ABSTRACT: In this study, an iterative three-dimensional (3-D) inversion algorithm was developed for the analysis of gridded gravity data. Vertical sided prismatic ...
International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
3-D Gravity Inversion: A Comparison of Decomposition-Based and Gradient-Based Solvers Ekinci Y.L. Çanakkale Onsekiz Mart University, Department of Geophysical Engineering, Çanakkale
ABSTRACT: In this study, an iterative three-dimensional (3-D) inversion algorithm was developed for the analysis of gridded gravity data. Vertical sided prismatic bodies were used as basic models in the forward solution part of the inversion scheme. Non-linear optimization techniques were employed in order to estimate the model parameters. The partial derivatives of data with respect to the user defined initial model parameters were computed by using numerical differentiation with forward difference approximation. The inversion of synthetically produced gravity data were performed by means of various solvers such as singular value decomposition (SVD), cholesky decomposition (CholD) and conjugate gradient (CG). The results were examined to compare the efficiency of these solvers in terms of solution power. Key words: Gravity data, three-dimensional inversion, prismatic bodies, solvers.
INTRODUCTION A general approach for analyzing the gridded gravity data is to consider vertical sided prismatic bodies for the causative bodies. The geological structures such as intrusive plugs and uplifted basement fault blocks can be approximated by vertical sided prismatic models [1]. Thus the contour or image map of the gravity data may be interpreted by using single or multiple prismatic bodies. The gravity anomaly equation of prismatic model is non-linear in its parameters and many techniques can be employed in order to determine the model parameters. The model parameters of the prismatic body such as; the horizontal distances from the origin to the faces of the prism parallel to the x- and y- axes, the depths to the top and bottom of the prism and the density contrast of the prism can be adjusted from an initial model either manual by trial and error or automatically by using various inversion techniques. In the inversion schemes various system solvers can be employed. One of the most employed solvers in the inversion algorithm is the least-squares solution with singular value decomposition (SVD). It is mathematically robust and numerically stable. It also provides other vital information on the state of the model and data thus enabling model resolution and covariance studies [2]. The other decomposition-based solver for the solution of the normal equations is the Cholesky decomposition (CholD) procedure and it can be easily applied if the Jacobian matrix is symmetric and positive definite [3]. Conjugate gradient solver is a gradient-based solution technique and uses vectors to solve the system of equations and matrix inversions are not perfomed [4]. In the scope of this work, these system solvers were compared in terms of accuracy and solution power by using synthetically produced gravity data. ANOMALY EQUATION In order to approximate a volume of mass, considering a rectangular prism is one of the simple ways. The gravitational attraction of a single rectangular prism can be calculated by integration over the limits of the prism and one can employ the following equation [5]:
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International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
z
z2 y2 x2
g
z1 y1 x1
x y z 2
2
2
3 2
dxdydz
(1)
where, is the density contrast and is the gravitational constant. After the simplifications of preceding integral the following result is provided [6]: 2 2 2 xy g ijk zk arctan i j xi log( Rijk y j ) y j log( Rijk xi ) zk Rijk i 1 j 1 k 1
(2)
where,
Rijk ( xi2 y 2j zk2 ) ijk (1)i (1) j (1)k
(3)
SVD SOLVER As mentioned earlier, gravity anomaly equation of prismatic model is non-linear in its parameters. The unknown parameters can be estimated by least squares solution. The parameter correction vector or model updates for least squares solution is given as:
p J T J J T g 1
(4)
where, g is the discrepancy vector between the observed and calculated gravity data, J is the sensitivity or Jacobian matrix. If we introduce a damping ( ) to the system of equations and employ SVD solver, parameter correction vector ( p ) are calculated as follows [7]:
j T p Vdiag 2 U d 2 j
(5)
CholD SOLVER The system of simultaneous equations can be written in matrix notation as follows:
Ax b
(6)
where, A is the coefficient matrix with known elements, x is the column matrix of unknown parameter increments to be solved for and b is the column matrix with known elements. These may be given as follows [1]:
T (i, j ) T (i, j ) (1 kl ), Pl Pk j 1
Mx My
A i 1
x dPk Mx My
k 1...N
k 1...N
b Tobs (i, j ) Tcal (i, j ) i 1 j 1
l 1...N
(7)
(8)
T (i, j ) , Pl
l 1...N
(9)
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International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
where, M x and M y are the number of observation points, N is total number of the unknown parameters, is damping factor, Pk represents one of the parameters of each prism, T / Pl represents the derivative of the anomaly with respect to Pl , dPk represents the increment or decrement of the kth parameter to be solved for and finally to be added to the initial parameters [1]. Matrix A is always symmetric and made positive definite by multiplying its diagonal elements by ( 1 ) [1, 3]. CG SOLVER The conjugate gradient (CG) solver can be used for the numerical solution of symmetric and positive-definite linear systems. Vectors are used to solve the system of equations. The solution of the normal equation given in Eq. (10)
AT A x AT b
(10)
is the critical points of the following equation [8].
b Ax b Ax 2
T
b Ax
(11)
The conjugate gradient method is an iterative method, it can be applied to large sparse systems without matrix inversion. The details of the method can be found in [8, 9]. SYNTHETIC TESTS In order to compare the efficiency of the system solvers in inversion procedure, noise-free synthetic models were used. Three distinct prismatic bodies with a density contrast of 100kg/m3 were used as disturbing bodies. Model parameters of the prismatic bodies are given in Table 1. The cross-sectional plan view of the bodies and the gravity anomaly map produced by using the Eq. (2) are shown in Fig. 1 and 2, respectively. The inversion algorithms were developed in MATLAB format. Theoretical noise-free tests were performed by means of 3 different solvers with the same initial model parameters. Iteration numbers were set to 30 for 3 cases. It was aimed to examine the solution powers of the solvers in terms of accuracy and convergency. The partial derivatives of anomalies of prismatic bodies were computed by numerical differentiation. Because computation of the Jacobian matrix involves repeated use of Equation (2) in all data points for all model parameters, much longer processing time is required in comparison with the analytical methods. For 50x50 grid points and 3 bodies with 21 model parameters (7 for each), 30 iterations need more than 1.5 million forward computations. Tests showed that it requires ~32 minutes per iteration (on a 2.4 Ghz processor with 3 Gb of RAM). In order to overcome this problem, vectorizing loops and preallocating arrays were used in MATLAB environment and computing time was minimized to acceptable levels (~3 seconds per iteration). The results of the tests are given in Table 1 and the error function between the observed and calculated data are shown in Fig. 3. It was observed that the inversion algorithm with SVD solver converged faster to real model parameters and reduces the error function very rapidly. Similarly, the inversion algorithm with CholD solver decreases the error function rapidly but the solutions were not converged at the same degree of accuracy. On the other hand, both decreasing the error function and converging the global minimum is more slowly in CG algorithm.
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International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
Figure 1: The plan view of the disturbing prismatic bodies (not to scale).
Figure 2: Gravity anomaly map produced by 3 distinct prismatic bodies.
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International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
Figure 3: The errors between the observed and calculated gravity data. CONCLUSIONS Theoretical noise-free small scale applications were performed for 3-D inversion of gravity data in order to test the solution power of the various solvers. It was observed that the inversion algorithm with SVD solver found the global minimum more rapidly in comparison with the other ones. For small scale problems it can be easily applied but it must be noted that the damping factor should be carefully estimated. Although the error function decreased very rapidly by using CholD algorithm, the desired convergence was not obtained. On the other hand, the convergence to desirable results by using the CG algorithm is more slowly than the others. The advantage of CG algorithm is that the matrix inversions are not needed and thus there is no risk of the singularity of the Jacobian matrix. Additionally, it can be applied to large sparse systems. Therefore, the use of CG solver is logical for large scale gravity data. However reasonable solutions need more iterations than the other solvers. In this case it was observed that more than 300 iterations are adequate for acceptable convergence (not shown). 271
International Symposium on Engineering and Architectural Sciences of Balkan, Caucasus and Turkic Republics, Süleyman Demirel University, October 22-24, 2009 Isparta, TURKEY
Table 1: Real, initial and calculated model parameters.
Real Parameters Initial Parameters SVD Solutions CholD Solutions CG Solutions
Prism No 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
x1 (km) 9.00 23.00 37.00 8.00 24.00 33.00 9.16 23.16 36.79 8.68 23.02 35.38 8.48 23.91 33.20
x2 (km) 11.00 27.00 43.00 12.00 30.00 44.00 10.81 26.82 43.23 11.31 26.97 44.62 11.40 26.20 47.05
y1 (km) 9.00 23.00 37.00 7.00 21.00 35.00 8.98 23.17 36.79 8.69 23.06 35.39 8.20 23.88 33.20
y2 (km) 11.00 27.00 43.00 13.00 28.00 45.00 11.01 26.81 43.22 11.30 26.92 44.61 11.73 26.23 47.05
h1 (km) 5.00 10.00 15.00 5.00 5.00 5.00 4.85 9.73 15.22 4.77 9.86 13.14 5.07 7.59 14.85
h2 (km) 7.00 14.00 21.00 10.00 10.00 10.00 7.22 14.62 20.43 7.12 14.27 22.94 5.72 23.81 15.80
Density (kg/m3) 100.00 100.00 100.00 100.00 100.00 100.00 99.95 99.84 99.02 48.68 96.02 25.85 99.95 99.87 98.49
REFERENCES
[1] Rao, B.D. and N.R. Babu, 1991. “A rapid method for three-dimensional modeling of magnetic anomalies” Geophysics, Vol. 56, pp. 1729-1737. [2] Meju, M.A., 1994. Geophysical Data Analysis: Understanding Inverse Problem Theory and Practice. Society of Exploration Geophysics. [3] Bhattacharyya, B.K., 1980. “A generalized multibody model for inversion of magnetic anomalies” Geophysics, Vol. 45, pp. 225-270. [4] Candansayar, M.E., 2008. “Two-dimensional inversion of magnetotelluric data with consecutive use of conjugate gradient and least squares solution with singular value decomposition algorithms” Geophysical Prospecting, Vol. 56, pp. 141–157. [5] Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications. Cambridge University Press. [6] Plouff, D., “Gravity and magnetic fields of polygonal prisms and application to magnetic terrain correction” Geophysics, Vol. 41, pp. 727-741. [7] Başokur, A.T., 2002. Inversion of Linear and Non-linear Problems. UCTEA, Chamber of Geophysical Engineers Press (in Turkish). [8] Oruç, B., 2006. Modeling in Geophysics with Theory and Examples. Kocaeli University Press (in Turkish). [9] Zhdanov M.S. 2002. Geophysical Inverse Theory and Regularization Problems. Elsevier.
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