplacians) and the reported syntheticexample showed a direct correlation between the resistivitiesand seismic velocities, which may not always be the case infield situations. This leads us to ask the question: Is itpossible to develop a generalised quantitative criterion forevaluating resistivity-velocity images for congruency orsimilarity in complex environments with heterogeneousmaterials? In any case, both the petrophysical and structuralapproaches are somewhat restrictive and as yet, there is nopublished technique that equably and adequately satisfies thedata from both methods and is flexible enough to handlepossible dissimilarities in the distribution of their relevantpetrophysical properties. This poses the question, can joint2D inversion really offer a means for improved character-isation of heterogeneous materials? The main thrust of thispaper is to address some of the issues highlighted aboveusing joint structural inversion incorporating the concept ofresistivity-velocity cross-gradients.
2. Methodology for Joint 2D Resistivity-SeismicInversion
2.1. Problem Formulation With aCross-Gradients Function
[5] There are many algorithms for inverting seismic or dcresistivity data [e.g., Zelt and Barton, 1998; Perez-Flores et
GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 13, 1658, doi:10.1029/2003GL017370, 2003
1Also at Division de Ciencias de la Tierra, CICESE, Mexico.
Copyright 2003 by the American Geophysical Union.0094-8276/03/2003GL017370$05.00
1 -- 1
al., 2001]. The main difference is that we adopt a jointproblem formulation with a novel resistivity-velocity cross-gradients function (Gallardo and Meju, manuscript inpreparation, 2003) in order to provide the required effectivelink between the dc resistivity model and the seismicvelocity model. We use the cross product of the gradientsdefined as
!t x; y; zð Þ ¼ rmr x; y; zð Þ � rms x; y; zð Þ ð1Þ
where mr and ms refer to the logarithm of resistivity and theP-wave slowness respectively. In the two-dimensional case,!t x; y; zð Þ always points in the strike direction, i.e., will betreated as a scalar t. It is incorporated as part of the objectivefunction, viz: Minimize
� mr;msð Þ ¼ dr � f r mrð Þds � f s msð Þ
��������2
C�1dd
þ arDmr
asDms
��������2
þ mr �mRr
ms �mRs
��������2
C�1RR
subject to t mð Þ ¼ 0: ð2Þ
[6] Here, d represents the vector of observed data(logarithm of apparent resistivity, dr and seismic travel-times, ds), m = [mr: ms]
T is the vector of the modelparameters, f is the theoretical model response, D is thediscrete version of a smoothing operator acting on m,mR = [mRr: mRs]
T is an a priori model, Cdd is thecovariance of the field data (assumed diagonal, i.e., fullyuncorrelated data), CRR is the covariance of the a priorimodel (assumed diagonal) and ar and as are weightingfactors to control the level of smoothing of the resistivityand seismic models.[7] The cross-gradients criterion requires the problem to
satisfy the condition t(m) = 0, where any spatial changesoccurring in both resistivity and velocity must point in thesame or opposite direction irrespective of the amplitude. Ina geological sense this implies that if a boundary exists, thenit must be sensed by both methods in a common orientationregardless of the amplitude of the physical propertychanges. An additional flexibility of the technique is thatthe cross-gradients constraint is also satisfied when eitherrmr or rms vanishes in some part of the model, thusgiving the models the possibility of admitting a geologicalboundary which has a significant change only in theelectrical resistivity or seismic velocity of the adjoiningrocks.
2.2. Description of the Inversion Scheme
[8] In our iterative 2D inversion approach, the sub-surface model is discretised into rectangular cells ofvariable sizes optimized according to the natural sensi-tivity of each particular set of resistivity and seismicmeasurements. The dc forward problem is solved usingthe fast approximate scheme of Perez-Flores et al.[2001]. For the seismic forward problem, we implementedthe technique of progressive finite differences of Vidale[1990] incorporating the main features described in Zelt andBarton [1998], which allow us to efficiently computeaccurate travel times.[9] We define the discrete version of equation (1) and the
corresponding derivatives using the elements of the 3-cells
scheme depicted in Figure 1. For the along-strike compo-nent this simplifies to:
t ffi 4
�x�zmrc msb � msrð Þð þmrr msc � msbð Þ þ mrb msr � mscð ÞÞ
ð3Þ
where the quantities �x, �z, mrc, mrb etc are defined inFigure 1.[10] Using a first-order Taylor series expansion, equation
(2) is equivalent to
min 2nT2mþmTN1m� �
subject to t m0ð Þ þ Bm� Bm0 ¼ 0 ð4Þ
where
N1 ¼AT
r C�1ddrAr þ a2
rDTDþ C�1
RRr 0
0 ATsC
�1ddsAs þ a2
sDTDþ C�1
RRs
24
35
and n2 ¼AT
r C�1ddr dr � f r m0rð Þ þ Arm0rf g þ C�1
RRrmRr
ATsC
�1dds ds � f s m0sð Þ þ Asm0sf g þ C�1
RRsmRs
24
35;
Ar, As and B are the respective partial derivatives of fr, fs, and t
evaluated at the initial model, m0 = [m0r: m0s]T. The jacobian
matrix for seismics, As is computed using ray tracing as suggestedby Vidale [1990] and Zelt and Barton [1998] while thejacobian matrix for dc resisitivity Ar is computed using thecode of Perez-Flores et al. [2001].[11] The solution to (4) used in our iterative scheme is
given by:
m ¼ N�11 n2 � N�1
1 BT BN�11 BT
� �1BN�1
1 n2 � Bm0 þ t m0ð Þ �
: ð5Þ
In the regularised solution process, the weighting factors areinitially assigned large values which are then graduallyreduced in subsequent iterations until the data are fitted tothe required level. The joint inversion is initiated using ahalf-space model in the absence of reliable a prioriinformation. The implemented cross-gradients criterionand regularisation measures ensure that the resolutioncharacteristics of the individual data sets are fully exploitedin the search for structurally linked models. Note that withthe cross-gradients criterion, there is no need to define or
Figure 1. Definition of the resistivity-velocity cross-gradients function and its derivatives on a rectangular griddomain. For a 2D grid extending in the x and z directions,with each grid element characterised by logarithm ofresistivity mr, and seismic slowness ms, the function t isdefined at the centre of a given element (marked x)considering the parameters from two elements it is incontact with to its right (subscripted r) and bottom(subscripted b).
1 - 2 GALLARDO AND MEJU: CHARACTERIZATION OF HETEROGENEOUS NEAR-SURFACE MATERIALS
assume ab initio any interdependence of resistivity andseismic velocity which could bias the inverse solution. Thecomputational details and validation tests of our inversioncode are given elsewhere (Gallardo and Meju, manuscriptin preparation, 2003).
3. Near-Surface Characterization by JointData Inversion
[12] Field data sets from collocated dc resistivity andseismic refraction experiments have recently been sepa-rately inverted and the resulting models correlated todefine a resistivity-velocity relation [see Meju et al.,2003]. We have chosen the same data sets for simultaneousinversion using our algorithm in order to gauge the potentialof the joint 2D approach for improved near-surfacecharacterization. We expect the joint inversion of the datato define more accurately the main geological andpetrophysical features of the subsoil. The seismic refractiondata consist of first-arrivals from 4 shot points recordedusing geophones placed at 2 m intervals over a total spreadlength of 166 m (Figure 2a). The dc resistivity data consistof six in-line Schlumberger soundings with half-currentelectrode spacings (AB/2) ranging from 1.5 to 90 m(Figure 2b). The test area is a buried hillside composed of ahighly fractured granodiorite rock-mass (deeply erodedyielding an irregular surface and) successively overlain by aheterogeneous mudstone sequence and glacial drift deposits[Meju et al., 2003].
[13] The optimal seismic velocity and resistivity imagesderived by joint inversion of the field data sets using initialhalf-space models are shown in Figure 3. The fit betweenthe model responses and the field data (see Figures 2aand 2b) is about 1 rms error which is in accord with theexpected level of noise in our data. The reconstructeddistributions of model parameters show structural similar-ities and hence good spatial correlation of velocity andresistivity. Note that the adopted cross-gradients criterionserves for geologic structural control but does not force thetwo models into conformity if not justified by the field data.Note: (i) the zone of coincident low velocities and highresistivities in the top 3 m where there are glacial driftdeposits; (ii) the intermediate zone (3–20 m depth)
Figure 2. Fit between observed geophysical data at a testsite and those computed for the optimal models from jointresistivity-seismic inversion. (a) seismic traveltime data and(b) dc apparent resistivity data. The dc sounding positionson the survey line are indicated in the top righthand cornersof the relevant plots. The cross symbol represents field data.The solid line is the computed response of the optimalmodels given in Figure 3.
Figure 4. Relationship between logarithmic resistivity andvelocity models presented in Figure 3. The samplingpositions are the coincident grid cells in the 2D models.Samples (pixels) falling on the same trend are assigned thesame numerical symbol.
Figure 3. Optimal 2D joint velocity and resistivity modelsderived for the field data shown in Figure 2. The velocitymodel is shown at the top and the resistivity model at thebottom. The fit to the field recordings is shown in Figure 2.
GALLARDO AND MEJU: CHARACTERIZATION OF HETEROGENEOUS NEAR-SURFACE MATERIALS 1 - 3
showing a heterogeneous character and undulatory basepossibly consisting of subcropping basement blocks andminor troughs filled by mudstone or clayey materials; and(iii) the basal section of high velocity and high resistivitypossibly corresponding to hard granodiorite bedrock. As aconsequence of the cross-gradients structural constraint, thedistribution of possible rock-contacts is concordant in bothmodels.[14] To ascertain the resistivity-velocity relationship for
the reconstructed models, we have plotted in Figure 4 thesetwo physical parameters for all the coincident samplingpositions (grid cells) in the joint inversion models. This plotshows a structural feature that was not apparent in theresults of separate inversion reported previously in Mejuet al. [2003]; the joint inversion result appears to suggestdistinct sub-groupings within these near-surface materials.The trend reversal yielding the v-shaped curves may be aconsequence of water saturation (water table) or a naturaldivide between unconsolidated and consolidated materials[cf. Meju et al., 2003]. To understand these trends, wehave mapped all the data pairs defining a given segment inFigure 4 onto a distinct zone on the ground (Figure 5),considering only those parts of the model that are wellconstrained by the field data. This yields a single consistentstructural picture of the near-surface, and limited sensitivityanalysis (see sample seismic ray paths in Figure 5) wouldsuggest that this structural model is consistent with the fielddata. In Figure 5, notice that the seismic rays are criticallyrefracted at the top of the structural units mapped as zones 3and 4, defining the possible boundary between consolidatedand unconsolidated materials. Joint inversion thus appearsto offer a means for improved characterization of suchcomplex near-surface environments. It remains to addressthe question: If the contention that electrical resistivity andseismic images are both influenced by the same geologicalstructures is correct, and there is some structural control onthe distribution of petrophysical properties in the subsur-face, can we accurately deduce any petrophysical featurefrom joint structural inversion models? It is possible thatFigures 4 and 5 have petrophysical implications butadditional information or model transformation [e.g.,Berryman et al., 2000] may be required in order to extract
porosity, permeability or water saturation. This is beyondthe scope of this paper.
4. Conclusion
[15] The incorporation of cross-gradients criterion in 2Dinversion leads to geologically meaningful solutions byimproving the structural conformity between the velocityand resistivity images without forcing or assuming the formof the relationship between them. The cross-gradients cri-terion also allows the delineation of those subsurfacefeatures for which only one of the geophysical techniquesis sensitive, leading to a better structural characterization.[16] The application of joint 2D inversion with cross-
gradients to field data from collocated seismic and dcresistivity experiments has led to an improved characteriza-tion of near-surface heterogeneous materials. This studysuggests that unconsolidated and consolidated (or possiblyunsaturated and saturated) materials may be sub-classified onthe basis of their resistivity-velocity relationship evincedfrom joint inversion, a feature that was not observed in aprevious study using separate inversion models. The cross-gradients approach adopted here can also be used for 3Dproblems and for any combination of independent geophys-ical methods.
[17] Acknowledgments. We are grateful to SUPERA-ANUIES inMexico for a scholarship award to L. Gallardo. We thank K. Whaler andtwo anonymous referees for their very useful comments. We thank ColinZelt and Marco A. Perez-Flores for making available their codes.
ReferencesBerryman, J. G., P. A. Berge, and B. P. Bonner, Transformation of seismicvelocity data to extract porosity and saturation values for rocks, J. Acous-tical Soc. of America, 107, 3018–3027, 2000.
Hering, A., R. Misiek, A. Gyulai, M. Dobroka, and L. Dresen, A jointinversion algorithm to process geoelectric and surface wave seismic data.Part I: basic ideas, Geophys. Prosp., 43, 135–156, 1995.
Kozlovskaya, E., Theory and application of joint interpretation of multi-method geophysical data, PhD dissertation, Univ. Oulu, 2001.
Manglik, A., and K. Verma, Delineation of sediments below flood basalt byjoint inversion of seismic and magnetotelluric data, Geophys. Res. Lett.,25, 4015–4018, 1998.
Meju, M. A., L. A. Gallardo, and A. K. Mohamed, Evidence for correlationof electrical resistivity and seismic velocity in heterogeneous near-surfacematerials, Geophys. Res. Lett., 30(7), 10.1029/2002GL016048, 2003.
Perez-Flores, M. A., S. Mendez-Delgado, and E. Gomez-Trevino, Imaginglow-frequency and dc electromagnetic fields using a simple linearapproximation, Geophysics, 66, 1067–1081, 2001.
Rudman, A. J., J. F. Whaley, R. F. Blake, and M. E. Biggs, Transformationof resistivity to pseudovelocity logs, AAPG Bull., 59, 1151–1165, 1975.
Vidale, J. E., Finite-difference calculation of traveltimes in three-dimen-sions, Geophysics, 55, 521–526, 1990.
Zelt, C. A., and P. J. Barton, Three-dimensional seismic refraction tomo-graphy: a comparison of two methods applied to data from the FaeroeBasin, J. Geophys. Res., 103, 7187–7210, 1998.
Zhang, J., and F. Dale Morgan, Joint seismic and electrical tomography,Proc. EEGS Symposium on Applications of Geophysics to Engineeringand Environmental Problems, Keystone, Colorado, 391–396, 1996.
�����������������������L. A. Gallardo and M. A. Meju, Department of Environmental Science,
Lancaster University, Lancaster, LA1 4YQ, UK. ([email protected];[email protected])
Figure 5. Interpretative model showing subsurface dis-tribution of well constrained resistivity-velocity pointscharacterising the respective trends or segments of thecurves shown in Figure 4.
1 - 4 GALLARDO AND MEJU: CHARACTERIZATION OF HETEROGENEOUS NEAR-SURFACE MATERIALS
Paper 3
Joint two-dimensional DC resistivity and seismic travel time
inversion with cross-gradients constraints
by Luis A. Gallardo and Max A. Meju
94
Joint two-dimensional DC resistivity and seismic travel time inversion
with cross-gradients constraints
Luis A. Gallardo1 and Max A. MejuDepartment of Environmental Science, Lancaster University, Lancaster, UK
Received 31 July 2003; revised 29 December 2003; accepted 8 January 2004; published 27 March 2004.
[1] It is now common practice to perform collocated DC resistivity and seismic refractionsurveys that complement each other in the search for more accurate characterization of thesubsurface. Although conventional separate DC resistivity and seismic models can bediagnostic, we posit that better results can be derived from jointly estimated models. Wemake the assumption that both methods must be sensing the same underlying geology andhave developed an innovative resistivity-velocity cross-gradients relationship to evaluatethe structural features common to both methods. The cross-gradients function isincorporated as a constraint in a nonlinear least squares problem formulation, which issolved using the Lagrange multiplier method. The resultant iterative two-dimensional(2-D) joint inversion scheme is successfully applied to synthetic data (serving asvalidation tests here) and to field data from collocated DC resistivity and seismicrefraction profiling experiments and also compared to conventional separate inversionresults. The joint inversion results are shown to be superior to those from separate 2-Dinversions of the respective data sets, since our algorithm leads to resistivity and velocitymodels with remarkable structural agreement. INDEX TERMS: 0902 Exploration Geophysics:
Computational methods, seismic; 0925 Exploration Geophysics: Magnetic and electrical methods; 0935
Exploration Geophysics: Seismic methods (3025); 3260 Mathematical Geophysics: Inverse theory;
KEYWORDS: joint inversion, DC resistivity, seismic refraction
Citation: Gallardo, L. A., and M. A. Meju (2004), Joint two-dimensional DC resistivity and seismic travel time inversion with cross-
gradients constraints, J. Geophys. Res., 109, B03311, doi:10.1029/2003JB002716.
1. Introduction
[2] The need for improved characterization of the nearsurface has led to an increase in the popularity of collocatedresistivity and seismic profiling surveys [e.g., Scott et al.,2000; Meju et al., 2003]. However, the experimental datahave so far been interpreted using separate two-dimensional(2-D) inversion schemes for each method leading some-times to models that are not in good accord. It is conven-tional practice to perform 2-D inversion of either DCresistivity or seismic refraction data and then use the resultto constrain the inversion of the data for the other method[e.g., Scott et al., 2000] but the resulting model from this‘‘sequential inversion’’ [cf. Lines et al., 1988] is commonlybiased toward the input model. A better model can beobtained by simultaneous fitting (i.e., joint inversion) ofthe combined data sets from the different methods.[3] A great deal of work has been done on multidimen-
sional inversion of complementary geophysical measure-ments that sense the same physical properties [e.g., Sasaki,1989]. Multidimensional joint inversion of disparate oruncorrelated data from methods based on fundamentally
different physical properties, for which there is no estab-lished analytical relationship, is much less studied.[4] In recent times, there have been different approaches
to 2-D joint inversion of disparate data with varying degreesof success. We may classify them under two differentphilosophical approaches: (1) Inversion methods involvingthe use of petrophysical or hydrological characteristics torelate two different geophysical properties [e.g., Berge etal., 2000]. For instance, water saturation and porosity havebeen assumed to provide a link between resistivity andseismic velocity in porous media [see, e.g., Tillmann andStocker, 2000]. (2) Inversion approaches involving the useof structural attributes (e.g., boundaries of geological tar-gets) as a common factor between two geophysical models[e.g., Lines et al., 1988; Haber and Oldenburg, 1997; Musilet al., 2003; Gallardo-Delgado et al., 2003]. Examples ofstructurally based algorithms for 2-D joint resistivity andseismic inversion are those of Zhang and Morgan [1996]and Haber and Oldenburg [1997]. Their algorithms aim toenhance the common boundaries given by the largestchanges in the estimated parameters as measured by aLaplacian operator, concentrating on the magnitude of thechanges and losing the direction-dependent information.[5] In this paper, we address the problem of how to relate
effectively the physical properties sensed by DC resistivityand seismic refraction methods using a 2-D joint structuralinversion approach. We provide a generalized framework formultidimensional joint inversion of disparate data sets as a
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B03311, doi:10.1029/2003JB002716, 2004
1Also at Division de Ciencias de la Tierra, Centro de InvestigacionCientifica y Educacion Superior de Ensenada, Mexico.
Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JB002716$09.00
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follow-up to the algorithm proposed by Gallardo and Meju[2003] for joint 2-D DC resistivity and seismic P-wave traveltime data inversion. We also undertake experiments thatdemonstrate the effectiveness of the inversion procedure.First, we define our concept of a structural link and how weincorporate it in a least squares based objective function andhow the problem is solved iteratively. Then the main ele-ments of the algorithm are validated using data from alter-native conventional forward modeling codes and thenapplied to field data from a site characterized by heteroge-neous materials.
2. Definition of the Resistivity-VelocityCross-Gradients Function
[6] The philosophy in our approach to joint 2-D inversionis that even in the absence of an analytical relationshipbetween the physical properties exploited by differentgeophysical methods, we expect a degree of structuralsimilarity in the images that they furnish. In general, theproperties of the subsurface vary with position and thesevariations can occur in any direction. Thus, at any position,the changes can be characterized in terms of two attributes,the intensity or magnitude and the specific direction. Acommonality of distribution of these changes determineswhether or not electrical resistivity and seismic velocityimages are perceived as being structurally similar. Theseattributes can be represented mathematically by the vectorfield of the gradients of the electrical and seismic properties.The key issue here is the following: Can these attributes beused to develop a generalized mathematical scheme forquantifying the structural similarity between resistivityand seismic models of the heterogeneous subsurface?[7] The structural differences between seismic velocity
and electrical resistivity images may be measured using anangular function such as
q x; y; zð Þ ¼ cos�1 rmr x; y; zð Þ � rms x; y; zð Þrmr x; y; zð Þj j rms x; y; zð Þj j
� �; ð1Þ
or related functionals such as 1/(1 � cos q), wherermr(x, y, z) and rms(x, y, z) are the resistivity and seismicproperty gradients, respectively. However, these nonlinearfunctions have discontinuities arising from their angularnature and singularities in areas where rmr(x, y, z) orrms(x, y, z) vanish, as in homogeneous materials or localmaxima or minima of the physical properties. Instead, thepreferred approach in this paper is to use the cross-gradientsfunction [see Gallardo and Meju, 2003], which is given by
~t x; y; zð Þ ¼ rmr x; y; zð Þ � rms x; y; zð Þ: ð2Þ
This nonlinear second-order function has no problems ofdiscontinuity or singularity other than those particular to theadopted model parameterizations.[8] Using this cross-gradients function, we deem the
resistivity and seismic models to be structurally identicalif ~t(x, y, z) vanishes everywhere, i.e., ~t(x, y, z) = ~0, as itimplies full collinearity of simultaneous changes on theresistivity and seismic parameters.
[9] In the two-dimensional case of interest, the x- andz-components of ~t vanish. Thus we are interested in itsy-component, henceforth referred to as t:
t x; zð Þ ¼ @mr x; zð Þ@z
� �@ms x; zð Þ
@x
� �� @mr x; zð Þ
@x
� �@ms x; zð Þ
@z
� �:
ð3Þ
[10] We estimate the derivatives in equation (3) usingforward differences (see Figure 1) yielding the formula[Gallardo and Meju, 2003]
t ffi 4
DxDzmrc msb � msrð Þð þ mrr msc � msbð Þ þ mrb msr � smscð ÞÞ;
ð4Þ
where the second subscript c, b, or r on mr (logarithm ofresistivity) or ms (slowness) denotes center, bottom or rightcell in our heterogeneous 2-D grid as depicted in Figure 1.Dx and Dz are the horizontal and vertical dimensions of thecells (Figure 1) and serve to normalize for grid differencesin equation (4).[11] In the rest of this paper, we will develop and evaluate
an algorithm that is underpinned by the cross-gradientsconcept so as to gauge the effectiveness of t as a structurallink between DC resistivity and seismic refraction models inmultidimensional joint inversion.
3. Regularized Least Squares Inversion WithCross-Gradients Constraint
[12] The conventional regularized inverse problem for-mulations for separate two-dimensional seismic or DCresistivity inversion [e.g., Loke and Barker, 1995; Zeltand Barton, 1998; Perez-Flores et al., 2001] involve leastsquares minimization of data misfit and smoothness con-straints. The smoothness measures help to overcome the
Figure 1. Basic 2-D grid used to represent our models ofthe subsurface. The three-cell scheme used to define thediscrete version of the cross-gradients at any cell position inthe model is also depicted.
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problems of nonuniqueness and instability that bedevilgeophysical inversion methods and can be combined withthe cross-gradients concept to great effect in problemformulation. We therefore define our objective function as[cf. Gallardo and Meju, 2003]:
min F�mr;ms
� �¼ dr � f r mrð Þ½ �T C�1
rr dr � f r mrð Þ½ �
þ ds � f s msð Þ½ �T C�1ss ds � f s msð Þ½ �
þ a2rm
Tr D
TDmr þ a2sm
TsD
TDms
þmr �mRr
ms �mRs
� �TC�1
RR
mr �mRr
ms �mRs
� ��
subject to t mr;msð Þ ¼ 0: ð5Þ
Here, dr (logarithm of apparent resistivity) and ds (seismictravel times) are the observed data, fr(mr) are the computedapparent resistivities, fs(ms) are the computed travel times,Crr is the covariance matrix of the resistivity data (which areassumed to be uncorrelated) and Css is the covariancematrix of the observed travel times (also assumeduncorrelated). t(mr, ms) contains the cross-gradients for allthe cells making up the model. D is the smoothness matrix,ar and as are weighting factors that define the level ofsmoothness required in the models, andmRr andmRs are thea priori model parameters with covariance CRR. Thesuperscripts T and �1 denote matrix transpose and matrixinverse, respectively. Note that reliable prior knowledge ofpetrophysical relationships linking resistivity and seismicparameters [e.g., Berge et al., 2000; Kozlovskaya, 2001;Meju et al., 2003] can be incorporated into the objectivefunction via the a priori model parameters (mRr and mRs)and off-diagonal elements of their covariance matrix CRR,but this issue will not be addressed in this paper.[13] Our objective function (5) is nonlinear since the DC
resistivity and seismic forward problems as well as thecross-gradients constraint are nonlinear. Our solution toequation (5) is thus achieved by linearization. For the DCresistivity forward problem, we adopt the approach ofPerez-Flores et al. [2001] founded on the nonlinear integralequations for electromagnetic inverse problems of Gomez-Trevino [1987]. In this approach, the DC resistivity forwardcomputation simplifies to a linear problem of the form
f r mrð Þ ffi Armr; ð6Þ
where Ar defines the Jacobian matrix, evaluated using theformulas given by Perez-Flores et al. [2001]. Note that Ar isindependent of any particular mr thus it is computed onlyonce and is recurrently used in later calculations. We haveadopted this approximate model for the 2-D DC resistivityforward problem because of its computational speed;however, a more rigorous 2-D forward model can be usedin the cross-gradients inversion approach.[14] The relationship between seismic travel time and
slowness in the functional fs(ms) is nonlinear since thetrajectory of the seismic ray path in the subsurface dependsupon the slowness [see Hole, 1992], but it can be linearizedby considering a small perturbation of the slowness about areference slowness model, m0s. The resulting linearizedexpression is
f s msð Þ ffi f s m0sð Þ þ As ms �m0sð Þ: ð7Þ
The travel time computation is carried out using ourimplementation of the progressive finite differences methodof Vidale [1990], incorporating some of the improvementsdeveloped by C. Zelt [Zelt and Barton, 1998]. The Jacobianmatrix As, composed by the fraction of distance of every raypath in each model cell, is computed efficiently by raytracing through the velocity field generated during theforward modeling process.[15] Similarly, the linearization of the cross-gradients
constraint in equation (5) is accomplished using a first-order Taylor expansion (neglecting higher orders), namely,
t mr;msð Þ ffi t m0r;m0sð Þ þ B
mr �m0r
ms �m0s
0@
1A: ð8Þ
In this case, we require a reference resistivity model m0r,a reference slowness model m0s and the derivatives of twith respect to the model parameters given by B. Therelevant expressions for computing B are obtained fromthe 2-D discrete version of t given in equation (4). Theseare
@t
@mrc
ffi 4
DxDzmsb � msrð Þ; @t
@msc
ffi 4
DxDzmrr � mrbð Þ;
@t
@mrr
ffi 4
DxDzmsc � msbð Þ; @t
@msr
ffi 4
DxDzmrb � mrcð Þ;
@t
@mrb
ffi 4
DxDzmsr � mscð Þ; and @t
@msb
ffi 4
DxDzmrc � mrrð Þ:
ð9Þ
[16] Using equations (6), (7), and (8), the linearizedweighted equivalent of equation (5) is stated as
min
�FL mr;msð Þ ¼ 1
b2dr � Armr½ �TC�1
rr dr � Armr½ �
þ 1
b2ds � f s m0sð Þ½ �As ms �m0sð Þ�T
C�1ss ds � f s m0sð Þ½ �As ms �m0sð Þ�
þ a2rm
Tr D
TDmr þ a2sm
TsD
TDms
þmr �mRr
ms �mRs
� �TC�1
RR
mr �mRr
ms �mRs
� ��
subject to t m0r;m0sð Þ þ Bmr �m0r
ms �m0s
� �¼ 0; ð10Þ
where b is an auxiliary damping factor.[17] The solution of equation (10) is determined, using
Lagrange multipliers [see e.g., Menke, 1984; Tarantola,1987], by solving the system of equations:
@
@mi
FL þ 2Xnj¼1
lj
X2nk¼1
bj;k mk � m0kð Þ þ t m0ð Þj
" #( )¼ 0
for i ¼ 1; 2n
ð11Þ
X2nj¼1
bp;j mj � m0j
� �þ t m0ð Þp¼ 0 for p ¼ 1; n; ð12Þ
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where lj are the required Lagrange multipliers, bj,k are thecoefficients of B and n refers to the total number of cells inthe rectangular model grid. In these expressions mi (i = 1, n)are the elements of mr, mi (i = n + 1, 2n) are the elements ofms and the zero subscript refers to the initial model andthereafter to the previous iterate.[18] Using matrix notation, let us define the variables
m ¼mr
ms
24
35; m0 ¼
m0r
m0s
24
35
and + = {li} where i = 1, n. The solutions to equations (11)and (12), after some algebra, are
L ¼ BN�11 BT
� ��1BN�1
1 n2 � Bm0 þ t0� �
; ð13Þ
m ¼ N�11 n2 � N�1
1 BTL: ð14Þ
Here,
N1 ¼
1
b2AT
r C�1rr Ar þ a2
rDTDþ C�1
RRr 0
01
b2AT
sC�1ss As þ a2
sDTDþ C�1
RRs
2664
3775
and n2 ¼
1
b2AT
r C�1rr drf g þ C�1
RRrmRr
1
b2AT
sC�1ss ds � f s m0sð Þ þ Asm0sf g þ C�1
RRsmRs
2664
3775
are the matrices of the normal equations for the regularizedproblem, and t0 is t(m0r, m0s) in equation (8). The first termon the right-hand side of equation (14) corresponds to theregularized (but structurally unlinked) least squares solutionwhile the second term (N1
�1BT+) is the linking contribution
from the cross-gradients constraint. Note that N1 should bepositive definite to use the above equations and this isassured by the use of Twomey-Tikhonov-type [Tikhonovand Arsenin, 1977; Twomey, 1977] derivative regularizationmeasures and the covariances CRRr and CRRs of the a priorimodels mRr and mRs, respectively.[19] In the iterative solution process, the search for the
optimal solution is started with an initial model m0 (notnecessarily the same asmR and preferably a smooth model),which is then updated at every iterative step. Importantconsiderations are the stability and convergence character-istics of the algorithm and the role played by the regular-ization factors (ar and as) and auxiliary damping factor b.The main purpose of the regularization is to stabilize theprocess and avoid local minima. Note that in equation (14)the values of the linearized cross-gradients are effectivelyequal to zero. However, the original (i.e., nonlinear) con-straint in equation (5) is dependent on the convergence ofthe model; it will tend toward zero as the model convergesto a stable solution. We carried out comparative tests inwhich b, ar, and as were initially assigned large values andthen decreased in successive iterations and found thatsatisfactory convergence was obtained when predeterminedvalues of ar and as are held fixed while b was allowed todecrease (until its threshold value of unity is reached). Weuse a two-stage minimization process; the main iteration fits
the data as b is varied while a substage minimizationensures structural similarity by seeking the solution satisfy-ing the cross-gradients constraint for a constant b.[20] To track the evolution of the misfit at every iterative
step the rms values of the normalized residuals of the fitteddata are computed as
rmsr ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidr � f r mrð Þ½ �T C�1
ddr dr � f r mrð Þ½ �nr
s; ð15Þ
rmss ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffids � f s msð Þ½ �T C�1
dds ds � f s msð Þ½ �ns
s; ð16Þ
where nr and ns are the number of DC resistivity and seismictravel time data, respectively. The convergence of the mainiterative process is based on reducing equations (15) and(16) to their desired values of unity, while that of thesubstage minimization is given by the relative differencesbetween the parameters of the models at two consecutiveiterations
convr %ð Þ ¼ 100
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1
mi � m0ið Þ2
m20i þ e
n
vuuut; ð17Þ
convs %ð Þ ¼ 100
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2ni¼nþ1
mi � m0ið Þ2
m20i þ e
n
vuuuut; ð18Þ
where the parameters involved are described in equations(11) and (12) and e is a small positive number that serves toprevent division by zero.
4. Separate Versus Joint 2-D Inversion ofSynthetic Data
[21] We tested our implemented seismic forward model-ing algorithm and the adopted DC resistivity code using thesimple 2-D model depicted in Figure 2. The model consistsof two rectangular boxes embedded in a half-space andhaving the following properties:
Half -space r ¼ 100 ohm m; Vp ¼ 1000 m=s;
Box 1 r ¼ 10 ohm m; Vp ¼ 2000 m=s;
Box 2 r ¼ 1000 ohm m; Vp ¼ 2000 m=s:
[22] The values of the model parameters were chosen sothat one box had host-target contrasts of the same sign forboth resistivity and velocity and the other box had oppositecontrasts. The model was discretized into rectangular cellsof variable sizes in the area of interest, as shown in Figure 2,and the bordering zones were padded with cells of increas-ing dimensions up to a lateral and depth extent of 500 m, anecessary feature for the DC resistivity modeling.
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[23] We computed the resistivity response of the model,using the forward codes of Perez-Flores et al. [2001] andthat of Dey and Morrison [1979], for a dipole-dipole arrayrecording from position �100 up to +100 m with anelectrode spacing (a) of 10 m and interval separation (na)of 10 m up to 180 m. The maximum differences betweenthe computed responses were no more than 4.8% in thelogarithm of apparent resistivity, which we consider appro-priate for the level of noise expected in typical field data.Henceforth, we use the data depicted in Figure 3 generatedusing the Dey and Morrison [1979] code as the synthetictest data for inversion.[24] In the case of the forward computations of seismic
P-wave travel times, the responses of the test model wereobtained for a deployment consisting of reversed profilesrunning from �100 m up to 100 m with geophones locatedevery 5 m and seismic sources every 50 m along the profile.The synthetic data from our code are compared with thoseobtained using C. Zelt’s code in Figure 4a. The two datasets are in accord, any minor differences arising mainlyfrom truncating and rounding off the numerical data values.We plotted the ray paths in Figure 4b, which resulted fromthe direct, critically refracted and diffracted waves. The raytracing in the seismic model reflects, in the conventionalprocedure, the sensitivity of the synthetic data to thedifferent cells in the model. Notice that the rays are
Figure 2. Theoretical test model consisting of tworectangular boxes embedded in a uniform host medium.(a) Resistivity model. (b) Seismic velocity model. The plottedcells correspond to those used to define our test model.
Figure 3. Dipole-dipole resistivity response for the testmodel computed using the code of Dey and Morrison[1979] constituting our synthetic data for 2-D inversion.
Figure 4. (a) Computed travel time responses for the testmodel using the seismic code implemented in our inversionscheme (crosses) and using the code of C. Zelt [Zelt andBarton, 1998] (solid lines). (b) Ray tracing showing thecoverage associated with the test data.
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concentrated in the upper surface of the boxes. Thus thesynthesized data will have the necessary sensitivity todefine the upper part of the boxes but not their depth extent.[25] We inverted the synthetic data without adding ran-
dom noise, but uniform standard deviations of 1% and0.1 ms were used for the data covariance matrices for theapparent resistivity and travel times, respectively. The datawere first inverted simultaneously but without the cross-gradients constraint, thus mimicking the conventional ap-proach of separately inverting the resistivity and seismicdata sets. The inversion process was then repeated incorpo-rating the cross-gradients constraint ( joint inversion proper)with the same regularization parameters for comparison.
4.1. Simultaneous Inversion of Synthetic Data WithoutCross-Gradients Constraint
[26] For the separate inversion, the process was initiatedusing a half-space model with a resistivity of 100 ohm m anda seismic velocity of 1000 m/s. The multiplication factor bwas selected as 31.62, as larger values ended in flat modelsand it was set to decrease in 6 steps down to 1 (the thresholdvalue required to fit the data at the level defined by theirstandard deviations). The final seismic and resistivity mod-els after six iterations did not change by more than 2% fromthose of the penultimate iterate, based on equations (17) and(18), and fit the data satisfactorily according to equations (15)and (16). The misfits of the final models are rmsr = 0.403 forthe resistivity data and rmss = 1.329 for the seismic data. Thefit of the responses of these optimal models to the test data isshown in Figures 5a and 5b.[27] The optimal resistivity and velocity models obtained
are shown in Figures 6a and 6b. Both techniques recoveredparts of the original model as allowed by their respectivedata coverage, the regularization (smoothness) measures,and model errors (differences between responses from theimplemented forward modeling codes and those used togenerate the test data). Note that the resultant resistivitymodel recovers the gross structure of the test model betterthan the seismic model as the DC resistivity data containmeasurements from large enough electrode spacing to sensedeeper structures enabling the definition of the bottom ofthe boxes. The detailed sampling of the top of the boxes bythe seismic rays resulted in a relatively flat structure thatresembles better the upper parts of the boxes and theirlateral extents. However, the basal parts and vertical wallsof the boxes are poorly resolved in the seismic model andare mainly controlled by the combined effects of smooth-ness measures and the a priori model.
4.2. Joint Inversion of Synthetic Data
[28] To evaluate the performance of our joint cross-gradients inversion algorithm, we inverted the same datasets using the same initial models (m0 = mR = half-space)and regularization factors (ar, as, b). The models recoveredafter 6 iterations satisfied the target misfit with rmsr = 0.535and rmss = 1.989 and the prescribed convergence criterion(convr < 2% and convs < 2%). The misfit values aresomewhat larger than those of the separate inversion exper-iment because of the cross-gradients constraint. However,the data fit of the optimal models is not significantlydifferent from that of Figures 5a and 5b and is thereforenot shown here. The evolution of the rms misfit and the
multiplication factor b at every iteration of the joint inver-sion process is shown in Figure 7.[29] The final resistivity model obtained by joint inver-
sion is shown in Figure 6c, and the corresponding seismicvelocity model is shown in Figure 6d. These jointly recon-structed resistivity and velocity models show improvedfeatures of interest. These are as follows: (1) The structuralaspect of the boxes (defined by the shapes of the contours)is very similar in both models, as required by our jointinversion algorithm. (2) The bottom of each box nowbecomes defined in the seismic model, this being an indirectcontribution propagated from the resistivity model by thecross-gradients constraint. (3) The top and lateral extent ofthe boxes are better delineated; the resistivity model hasimproved on the round shape in the separate inversion
Figure 5. (a) Computed response (contours) for theresistivity model obtained from separate data inversion.The point values are the corresponding relative differences inpercent (compare Figure 3). (b) Comparison of travel timetest data (crosses) and computed travel times (solid lines)from the separate inversion model. Selected residual values(in ms) are annotated above their corresponding positions.
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(compare Figure 6a) while the slanted top margin of theseismic model (compare Figure 6b) has become horizontal,yielding the desired shape of the boxed structure with onlyminor vertical smearing. This smearing may have resultedfrom the differences of the adopted DC resistivity approachpropagated into the estimated model. (4) The values of theseismic velocity in the boxes are closer to the true valuesthan those of the model from separate inversion (compareFigure 6b).[30] We can compare the computed values of the cross-
gradients function for the two optimal models from theseparate 2-D inversion (Figure 8a) with those for the jointinversion models (Figure 8b). Figure 8a reveals zones ofstructural incongruence in the separate inversion models(see Figures 6a and 6b). It is pertinent to mention that thecross-gradients function tends to decay to zero in areas farfrom those jointly constrained by both sets of data. Thisimplies that one insensitive method will allow its counter-part to continue defining the proper model on its own untilneither data set constrains the model, whereafter it graduallytends toward the a priori (usually flat) model. The values ofthe cross-gradients function for the joint 2-D models(Figure 8b) are more than 1 order of magnitude closer tozero than those computed using the results of separateinversion and more randomly distributed. This suggests thatthe cross-gradients constraint leads to joint inversion models
Figure 6. Recovered models for the hypothetical two-box example. Shown are (a) resistivity and(b) seismic velocity models from separate inversion. Also shown are the (c) resistivity model and(d) seismic velocity model from joint inversion.
Figure 7. RMS values of the normalized residuals at everyiteration for the test experiment with joint inversion.
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with enhanced common structural attributes relative to theconventional separate 2-D inversion schemes.
5. Joint Inversion of Field Data
[31] We have inverted field data sets from collocated DCresistivity and seismic refraction experiments along a 200 mlong profile at a test site in Quorn in England. Theexploration target is the upper 40 m of the subsurface.The geology consists of a highly fractured and erodedgranodiorite bedrock that is successively overlain by Mer-cian Mudstone and heterogeneous glacial drift deposits[Meju et al., 2003]. The resistivity data consist of sixvertical electric soundings (VES) using the Schlumbergerconfiguration with maximum current electrode spreadlength (AB/2) of 90 m. The seismic data consist of a setof forward and reversed refraction profiles over a 165 mtransect with a 2 m geophone spacing and 4 shot points. Asobservational errors were not available for the Quorn datasets, uniform standard deviations of 5% and 1 ms were usedfor the data covariance matrices for the apparent resistivityand travel times, respectively.[32] We started our joint inversion of the Quorn data
using an initial model with constant resistivity and seismic
velocity. We selected a half-space resistivity of 200 ohm m,this being the average value of the observed apparentresistivity data. The initial seismic velocity was set to5000 m/s, this being typical of the velocity of granodiorites.Thus the slowness model ms is an order of magnitudedifferent from mr for the starting models.[33] The subsurface was discretized using irregular cells
with a finer mesh in the areas sampled by the field measure-ments and padded with thicker cells outside the areas ofinterest. To make the most of the resolution capability of thedata and enhance the expected geological structure (espe-cially at shallow depth), we selected an unnormalized dis-crete Laplacian operator (D), which is independent of the cellspacing. This allows the models to produce horizontallyelongated structures near the surface. The factors ar = 1and as = 10, selected after some trials, define the relativeweight of this operator. The difference in these values iscomparable to the 1 order of magnitude difference in theinitial resistivity and velocity model parameters, which wefind appropriate to keep the level of smoothness in bothmodels similar. In this example we found it appropriate tovary the multiplication factor b from 100 to 1 in 8 predeter-mined steps. To obtain the maximum structural concordancebetween the models, each substage minimization was re-quired to continue for 10 iterations unless a global conver-gence (convr and convs) of 2% is achieved.[34] Figure 9 shows the convergence characteristics of the
algorithm for 8 iterations for these field data sets. Notice
Figure 8. Mapping of the cross-gradients function.(a) Results for the models obtained by separate inversion(Figures 6a and 6b). (b) Results for the models obtained bythe cross-gradients joint inversion (Figures 6c and 6d). Seecolor version of this figure in the HTML.
Figure 9. RMS values of the normalized residuals andmultiplication factor b per iteration for the joint inversionexperiment for Quorn data. Note the gradual decrease of themisfit at each iteration as b is varied. The inset shows theconvergencemeasures (equations (17) and (18)) and the trendof the root-mean-square values of the cross-gradientsfunction (i.e., substage minimization) for the last iteration.
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that the rmsr and rmss values per iteration were graduallyreduced as the b factor was varied. Figure 10 shows theresistivity and seismic models obtained for each iteration.Note that this gradual process of joint solution reconstruc-tion not only reduced the possibility of being trapped in
local minima but also controlled the development of im-portant common features in both models. As a result, thefinal resistivity and seismic models (at iteration 8) show aremarkable structural resemblance, which can be gleanedfrom the shape of the contours in Figure 10.
Figure 10. Evolution of the joint inversion process. Shown are the resultant resistivity and velocitymodels for each iteration. Note the gradual development of common structural features in both sets ofmodels during the process. See color version of this figure in the HTML.
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[35] Figures 11a and 11b show the fit obtained for theresistivity and seismic data in the final iteration of this jointinversion exercise. The normalized values of the residualsfor the final iteration were rmsr = 1.004 for the apparentresistivity data and rmss = 0.864 for the travel time data.This suggests that the data were fitted to the expected levelseven though there were no actual field data errors available.Note that there are some larger misfits in particular seg-ments of the travel time data that can only be fitted by morecomplex models or are manifestations of 3D effects on thedata. In the final iteration, some parts of the models stillshow slight differences from that of the previous iterates(defining a convergence factor of 3%), but they do notsubstantially change the fit of the data and will onlymarginally affect the cross-gradients function.[36] The same field data sets were previously inverted
separately by Meju et al. [2003] and the resultant modelswere interpreted as having similar structural features. How-ever, we show that there are subtle differences in structurebetween the separate models of Meju et al. [2003] asevidenced by the values of the computed cross-gradientsfunction of their models (Figure 12a). The largest values of tin the top 2 meters are due to the high gradients shown byboth sets of model parameters (from 4 to 1 in the log ofresistivity and from 3 to 0.1 s/km in slowness). On the otherhand, the smallest t values at the bottom of the resistivityand seismic models relate to less abrupt gradients (2 to 3 in
log of resistivity and 0.5 to 0.2 s/km in slowness). Incomparison, Figure 12b shows the cross-gradients valuesof our jointly inverted models. Note the similar values of tin both the top and the middle portions of the model wherethere is good data coverage. However, there are somedifferences between the resistivity and velocity images inthe bottom part of our joint models. These correspond toareas with the smallest gradients of the slowness and logresistivity parameters and where the data coverage is low. Inaddition to the gained structural conformity of the cross-gradients inversion models, they can be shown to highlightimportant resistivity-velocity trends [see Gallardo andMeju, 2003], and can complement petrophysical measure-ments for improved subsurface characterization.
6. Conclusions
[37] We have demonstrated in this paper that the cross-gradients function offers a quantitative means to evaluatestructural similarities between two smooth images. Thecross-gradients function can be applied to geophysicalimages from heterogeneous geological environments, andcan be used to provide a link between two seeminglydisparate geophysical models. We have incorporated it asa constraint into a simple 2-D joint inversion procedure forDC resistivity and seismic travel time data from collocatedsurveys.[38] The results from inversion experiments using syn-
thetic and field data revealed several interesting features ofour joint inversion procedure. The algorithm yields resis-tivity and seismic models that are consistent with theexperimental data and have improved structural similarity.Importantly, this conformity is reached without forcing or
Figure 11. The fit between the Quorn field data and theresponses of the seismic and resistivity models resultingfrom joint inversion. (a) DC apparent resistivity data. Thepositions of the DC soundings on the survey line areindicated in the top right-hand corners of the plots.(b) Seismic travel time data. In both cases, the crosssymbols represent field data, while the solid lines representthe computed response of the final models of Figure 10.
Figure 12. Comparison of cross-gradients values com-puted for the (a) separate inversion and (b) joint inversion ofthe Quorn field data. See color version of this figure in theHTML.
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assuming the form of the relationship between electricalresistivity and seismic velocity. The algorithm is alsoflexible in the sense that it can admit boundaries that areonly constrained by one method and not the other. Theinversion scheme is robust since it showed adequate con-vergence characteristics. We conclude that our algorithm isan effective procedure for jointly interpreting resistivity andseismic data sets from complex two-dimensional environ-ments. A drawback of our present inversion scheme is that ituses an approximate 2-D resistivity forward model forcomputational efficiency. For an alternative and more rig-orous approach we recommend using the approximatemethod for initial convergence and then switching to aproper resistivity finite difference solution for the finaliterations.[39] Although we have demonstrated the cross-gradients
method for the 2-D case and using only resistivity andseismic data, the method should also be extendable to anyother geophysical methods and in three-dimensions.
[40] Acknowledgments. We acknowledge SUPERA-ANUIES inMexico for a scholarship award to Luis A. Gallardo. We also thank ColinZelt and Marco A. Perez-Flores for making their codes available. We aregrateful to K. Whaler, S. K. Park, and J. Berryman for their incisivecomments, which improved the clarity of this paper.
ReferencesBerge, P. A., J. G. Berryman, H. Bertete-Aguirre, P. Bonner, J. J. Roberts,and D. Wildenschild (2000), Joint inversion of geophysical data for sitecharacterization and restoration monitoring, LLNL Rep. UCRL-ID-128343, Proj. 55411, Lawrence Livermore Natl. Lab., Livermore, Calif.
Dey, A., and H. F. Morrison (1979), Resistivity modelling for arbitrarilyshaped two-dimensional structures, Geophys. Prospect., 27, 106–136.
Gallardo, L. A., and M. A. Meju (2003), Characterization of heterogeneousnear-surface materials by joint 2D inversion of DC resistivity and seismicdata, Geophys. Res. Lett., 30(13), 1658, doi:10.1029/2003GL017370.
Gallardo-Delgado, L. A., M. A. Perez-Flores, and E. Gomez-Trevino(2003), A versatile algorithm for joint 3-D inversion of gravity and mag-netic data, Geophysics, 68, 949–959.
Gomez-Trevino, E. (1987), Nonlinear integral equations for electromag-netic inverse problems, Geophysics, 52, 1297–1302.
Haber, E., and D. Oldenburg (1997), Joint inversion: A structural approach,Inverse Problems, 13, 63–77.
Hole, J. A. (1992), Non-linear high-resolution three-dimensional travel timetomography, J. Geophys. Res., 97, 6553–6562.
Kozlovskaya, E. (2001), Theory and application of joint interpretation ofmultimethod geophysical data, Ph.D. dissertation, Univ. of Oulu, Oulu,Finland.
Lines, L. R., A. K. Schultz, and S. Treitel (1988), Cooperative inversion ofgeophysical data, Geophysics, 53, 8–20.
Loke, M. H., and R. D. Barker (1995), Least-squares deconvolution ofapparent resistivity pseudosections, Geophysics, 60, 1682–1690.
Meju, M. A., L. A. Gallardo, and A. K. Mohamed (2003), Evidence forcorrelation of electrical resistivity and seismic velocity in heterogeneousnear-surface materials, Geophys. Res. Lett., 30(7), 1373, doi:10.1029/2002GL016048.
Menke, W. (1984), Geophysical Data Analysis: Discrete Inverse Theory,Academic, San Diego, Calif.
Musil, M., H. R. Maurer, and A. G. Green (2003), Discrete tomography andjoint inversion for loosely connected or unconnected physical properties:Application to crosshole seismic and georadar data sets, Geophys. J. Int.,153, 389–402.
Perez-Flores, M. A., S. Mendez-Delgado, and E. Gomez-Trevino (2001),Imaging low-frequency and DC electromagnetic fields using a simplelinear approximation, Geophysics, 66, 1067–1081.
Sasaki, Y. (1989), Two-dimensional joint inversion of magnetotelluric anddipole-dipole resistivity data, Geophysics, 54, 254–262.
Scott, J. B. T., R. D. Barker, and S. Peacock (2000), Combined seismicrefraction and electrical imaging, paper presented at 6th Meeting of En-vironmental and Engineering Geophysics, Environ. and Eng. Geophys.Soc., Bochum, Germany.
Tarantola, A. (1987), Inverse Problem Theory, Elsevier Sci., New York.Tikhonov, A. N., and V. Y. Arsenin (1977), Solutions of Ill-Posed Problems,John Wiley, Hoboken, N. J.
Tillmann, A., and T. Stocker (2000), A new approach for the joint inversionof seismic and geoelectric data, paper presented at 63rd EAGE Conferenceand Technical Exhibition, Eur. Assoc. of Geosci. and Eng., Amsterdam.
Twomey, S. (1977), An Introduction to the Mathematics of Inversion inRemote Sensing and Indirect Measurement, Elsevier Sci., New York.
Vidale, J. E. (1990), Finite-difference calculation of traveltimes in three-dimensions, Geophysics, 55, 521–526.
Zelt, C. A., and P. J. Barton (1998), Three-dimensional seismic refractiontomography: A comparison of two methods applied to data from theFaeroe Basin, J. Geophys. Res., 103, 7187–7210.
Zhang, J., and F. D. Morgan (1996), Joint seismic and electrical tomography,paper presented at EEGS Symposium on Applications of Geophysics toEngineering and Environmental Problems, Environ. and Eng. Geophys.Soc., Keystone, Colo.
�����������������������L. A. Gallardo and M. A. Meju, Department of Environmental Science,
Lancaster University, Lancaster LA1 4YQ, UK. ([email protected];[email protected])
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Paper 4
Joint 2d cross-gradients inversion of magnetotelluric and seismic
refraction data: Implication for structural and lithological
classification
by Luis A. Gallardo and Max A. Meju
106
Joint 2D cross-gradients imaging of magnetotelluric and seismic
refraction data: Implication for structural and lithological
classification
Luis A. Gallardo1 and Max A. Meju
Department of Environmental Science, Lancaster University, Lancaster, UK
1Also at División de Ciencias de la Tierra, CICESE, Mexico.
Abstract. We present a joint 2D inversion approach for imaging collocated
magnetotelluric (MT) and seismic refraction data with cross-gradients constraints. We
apply the scheme to near-surface field data to determine the resistivity-velocity
interrelationship in order to test the consistency of a recently proposed hypothesis on
subsurface ‘lithofacies’ or structural classification. The MT-seismic relationship is
found to be in excellent accord with that derived previously for dc resistivity and
seismic data set at the test site. We therefore suggest that joint MT-seismic structural
classification is feasible and can also be applied to deep imaging studies.
1. Introduction
Collocated magnetotelluric (MT) and seismic experiments are becoming increasingly
popular in studies of the near-surface down to mantle depths (Jones, 1998; Meju et al.,
2003). However, the resulting experimental profile data are typically interpreted
separately since no practical schemes exist for multi-dimensional joint inversion of
107
MT and seismic data. This commonly results in models that are plagued by non-
uniqueness due to the band-limited nature of experimental data, measurement errors
and geological heterogeneity. Consequently, it is often difficult to obtain structurally
consistent MT and seismic velocity models for coincident survey lines and this has
somewhat hindered the development of an objective and generic method for
lithological or structural classification based on MT and seismic models, such as was
recently proposed for joint seismic and dc resistivity inversion models (Gallardo and
Meju 2003).
We believe that the current ambiguity associated with separate data interpretation can
be significantly reduced by the appropriately coupled or joint 2D inversion of MT
impedance and seismic travel-time data from collocated experiments. This will result
in better characterization of heterogeneity and lead to improved understanding of the
resistivity-velocity interrelationships in complex subsurface materials (Meju et al.,
2003), necessary for lithological or structural classification as recently proposed for
near-surface dc resistivity and seismic refraction studies (Gallardo and Meju, 2003).
To our knowledge, MT and seismic data have only been jointly inverted in 1D by
Manglik and Verma (1998) and only using synthetic examples. Pilot studies of joint
2D dc resistivity and seismic refraction inversion have shown that coupled data
inversion incorporating the cross-products of the resistivity and seismic velocity
gradients as constraints, provides an effective structural linkage between the velocity
and resistivity models (Gallardo and Meju, 2003, 2004; Gallardo et al., 2005). It also
leads to more accurate models and improves the structural conformity between the
resistivity and velocity images without forcing or assuming the form of the
relationship between them (Gallardo and Meju 2003,2004).
108
In this paper, we present an imaging algorithm that couples MT and seismic refraction
methods in 2D. One hypothesis to test here is that coupled inversion can find
equivocal solutions that consistently explain the subsurface property distributions
since seemingly disparate data from collocated geophysical experiments essentially
sense aspects of the same subsurface. Our final goal is to provide a means to
determine the 2D MT resistivity and seismic velocity interrelationships and prove
their consistency at a field site used by Gallardo and Meju (2003) to develop their
‘lithofacies’ classification hypothesis based on joint inversion of near-surface dc
resistivity and seismic data. The development of a 2D joint MT resistivity and seismic
inversion algorithm based on the cross-gradients approach (Gallardo, 2004) is an
important strategy to achieve the set goal. The significance of developing this
algorithm derives from the wide range of probing depths potentially covered by both
methods. This will facilitate the deduction of structural and lithological patterns from
near-surface to mantle depths, necessary for understanding fundamental geological
processes occurring in the Earth’s crust and mantle.
2. Methodology for Joint MT-Seismic Inversion
Since there is no established analytical relationship between MT resistivity and
seismic velocity valid for heterogeneous subsurface environments, we seek models
that satisfy the cross-gradients structural constraint of Gallardo and Meju (2003,2004).
The developed joint inversion approach exploits the commonality of the structural
features of MT resistivity and seismic velocity models.
109
The structural similarity is appraised by the relative direction of the resistivity and
velocity changes as measured by the 2D version of the cross-gradients function
(Gallardo and Meju, 2004), given by the scalar function:
( ) ⎟⎠⎞
⎜⎝⎛
∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
−⎟⎠⎞
⎜⎝⎛
∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
=z
zxmx
zxmx
zxmz
zxmzxt ),(),(),(),(, SMTSMT , (1)
where mMT(x,z) (logarithm of resistivity) and mS(x,z) (slowness) furnish the 2D MT
resistivity and the seismic velocity images, respectively. The x direction runs
horizontally across the model and z points downward. The simplicity of this formula
makes the cross-gradients easy to incorporate as constraints in joint inversion schemes
for multiple data sets and also compatible with other solution constraints or
regularization measures (Gallardo, 2004; Gallardo et al., 2005).
Following Gallardo and Meju (2004), the two-dimensional model of the subsurface is
discretized into a group of rectangular blocks that extend to infinity along strike, each
block with a constant resistivity and seismic velocity. The objective function is
defined as:
[ ] [ ]
[ ] [ ]}
{
R1
RRT
R
2S
2MTTT1
ddT
)()()(
mmCmm
DmI0
0IDmmfdCmfdm
−−+
⎥⎦
⎤⎢⎣
⎡+−−=Φ
−
−
αα
min
subject to . (2) 0m =)(t
Here, m is the combined vector of the model parameters for MT (logarithm of
resistivity) and seismics (slowness), the vector d contains the MT data (logarithm of
110
MT apparent resistivity and phase) and the seismic first arrivals with covariance
matrix Cdd (assumed diagonal), and f(m) is the computed MT and seismic responses
of the model m. t(m) contains the cross-gradients values for all the cells making up
the model, D is a smoothness matrix weighted by the values αMT and αS that define
the level of smoothness required in the models, and mR is the vector of a priori model
parameters with covariance CRR. Note that it is not assumed, at the outset, any
correlation between the MT and seismic parameters, for instance through the use of
the factors αMT and αS or the off-diagonal values of the covariance matrices (Cdd and
CRR), but instead the cross-gradients constraint is used to provide the required link
between resistivity and velocity.
The non-linear objective function in (2) is linearized and minimized iteratively. This
involves defining initial MT and seismic models and applying first-order Taylor series
expansion of their responses. The two-stage minimization process described in
Gallardo and Meju (2004) is incorporated to perform the required iterations. This
procedure requires a gradual fit to data in multiple steps, thus necessitating the use of
fast forward and derivative computation schemes. For computational efficiency, the
algorithm is based on the progressive finite differences scheme of Vidale (1990) for
the computation of travel times and ray tracing for the derivatives, an approach that
has proved to be fast (see Gallardo and Meju, 2004). In the MT case, the forward and
derivative formulation approach of Smith and Booker (1991) is implemented since it
is a fast procedure. We are aware of the limited accuracy of the Smith and Booker’s
approach but it will suffice for demonstrating the feasibility of joint multi-dimensional
MT-seismic inversion with cross-gradients constraint. Unlike the inversion scheme
proposed by Smith and Booker (1991), the MT inversion scheme implemented here
111
does not include any interpolation of conductivity between sounding sites other than
that implicit in the objective function; i.e., any feature between two consecutive MT
sites results from the smoothness constraint and from that shared by the seismic model
via the cross-gradients constraint.
In the iterative minimization process, the updated model for every step is given by
(Gallardo and Meju, 2003):
[ ].)()( 0021
11T1
1T1
121
1 mBmnBNBBNBNnNm t+−−= −−−−− , (3)
where N1 and n2 are the regularized least-squares normal matrices, B is the matrix of
the derivative of the cross-gradients and m0 is the initial model (or previous iterate).
Further details of the cross-gradients algorithm are given in Gallardo and Meju
(2004). The resulting MT-seismic imaging scheme was extensively validated using
synthetic data from models mimicking regional profiling of the deep crust and mantle
and found the results very satisfactory (Gallardo, 2004). Field data from near-surface
profiling experiments (Meju et al., 2003) are available and it is instructive to apply the
developed joint imaging scheme to these data for which independent interpretative
models exist for comparison.
112
3. Joint inversion of field data and subsurface structural classification
Ideally, joint inversion of collocated seismic refraction and either electrical or
electromagnetic data sets should yield the same results, providing they have similar
depth of penetration and lateral resolution. The collocated dc resistivity soundings
(using 30 m station spacing) and seismic refraction data have been jointly inverted in a
previous study and the results will serve for evaluating the present MT-seismic
analysis as well as the analysis of the concept of structural/lithological classification
proposed by Gallardo and Meju (2003).
3.1. The field measurements
We selected the audiofrequency magnetotelluric (AMT) and seismic refraction data
from the Quorn field site studied by Meju et al. (2003). This site is characterized by a
highly fractured granodioritic basement and heterogeneous sedimentary cover
consisting of mudstone and glacial till. The seismic survey employed 2 m geophone
spacing and four shot points along a 165 m transect that is orthogonal to the known
local geological strike. The AMT data of interest in this study were recorded using 15
m long electric dipoles aligned in the seismic profiling direction and a magnetic
induction coil placed in the orthogonal direction with a 15 m station interval. A total
of 13 AMT stations were recorded over the frequency range 80 KHz to 1 Hz over a
distance of 195 m on the seismic line. These AMT data correspond to the TM mode
measurements. The apparent resistivity data show errors ranging from 1 to 10 percent
and are deemed acceptable but the phase information was much noisier and hence
have not been emphasized in the present study. All the AMT apparent resistivity data
113
were corrected for static shift using coincident transient electromagnetic soundings
[e.g. Figure 1].
10
100
1000
10000
100000
1 10 100 1000 10000
Half electrode array length, L (m)
App
. res
istiv
ity (O
hm.m
)
DC-xy
DC-yx
TEM
MT-xy
MT-yx
Figure 1. Example of AMT data from Quorn. At each station, soundings were made in the survey
line direction (xy) and orthogonal to it (yx). Dc resistivity and TEM data were also recorded at
each station. All the data are shown in an alternative format (Meju 2005) as a function of their
equivalent electrode spread-length to emphasize the coverage of the near-surface, relevant to
later comparisons with dc resistivity results. The TEM data were used for assessment and
correction of electrical static shift. Note that xy and yx sounding curves are similar in shape.
3.2. Joint inversion of AMT and seismic refraction data
Each AMT datum was assigned the corresponding observational error from the actual
field survey but the seismic data were assigned a standard deviation of 1ms. The
subsurface model was discretized into rectangular cells exactly coincident to those
used in Gallardo and Meju (2003) to permit a direct comparison. Unlike the joint dc-
114
seismic inversion in Gallardo and Meju (2003) that involved only smooth initial
models, we started the AMT-seismic inversion process with layered models with
increasing velocity and resistivity with depth. This will reduce the number of steps
required in the iterative process as well as test the robustness of the algorithm. The
joint inversion process was allowed initial target normalized misfits (β) of 10 and
gradually reduced to 1 as recommended in Gallardo and Meju (2004). The smoothing
factors were selected after some experimentation and we found that stable results were
obtained using αMT=αS=100.
The final models obtained for a target normalized misfit of 1 are shown in Figure 2.
The models obtained matched the seismic data well (to normalized rms misfit of 0.97)
but less satisfactorily for the MT data which have a misfit greater than 1. The
distributions of the normalized residuals for AMT and seismic data are presented
respectively in Figures 3a and 3b. The images recovered show geometrical similarity
as required by the cross-gradients constraint. These models show some refinements
when compared to the results of separate AMT and seismic inversion given in Meju et
al. (2003). The structural differences can be quantified by computing the cross-
gradients values for the separate inversion models (Figure 4a) and joint inversion
models (Figure 4b).
115
0 20 40 60 80 100 120 140 160 180 200Distance (m)
40
20
0
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h(m
)
0.0 1.0 2.0 3.0 4.0
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0 20 40 60 80 100 120 140 160Distance (m)
40
20
0
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h(m
)Log resistivity (ohm m)
Velocity (m/s)
a)
b)
Figure 2. a) AMT resistivity model and b) seismic velocity model derived from joint inversion of
apparent resistivity and seismic traveltime data from a collocated profile at the Quorn test site.
The inverted triangles denote the AMT sounding positions while the arrows show the positions of
the seismic shot points.
3.2. Implication for structural or lithological classification
To gauge the reliability of the MT contribution in the joint inversion models of Figure
2, it is instructive to compare our resistivity model with that obtained by Gallardo and
Meju (2003) given that the same seismic data were used in both joint inversion
experiments. The computed differences between both resistivity models are shown in
Figure 5. This figure shows that there are no significant differences between them in
the depth interval 5 to 20m relatively equally sampled by both the dc resistivity and
AMT methods. There is a notable difference in the resistivity models beyond about
30m depth (e.g. see profile positions 20 m and 80 m) not constrained by the dc
resistivity data and at depths shallower than about 3 m that are not imaged by the
present AMT data (cf. Figure 1). We will limit our evaluation of AMT-seismic
116
interrelationships to the depth interval sampled by both dc resistivity and AMT
methods but posit that the analysis can be generalized to greater depths.
-1.4-1.52.24.58.0
1.70.70.80.6
2.72.9
-1.90.41.35.34.9
3.60.9-1.4-0.2-0.21.1-0.2
-6.8
0.63.14.52.61.23.7-2.1
8.68.6
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0.01.93.75.14.71.2-1.8-0.7-2.0-0.21.9
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0.3-4.6-4.2-0.42.43.90.3-1.7
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4.72.15.6
2.54.54.55.72.30.60.91.11.20.1
-2.9
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1.0-3.7
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-1.3-3.0-26.8-17.3-7.23.5-1.8-8.7-11.4-0.6
-6.7-0.8
-3.0-1.7-2.7-1.24.03.83.31.39.71.11.61.71.0
-1.0-2.00.63.76.95.53.73.71.31.81.01.33.1
-7.1-3.8-12.8-4.7-0.5-0.41.31.2-2.8-1.4-0.1-2.2-3.71.8
-9.6-9.5-8.3-5.0-3.5-2.6-4.6-7.2-2.7-2.2
-1.7-4.0
-5.2-4.0-2.21.42.43.01.6-2.3-0.32.35.4-1.5-2.3
0 20 40 60 80 100 120 140 160 180 200Distance (m)
1 2 3
Log apparent resistivity (ohm m)
a)
0 20 40 60 80 100 120 140 160Distance (m)
0
10
20
30
40
50
60
70
Tim
e(m
s)
b)
Figure 3. a) Pseudosection of the observed AMT apparent resistivity data (coloured background
image) with superimposed values of normalized residuals (percent misfit with computed
responses of the joint inversion model of Figure 2a) for each inverted frequency for all sites along
the profile. b) Comparison of observed travel time data (show with standard deviation as grey
bars) and the travel times computed for the joint inversion model of Figure 2b (solid lines).
117
0 20 40 60 80 100 120 140 160Distance (m)
40
20
0
Dept
h(m
)
0 20 40 60 80 100 120 140 160Distance (m)
40
20
0
Dept
h(m
)
-0.010 -0.005 0.000 0.005 0.010
a)
b)
Figure 4. Cross gradients values computed from MT and seismic models a) obtained by separate
inversion, and b) obtained by joint inversion of the same data sets for the Quorn profile.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0 20 40 60 80 100 120 140 160 180 200Distance (m)
40
20
0
Dept
h(m
)
Figure 5. Differences between model resistivities obtained by joint dc-seismic inversion (Gallardo
and Meju, 2003) and joint AMT-seismic inversion (Figure 2a). The significant differences occur
in areas without joint coverage by both methods.
In Figure 6 is shown the AMT resistivity and seismic velocity interrelationships as
deduced from the joint inversion models (Figure 2). Note the correlatable trends and
the excellent agreement with those previously determined by Gallardo and Meju
(2003) based on joint dc resistivity-seismic inversion. In Figure 6, the AMT
118
resistivity-velocity trend (dark crosses) is associated with the deepest structural
features. The AMT-seismic joint inversion results thus not only confirm the existence
of such trends but also demonstrate the structural consistency that can be achieved and
the resistivity-velocity interrelationships to be expected from joint inversion of AMT
and seismic refraction data. It can be expected that the same deductions would apply
to models from deep regional MT and seismic surveys.
100 1000 10000Velocity (m/s)
1
10
100
1000
10000
Resis
tivity
(ohm
m)
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th (m
)
Figure 6. Comparison of resistivity and seismic velocity cross-plots for the Quorn profile. The
grey dots correspond to those from dc-seismic joint inversion (Gallardo and Meju, 2003). The
dark crosses are from the AMT-seismic joint inversion results. The inset shows the actual spatial
distribution of the model cells (pixels) corresponding to the trend. Note the coincidence of the
AMT and dc trends for velocities greater than 1500 m/s.
119
4. Conclusions
The cross-gradients technique has been extended to the inverse problem of
simultaneous interpretation of MT and seismic travel time data in two-dimensions. We
applied the method to a set of field data from a test site, where comparative data are
available, and obtained results that are in excellent agreement with those obtained
independently from joint 2D inversion of collocated dc resistivity and seismic data.
The joint inversion models recovered consistent structural features of the subsurface.
We suggest that joint inversion of collocated MT-seismic profiles and the use of the
results in structural or lithological classification will lead to improved subsurface
characterization in complicated geological terrains and should be seen as the way
forward in subsurface studies.
Acknowledgments. We are grateful to SUPERA-ANUIES for a scholarship award to
Luis A. Gallardo. We thank J. T. Smith for making available his forward modeling
code.
References
Gallardo, L. A. (2004), Joint two-dimensional inversion of geoelectromagnetic and
seismic refraction data with cross-gradients constraint, PhD Dissertation,
Lancaster University, UK.
Gallardo, L. A., and M. A. Meju (2003), Characterization of heterogeneous near-
surface materials by joint 2D inversion of dc resistivity and seismic data,
Geophys. Res. Lett., 30(13), 1658, doi:10.1029/2003GL017370.
120
Gallardo, L. A., and M. A. Meju (2004), Joint two-dimensional DC resistivity and
seismic travel time inversion with cross-gradients constraints, J. Geophys. Res.,
109, B03311, doi:10.1029/2003JB002716.
Gallardo, L. A., M. A. Meju, and M. A. Pérez-Flores (2005), A quadratic
programming approach for joint image reconstruction: mathematical and
geophysical examples, Inverse Problems, 21, 435-452, doi:10.1088/0266-
5611/21/2/002.
Jones, A.G. (1998), Waves of the future: Superior inferences from collocated seismic
and EM experiments, Tectonophysics, 286, 273-298.
Meju, M.A. (2005), Simple relative space-time scaling of electrical and
electromagnetic depth sounding arrays: Implications for electrical static shift
removal and joint DC-TEM data inversion with the most-squares criterion,
Geophys. Prosp. (in press).
Meju, M.A., L. A. Gallardo and A.K. Mohamed (2003), Evidence for correlation of
electrical resistivity and seismic velocity in heterogeneous near-surface
materials, Geophys. Res. Lett., 30(10), 10.1029/2002GL016048.
Smith, J. T., and J. R. Booker (1991), Rapid Inversion of Two- and Three-
Dimensional Magnetotelluric Data, J. Geophys. Res., 96, 3905-3922.
Vidale, J. E. (1990), Finite-difference calculation of traveltimes in three-dimensions,
Geophysics, 55, 521-526.
121
Paper 5
A Quadratic Programming Approach for Joint Image
Reconstruction: Mathematical and Geophysical Examples
by Luis A. Gallardo, Max A. Meju and Marco A. Pérez-Flores
122
INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS
Inverse Problems 21 (2005) 435–452 doi:10.1088/0266-5611/21/2/002
A quadratic programming approach for joint imagereconstruction: mathematical and geophysicalexamples
Luis A Gallardo1,2, Max A Meju1 and Marco A Perez-Flores2
1 Department of Environmental Science, Lancaster University, Lancaster LA1 4YQ, UK2 Division de Ciencias de la Tierra, CICESE, Km 107 Carretera Tijuana-Ensenada,Ensenada BC 22860, Mexico
E-mail: [email protected]
Received 29 June 2004, in final form 16 December 2004Published 21 January 2005Online at stacks.iop.org/IP/21/435
AbstractAlthough a comparative analysis of multiple images of a physical targetcan be useful, a joint image reconstruction approach should provide betterinterpretative elements for multi-spectral images. We present a generalizedimage reconstruction algorithm for the simultaneous reconstruction of band-limited images based on the novel cross-gradients concept developed forgeophysical imaging. The general problem is formulated as the search forthose images that stay within their band limits, are geometrically similar andsatisfy their respective data in a least-squares sense. A robust iterative quadraticprogramming scheme is used to minimize the resulting objective function.We apply the algorithm to synthetic data generated using linear mathematicalfunctions and to comparative geophysical test data. The resulting imagesrecovered the test targets and show improved structural semblance between thereconstructed images in comparison to the results from two other conventionalapproaches.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Simultaneous analysis of multiple observations on a physical system will lead to superiorinferences in comparison with analysis based on single observations. For example, it isknown in medical imaging that simultaneous analysis of images derived from diverse sources(ultrasound, PET scans, magnetic resonance imaging, x-ray CT) is useful for improveddiagnosis (e.g. Shung et al 1992, Webb 1988). Similarly, detailed characterization of theEarth’s surface is derived from the fusion of Pan images and MS images at differentresolutions from Earth resource satellites and airborne sensors (Zhang 2004). However, the
0266-5611/05/020435+18$30.00 © 2005 IOP Publishing Ltd Printed in the UK 435
436 L A Gallardo et al
conventional simultaneous analysis of multiple images has been mostly confined to visualcomparison and numerical manipulations in which the images may not mutually benefit fromeach other. To obtain mutual benefits, the multiple images involved should be processedsimultaneously, guided by common features in a mathematically constrained process hereintermed joint image reconstruction (JIR). The main difficulty faced hitherto in conventionalsimultaneous analysis arises from the dissimilar nature of the physical processes underpinningsome of the imaging techniques. For instance, an electromagnetic experiment based on adiffusive process is generally not related to an acoustic experiment performed in the same placeexcept that both may sense a common physical target. Interestingly, it is beginning to emergefrom geophysical imaging that experimental data from methods based on fundamentallydifferent physical processes can be simultaneously inverted with structural constraints to imagesubsurface targets (Zhang and Morgan 1996, Haber and Oldenburg 1997, Gallardo and Meju2003, 2004, Gallardo 2004). It is therefore opportune to investigate the possibility of extendingthe so-called structurally based joint inversion techniques in geophysics to other imagingproblems.
There are some drawbacks with the existing structurally based inversion techniquesthat limit their straightforward application to many imaging problems. For example, thejoint inversion schemes of Zhang and Morgan (1996) and Haber and Oldenburg (1997) relyon heavy smoothing assumptions for their stability, yielding only very smooth geophysicalimages. In comparison to smoother images derived from such geophysical applications, someother imaging problems (e.g. satellite imagery) sometimes require highly defined featureswith sharp contrasts to be resolved, posing a special challenge for the image processor. Thisproblem is reduced in the joint inversion approach introduced in Gallardo and Meju (2003).This approach encourages parallelism in the corresponding property changes of two images(structural-similarity) by using Lagrange multipliers and shows no restriction in the value ofthe physical property itself, which can be any real number. However, a notable differencein image applications in general is that many of the reproduced images are of band-limitednature, where the elementary pixels that compose the images are limited between certain values(for instance from 0 to 1 in reflectance or 0 to 255 in 8-bit tones). These values constitutecomplementary information that constrains the images and can help us to resolve ambiguitiesin their reconstruction. The use of property band-limits as a priori information to improvegeophysical images is shown for instance in the discrete tomography method of Musil et al(2003).
Given the differences in the constituent parameters of various other targets of imaginginterest (e.g. their band-limits) and their particular responses to source signals, there is a needfor the development of further specialized algorithms for JIR. We consider that the recentlydeveloped cross-gradients approach (Gallardo and Meju 2003, 2004) is more appropriate forgeneralized JIR because it does not influence the pixel values directly and it defines sharperimages better as shown in the synthetic examples of Gallardo (2004) and Gallardo and Meju(2004). In this paper, we will go one step further and show how this approach can be extendedto the problem of multiple JIR for band-limited images by using quadratic programmingtechniques and show the advantages of this scheme using synthetic data.
2. Formulation of the joint image reconstruction problem
To fully develop our concept of JIR, first we need to quantify the common features that make usperceive two images as being similar and then combine them with the conventional principleof image reconstruction based on a quantitative objective function. The resulting JIR objective
A quadratic programming approach for joint image reconstruction 437
function must be minimized using an appropriate numerical-computational scheme. Thesesteps are developed and discussed in the following subsections.
2.1. Cross-gradients function for image appraisal
As discussed in Haber and Oldenburg (1997) and in Gallardo and Meju (2004), the perceptionof geometrical similarities between disparate images is linked to the distribution of the propertychanges, rather than the magnitude of the properties themselves. For instance, in the field ofmedical image registration these property changes are used as a measure of the morphologyof an image. The comparison of the morphology of two medical images provides a means tooverlay the images based on the congruence of their depicted shapes (e.g. Droske and Rumpf2004, and references therein). We therefore use the scalar functions p(x, y) and q(x, y)
to define two tomographic images in �2 and concentrate on their geometrical features asmeasured by the vector fields of their gradients ∇p(x, y) and ∇q(x, y). We follow Gallardoand Meju (2004) and assume that the commonality in the direction of these vector fields isa good indicator of the geometrical similarity between both images. This commonality canbe mathematically measured by the cross-gradients function (t) (Gallardo and Meju 2003)given by
t (x, y) = ∇p(x, y) × ∇q(x, y). (1)
For two-dimensional images, this vector function has only a perpendicular (z) componentand we will refer to that component as a scalar function t (x, y). Alternative functions toevaluate the geometrical similarity between two images have been proposed by Droske andRumpf (2004). However, they base their functions on the normalized gradients of the images(Gauss map), which is completely independent of the magnitude of the physical changesand show singularities in areas with vanishing image gradients. Although, some other non-normalized geometrical measures similar to the cross-gradients function can also be formulated(see Gallardo and Meju (2004)), we prefer the cross-gradients function because it has someadditional advantages in mathematical and computational terms as discussed in detail byGallardo (2004).
To illustrate the geometrical meaning of this function we compare two smooth images(figures 1(a) and (b)), which we assume correspond to the same target but were derived fromdifferent physical experiments. The geometrical differences between the images accordingto the cross-gradients approach are dictated by the magnitude of the mutually perpendicularcomponents of their gradient fields, which have a value of zero in the regions where bothgradients are collinear. Figure 1(c) is the corresponding map of the cross-gradients valuescomputed from the images. In this figure, the largest values of t coincide with the areasof maximum differences in the geometrical aspect of the images of figures 1(a) and (b).Mathematically, the full coincidence of the direction of the vector field of the gradients ofthese images is reflected by the values of t equal to zero, a fact that is exploited in thecross-gradients approach in Gallardo and Meju (2003, 2004). Following the same logic, weincorporate this function as part of an objective function for JIR.
2.2. Objective function for joint image reconstruction
To formulate the concept of JIR mathematically, we define all the desirable features of theimages and summarize them in an objective function.
First, we consider two sets of ndf and ndg data from independent experiments performedon the images. These experiments are represented by the functions f (p) and g(q) in the form
df i = fi(p) + ef i i = 1, . . . , ndf
438 L A Gallardo et al
Figure 1. Illustration of the concept of geometrical similarity between two images using schematicp (a) and q (b) images, both of arbitrary units from 0 to 10. The coincident vectors represent thegradients of the properties in corresponding zones. The corresponding vectors at positions 1and 2 in both images, have significant amplitudes but point in different directions, implying nostructural similarity. The vectors at position 3 also have significant amplitudes and point in exactlyopposite directions, implying structural similarity. (c) Contour map of the calculated values ofthe cross-gradients function for the p and q images. The largest positive or negative values of thecross-gradients are found in those areas with least structural similarity.
and
dgj = gj (q) + egj j = 1, . . . , ndg (2)
where df and dg are the actual data, which are predictable from their respective imageparameters but are individually subject to independent (uncorrelated) errors, ef and eg. Wedo not assume ab initio that there is any correlation between df and dg and define the firstelement (φd ) of our objective function in a least-squares sense as
φd(p, q) = ndf (rmsf )2 + ndg(rmsg)2, (3)
where
rmsf =√√√√ 1
ndf
ndf∑i=1
(df i − fi(p)
σf i
)2
(4)
and
rmsg =√√√√ 1
ndg
ndg∑j=1
(dgj − gj (q)
σgj
)2
. (5)
In these expressions σf and σg are the measured or assumed standard deviations of thecorresponding data, which will give the appropriate weight to each datum and keep the globalstatistical meaning of (3).
To provide the linking feature of the images required for JIR, we adopt the cross-gradientsfunction (equation (1)) and make the assumption that two images are geometrically identicalif the coincident property changes in both images are collinear. This is conveniently measuredby the cross-gradients constraint (Gallardo and Meju 2003, 2004) given by
t (x, y) = 0. (6)
A quadratic programming approach for joint image reconstruction 439
Equations (3) and (6) are combined with other more conventional regularization terms toformulate an objective function of the form
min
{φ = φd(p, q) +
∫ ∫M
((∇2p(x, y))2
α2p(x, y)
+(∇2q(x, y))2
α2q(x, y)
)dx dy
+∫ ∫
M
((p(x, y) − pR(x, y))2
σ 2Rp(x, y)
+(q(x, y) − qR(x, y))2
σ 2Rq(x, y)
)dx dy
}.
Subject to: t (x, y) = 0,
pmin(x, y) � p(x, y) � pmax(x, y)
and
qmin(x, y) � q(x, y) � qmax(x, y). (7)
The second term in the right-hand side of equation (7) relates to a condition of smoothness(Tikhonov and Arsenin 1977, Twomey 1977), where both images are required to minimizetheir curvature as measured by the Laplacian of the images. This regularization measureis weighted by the functions αp(x, y) and αq(x, y), which could play the statistical role ofthe standard deviations of the Laplacian operator as applied to the images. Although theseweighting functions can be set variable to facilitate the reconstruction of images with specifictrends in selected areas (for instance, geophysical images with a ‘stratified’ aspect), they arepreferably assumed constant for the whole space defined by the model M, in the form ofsmoothing factors (Constable et al 1987). The third term in the objective function (φ) inequation (7) is used to control the closeness to the a priori images pR(x, y) and qR(x, y) in aridge-regression-type constraint (Hoerl and Kennard 1970, Marquardt 1970) where σRp(x, y)
and σRq(x, y) are the estimated or assumed standard deviations for the a priori images. Theregularization terms are useful not only to control those parts of the images that are not wellconstrained by the data but also to provide stability in the computational implementation andminimization of this objective function. The ridge-regression-type term in equation (7) ismainly reserved for padding, i.e. to overcome those parts of the images that are not coveredby the data and may fall into the null space of the Laplacian functions (e.g. constant values).Examples of this situation are the areas at the borders of ray tomographic images, which do notusually cover the borders of the images, and any other padding pixels that could be requiredto make both images spatially coincident (i.e. equal M).
In general, the cross-gradients constraint (t (x, y) = 0) should guarantee geometricalsimilarities between any given images, i.e. p(x, y) and q(x, y), in the form of commonproperty changes between them. In addition, the fact that equation (6) is also satisfied wheneither ∇p(x, y) or ∇q(x, y) equals zero allows the admission of a property change which issensed by only one data set. This is important in complementary phenomena, for instance,both x-rays and MRI can distinguish bone from soft tissues but different types of soft tissueswill only appear as shadows in conventional radiography (e.g. Li et al 2003). This conditionis also important in areas without simultaneous coverage of both data sets; in this case, ifthe image without coverage is properly padded with very smooth or constant image values(e.g. using smoothness constraints) then t (x, y) will always vanish and the other image isreconstructed on its own. In more practical terms, this condition allows the application of JIReven on images with different coverage.
Added to the cross-gradients constraint, we include inequality constraints that keep thevalues of the images within feasible limits defined by the values pmin, pmax, qmin and qmax.These constraints not only keep the physical sense of the images (e.g. we cannot expect
440 L A Gallardo et al
reflectivity coefficients out of the range [0,1] or negative acoustic velocities) but also combinewith the data and geometrical constraints to reduce the space of possible image-solutions.Conveniently, the inequality constraints can be reduced to equality constraints to accountfor individual pixel values that can be directly provided by control points for defined (x, y)
positions.Alternative objective functions similar to (7) can be proposed but they must be carefully
selected and tested. For instance, earlier experiments performed by Gallardo (2004) showedthat the minimization of an objective function that includes the rms values of the cross-gradients can be unstable and the models obtained could not satisfy the data. Similarly, theobjective function in equation (7) is defined for continuous functions and has integrative terms.However, we should pay special attention to mathematical singularities or discontinuities inthe images, where the derivatives used in the Laplacian or the cross-gradients operators may beinfinite, as well as to the stability of the solution process. In the discrete problem this impliesnot only the selection of an appropriate computational algorithm but also the design of anappropriate discrete formulation and sampling that satisfy, for instance, the inf–sup condition(see Brezzi and Fortin (1991)) and the adequate selection of matrix conditioning (damping).
3. Quadratic programming solution for band-limited images
The solution of equation (7) leads to a nonlinear optimization problem with second-orderconstraints. Although there are some computational algorithms developed for this type ofapplication (e.g. Gill et al 1986b), the solution of equations similar to (7) is usually reducedto an iterative process of sequential linear optimization with first-order inequality constraints(using simplex-type schemes). In any case, it is required that the functions f (p), g(q) as wellas t (p, q) are Frechet differentiable. For this we divide our images into multiple sub-regionsand use constant functions to describe the properties within the elements. For simplicity, wediscretize each two-dimensional image in n coincident rectangular cells and use the vectorsp = {pi} and q = {qi} for i = 1, . . . , n to include the individual p and q values that cover thewhole space of the images.
The functions f (p), g(q) and t (p, q) are linearized using a first-order Taylor expansionof the form
fi(p)∼= fi(p0) +n∑
k=1
∂fi(p)
∂pk
∣∣∣∣p=p0
· (pk − p0k) i = 1, . . . , ndf , (8)
gj (q)∼= gj (q0) +n∑
k=1
∂gj (q)
∂qk
∣∣∣∣q=q0
· (qk − q0k) j = 1, . . . , ndg (9)
and
tl(p, q)∼= tl(p0, q0) +n∑
k=1
{∂tl(p, q)
∂pk
∣∣∣∣p=p0q=q0
· (pk − p0k) +∂tl(p, q)
∂qk
∣∣∣∣p=p0q=q0
· (qk − q0k)
}
l = 1, . . . , n (10)
where ∂fi (p)
∂pk, ∂gj (q)
∂qk, ∂tl (p,q)
∂pkand ∂tl (p,q)
∂qkare the p, q and t Frechet derivatives respectively, and
the 0 indexed terms refer to previously reconstructed images. The corresponding discreteexpressions for ∂tl (p,q)
∂pkand ∂tl (p,q)
∂qkfor a three-cell scheme using finite differences can be
found in Gallardo and Meju (2004) whereas ∂fi (p)
∂pkand ∂gj (q)
∂qkdepend on the particular imaging
techniques. Note that in a linear case, the first two linearized expressions (equations (8) and
A quadratic programming approach for joint image reconstruction 441
(9)) are not necessary, however the linearized form of t is still required to reduce the quadraticconstrained problem of equation (7) to a linear constrained problem.
The use of the linearized functions (8), (9) and (10) implies that the objective function (7)should be solved in iterative steps. Denoting the two images from a previous iteration as p0
and q0, the updated images p and q are computed by solving the linear problem:
min{φL = [df − f(p0) − F(p0) · (p − p0)]
T C−1ddf [df − f(p0) − F(p0) · (p − p0)]
+ [dg − g(q0) − G(q0) · (q − q0)]T C−1
ddg[dg − g(q0) − G(q0) · (q − q0)]
+ pT LT C−1LLpLp + qT LT C−1
LLqLq + [p − pR]T C−1RRp[p − pR]
+ [q − qR]T C−1RRq[q − qR]
}subject to t(p0, q0) + T(p0, q0) ·
[p − p0
q − q0
]= 0,
pmin � p � pmax and qmin � q � qmax. (11)
In these expressions the data and mathematical functions are represented by their respectivevectors, namely:
df = {df i}, dg = {dgj }, f(p0) = {fi(p0)}, g(q0) = {gj (q0)},
F(p0) ={
∂fi(p)
∂pk
∣∣∣∣p=p0
}, G(q0) =
{∂gj (q)
∂qk
∣∣∣∣q=q0
},
t(p0, q0) = {tk(p0, q0)}, T(p0, q0) ={
∂tl(p, q)
∂pk
∣∣∣∣p=p0q=q0
,∂tl(p, q)
∂qk
∣∣∣∣p=p0q=q0
},
for i = 1, . . . , ndf , j = 1, . . . , ndg , l = 1, . . . , n and k = 1, . . . , n. The standard deviationsreferred to in equation (7) are all included in the covariance matrices in equation (11)using the notation: diag(Cddf ) = {
σ 2f i
}, diag(Cddg) = {
σ 2gj
}, diag(CLLp) = {
α2pl
},
diag(CLLq) = {α2
qk
}, diag(CRRp) = {
σ 2Rpk
}and diag(CRRq) = {
σ 2Rqk
}, for i = 1, . . . , ndf ,
j = 1, . . . , ndg and k = 1, . . . , n. For simplicity, we assume that the off-diagonal elementsof these matrices are zero. The a priori images are denoted by pR and qR, and they can serveas p0 and q0 but this is not mandatory. The matrix L corresponds to the discrete version ofthe Laplacian operator given by equation (7) for which we use finite differences and a centredfive-point scheme. The subscripts ‘min’ and ‘max’ refer to the minimum and maximum valuesof the p and q pixels of the images that define their band limits.
The solution to equation (11) for every iterative step reduces to a quadratic programmingproblem with linear inequality constraints. A convenient means to solve this problem is thequadratic programming algorithm of Gill et al (1986a). This algorithm exploits the convexityand treats the singularities for constrained least-squares problems (Gill et al 1986a) and canalso classify and propose solutions whether they are unique (global), weak (unstable) orunbounded, thus warning about the stability of the process. This algorithm has also provedsuccessful in similar constrained problems in geophysical applications (Perez-Flores et al2001, Gallardo-Delgado et al 2003).
The algorithm of Gill et al (1986a) aims to solve for x by
min{cT x + 1
2 xT Qx}
subject to
[xmin
bmin
]�
[x
Bx
]�
[xmax
bmax
].
(12)
For the present JIR problem, x = [ pq
], xmin = [ pmin
qmin
], xmax = [ pmax
qmax
]and the appropriate
expressions for c, Q, B and b are
442 L A Gallardo et al
c = 1
β2 FT C−1ddf [df − f(p0) + Fp0] + C−1
RRppR
1β2 GT C−1
ddg[dg − g(q0) + Gq0] + C−1RRqqR
, (13)
Q = 1
β2 FT C−1ddf F + LT C−1
LLpL + C−1RRp 0
0 1β2 GT C−1
ddgG + LT C−1LLqL + C−1
RRq
, (14)
B = T(p0, q0) (15)
and
bmin = bmax = T(p0, q0) ·[
p0
q0
]− t(p0, q0). (16)
We found that the two-stage minimization process described in Gallardo and Meju (2004)also furnishes an appropriate iterative procedure for JIR, which we summarize in figure 2.The main stage of this process ensures the simultaneous gradual fit of the data at each iterationby establishing a target misfit and conditioned by the auxiliary damping factor β, whilethe sub-stage iterations seek the convergence of the images to satisfy the cross-gradientsconstraint thereby ensuring geometrical conformity between the images. Note that while thelinearized version of the cross-gradients constraint (in equation (11)) is satisfied at each sub-stage step, the corresponding values of the cross-gradients function are reduced gradually. Thegradual image evolution implicit in the two-stage process normally requires a large number ofiterations so that it may be inefficient to incorporate complicated computational schemes. Thusthe quadratic programming methodology can make the JIR process slower than the alternativeLagrange-multiplier solution provided in Gallardo and Meju (2004). Another computationaldifficulty is the increased memory requirements of the algorithm to process images with morepixel resolution, compared with that required for an equivalent processing using the Lagrangemultiplier approach proposed in Gallardo and Meju (2004), as shown in figure 3. Nevertheless,the relative disadvantages of time and memory involved in the present approach should beweighed against the possibility of production of more accurate band-limited images withphysical significance as demonstrated in the following sections.
4. Application to linear mathematical functions
To test the algorithm we define two images divided into n = 280 cells of dimensions 10 ×5 units. Each image has a constant background value of p = 0 and q = 0.1 and two squareblocks immersed in the images with different p and q values as depicted in figures 4 and 5,respectively.
To generate synthetic data sets we selected a mathematical function h(s) based on anormalized area-averaging Gaussian function given by
h(s) =∫∫
M e−(r−rc )2
δ2 s(x, y) dx dy∫∫M e
−(ρ−rc )2
δ2 dς dξ
(17)
where r =√
(x − xc)2 + (y − yc)2, ρ =√
(ς − xc)2 + (ξ − yc)2 and s(x, y) represents theimage parameters (p or q). This function gives more weight (data sensitivity) to the areas nextto the circumference of radius rc centred on (xc, yc). The δ value gives the function decay,spreading radially from the circumference.
A quadratic programming approach for joint image reconstruction 443
Figure 2. Flow chart showing the main features of the two-stage quadratic programming algorithmfor JIR using cross-gradients.
For the function
f (p) =∫ ∫
Mwf (x, y)p(x, y) dx dy (18)
we used equation (17) with the value of rc held fixed at zero and varying the (xc, yc)
positions as well as the decay δ. This function conforms to a circular Gaussian bell thatmay resemble the effects associated with diffusive phenomena (like those measured by low-frequency electromagnetic imaging deployments) where (xc, yc) can represent the position ofthe sensor or source and δ the coverage of the diffusive spreading. An example of the wf (x, y)
function (i.e. sensitivity of this type of data to the image) is shown in figure 4.Similarly, we defined
g(q) =∫ ∫
Mwg(x, y)q(x, y) dx dy (19)
from equation (17) using a fixed (small) value for δ and varying both the position (xc, yc) andradius rc. This will produce narrow averaging areas that can emulate the data sensitivity for
444 L A Gallardo et al
Figure 3. Comparison of the memory requirements for storage of the Hessian matrix (Q inequation (14)) and the workspace for the simultaneous processing of two images with n pixels,using the proposed quadratic programming approach (a) and the Lagrange multiplier approach ofGallardo and Meju (2004) (b).
Figure 4. Mesh design for the test p image discretized by the cells indicated by dashed lines.The thick lines show the double-box structure of the image and enclose areas with constant pvalue (annotated). Two examples of the area-averaging function wf (x, y) (equation (18)) for thesensor–receiver indicated with double arrows at position 1 (using δ = 3) and position 2 (usingδ = 40) are also illustrated.
ray-tomography experiments using beam propagation paths (cf Michelena and Harris 1991).This function will then furnish the q image. The combination of position (xc, yc) and theradius rc defines indirectly the position of hypothetical sources and sensors as well as thepenetration of the curved rays. An example of the wg(x, y) function (sensitivity of this typeof data) is shown in figure 5.
We computed 45 df (p) data using the image in figure 4 and the mathematical functionf (p) of equation (18). For this function we selected a fixed yc position in the top margin
A quadratic programming approach for joint image reconstruction 445
Figure 5. Mesh design for the test q image discretized by the cells indicated by dashed lines. Thethick lines correspond with those of figure 4; they show the double-box structure of the imageand enclose areas with constant q value (annotated). Two examples of the trajectory-averagingfunction wg(x, y) (equation (19)) for the sensor–receiver arrays indicated with arrows and invertedtriangles, at positions 1–1′ and 2–2′, are also illustrated.
of the image and set the xc position at five different places (–60, –30, 0, 30, 60). For eachxc position we used nine δ values distributed in the range [3, 40]. We degraded these datawith normally distributed errors of zero mean and standard deviation 0.05 units. Similarly, wecomputed 51 dg(q) data using the parameters q of the image in figure 5 and the function g(q)
given by equation (19). We selected appropriate (xc, yc) and rc parameters for each datum tomake it coincident with a receiver–source position at the top of the model and a maximumdepth of penetration of one third of the source–receiver separation. We set three sourcesat profile positions –80, 0 and 80 as well as 17 receivers equally spaced every 10 units.The ray penetration achieved ranges from 3 to 53 units and will cover the box-shapedheterogeneities. The decay parameter δ was fixed at δ = 7.9 which equals the rms valueof the cell dimensions used in the model. To these data were added normally distributed errorsof zero mean and standard deviation up to 30% for the arrays with 10 unit spread arrays whichgradually decreased down to 1% for those arrays with 160 unit spreads.
We used the developed algorithm to reconstruct the images that will reproduce thesesynthetic data. We performed three comparative tests: the first test used the conventionalmethod of independent reconstruction of the images; the second test employed the quadraticprogramming approach using cross-gradients constraints but no inequality constraints thusresembling the Lagrange-multiplier solution given in Gallardo and Meju (2004); and the thirdtest employed the full quadratic programming scheme including inequality constraints. Inall the tests, the regularization parameters were held fixed to validate the comparison, eventhough for the separate reconstruction test, there was no need of iterative steps since f (p) aswell as g(q) are linear.
For all the tests we started the reconstruction process from images of flat backgroundswith constant pR = {0} and qR = {0.1}, and selected constant regularization (or damping)factors αp = αq = 0.01 and σRp = σRq = 10, these being the values selected after sometrials. For the experiments of JIR we follow the two-stage minimization process described,setting normalized misfit targets that gradually decrease from β = 31.6 to β = 1 in six stepsthat are equally spaced logarithmically. In the third test, we assumed that feasible values of
446 L A Gallardo et al
Figure 6. Images recovered in the test experiments: (a), (b) conventional separate estimation ofthe p and q images of figures 4 and 5. (c), (d) Joint image reconstruction for the p and q imagesusing only cross-gradients constraints. (e), (f ) Joint image reconstruction for the p and q imageswith both cross-gradients and inequality constraints. Note the similarity in the geometrical aspectof the images recovered by JIR in contrast to those obtained by separate image reconstructionand the enhancement when the inequality constraints are added. The hatched zones indicate thevalues beyond the selected upper or lower bounds. The arrows and inverted triangles show the‘sensor–receiver’ positions for the synthetic data as explained in figures 4 and 5.
p and q should exist between the margins −1 � p � 1 and 0.1 � q �1 for every pixel andimposed these values as inequality constraints in the image reconstruction process.
For comparison, the best-fitting images recovered in the separate and joint reconstructiontests are presented in figures 6(a)–(f). The images of figures 6(a) and (b) were obtained byseparate estimation and they barely recovered the structure of the boxes as allowed by therespective data coverage. The spacing of the data locations was reflected as several highfrequency changes at the top of the images while the effect of the regularization measures(smoothness and a priori flat image) are evident at the borders of both images. However, thedifferences in the structures delineated and the q values obtained outside their feasible limitsare noticeable. The images resulting from the second test (joint inversion without inequality
A quadratic programming approach for joint image reconstruction 447
Figure 7. Cross-gradients values for the images of figures 6(a) and (b). The largest geometricaldifferences in the figures are reflected by the largest values of the cross-gradients.
constraints) are shown in figures 6(c) and (d). These resolved the gross features of the imagesbut also show common features that enhance the target double-box structure of the images.Some high-frequency features at the top of the images are reduced (cf figures 6(a) and (b));however, there are still some undesirable padding structures common to both images and thevalues of q exceeded their feasible limits in some cells. The images resulting from the fullquadratic programming approach with inequality constraints are shown in figures 6(e) and (f).In this case, the recovered images not only show common features but also have enhancedthe true target structure eliminating the high-frequency features and most of the undesirableundulating padding features. As stipulated in this approach, all the values (especially theq values) were kept within physically meaningful limits and this contributed to the improvementachieved in both images.
To assess the geometrical similarities between corresponding pairs of images we plottedthe values of the cross-gradients for the separate images of figures 6(a) and (b) in figure 7.This figure shows the influence of the differences in the high (shallow) and low (padding)frequency contrasts around the edges of the boxes. In contrast, the values of the cross-gradientsfor the jointly reconstructed images are more randomly distributed and their small rms values(9.15 × 10−7 for the second test and 1.54 × 10−6 for the third test) make them indiscernibleat the scale of the corresponding values for the separate images.
In the full JIR approach using inequality constraints, the two-stage process to constructthe images showed robustness as it constrained the individual images for every step to staywithin the margins imposed. As requested, at every step the rms level of misfit was reducedaccording to the values set, as shown in figure 8(a), and the differences between both imageswere gradually reduced to achieve the sought geometrical resemblance. The predicted datamatched the test data to normalized misfits of rmsf = 0.87 and rmsg = 1.05 and the globaldistribution of the normalized residuals is shown in figure 8(b).
5. A geophysical example
We tested the developed quadratic programming algorithm for JIR using direct current(dc) electrical resistivity and seismic refraction geophysical experiments to produce two-dimensional geophysical images of the subsurface. We selected these two geophysical imagingtechniques because they are based on different physical phenomena and are highly popular
448 L A Gallardo et al
Figure 8. (a) Rms normalized misfit obtained at every main-iterative step. (b) Distribution of thefinal global normalized residuals.
for engineering and groundwater applications (e.g. Riley 1993), thus providing an appropriatereference framework to evaluate the attributes of the developed JIR approach.
The geophysical images were parametrized using the logarithm of electrical resistivity(in m) as the p parameter and elastic wave slowness (in s km−1) as the q parameter. Wedivided each image into n = 1740 rectangular cells of varied dimensions as illustrated infigure 9 but both images were extended 400 m beyond the plotted limits to allow the naturaldecay of the electrical fields. The synthetic images (or test models) were generated fromappropriately chosen values of p and q distributed so as to yield the double-box structureshown in figure 9. We avoided the occurrence of any direct dependence between the imageparameters ( p and q) other than the common structure of the target and deliberately made thisexample similar to that of Gallardo and Meju (2004) for comparison purposes.
To account for the electrical imaging we selected the approach developed by Perez-Floreset al (2001) founded on the nonlinear integral equations of Gomez-Trevino (1987) since itsimplifies the otherwise nonlinear function f (p) to a linear form Fp, and directly providesthe required Frechet derivatives. An example of this function for a specific electrode arrayis shown in figure 9. For the seismic imaging, we implemented the forward-finite differencetechnique of Vidale (1990) which solves the eikonal equation to produce the travel time field(g(q)) for a seismic source. We combined this methodology with the conventional ray-tracingtechnique to compute the Frechet derivatives (G). An example of the ray coverage for asource located at profile position −100 m is also illustrated in figure 9. The mathematical andnumerical details of both techniques are given in the above-mentioned papers, and we willonly focus on their numerical application for JIR using quadratic programming.
We used the respective test models to generate the relevant test data. A total of 171apparent resistivity data (df ) were computed for several electrode array deployments on thehypothetical Earth’s surface. We used dipole–dipole electrode arrays similar to that illustratedin figure 9, with fixed electrode spacing (a = 10 m) and varied dipole–dipole separations(na = 10 m to 160 m) all distributed along the plotted profile. The coverage of these datashould suffice for imaging the target structures as exemplified by the sensitivity values for theelectrode array shown in figure 9. Our seismic data (dg) consisted of 200 records of the time
A quadratic programming approach for joint image reconstruction 449
Figure 9. Synthetic model for generating geophysical test data. The model is discretized intorectangular cells indicated by dashed lines. The numbers in the model represent the values of thelogarithm of resistivity parameter ( p) and the slowness parameter (q). The thick solid lines definea two-box structure with constant p and q values (annotated). The colour tones show the sensitivityfunction (Frechet derivative values F ) for the four-electrode (dipole–dipole) array computed bythe approach of Perez-Flores et al (2001). The solid lines illustrate the sensitivity of a seismicdeployment with source at position −100 m and receivers every 10 m using conventional raytracing.
taken for the fastest wavefront to reach different receivers (first arrival times). The elasticwaves were generated from five seismic sources distributed along the hypothetical Earth’ssurface at profile positions −100, −50, 0, 50 and 100 m. We used 40 collinear surface-basedreceivers set every 5 m covering also the plotted profile. For completeness, the syntheticapparent resistivity and travel time data were degraded with normally distributed errors ofzero mean and standard deviation 1% and 0.1 s, respectively.
To perform the JIR of the geophysical images we selected constant regularizationparameters (αp = 31.62, αq = 316.2, σRp = 10 and σRq = 3) after several experiments. Westarted the reconstruction process from the flat a priori images (pR = {2} and qR = {1}) andassigned the band limits 0 � p � 4 and 0.1 � q � 3.125 as the extreme but feasible values.We assigned standard deviation of 1% for the apparent resistivity data (σf ) and 0.1 s for thetravel times (σg) to account for the data errors and performed the JIR using initial β value of31.62, which was set to decrease to 1 in six main-stage iterative steps.
The algorithm converged to the electrical and seismic images shown in figures 10(a) and(b), which are geometrically similar. Both images define the original double-box structure(depicted in figure 9) better than the images obtained from the corresponding separate inversionexperiments (cf Gallardo and Meju (2004), figures 7(a) and (b)). The reconstructed images alsoturned out to be similar to those obtained using Lagrange multipliers (cf Gallardo and Meju(2004), figures 7(c) and (d)) due to the relatively large separation specified for the inequalitybounds. The overly large q values attained in figure 10(b) resulted from the production of asmooth model as optimal. This illustrates the role played by the complementary regularizationmeasures when the data, the cross-gradients and the inequality constraints become insufficientto constrain the target image. Regarding the data fit, the theoretical responses of the models fit
450 L A Gallardo et al
(a)
(b)
Figure 10. (a) Resistivity and (b) seismic images obtained by JIR using quadratic programming.
the test data to normalized misfits of rmsf = 0.815 and rmsg = 1.300 and figure 11 shows thatthe corresponding normalized residuals are Gaussian distributed. In general, the reconstructedgeophysical images satisfy the conditions set in the objective function: they are similar, staybetween their feasibility limits and satisfactorily justify the data.
For this particular example, the computational cost of the quadratic programming approachis illustrated in figures 12(a) and (b). The advantage of the quadratic programming approachover the Lagrange multiplier algorithm of Gallardo and Meju (2004) is the stability ofthe iterative process. This can be seen in the convergence of the respective rms values(cf figures 12(a) and (b)). This arises because the geophysical images at each sub-stageiteration were kept within the feasible limits thus avoiding major computational difficultiessuch as violations of the causality principle for the seismic wave propagation, particularly thatdue to unrealistic zero or negative slowness values. However, the process also showed the highcomputational cost of increasing the processing time by one order of magnitude. Though wedid not explore this field further, it is likely that the incorporation of computational strategiesapplied to large-scale problems may help us to reduce this inconvenience.
A quadratic programming approach for joint image reconstruction 451
Figure 11. Distribution of the final normalized residuals obtained for the geophysical test example.For this example the global normalized rms misfit attained is 1.1.
Figure 12. Comparison of convergence trends and computing time for all the sub-stage iterativesteps in the image reconstruction approaches for the geophysical example: (a) using Lagrangemultiplier method (Gallardo and Meju 2004), (b) using the quadratic programming approach.Note the differences in stability and convergence rate for the two approaches.
6. Conclusions
We have presented a robust solution methodology for joint image reconstruction (JIR) that canbe used to simultaneously process multiple imaging experiments of any nature performed overa band-limited target. The experimental results demonstrate that the multiple images derivedusing JIR not only reproduce the experimental data satisfactorily but also have enhancedcommon geometrical features, and are superior to conventional separately reconstructedimages. The synthetic experiments also demonstrate the advantages of including band limitsin the process of image reconstruction. These inequality constraints prevent the recovery ofunrealistic images, improve the overall geometrical features and provide further stability to
452 L A Gallardo et al
the JIR process by helping avoid major computational difficulties. Considering the quadraticprogramming approach adopted here, it can be expected that the time consumed in the processof JIR using quadratic programming techniques and inequality constraints will normally exceedthat of a Lagrange constrained algorithm. However, this can be balanced by the benefits ofthe inequality constraints when dealing with band-limited images.
Acknowledgments
The authors acknowledge the thoughtful comments of two anonymous referees that greatlyimproved this manuscript. LAG acknowledges a postgraduate research award from SUPERA-ANUIES, Mexico.
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