Joint Inversion of Multiple Data Sets for Upper Mantle Velocities

in the Southern Rio Grande Rift

Anibal Sosa1,4, Lennox Thompson2, Aaron A. Velasco2, and Rodrigo Romero3

1 Computational Science Program, 2 Department of Geological Sciences, 3 Department of Computer Science,The University of Texas at El Paso, El Paso, TX, 4 Universidad Icesi, Cali - Colombia.

usosaaguirre@miners.utep.edu

Abstract

We propose a novel approach for joint inversion of multiple geophysical data sets to charac-terize velocity structure of the upper mantle in the southern Rio Grande Rift region. The in-verse problem is posed as in nonlinear programming and solved with interior-point methods.We introduce physical bounds over the model parameters and a measure of differences ingeological structure. Bound and structural constraints introduced as a priori information tobetter estimate the physics of each dataset, has shown to improve the numerical resultswhile reducing computational efforts in finding regularization parameters. We will presentinitial results revealing 3-D velocity structure that suggests continuation of deformation andextension of the Rift.

1. Background & Motivation

The southern terminus of the Rio Grande Rift (RGR) region (Figure 1) has been poorly de-fined in the geologic record, with few seismic studies providing information on the deeperrift structure. Important questions related to tectonic and lithospheric activity of the RGRremain under debate:• Is the RGR still active? What is the role of the mantle in rift formation?

We address these geological questions by using the available seismic data from EarthScopeTransportable Array (USArray). Moreover, we also respond to the following question:•How can multiple geophysical data sets with different sensitivity and resolution domains

be integrated to characterize Earth structure ?

We use a constrained optimization approach for joint inversion of seismic data, to constructnew models of 3-D S−wave velocity structure in the upper mantle of the southern RGR.• Advantage: complementary data improves final (velocity) model resolution, and a

Bayesian kriging interpolation scheme allow us to introduce error estimates.•Disadvantages: data level of influence and selection of regularization parameters.

Figure 1: SRGR topography, tectonicprovinces, and stations used (whitetriangles) mostly from USArray.

Figure 2: Topography map and plot ofall the seismicity from 1975 to 2012.The size of the red circles indicatesthe event magnitude.

2. Problem Formulation

Our goal: Use a 1-D constrained optimization approach for joint inversion of observedground motion measurements, y = (y1, ..., ym), to estimate S-wave velocities x = (x1, ..., xn).

2.1 Forward & Inverse Problems• Forward problem : Given a velocity model x ∈ Rn+ evaluate F (x) = y, where F : Rn+ →Rm is a nonlinear forward operator and y ∈ Rm is the estimated data.• Inverse problem : Given some measurements y, find the unknown model x such thatF (x) ≈ y, i.e. solve a nonlinear LS Problem,

minx‖F (x)− y‖2 = min

x

m∑i=1

(Fi(x)− yi)2 . (m� n) (1)

2.2 Unconstrained vs Constrained Joint Inversion Schemes

Unconstrained Joint Inversion

•Widely used by the geophysicalcommunity.•Robust algorithms for solving (1),

i.e. TSVD.• Key idea: To penalize constraint vi-

olations.•Requires computation of adequate

regularization parameters.

Constrained Joint Inversion

• Apply techniques from the opti-mization field for joint inversion.

• Powerful algorithms for solvinglarge-scale problems.

• Key idea: To keep iterates feasiblewith inequality constraints.

•Requires a good starting modeland robust equation solvers.

3. Methodology

3.1 1-D Constrained Joint InversionWe apply our joint inversion approach to two complementary data sets, receiver functionsand surface waves group dispersion [Julia et al., 2000], and use Primal-Dual Interior-Point(PDIP) methods as a solver [Sosa et al., 2012].• Interior-point methods were introduced by Karmakar in 1984 as an alternative algorithm

for solving linear programming problems:

min cTx (Primal) max bTz (Dual)s.t. Ax = b, x ≥ 0 s.t. ATz + s = c. s ≥ 0

• Successful methods for solving linear and nonlinear programming problems. Two mainapproaches:– Logarithmic Barrier Method − Primal-Dual Method.

We apply a PDIP methodology to solve a constrained formulation of problem (1):

minx

12‖F (x)− y‖2W

s. t. g(x) ≥ 0,(2)

where g(x) =

x− cmincmax − x

nγ − 12‖Lx‖

2

, γ ∈ (0, cmax − cmin) allows us to add appropriate physical

bounds over the model x, a structural constraint by using a first order discrete derivativeoperator L, and to equalize the contribution of each data set while accounting for their influ-ence with a weighted diagonal matrix W .• In this methodology we define a Lagrangian function associated to (2)

`(x, z) =1

2||F ′(xk)x + r(xk)||2 − g(x)Tz, (g(x), z) > 0, (3)

where r(xk) = F (xk)− y − F ′(xk)xk. The Newton’s system associated to (3) is reduced to:[−F ′(xk)TF ′(xk) + z2n+1L

TL ∇g(x)T

∇g(x) Z−1G(x)

] [∆x∆z

]=

[−∇x`(x, z)

µZ−1e− g(x)

](4)

Z = diag(z), G(x) = diag(g(x)), z is a lagrange multiplier, and µ is a perturbation parameter.

• The symmetric system in (4) can be solved with a direct method (LU decomposition) oran inexact method (CG algorithm).

4. Joint Inversion Results

•We perform 1-D joint inversions using our PDIP approach independently for 147 USArrayand other seismic stations down to 420 km depth.•We interpolate these 1-D velocity models with a Bayesian kriging scheme, to create 3-D

crustal and upper mantle structure images for the three profiles in Figure 2.•We constrain the velocity to 4− 5 km/s to highlight features deep in the upper mantle.

Figure 3: Cross-section A-A’ at latitude 34 ◦ shows a clear distinction between the ColoradoPlateau (CP), Socorro Magma Body (SMB) at the center of the RGR, and the Great Plains(GP). We find that near the CP, there is a low velocity lid with high velocities beneath thisprovince, consistent with other studies.

Figure 4: Cross-section B-B’ coincides with longitude 107 ◦ and intersects cross section A-A‘ and passes through SMB. The SMB seems to have slow velocities between the ColoradoPlateau and Great Plains.

Figure 5: Cross-section C-C’ coincides with the southern part of the LA RISTRA passiveexperiment. Seismically fast mantle underlies the RGR and relatively slow mantle is seenbeneath the Socorro Magma Body and Colorado Plateau.

Figure 6: Full 3-D velocity structure perspective of the RGR view from N-W. We identifylow velocities associated to the southernmost part of the RGR. The upper 200 − 300 km ofmantle beneath the magmatically and tectonically active RGR and Basin & Range (B&R) isseismically distinct from the mantle beneath the stable Colorado Plateau and Great Plains.

5. Conclusions

•We present a robust approach that connects a constrained joint inversion algorithm with aKriging interpolation scheme for high resolution imaging of Earth structure. This approachallows us to combine 1-D velocity models to produce 3-D images of the Earth.•We find two fast seismic anomalies beneath the central Colorado Plateau and the Great

Plains respectively, and a third anomalously low velocity within the upper mantle at theSocorro Magma Body.•We can identify the boundaries between the provinces of B&R, CP, GP and RGR. More-

over, we hypothesize that the SMB might originate from a deep plume source probablylocated outside our region of study.• The southern most part of the RGR dies out in El Paso, but this region remains unresolved

due to lack of data for the northern most portion of Mexico.

Future work will include:• Implementation of a stronger structural constraint to remove noisy components during the

inversion, and a grid continuation scheme for modeling resolution.• Include other compatible data sets, e.g. gravity and delay travel time data, to constrain

further the inversion process and increase the resolution of the mantle.

6. Acknowledgments

This research project is based on work supported by the National Science Foundation under Grant No.0734825, CAHSI funded by NSF Grant No.CNS-1042341 and the Program in Computational Science at UTEP.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not neces-sarily reflect the views of the National Science Foundation.

References

[1] J. Julia, C. J. Ammon, R. Herrmann, and M. Correig. Joint inversion of receiver function and surface wavedispersion observations. Geophysics Int. J., 142: 99-112. 2000.

[2] M. Maceira and C.J. Ammon, Joint inversion of surface wave velocity and gravity observations and itsapplication to central Asian basins s-velocity structure. Journal of Geophysical Research, 114, 2009.

[3] A. Sosa, L. Thompson, A. A. Velasco, R. Romero and R. Herrmann. 3-D Structure of the southern RioGrande Rift from 1-D constrained joint inversion of receiver functions and surface wave dispersion, Journalof Geophysical Research (To be submitted).

[4] L. Thompson, A. A. Velasco and M. Hussein. Geophysical Constraints on the Crustal Structure of thesouthern Rio Grande Rift. Journal of Geophysical Research (To be submitted).

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