High-accuracy practical spline-based 3D and 2D integral ...web.mst.edu/~liukh/publications/Wang_Gao_GeopPros_2012_spline.pdfGeophysical Prospecting doi: 10.1111/j.1365-2478.2011.01026.x

Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.01026.x

High-accuracy practical spline-based 3D and 2D integraltransformations in potential-field geophysics

Bingzhu Wang1,2∗, Stephen S. Gao1, Kelly H. Liu1 and Edward S. Krebes3

1Department of Geological Sciences and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA,2GEOTOP-UQAM, Montreal, QC H3C 3P8, Canada, and 3Department of Geoscience, University of Calgary, Calgary,Alberta T2N 1N4, Canada

Received October 2010, revision accepted September 2011

ABSTRACTPotential, potential field and potential-field gradient data are supplemental to eachother for resolving sources of interest in both exploration and solid Earth studies. Wepropose flexible high-accuracy practical techniques to perform 3D and 2D integraltransformations from potential field components to potential and from potential-fieldgradient components to potential field components in the space domain using cubicB-splines. The spline techniques are applicable to either uniform or non-uniform rect-angular grids for the 3D case, and applicable to either regular or irregular grids forthe 2D case. The spline-based indefinite integrations can be computed at any point inthe computational domain. In our synthetic 3D gravity and magnetic transformationexamples, we show that the spline techniques are substantially more accurate thanthe Fourier transform techniques, and demonstrate that harmonicity is confirmedsubstantially better for the spline method than the Fourier transform method andthat spline-based integration and differentiation are invertible. The cost of the in-crease in accuracy is an increase in computing time. Our real data examples of 3Dtransformations show that the spline-based results agree substantially better or bet-ter with the observed data than do the Fourier-based results. The spline techniqueswould therefore be very useful for data quality control through comparisons of thecomputed and observed components. If certain desired components of the potentialfield or gradient data are not measured, they can be obtained using the spline-basedtransformations as alternatives to the Fourier transform techniques.

Key words: Integral transformations, Cubic B-splines, Gravity, Magnetic, Gravitygradients

INTRODUCTION

The vertical components of the gravity acceleration field andthe total-field magnetic data are traditionally measured andthe potential-field gradients are increasingly observed for awide scope of studies in exploration (Nabighian et al. 2005a,b). Horizontal derivatives are used for detecting edges ofsource bodies (Blakely and Simpson 1986; Grauch and Cordell1987). Both horizontal and vertical derivatives are needed

∗E-mail: [email protected]

for determining the lateral location and depth of single ormultiple simple sources with Euler deconvolution (Thompson1982; Reid et al. 1990; Ravat 1996; Nabighian and Hansen2001; Hansen and Suciu 2002; Ravat et al. 2002; FitzGerald,Reid and McInerney 2004), the 2D analytic signal (Nabighian1972, 1974) and the 3D total gradient techniques.

Gravity, magnetic field and gradient data are also appliedto solid Earth studies, such as crustal structure (Behrendt,Meister and Henderson 1966; Thomas, Grieve and Sharpton1988; Allen and Hinze 1992; Bosum et al. 1997; Cochranet al. 1999; Berrino, Corrado and Riccardi 2008), the

C© 2012 European Association of Geoscientists & Engineers 1

2 B. Wang et al.

lithosphere (Martinez, Goodliffe and Taylor 2001; Hebertet al 2001; Jallouli, Mickus and Turki 2002; Abers et al.

2002; Fischer 2002), continental rift studies (Martinez et al

2001; Abers et al. 2002; Mickus et al. 2007), impact structure(Ravat et al. 2002), geothermal models (Bektas et al. 2007;Purucker et al. 2007), seismicity (Kostoglodov et al. 1996),bathymetry (Wang 2000) and volcanic island studies (Carboet al. 2003; Blanco-Montenegro et al. 2005).

Potential, Potential Field (PF) and Potential-field Gradient(PG) data are supplemental to each other for resolving sources.Transformations facilitate data comparison and provide moremeans for interpretation by transforming the measured datainto other forms of data. The fast Fourier transform (FT) isused widely in potential-field geophysics (e.g. Blakely 1996;Sandwell and Smith 1997; Mickus and Hinojosa 2001; Carboet al. 2003). In theory, the Fourier transform techniques couldbe applied to perform potential-field transformations. How-ever, the results of the Fourier transform techniques are oftenless accurate than one might like, and the Fourier transformtechniques are only applied to regular grid points (e.g., Ricardand Blakely 1988; Wang 2006).

High-accuracy spline-based techniques of 3D and 2Dpotential-field upward continuation (Wang 2006) and poten-tial field and gradient component transformations and deriva-tive computations (Wang, Krebes and Ravat 2008) have beendeveloped. In this paper, we propose flexible high-accuracytechniques to perform 3D and 2D integral transformationsfrom PF components to potential and from PG componentsto PF components in the space domain with cubic B-splines.Using synthetic 3D gravity and magnetic transformation ex-amples, we find that the spline techniques are substantiallymore accurate than the Fourier transform techniques, anddemonstrate that harmonicity is confirmed substantially bet-ter for the spline method than the Fourier transform methodand that spline-based integration and differentiation are in-vertible. Real data examples of 3D transformations showthat the spline-based results agree substantially better (fromgravity-gradient components to gravity) or better (betweengravity-gradient components) with the observed data than dothe Fourier-based results. For synthetic or real-data examples,relative root mean square errors between the computed valuesand the corresponding exact or observed values are taken tomeasure the accuracy.

POTENTIAL , POTENTIAL F IELD ANDPOTENTIAL-F IELD GRADIENT

Let U(x, y, z) be the potential, f(x, y, z) be the potential field,and D(x, y, z) be the potential-field gradient tensor. For con-

venience, we use a uniform notation with U(x, y, z) and itsderivatives in this paper.

A potential field component equals to the correspondingpartial derivative of the potential:

fi (x, y, z) = Ui (x, y, z) = ∂U∂i

, i = x, y, z. (1)

Analogously, a potential-field gradient component equals tothe corresponding second-order partial derivative of the po-tential:

Di j (x, y, z) = fi j (x, y, z) = ∂ fi

∂ j= Ui j (x, y, z) = ∂2U

∂i∂ j,

i, j = x, y, z. (2)

Any potential field f(x, y, z) is conservative and curl free, i.e.∇ × f(x, y, z) = 0, so that

Uji (x, y, z) = Ui j (x, y, z), i, j = x, y, z, i �= j . (3)

In the region outside the sources any potential field f(x, y, z)is divergence free, i.e. ∇ · f(x, y, z) = 0. Therefore Laplace’sequation holds

Uzz(x, y, z) = −[Uxx(x, y, z) + Uyy(x, y, z)]. (4)

Considering equations (3) and (4), for the 3D case only five(e.g., Uxx,Uxy,Uxz,Uyy,Uyz) of the nine components of thegradient tensor D(x, y, z) are independent. Analogously, forthe 2D case only two (e.g., Uxx,Uxz) of the four componentsof the gradient tensor D(x, z) are independent.

CALCULATING HORIZONTAL INDEFINITEINTEGRALS W ITH C UBIC B-SPL INES

The 2D case

Approximate V(x) = Uxk, k = x, z or V(x) = Ux withsplines, satisfying conditions (A7) and (A8). The interpola-tion coefficients {Ci} can be determined (Appendix A). Wethen have

∫V(x)dx =

N+1∑i=−1

Ci N−1i (x), (5)

where N−1i (x) is given by equation (A3) in Appendix A.

The 3D case

Approximate V(x, y) = Uxk, k = x, y, z or V(x, y) = Ux withsplines, satisfying conditions (B3) through (B6). The interpo-lation coefficients {Ci, j} can be determined (Appendix B). We

C© 2012 European Association of Geoscientists & Engineers, Geophysical Prospecting, 1–16

Spline-based potential-field integral transformations 3

then have

∫V(x, y)dx =

Nx+1∑i=−1

Ny+1∑j=−1

N−1i (x)Ci, j Nj (y). (6)

Similarly, approximating V(x, y) = Uyk, k = x, y, z orV(x, y) = Uy with splines, satisfying conditions (B3) through(B6). We then have

∫V(x, y)dy =

Nx+1∑i=−1

Ny+1∑j=−1

Ni (x)Ci, j N−1j (y), (7)

where Ni (x), N−1i (x) are given by equation (A3) in Appendix

A, and Nj (y), N−1j (y) are similarly obtained for subscript j .

TRANSFORMATIONS FROM PFCOMPONENTS T O POTENTIAL

2D transformations from Ux to ULet V(x) be Ux(x) and use equation (5) to obtain U(x).

3D transformations from Ux or Uy to ULet V(x, y) be Ux(x, y) or Uy(x, y) and use equation (6) or (7)to obtain U(x, y).

TRANSFORMATIONS FROM PGCOMPONENTS T O PF C OMPONENTS

2D transformations from Uxx to (Ux, Uz)Let V(x) be Uxx(x) and use equation (5) to obtain Ux(x).Knowing Ux(x), calculate Uz(ξ ) (Wang et al. 2008) from

Uz(ξ ) = 1π

∫ ∞

−∞

Ux(x)ξ − x

dx. (8)

2D transformations from Uxz to (Ux, Uz)Let V(x) be Uxz(x) and use equations (5) to obtain Uz(x).Knowing Uz(x), Ux(ξ ) can be calculated (Wang et al. 2008)from

Ux(ξ ) = − 1π

∫ ∞

−∞

Uz(x)ξ − x

dx. (9)

3D transformations from Uxy to (Ux, Uy, Uz)Consider equations (6) and (7). Let V(x, y) be Uxy(x, y), thenUx(x, y) and Uy(x, y) can be calculated. Knowing Ux(x, y) andUy(x, y), calculate Uz(ξ, η) (Wang et al. 2008) from

Uz(ξ, η) = 12π

∫ ∞

−∞

∫ ∞

−∞

(ξ − x)Ux(x, y) + (η − y)Uy(x, y)

[(x − ξ )2 + (y − η)2]3/2

dxdy.

(10)

3D transformations from Uxz or Uyz to (Ux, Uy, Uz)Let V(x, y) be Uxz(x, y) or Uyz(x, y) and use equation (6) or(7) to obtain Uz(x, y). Knowing Uz(ξ, η), calculate Ux(ξ, η) andUy(ξ, η) (Wang et al. 2008) from

Ux(ξ, η) = − 12π

∫ ∞

−∞

∫ ∞

−∞

(ξ − x)Uz(x, y)[(x − ξ )2 + (y − η)2]3/2

dxdy, (11)

Uy(ξ, η) = − 12π

∫ ∞

−∞

∫ ∞

−∞

(η − y)Uz(x, y)[(x − ξ )2 + (y − η)2]3/2

dxdy. (12)

3D transformations from (Uxx, Uyy) to (Ux, Uy, Uz)Let V(x, y) be Uxx(x, y) and use equation (6) to obtainUx(x, y). Let V(x, y) be Uyy(x, y) and use equation (7) to ob-tain Uy(x, y). Knowing Ux(x, y) and Uy(x, y), compute Uz(ξ, η)using equation (10).

EVALUATION OF INFINITE INTEGRALSAND D OUBLE INTEGRALS

Equations (8 and 9) are similar infinite integral relations.When the computational domain D1 = {x|a ≤ x ≤ b} is wellbeyond the lateral extent of all sources of interest, approxi-mate each infinite integral with a definite integral and evaluateit using the spline technique. To avoid a singularity, the sam-pling point (ξ ) must not coincide with a spline knot (x). Thecentre of each interval of the spline grid is an ideal locationfor a sampling point. For a given point (ξ ), approximate thewhole of the integrand in equation (8), e.g., with splines, theinterpolation coefficients {Ci} can be determined (AppendixA). Thus, we have

Uz(ξ ) = 1π

N+1∑i=−1

Ci (ξ )[N−1i (b) − N−1

i (a)], (13)

where N−1i (·) is given by equation (A3) in Appendix A.

Equations (10 and 12) are similar infinite double in-tegral relations. When the computational domain D2 ={(x, y)| a ≤ x ≤ b, c ≤ y ≤ d } is well beyond the lateral extentof all sources of interest, approximate each infinite double in-tegral with a definite double integral and evaluate it using thespline technique. To avoid a singularity, the sampling point(ξ, η) must not coincide with a spline knot (x, y). The centreof each rectangular unit of the spline grid is an ideal locationfor a sampling point. For a given point (ξ, η), approximate thewhole of the integrand in equation (10), e.g., with splines, theinterpolation coefficients {Ci, j} can be determined (Appendix


4 B. Wang et al.

B). Thus, we have

Uz(ξ, η) = 12π

Nx+1∑i=−1

Ny+1∑j=−1

[N−1i (b) − N−1

i (a)]Ci, j (ξ, η)

× [N−1j (d) − N−1

j (c)], (14)

where N−1i (·) is given by equation (A3) in Appendix A, and

N−1j (·) is similarly obtained for subscript j .

SY NTHETIC EX A MPL ES

3D gravity example of Uz obtained from noisy Uxz with thespline technique and comparison with the Fourier transformtechniqueIn equation (6), let V(x, y) = Uxz, calculate Uz. The 3D sourcesare four solid spheres with densities and geometrical parame-ters listed in Table 1. Figure 1 shows the effectiveness of com-puting Uz from Uxz in the noisy situation. Figure 1(f) showsthe Uxz data contaminated by random noise with a zero meanand a standard deviation of 0.65 mGal/km. Figure 1a showsthe exact theoretical Uz map. The Uz map obtained using thespline technique is shown in Fig. 1(b). In order to improvethe performance of both the spline technique and the Fouriertransform technique, expand the original grids with 32 datapoints on each side of the computational domain and taperthe expanded parts, but just keep the computed Uz valueson the original grids. The Uz map obtained using the splinetechnique from the expanded Uxz data is shown in Fig. 1(c).The Uz map obtained using the Fourier transform technique isshown in Fig. 1(d). Fig. 1(e) shows the Uz map obtained usingthe Fourier transform technique from the expanded Uxz data.

In order to measure the difference between the computedvalues Vci and the corresponding exact or observed values Vti ,define the RE (relative root mean square error) as

RE =

√1N

N∑i=1

(Vci − Vti )2

(Vti )max − (Vti )min. (15)

Table 1 Source parameters of the four solid spheres in Fig. 1.The centre of a sphere is at (x0, y0, z0). The radius of thesphere is a and its density is ρ

Sphere x0(km) y0(km) z0(km) a(km) ρ(kg/.m3)

1 0.0 0.0 6.0 3.8 12002 10.0 0.0 1.6 1.5 15003 −5.0 8.7 1.6 1.5 15004 −5.0 −8.7 1.6 1.5 1500

Compared with Fig. 1(b), Fig. 1(d) is significantly less accu-rate. Compared with Fig. 1(c), Fig. 1(e) is significantly lessaccurate and based on Fig. 1(e) one can hardly identify thelarge source located at the centre. So, the results computedwith the spline technique agree substantially better with theexact data than do the results computed with the Fouriertransform technique. This statement is verified by comparingthe REs shown in Table 2.

The pertinent computation times of this example with a2.80 GHz laptop computer for different grid sizes and differ-ent methods are listed in Table 3. The extra time overhead ofthe spline method is the penalty for its increased accuracy.

Let the grid dimension be N, and the computation time beY. For the spline method, Y(N)=C1(N)N2.9. For the Fouriertransform method, Y(N)=C2(N)N2 log10N. The boundednessof C1(N) and C2(N) (Table 3) suggests the polynomial O(N2.9)growth for the spline method and the usual O(N2 log10N)growth for the Fourier transform method.

3D example of magnetic potential computed from a noisyx-component of magnetic induction with the spline techniqueand comparison with the Fourier transform technique

The 3D source is a solid sphere, whose magnetic and geometri-cal parameters are the following: inclination, declination andintensity of magnetization of the sphere are 60 degree, 20 de-gree and 1 A/m, respectively; the centre of the sphere is at (0,0, 1.2 km) and the radius of the sphere is 0.5 km. Analyticalmagnetic potential (Um) and x-component of magnetic induc-tion (Bx) can be calculated (Blakely 1996). Um = − 1

μ0

∫Bxdx

where μ0 is permeability of free space. In equation (6), letV(x,y) = Bx(x,y) and calculate Um. Figure 2 shows the ef-fectiveness of computing Um from Bx in the noisy situation.The Bx data contaminated by random noise with a zero meanand a standard deviation of 0.5 nT are shown in Fig. 2(f).Figure 2(a) shows the exact theoretical Um map. The Um mapobtained using the spline technique is shown in Fig. 2(b). Inorder to improve the performance of both the spline techniqueand the Fourier transform technique, expand the original gridswith 32 data points on each side of the computational domainand taper the expanded parts, but only keep the computedUm values on the original grids. The Um map obtained us-ing the spline technique from the expanded Bx data is shownin Fig. 2(c). Figure 2(d) shows the Um map obtained usingthe Fourier transform technique. The Um map obtained us-ing the Fourier transform technique from the expanded Bx

data is shown in Fig. 2(e). Compared with Fig. 2(b), Fig. 2(d)is significantly less accurate. So is Fig. 2(e), compared with



Figure 1 The 3D gravity example of Uz obtained from noisy Uxz data using the spline technique and the Fourier transform technique. Thesources are four solid spheres with densities and geometrical parameters listed in Table 1. Data spacing is 0.5 km in both the x and y directions.(a) Uz (analytical). (b) Uz from Uxz (spline). (c) Uz from expanded Uxz(spline). (d) Uz from Uxz (FT). (e) Uz from expanded Uxz (FT). (f) NoisyUxz (contaminated by random noise with a zero mean and a standard deviation of 0.65 mGal/km).

Fig. 2(c). Therefore, the results computed with the spline tech-nique agree substantially better with the exact data than dothe results computed with the Fourier transform technique.The REs shown in Table 4 strongly support this statement.

3D gravity example of Uxz obtained fromrectangular-gridded Uzz with the spline technique

This is a 3D gravity example of Uxz recovered from exact andnoisy rectangular-gridded Uzz data using the spline technique

Table 2 REs between panels I, I = (b), (c),(d), (e) and panel (a) in Fig. 1

Panel (b) (c) (d) (e)

RE (%)9 8.72 7.29 19.13 10.29

(Wang et al. 2008). The 3D sources are five solid spheres withdensities and geometrical parameters listed in Table 5. Thedata spacing is 0.15 km in the x direction and 0.10 km in they direction. Figure 3(d) shows the exact theoretical Uzz data.The Uxz map obtained using the spline technique from theexact Uzz data is shown in Fig. 3(b). Figure 3(e) shows the Uzz

data contaminated by random noise with a zero mean and astandard deviation of 0.41 mGal/km. The Uxz map obtainedusing the spline technique from the noisy Uzz data is shown inFig. 3(c). Compared with the exact Uxz (Fig. 3a), the REs are0.37% and 0.62% for the exact and noisy cases in Fig. 3(b,c), respectively.

R E A L D A T A E X A M P L E S

The real data are the free air gravity and full tensor grav-ity gradient dataset (Uz and Uxx, Uxy, Uxz, Uyy, Uyz, Uzz)


6 B. Wang et al.

Table 3 Time complexity of the spline and Fourier transform methods

N (N×N grid) tspline (second) tFT (second) tspline (N2.9) tFT (N2log10N)

64 0.656 0.031 3.79e-06 4.19e-06128 4.672 0.141 3.62e-06 4.08e-06256 36.55 0.563 3.79e-06 3.57e-06

collected in the Gulf of Mexico. The dataset satisfies equa-tion (4) (Laplace’s equation) well. Using the spline-based tech-niques, the following transformations are performed.

3D gravity-gradient component transformations

Examples of these are shown in Fig. 4. The computed Uzz re-covered from the observed Uxz and Uyz (Fig. 5b) data using the

spline technique (Wang et al. 2008) and the Fourier transformtechnique are shown in Fig. 4(b) and 4(c), respectively. Com-pared with the observed Uzz data (Fig. 4a), the RE is 6.89%for the spline technique and 14.86% for the Fourier transformtechnique.

Figures 4(e,f) show the computed Uxz recovered fromthe observed Uzz data using the spline technique and theFourier transform technique, respectively. Compared with the

Figure 2 The 3D magnetic example of Um obtained from noisy Bx data using the spline technique and the Fourier transform technique. Thesource is a solid sphere centered at (0, 0, 1.2 km) with a radius of 0.5 km. Data spacing is 0.1 km in both the x and y directions. (a) Um(analytical).(b) Um from Bx (spline). (c) Um from expanded Bx (spline). (d) Um from Bx (FT). (e) Um from expanded Bx (FT). (f) Noisy Bx (contaminated byrandom noise with a zero mean and a standard deviation of 0.5 nT).



Table 4 REs between panels I, I = (b), (c),(d), (e) and panel (a) in Fig. 2

Panel (b) (c) (d) (e)

RE (%) 1.43 1.37 17.34 10.23

Table 5 Source parameters of the five solid spheres in Figs. 3,6, 7, 8 and 9. The centre of a sphere is at (x0, y0, z0). Theradius of the sphere is a, and its density is ρ

Sphere x0(km) y0(km) z0(km) a(km) ρ(kg/.m3)

1 0.0 0.0 1.72 1.56 10002 1.2 1.2 0.6 0.5 10003 −1.5 1.5 0.6 0.5 10004 −1.2 −1.2 0.6 0.5 10005 1.5 −1.5 0.6 0.5 1000

observed Uxz data (Fig. 4d), the RE is 10.45% for the splinetechnique and 10.57% for the Fourier transform technique.

The computed Uxy recovered from the observed Uxz datawith the spline technique and the Fourier transform techniqueare shown in Figs. 4(h, i), respectively. Compared with theobserved Uxy data (Fig. 4g), the RE is 9.41% for the splinetechnique and 12.40% for the Fourier transform technique.

The results computed with the spline technique agree betterwith the observed data than do the results computed with theFourier transform technique, although the differences in thisexample are not as pronounced as in other cases, e.g., betweenFigs. 2(c) and 2(e), or Figs. 5(g) and 5(h).

3D gravity Uz obtained from gravity-gradient componentsUxz and Uyz

An example is shown in Fig. 5. Figure 5(b) shows the ob-served Uyz data. The observed Uxz data are shown in Fig. 4(d).

Figure 3 The 3D gravity example of Uxz obtained from exact and noisy rectangular-gridded Uzz data using the spline technique. The sourcesare five solid spheres with densities and geometrical parameters listed in Table 5. Data spacing is 0.15 km in the x direction and 0.10 km inthe y direction. (a) Uxz (analytical). (b) Uxz from exact Uzz. (c) Uxz from noisy Uzz. (d) Uzz (analytical). (e) Noisy Uzz (contaminated by randomnoise with a zero mean and a standard deviation of 0.41 mGal/km).


8 B. Wang et al.

Figure 4 Real 3D examples of gravity-gradient component transformations. The data were collected in the Gulf of Mexico with data spacingof 0.5 km in both the x and y directions. (a) Uzz(observed). (b) Uzz from observed Uxz and Uyz (spline). (c) Uzz from observed Uxz and Uyz (FT).(d) Uxz (observed). See the observed Uyz data in Fig. 5(b). (e) Uxz from observed Uzz (spline). (f) Uxz from observed Uzz (FT). (g) Uxy (observed).(h) Uxy from observed Uxz (spline). (i) Uxy from observed Uxz (FT).

In equation (6), let V(x,y)=Uxz and calculate gravity Uz. Inequation (7), let V(x,y) = Uyz and calculate Uz again. Figures5(c, d) are the computed Uz obtained from Uyz with the splinetechnique and the Fourier transform technique, respectively.Figures 5(e, f) are the computed Uz obtained from Uxz withthe spline technique and the Fourier transform technique, re-spectively. Figure 5(g) shows the computed Uz with the splinetechnique averaged from panels (c) and (e). The computed Uz

with the Fourier transform technique averaged from panels(d) and (f) is shown in Fig. 5(h). Compared with the observedUz data (Fig. 5a), the REs are: (c) 15.07%, (d) 63.51%, (e)23.25%, (f) 79.27%, (g) 12.01%, (h) 65.34%.

Apparently, the results computed with the spline techniqueagree substantially better with the observed Uz data than dothe results computed with the Fourier transform technique.

HARMONICITY A N D IN V ER T I B I L I T YT E S T S

Harmonicity test using computed Uxx, Uyy and analytical Uzz

This is a 3D gravity example of harmonicity confirmation forthe spline technique. The sources are five solid spheres with

densities and geometrical parameters listed in Table 5. Dataspacing is 0.05 km in both the x and y directions. There arethree steps:

(I) compute potential U from the forwarded potential fieldUx; compute the first-order partial derivative Ux from thepotential U; then compute the second-order partial derivativeUxx from the Ux.

(II) compute potential U from the forwarded potential fieldUy; compute the first-order partial derivative Uy from the po-tential U; then compute the second-order partial derivativeUyy from the Uy.

(III) sum up the exact theoretical Uzz and the Uxx and Uyy

calculated in (I) and (II).Define the norm of an array of values Vi :

Norm = 1N

√√√√ N∑i=1

(Vi )2. (16)

Steps (I), (II) and (III) apply to both the spline techniqueand the Fourier transform technique. Figures 6(a) and 6(d)are analytical Uxx and Uyy, respectively. The spline-based Uxx

(Fig. 6b) and Uyy (Fig. 6e) agree substantially better with the



Figure 5 Real 3D example of gravity Uz obtained from gravity-gradient components Uxz and Uyz. The data were collected in the Gulf of Mexicowith data spacing of 0.5 km in both the x and y directions. (a) Uz (observed). (b) Uyz (observed). The observed Uxz data are in Fig. 4(d). (c) Uz

from Uyz (spline). (d) Uz from Uyz (FT). (e) Uz from Uxz (spline). (f) Uz from Uxz (FT). (g) Uz (spline), averaged from panels (c) and (e). (h) Uz

(FT), averaged from panels (d) and (f).

analytical results than do the Fourier transform-based Uxx

(Fig. 6c) and Uyy (Fig. 6f).The sum of Figs. 6(a), 6(d) and 6(g) (analytical Uzz), the

exact theoretical second-order derivatives, is exactly zero ev-erywhere, perfectly satisfying Laplace’s equation. The sumof Figs. 6(b), 6(e) and 6(g), shown in Fig. 6(h) (Norm 0.002mGal/km), is nearly zero, almost perfectly satisfying Laplace’sequation, which confirms harmonicity for the spline tech-

nique. The Uxx and Uyy are so accurately calculated withthe spline technique, Uzz can be reliably obtained using theLaplace’s equation (equation 4). Interestingly, one can iden-tify the four shallow sources shown as distinct anomalies onFig. 6(h).

The sum of Figs. 6(c), 6(f) and 6(g), shown in Fig. 6(i) (norm0.302 mGal/km), is not nearly as close to zero as the sum inFig. 6(h). i.e., harmonicity is substantially less well confirmed


10 B. Wang et al.

Figure 6 3D gravity example of harmonicity confirmation for the spline technique and comparison with the Fourier transform technique. Uxx andUyy are computed, Uzz is analytical. The sources are five solid spheres with densities and geometrical parameters listed in Table 5. Data spacing is0.05 km in both the x and y directions. (a) Uxx (analytical). (b) Uxx (spline). (c) Uxx (FT). (d) Uyy (analytical). (e) Uyy (spline). (f) Uyy (FT). (g)Uzz (analytical). (h) Uxx(spline) + Uyy(spline) + Uzz(analytical). (i) Uxx(FT) + Uyy(FT) + Uzz(analytical).

for the Fourier transform method than the spline method. Bothmethods have some edge effects. However, the edge problemis significantly less severe for the spline technique than theFourier transform technique.

Harmonicity tests using computed Uxx, Uyy, Uzz

These are further harmonicity tests for the spline techniqueand the Fourier transform technique. The sources are five solidspheres with densities and geometrical parameters listed inTable 5. Data spacing is 0.1 km in both the x and y directions.The following describes the procedure.

(I) Compute potential U from the forwarded potential fieldUx; compute the first-order partial derivative Ux from thepotential U; then compute the second-order partial derivativeUxx from the Ux.

(II) Compute potential U from the forwarded potential fieldUy; compute the first-order partial derivative Uy from the po-tential U; then compute the second-order partial derivativeUyy from the Uy.

(III) It takes the following steps to compute Uzz: ComputeUxy from the Ux obtained in (I), and compute Uyx from the Uy

obtained in (II); compute Uxz from Uxy and Uxx obtained in(I); compute Uyz from Uyx and Uyy obtained in (II); computeUzz from Uxz and Uyz.

(IV) Sum up the Uxx, Uyy and Uzz calculated in (I), (II) and(III).

(I), (II), (III) and (IV) apply to both the spline technique andthe Fourier transform technique. Figures 7(h, i) are the Uxy

maps computed with the spline technique and the Fouriertransform technique, respectively. Both are very close to



Figure 7 Intermediate steps of computing Uzz (Fig. 8): 3D gravity example to compare the spline technique and the Fourier transform techniquefor computing Uxz, Uyz, Uxy. The sources are five solid spheres with densities and geometrical parameters listed in Table 5. Data spacing is 0.1km in both the x and y directions. (a) Uxz (analytical). (b) Uxz (spline). (c) Uxz (FT). (d) Uyz (analytical). (e) Uyz (spline). (f) Uyz (FT). (g) Uxy

(analytical). (h) Uxy (spline). (i) Uxy (FT).

analytical Uxy as shown in Fig. 7(g). However, comparingwith analytical Uxz (Fig. 7a) and Uyz (Fig. 7d), the spline-basedUxz (Fig. 7b) and Uyz (Fig. 7e) are obviously substantially bet-ter than the Fourier transform-based Uxz (Fig. 7c) and Uyz

(Fig. 7f). Figures 8(a, b) are the Uzz maps and Figures 8(d,e) are the Uxx + Uyy + Uzz maps obtained using the splinetechnique and the Fourier transform technique, respectively.

(V) There is another alternative way to compute Uzz us-ing the Fourier transform technique: compute Uz from the Uobtained in (I); then compute Uzz from Uz. Figure 8f is theFourier transform-based Uxx + Uyy + Uzz map, where Uzz

(Fig. 8c) is the Uzz map computed through the alternativeway using the Fourier transform technique. Apparently, thisalternative approach is impractical to compute Uzz using theFourier transform technique.

The norms (equation 16) for Figs. 8(d, e, f) are 0.144,0.507, 165.232 mGal/km, respectively. i.e., harmonicity isagain substantially less well confirmed for the Fourier trans-form method than the spline method.

Invertibility test

This is a 3D gravity example to test the invertibility of spline-based integration and differentiation. The sources are fivesolid spheres with densities and geometrical parameters listedin Table 5. Data spacing is 0.1 km in both the x and y direc-tions. Figure 9(d) shows the exact theoretical Ux data. The Umap obtained from the exact Ux data using the spline-basedintegration is shown in Fig. 9(b); compared with the exactU (Fig. 9a), the RE is 5.99%. Figure 9(c) shows the Ux map


12 B. Wang et al.

Figure 8 3D gravity example to test harmonicity of the spline technique and comparison with the FT technique, Uxx(Fig. 6), Uyy(Fig. 6), Uzz

are all computed. The sources are five solid spheres with densities and geometrical parameters listed in Table 5. Data spacing is 0.1 km in boththe x and y directions. (a) Uzz (spline). (b) Uzz (FT). (c) Uzz (alternative FT). (d) Uxx + Uyy+Uzz (spline). (e) Uxx + Uyy + Uzz (FT; Uzz panelb). (f) Uxx + Uyy + Uzz (FT; Uzz panel c).

Figure 9. 3D gravity example to test whether spline-based integration and differentiation are invertible. The sources are five solid spheres withdensities and geometrical parameters listed in Table 5. Data spacing is 0.1 km in both the x and y directions. (a) U (analytical). (b) U obtainedfrom analytical Ux using spline-based integration. (c) Ux obtained from U in panel (b) using spline-based differentiation. (d) Ux (analytical).



obtained from the U as shown in Fig. 9(b) using the spline-based differentiation; compared with the exact Ux(Fig. 9d),the RE is only 0.01%.

We have obtained the potential from the field and recoveredthe field from the computed potential nicely, so that spline-based integration and differentiation are invertible.

CONCLUSIONS

Potential, potential field and potential-field gradient data aresupplemental to each other for resolving sources of inter-est. We advanced spline-based techniques for 3D and 2Dpotential-field upward continuation and potential field andgradient component transformations and derivative computa-tions in the previous studies. In this paper, we propose flexiblehigh-accuracy practical techniques to perform 3D and 2D in-tegral transformations from PF components to potential andfrom PG components to PF components in the space domainusing cubic B-splines. The spline techniques are applicable toeither uniform or non-uniform rectangular grids for the 3Dcase, and applicable to either regular or irregular grids for the2D case. The spline-based indefinite integrations can be com-puted at any point in the computational domain, as can thehorizontal derivatives.

In our synthetic 3D gravity examples (Uz obtained fromnoisy Uxz, Uxz obtained from rectangular-gridded Uzz, andthe demonstration that harmonicity is confirmed substantiallybetter for the spline technique than the Fourier transform tech-nique and spline-based integration and differentiation are in-vertible) and magnetic examples (Um obtained from noisy Bx),we have shown that the spline techniques are substantiallymore accurate and hence may provide better insights intounderstanding the sources than the Fourier transform tech-niques, and Uzz can be reliably obtained using the Laplace’sequation since the Uxx and Uyy are very accurately calculatedwith the spline technique. The cost of the increase in accuracyis some increase in computing time compared to the Fouriertransform technique. However, the speed is still fast, e.g., thespline-based computation of Uz from Uxz on a 128 by 128grid was performed within 5 seconds with a 2.80 GHz laptopcomputer. For the 3D gravity Uz obtained from Uxz example,the complexities of the computational time growth are poly-nomial O(N2.9) growth for the spline method and the usualO(N2 log10N) growth for the Fourier transform method.

Our real data examples of 3D transformations show that thespline-based results agree substantially better (from gravity-gradient components to gravity) or better (between gravity-

gradient components) with the observed data than do theFourier-based results.

The spline techniques would therefore be very useful fordata quality control through comparisons of the computedand observed components. If certain desired components ofthe potential field or gradient data are not measured, they canbe obtained using the spline-based transformations as alter-natives to the Fourier transform techniques.

ACKNOWLEDGEMENTS

We are grateful to editor Dr. Alan Reid, the associate editorand the reviewers for their very thoughtful and constructivecomments and suggestions. This work was partially supportedby the United States National Science Foundation awardsEAR-0739015 and EAR-0952064 and a grant from the Nat-ural Sciences and Engineering Research Council of Canadaheld by E.S. Krebes. We appreciate that John Mims providedthe real gravity gradient data. Map plotting was facilitatedwith the use of GMT software (Wessel and Smith 1995). Thisarticle is Geology and Geophysics contribution 39 of MissouriUniversity of Science and Technology.

REFERENCES

Abers G.A., Ferris A., Craig M., Davies H., Lerner-Lam A.L., MutterJ.C. and Taylor B. 2002. Mantle compensation of active metamor-phic core complexes at Woodlark rift in Papua New Guinea. Nature418, 862–865.

Allen D.J. and Hinze W.J. 1992. Wisconsin gravity minimum: So-lution of a geologic and geophysical puzzle and implications forcratonic evolution. Geology 20, 515–518.

Behrendt J.C., Meister L. and Henderson J.R. 1966. Airborne geo-physical study in the Pensacola Mountains of Antarctica. Science153, 1373–1376.

Bektas O., Ravat D., Buyuksarac A., BIlIm F. and Ates A. 2007.Regional geothermal characterisation of east Anatolia from aero-magnetic, heat flow and gravity data. Pure Applied Geophysics164, 975–998.

Berrino G., Corrado G. and Riccardi U. 2008. Sea gravity data in theGulf of Naples. A contribution to delineating the structural patternof the Phlegraean Volcanic District. Journal of Volcanology andGeothermal Research 175, 241–252.

Blakely R.J. 1996. Potential theory in gravity and magnetic applica-tions. Cambridge University Press.

Blakely R.J. and Simpson R.W. 1986. Approximating edges ofsource bodies from magnetic or gravity anomalies. Geophysics 51,1494–1498.

Blanco-Montenegro I., Gonzalez F.M., Garcia A., Vieira R. and Vil-lalain J.J. 2005. Paleomagnetic determinations on Lanzarote frommagnetic and gravity anomalies: Implications for the early history


14 B. Wang et al.

of the Canary Islands. Journal of Geophysical Research 110, B12102, 1–9.

Bosum W., Casten U., Fieberg F.C., Heyde I. and Soffel H.C.1997. Three-dimensional interpretation of the KTB gravity andmagnetic anomalies. Journal of Geophysical Research 102(B8),18307–18321.

Bracewell R.N. 1965. The Fourier transform and its applications.McGraw-Hill Inc.

Carbo A., Munoz-MartIn A., Llanes P., Alvarez J. and EEZ WorkingGroup. 2003. Gravity analysis offshore the Canary Islands from asystematic survey. Marine Geophysical Research 24, 113–127.

Cochran J.R., Fornari D.J., Coakley B.J., Herr R. and Tivey M.A.1999. Continuous near-bottom gravity measurements made with aBGM-3 gravimeter in DSV Alvin on the East Pacific Rise crest near9◦3 1′N and 9◦50′N. Journal of Geophysical Research 104(B5),10841–10861.

Fischer K.M. 2002. Waning buoyancy in the crustal roots of oldmountains. Nature 417, 933–936.

FitzGerald D., Reid A. and McInerney P. 2004. New discriminationtechniques for Euler deconvolution. Computers & Geosciences 30,461–469.

Grauch V.J.S. and Cordell L. 1987. Limitations of determining densityor magnetic boundaries from the horizontal gradient of gravity orpseudogravity data. Geophysics 52, 118–121.

Hansen R.O. and Suciu L. 2002. Multiple-source Euler deconvolu-tion. Geophysics 67, 525–535.

Hebert H., Deplus C., Huchon P., Khanbari K. and Audin L. 2001.Lithospheric structure of a nascent spreading ridge inferred fromgravity data: The western Gulf of Aden. Journal of GeophysicalResearch 106(B11), 26345–26363.

Jallouli C., Mickus K.L. and Turki M.M. 2002. Gravity constrainson the structure of the northern margin of Tunisia: Implicationson the nature of the northern African Plate boundary. GeophysicalJournal International 151, 117–131.

Kostoglodo V., Bandy W., Dominguez J. and Mena M. 1996. Gravityand seismicity over the Guerrero seismic gap, Mexico. GeophysicalResearch Letters 23, 3385–3388.

Martinez F., Goodliffe A.M. and Taylor B. 2001. Metamorphic corecomplex formation by density inversion and lower-crust extrusion.Nature 411, 930–934.

Mickus K.L. and Hinojosa J.H. 2001. The complete gravity gradienttensor derived from the vertical component of gravity: A Fouriertransform technique. Journal of Applied Geophysics 46, 159–174.

Mickus K., Tadesse K., Keller G.R. and Oluma B. 2007. Gravity anal-ysis of the main Ethiopian rift. Journal of African Earth Sciences48, 59–69.

Nabighian M.N. 1972. The analytic signal of two-dimensional mag-netic bodies with polygonal cross-section: Its properties and use forautomated anomaly interpretation. Geophysics 37, 507–517.

Nabighian M.N. 1974. Additional comments on the analytic sig-nal with two-dimensional magnetic bodies with polygonal cross-section. Geophysics 39, 85–92.

Nabighian M.N. 1984. Toward a three-dimensional automatic inter-pretation of potential field data via generalized Hilbert transforms:Fundamental relations. Geophysics 49, 780–786.

Nabighian M.N., Ander M.E., Grauch V.J.S., Hansen R.O., LaFehr

T.R., Li Y. et al. 2005b. Historical development of the gravitymethod in exploration. Geophysics 70, 63ND–89ND.

Nabighian M.N., Grauch V.J.S., Hansen R.O., LaFehr T.R., Li Y.,Peirce J.W. et al. 2005a. The historical development of the magneticmethod in exploration. Geophysics 70, 33ND–61ND.

Nabighian M.N. and Hansen R.O. 2001. Unification of Euler andWerner deconvolution in three dimensions via the generalizedHilbert transform. Geophysics 66, 1805–1810.

Nelson J.B. 1986. An alternative derivation of the three-dimensionalHilbert transform relations from first principles. Geophysics 51,1014–1015.

Purucker M., Sabaka T., Le G., Slavin J.A., Strangeway R.J. andBusby C. 2007. Magnetic field gradients from the ST-5 constella-tion: Improving magnetic and thermal models of the lithosphere.Geophysical Research Letters 34, L24306.

Ravat D. 1996. Analysis of the Euler method and its applicability inenvironmental magnetic investigations. Journal of Environmentaland Engineering Geophysics 1, 229–238.

Ravat D., Wang B., Wildermuth E. and Taylor P.T. 2002. Gradientsin the interpretation of satellite-altitude magnetic data: An examplefrom central Africa. Journal of Geodynamics 33, 131–142.

Reid A.B., Allsop J.M., Granser H., Millett A.J. and Somerton I.W.1990. Magnetic interpretation in three dimensions using Euler de-convolution. Geophysics 55, 80–91.

Ricard Y. and Blakely R.J. 1988. A method to minimize edge effectsin two-dimensional discrete Fourier transforms. Geophysics 53,1113–1117.

Sandwell D.T. and Smith W.H.F. 1997. Marine gravity anomaly fromGeosat and ERS 1 satellite altimetry. Journal of Geophysical Re-search 102(B5), 10039–10054.

Thomas M.D., Grieve R.A.F. and Sharpton V.L. 1988. Gravity do-mains and assembly of the North American continent by collisionaltectonics. Nature 331, 333–334.

Thompson D.T. 1982. EULDPH: A new technique for makingcomputer-assisted depth estimates from magnetic data. Geophysics47, 31–37.

Wang Y.M. 2000. Predicting bathymetry from the Earth’s gravitygradient anomalies. Marine Geodesy 23, 251–258.

Wang B. 2006. 2D and 3D potential-field upward continuation usingsplines. Geophysical Prospecting 54, 199–209.

Wang B., Krebes E.S. and Ravat D. 2008. High-precision potentialfield and gradient component transformations and derivative com-putations using cubic B-splines. Geophysics 73, I35–I42.

Wessel P. and Smith W.H.F. 1995. New version of the Generic Map-ping Tools released. AGU Eos Transactions 76, 329.

APPENDIX A: UNIVARIATE CUBICB-SPL INE INTERPOLATION

In the domain D1 = {x|a ≤ x ≤ b}, make a partition

x−3 < x−2 < x−1 < x0 = a < x1 < . . . < xN = b <

xN+1 < xN+2 < xN+3.



The univariate cubic B-splines, whose interior knots are{ xi , i = 0, 1, . . . , N }, can then be expressed as

S(x) =N+1∑i=−1

Ci Ni (x), (A1)

where Ci are interpolation coefficients. The unified formulafor interpolation, differentiation and integration is

Su(x) =N+1∑i=−1

Ci Nui (x), (A2)

where

Nui (x) = (xi+2 − xi−2)Bu

i (x), (A3)

and

Bui (x) = 3!

(3 − u)!

i+2∑k=i−2

(−1)uWk(xk − x)3−u+ , (A4)

Wk =i+2∏

m=i−2m�=k

1xk − xm

, (A5)

(xk − x)3−u+ =

{(xk − x)3−u, f or x ≤ xk

0, f or x > xk

. (A6)

The u in equation (A2) has the following meaning: when u is apositive integer, calculate the uth-order derivative; when u is anegative integer, calculate the |u|th-order integral; when u =0, do not calculate either derivative or integral, i.e. N0

i (x) =Ni (x).

V(x) is a function defined in the domain D1.{ V(i), i = 0, 1, . . . , N } are known values of V(x) at the in-terior knots. Use the univariate cubic B-splines (A1) to ap-proximate V(x) while satisfying the following conditions

S(xi ) = V(i), i = 0, 1, . . . , N, (A7)

∂S(x)∂x

= ∂V(x)∂x

, at x0, xN. (A8)

Substituting equation (A2) into equations (A7) and (A8), us-ing a difference quotient to replace ∂V(x)

∂x , and considering thelocalized nonzero characteristics of the cubic B-splines

Ni (xk) = 0, f or |i − k| > 1, (A9)

yields

YC = V. (A10)

where the dimensions are Y(N + 3, N + 3), C(N + 3, 1) andV(N + 3, 1).

From equation (A10) one obtains

C = Y−1V. (A11)

i.e., the interpolation coefficients are determined.

APPENDIX B: B IVARIATE CUBIC B-SPL INEINTERPOLATION

In the domain D2 = {(x, y)| a ≤ x ≤ b, c ≤ y ≤ d }, make apartition

x−3 < x−2 < x−1 < x0 = a < x1 < . . . < xNx = b

< xNx+1 < xNx+2 < xNx+3.

y−3 < y−2 < y−1 < y0 = c < y1 < . . . < yNy = d

< yNy+1 < yNy+2 < yNy+3

The bivariate cubic B-splines, whose interior knots are{ (xi , yj ), i = 0, 1, . . . , Nx , j = 0, 1, . . . , Ny }, can be ex-pressed as

S(x, y) =Nx+1∑i=−1

Ny+1∑j=−1

Ni (x)Ci, j Nj (y), (B1)

where Ci, j are interpolation coefficients. The unified formulafor interpolation, differentiation and integration is

Su,v(x, y) =Nx+1∑i=−1

Ny+1∑j=−1

Nui (x)Ci, j Nv

j (y), (B2)

where Nui (x) is shown in equation (A3), and Nv

j (y) is similarlyobtained for subscript j.

V(x, y) is a function defined in the domain D2.{ V(i, j), i = 0, 1, . . . , Nx, j = 0, 1, . . . , Ny } are known val-ues of V(x, y) at the interior knots. Use the bivariate cubicB-splines (B1) to approximate V(x, y) while satisfying the fol-lowing conditions

S(xi , yj ) = V(i, j), i = 0, 1, . . . , Nx, j = 0, 1, . . . , Ny,(B3)

∂S(x, y)∂x

= ∂V(x, y)∂x

, at (x0, yj ), (xNx, yj ),

j = 0, 1, . . . , Ny,

(B4)

∂S(x, y)∂y

= ∂V(x, y)∂y

, at (xi , y0), (xi , yNy),

i = 0, 1, . . . , Nx,

(B5)

∂2S(x, y)∂x∂y

= ∂2V(x, y)∂x∂y

, at (x0, y0), (x0, yNy), (xNx, y0),

(xNx, yNy). (B6)


16 B. Wang et al.

Substituting equation (B2) into equation (B3–B6), using dif-ference quotients to replace ∂V(x,y)

∂x , ∂V(x,y)∂y and ∂2V(x,y)

∂x∂y , andconsidering the localized nonzero characteristics of the cubicB-splines (A9), yields

XCY = V. (B7)

where the dimensions are X(Nx + 3, Nx + 3), C(Nx +3, Ny + 3), Y(Ny + 3, Ny + 3) and V(Nx + 3, Ny + 3).

From equation (B7) one obtains

C = X−1VY−1. (B8)

i.e., the interpolation coefficients are determined. Because ofthe localized feature of the cubic B-splines, a large matrix isdecomposed into two much smaller matrixes. The computa-tions are fast.


Documents

High-accuracy practical spline-based 3D and 2D integral ...web.mst.edu/~liukh/publications/Wang_Gao_GeopPros_2012_spline.pdfGeophysical Prospecting doi: 10.1111/j.1365-2478.2011.01026.x