APPLIED GEOPHYSICS, Vol.11, No.2 (June 2014), P. 139-148, 8 Figures. DOI: 10.1007/s11770-014-0440-6
Extrapolated Tikhonov method and inversion of 3D density images of gravity data* Wang Zhu-Wen1, Xu Shi1, Liu Yin-Ping1♦, and Liu Jing-Hua1 Abstract: Tikhonov regularization (TR) method has played a very important role in the gravity data and magnetic data process. In this paper, the Tikhonov regularization method with respect to the inversion of gravity data is discussed. and the extrapolated TR method (EXTR) is introduced to improve the fitting error. Furthermore, the effect of the parameters in the EXTR method on the fitting error, number of iterations, and inversion results are discussed in details. The computation results using a synthetic model with the same and different densities indicated that. compared with the TR method, the EXTR method not only achieves the a priori fitting error level set by the interpreter but also increases the fitting precision, although it increases the computation time and number of iterations. And the EXTR inversion results are more compact than the TR inversion results, which are more divergent. The range of the inversion data is closer to the default range of the model parameters, and the model features and default model density distribution agree well. Keywords: Gravity data inversion, 3D inversion, extrapolated Tikhonov regularization method, extrapolated Tikhonov parameter selection
Introduction Gravity and magnetic surveying is widely used in structural geology, mineral exploration, and engineering and environmental investigations. The 3D inversion of gravity data is critical for the quantitation interpretation because the density contrast distribution greatly increases the amount of information extracted from gravity data (Blakely, 1995; Li and Oldenburg, 1998; Zheng, 2005). The gravitational field expressed by the gravity data measured on the Earth’s surface, as we know, might be produced by the various equivalent subsurface density distributions bodies. When we inverse gravity data, the survey area is initially divided into a large number of adjacent prisms with contrasting density
and then the underground sources are rebuilt via the calculation of linear or nonlinear problems. The amount of observational data available for use in inversion is limited because the numbers of data are less than the discrete models of Earth; furthermore, they contain errors owing to differences in methods, equipment, and other factors. Therefore, geophysical data inversion does not produce unique solutions (Li and Oldenburg, 1996, 1998, 2003). To overcome the nonuniqueness of the inverse problem, various approaches have been used to introduce a priori information into gravity data inversion for obtaining unique solutions. For instance, the model parameters are inversed by taking the known density as the a priori information. We can inverse the thickness of the sedimentary basin layers using the known density
Manuscript received by the Editor June 17, 2013; revised manuscript received June 26, 2014. This study was supported by the National Scientific and Technological Plan (Nos. 2009BAB43B00 and 2009BAB43B01). 1. College of Geoexploration Science and Technology, Jilin University, Changchun 130026, China. ♦ Corresponding author: Liu Yin-Ping (Email:
[email protected]) © 2013 The Editorial Department of APPLIED GEOPHYSICS. All rights reserved.
139
Extrapolated Tikhonov method and inversion changes with depth. The above mentioned method can be used to effectively calculate a relatively flat sedimentary basement with limited a priori information, whereas it is difficult to establish a complex basement surface (Chai and Hinze, 1988; Chenot and Debeglia, 1990). Alternatively, others assume a known and constant density or magnetic susceptibility to inverse the geometric shape of the source. Shallow individual or multiple ore bodies can be thus identified but the resolution of deeper multiple sources is poor (Cordell and Henderson, 1968; Pedersen, 1977). In several studies, the density or magnetic distribution is determined by using the functional relation between the anomaly and the spatial distribution of the sources by adding additional a priori constraints and assuming a central location or direction for the source in the inversion. Therefore, horizontal and inclined dykes can be effectively inversed; however, complex ore bodies are difficult to recognize (Barbosa and Silva, 1994; Guillen and Menichetti, 1984; Last and Kubik, 1983). Overall, different types of a priori information are added to inversions of complex data. Models are fitted to the a priori information by using force-fitting methods, while observations and models meet the requirements of the fitting error level (Commer, 2011; Medeiros and Silva, 1996; Silva and Barbosa, 2006). The introduction of stochastic methods to the constrained minimization of smooth inversion gives better results, especially in terms of the computational requirements (Yao et al., 2003, 2007). Potential field data were better interpreted when inversion methods based on multiple observations were used (Guo et al., 2009; Meng et al., 2012). Joerge et al. (2013) present a new approach for computing an approximate solution of the Tikhonov regularized largescale total least-squares problem, they proposed an iterative method which solves a convergent sequence of projected linear systems and thereby builds up a highly suitable search space. Silvia et al. (2014) introduces a novel algorithm for the choice of the regularization parameter when performing the Arnoldi–Tikhonov method. To address the nonuniqueness in the inversion of gravity data, we discuss the theory of gravity inversion in detail and use the Tikhonov regularization (TR) and extrapolated TR (EXTR) methods to solve ill-conditioned problems. The selection of different criteria for the parameters has different effects on the inversion results when the abovementioned methods are used. Furthermore, the different range of the parameters within the same selection process has different effects on the inversion results and fitting error. In this study, the relation between the selection of parameters, the level of 140
fitting errors, and the number of iterations is discussed in detail. Finally, model test data and data statistics suggest that the selection of parameters has greater impact on the inversion results, and that the results with EXTR are more compact and accurate than the results from the TR inversion method.
Methodology Interpretations of model density differences focus on the region of the grid that comprises an underground abnormal source divided into multiline parallel cubic units with specified dimensions. Each grid of cubic cells with specific dimensions is regarded as a density unit. The density units are the parameters to estimate by using 3D inversion of gravity data in the coordinate system shown in Figure 1. We assume that the boundary coordinates of a grid of subsurface cubic cells along the x, y, and z axes is given by (x1, x2), (y1, y2), and (z1, z2), where x and y are two axes orthogonal to each other and parallel to the ground and the z axis is perpendicular to them. From the observation point r = (x, y, z), the integral form of the universal gravitation field in a single cubic grid cell is (Zhdanov, 1988) r1
g (r ) J ³³³ U (r ' ) r2
rc r dxcdycd z c , 3 rc r
(1)
where γ is the gravitational constant, ρ(r') is the density distribution difference in the grid of cubic cells at point r' = (x', y', z') in an abnormal body, g(r) is the gravitation at distance r between the measurement point and the anomaly source. The vertical component of the gravity anomaly of the infinitesimal volume dV' = dx'dy'dz' is gz(r) and thus equation (1) becomes
Fig.1 Schematic of subsurface grid
Wang et al.
gz
r1
J ³³³ U (r c) r2
zc z dxcdycdz c . 3 rc r
(2)
The discrete form of equation (2) is
J]
U
ª JU «D OQ G E E OQ G D ¬ DEF
§ D E G ·º , DF DUFWDQ¨ ¸» F © ¹¼ DEF
(3)
where d = |r'-r| is the distance between the measurement point and the discrete grid cells. In equation (3), a =x'-x, b =y'-y, and c =z'-z. The gravity anomaly is treated as the product of the discrete grid cell density with constant kernel function A; therefore, the compact form of equation (3) is Aρ =g ,
(4)
where A ∈RN×M is the kernel function matrix, ρ ∈ RN is the subsurface petrophysical properties distribution vector, g ∈ RM is the ground-observed gravity anomaly, and M and N represent the number of grid cells and abnormal data, respectively, in the data collection area. Depending on the size of the relation between M and N, three possible inverse problems might arise, i.e., the overdetermined, comfortable, and underdetermined. When M < N, the number of unknown parameters is greater than the number of underdetermined equation systems that we want to solve. Therefore, the 3D nature of the gravity inversion solution is unstable and nonunique, and thus the inversion is difficult to solve.
Inversionand parameter selection principles The ill-posed problems in the previous section can be solved by introducing a regularization parameter (Lawson and Hanson, 1974; Tikhonov, 1963, Tikhonov and Arsenin, 1977, Joerg and Heinrich, 2013). The TR expression in conjunction with equation (4) is
A
A OI U
T
A g,
A
T
A OI
where ρ is the density distribution vector to be solved, I is a unit matrix, and AT is the transpose of matrix AT A.
1
AT g .
(6)
Selection method for the sum of EXTR regularization parameters We discussed the use of the EXTR method in gravity data inversion in more detail elsewhere (Liu et al., 2013). This method is different from the TR method in that the calculation process needs two regularization parameters, namely α i and n. The selection of these parameters directly affects the morbid linear inversion; furthermore, whether the results are good or bad is directly related to the degree of agreement between the approximate and exact solutions. The primary theory of density inversion based on the EXTR method (Hamarik, 2007, 2008) are presented below. The density inversion formula based on the TR method is 1
A7 g ,
(7)
where T is the transpose of a matrix, α is the regular parameter of the TR method, and I has the same dimension as AT A with a unit matrix. The density inversion equation based on the EXTR method is
U n ,D (5)
The proposed inversion method has the following advantages. First, it does not require the conversion of data into frequency domains. Second, it does not require strictly gridded data and uses interpolation to invert the data by regularizing randomly distributed values. Third, it is acceptable to use a priori constraints, e.g., the density of a single cube or depth-weighted data. We use the values of maximum and minimum depth of the 3D density source as constraints in the inversion. If the inversion results exceed the range, we transform them to remain within the numerical range. If the density of a grid cell is known, its value will also be fixed in the inversion process.
UD D I A7 A
Gravity data inversion using TR method
T
Equation (5) is stabilized by increasing the eigenvalue of matrix A. Furthermore, a constant λ is added to the matrix of the diagonal elements. Thus, equation (5) transforms to
¦
n
i 1
d i U ai , d i
n
j
§ ¨ 1 ai 1, j zi ¨ © aj
1
· ¸ , i z j , ¸ ¹ (8)
where n is an extrapolated arbitrary constant, with values 2–10. According to the parameter selection principle, 141
Extrapolated Tikhonov method and inversion we discuss and define the corresponding parameters n D method, and calculate the corresponding EXTR approximate solution and fitting error. ρ ai is the TR method inversion density vector corresponding to the regular parameter ai. Generally, by determining one of the regular parameters a and n, the remaining become the regularization parameters to be solved. In this study, we first determine one group a 1,a 2,...,a N, where N is the total number of regular parameters and is known. Then, we determine a according to the fitting-error level and subsequently determine n(1 ≤ n ≤ N). First,
accurate value satisfy the following inequality
U n ,D U U n1,D U ˈ n 1,2,..., nME
and the attenuation infinite sequence α1, α2,... satisfies the following conditions f
¦D i 1
of q is equal to 16 2 , 4 2 , and 2; of course the q may be any suitable value. Second, we select the discrete principle to determine αD that meets the a priori error level from the parameter sequence. Finally, we analyze the TR approximate solution and the fitting error, and the corresponding EXTR approximate solution and the fitting error, with the same method that was used to obtain the parameter values. The method is based on the discrete parameter selection principle, where the first one that satisfies the condition ||Aρai - g|| =δ is the parameter αD, which is the first selected parameter value in the sequence αi and thus k = iD is the value that will be solved. To satisfy the above parameter selection principle, the EXTR approximate solution also needs to satisfy the following inequality. We assume that rn = Aρn,α - fδ and that C is constant and greater than one. Given the monotonic decreasing sequence formed by functions
n 1
f ˈ D n1 d const ¦ D i1 . i 1
Influence of regular parameters on fitting error and number of iterations In the following paragraphs, we consider the discrete parameter selection principle and discuss the fitting errors of the various parameters, the size, and the relation between the numbers of iterations. Tables 1 and 2 list the different q values calculated under different levels of fitting error and the required numbers of iterations. The statistics show that the smaller the q value is, the greater the required number of iterations. Therefore, to reduce the computation time, q is set equal to 2 to minimize the number of iterations but not the fitting error. Table 3 lists the fitting errors of the EXTR method calculated using different n values. It can be seen from the tables that the fitting error increases with the increasing n value; however, the fitting error is smaller than that of the TR method. Thus,
rn rn1 , rn1 , then if for all n
dD(n) = ||rn|| and d ME n
1 i
Therefore, the limited serial numbers nD and nME are obtained and the term n ∈{nD, nME} ensures that ||ρn,α-ρ|| approaches zero. Therefore, if equation (9) is established, we recognize that ρn,α is more accurate than ρn-1,α. Once the first αi satisfying the error level is ascertained, the value of n is also ascertained. Thus, the two EXTR parameters are known and using the EXTR inversion method, we can perform the gravity data inversion and obtain the approximate solution of density.
ai , and N = 500 , where q
we assume that aN = 1, D i 1
2 rn1
the inequality dD(n+1) < dME(n) < dD(n) always holds, the first serial numbers about αi that satisfy the inequalities dD(n) < Cδ and dME(n) < Cδ are nD and nME, respectively, where the serial number satisfies the inequality nD-1 ≤ nME ≤ nD. The second norm of the approximation and the
Table 1 Calculated error for different q values at different error levels
q value
10−1
10−2
10−3
10−4
10−5
10−6
q = 16 2
0.0978
0.0096
9.7589
9.8290
9.8940
9.9588
×10−4
×10−5
×10−6
×10−7
0.0092
9.7589
8.6311
9.0729
9.5365
0.0055
×10−4 6.9020 ×10−4
×10−5 8.6311 ×10−5
×10−6 5.3948 ×10−6
×10−7 6.7435 ×10−7
q=
4
2
q=2 142
0.0978 0.0832
(9)
Wang et al. it shows that the EXTR method minimizes the 3D inversion fitting error and improves the accuracy. Table 2 Number of iterations for different q values and different error levels
q value
10−1
10−2
10−3
10−4
10−5
10−6
q = 16 2
29
84
137
190
243
296
q=
2
8
22
35
49
62
75
q=2
3
7
10
13
17
20
4
Table 3 Error calculated with EXTR method for different error levels
N value
10−1
10−2
10−3
10−4
10−5
n=2
0.0025
1.0727
1.6846
2.6342
0
×10
×10
×10
2.8010
8.3152
1.645
8.5758
×10−4
×10−8
×10−10
×10−13
5.6020
1.2793
5.4953
7.3543
×10−5
×10−9
×10−13
×10−13
-
3.8749
2.6189
8.4911
6.4688
×10−11
×10−13
×10−13
×10−14
−5
n=3 n=5 n=5
−7
−9
0 0
Model test Two cuboids with same density model Figure 2 shows the two gravity anomalies from this model and Figure 3 show the model with different aspects of the imaging effects. The size of the model in the x-, y-, and z-directions is 200 × 200 × 100 m. The depth to the top of the buried cuboid is 100 m. The distribution features of the subsurface area are shown in Figure 3. The density of the cuboid is 1 g/ cm3. The observation area comprises a grid in the xand y-directions of 21 × 11 points, with 50 m interval spacing. In the inversion, we added 3% Gaussian noise to the observed data.
Fig.2 Model gravity anomaly of the synthetic model of the two cuboids with same density.
Fig.3 Different slices of gravity anomaly model of Figure 2. Model parameters: subsurface anormaly body size is 200 m × 200 m × 100 m. Slices through a 3-D density model composed of two cuboids with the same density in a uniform background. The depth to the top of the buried cuboid is 100 m.. The cuboids of size 200 m×200 m×100 m have the unit density of 1.0 g/cm3. The slices of x=300 m(a) and x=700 m(b) through different cuboids have the same size in the same view. The slices in the plane z=200 m(c) and y=250 m(d) indicate the regions with high density in different view. The gray scale indicates density in g/cm3.
143
Extrapolated Tikhonov method and inversion Figure 4 shows the imaging results based on the EXTR method. The black rectangle defines the default location of the model. Figures 4a–4c show different perspectives of the density distribution in the x-direction, which shows that the characteristics of the density distribution coincide with the default model for the slice x = 300 m and x = 700 m. Figure 4d–4f shows the different perspectives of the density distribution in the y-direction, which shows that the inversion results for the numerical size and distribution are in good agreement with the preset model. Figure 4g shows cross sections different from those in Figure 4a in the x-direction. Figure 4h shows slice maps different from those in Figure 4d in the y-direction. Figure 4i displays a cross section in the z-direction. These three smaller graphs demonstrate that the degree of convergence in the horizontal direction is better than in the vertical. The inversion results obtained with the classical TR method of the density model are shown in Figure 5. Figures 5a–5c and 5d–5f show the density distribution
from different perspectives and sections in the xand y-directions, respectively. Figure 5 shows that the inversion results well agree with the location and distribution of the default model. However, compared with the corresponding graphs in Figure 4, the inversion results produce a slightly larger density range for the slices x = 100 m and x = 900 m, i.e., the inversion results of the TR method are slightly more divergent than the results of the EXTR method. In Figure 5, in addition to the inversion results being more divergent, it can be seen that the identification of the location of the model boundary is also inferior compared with Figure 4. Therefore, by simple comparison of Figures 4 and 5, it is not difficult to see that the inversion results of the EXTR method are more accurate. However, because the EXTR method requires a greater number of iterations, the computation time is longer. The relevant statistics are shown in Table 4, where it can be seen that the fitting precision of the TR method cannot improve the inversion data without increased computation. Depending on
Fig. 4 Inversion density distribution of different slices for the model of Figure 2 based on EXTR method.
144
Wang et al. the required fitting precision, the number of iterations and computation time of the EXTR method differ. If high fitting precision is demanded, a corresponding increase in computation time and number of iterations will be required. This method is good for improving the inversion results.
Table 4 Errors in the calculated time, error, and numbers of iterations for EXTR and TR methods Method
Time (seconds) Fitting error Iterations
TR
4.717769
0.4226
1
EXTR
18.902032
0.00058378
21
Fig.5 Inversion density distribution of different slices for the model of Figure 2 based on TR method.
Two cuboids with different density model Figure 6 shows a combined model of gravity anomaly values for two cuboids of different density. Figure 7 shows the model with different aspects of imaging effects. The densities of the two cuboids are 0.6 and 1g/cm3, respectively. Except the densities, the model parameters and observations are the same as for the constant density model. Figure 8 shows the results for the variable density synthetic model based on the EXTR method. The black rectangle defines the theoretical model body position. Figures 8a-8c display different perspectives of the density distribution in the x-direction, which suggest
Fig.6 Model gravity anomaly of the two cuboids with different density.
145
Extrapolated Tikhonov method and inversion
Fig.7 Sections of synthetic model of Figure 6.
Fig.8 Inversion density distribution for synthetic model of Figure 6 based on EXTR method.
146
Wang et al. that the density distribution characteristics are close to the default model for the slices x = 300 m and x = 700 m. Figures 8d–8f shows different perspectives of the density distribution in the y-direction, which suggest that the distribution range and size of the inversion results for the numerical model are in good agreement with theory. Figures 8g and 8h shows different slices of the inversion results from those presented in Figure 8a and 8d along the x- and y-directions, respectively. The divergence of the inversion results is less. Figure 8i shows the inversion results for the z–direction. Compared with Figure 7c, the agreement between the inversion results and theoretical model is slightly better. Therefore, the inversion method is also effective for the variable density model.
Conclusions Based on the discussion on the parameter selection rules for the EXTR method, and with model testing, this paper analyzed and compared the influences of different parameter selection in EXTR method on the fitting errors, iteration numbers and inversion result in detail. And some conclusions can be described as follows: First, at the same level of fitting error, the smaller the q value, the greater the number of iterations and the longer the computation time is, and vise versa. Second, the greater the value of n, the faster the required fitting accuracy is. The inversion results for constant density show that the EXTR method is more accurate than the TR inversion method in modeling the density distribution characteristics, furthermore, under the corresponding fitting error level requirements the EXTR method can get more accurate result than that TR method. And the cost is the computation time greater than that in the TR method because the iteration time inevitably increases during inversion. the obtained statistical data and inversion results show that the proposed inversion method in this paper is more flexible and can produce an accurate density distribution.
References Barbosa, V. C. F., and Silva, J. B. C., 1994, Generalized compact gravity inversion: Geophysics, 59(1), 57–68. Blakely, R. J., 1995, Potential theory in gravity and magnetic applications: Cambridge University Press. Chai, Y., and Hinze, W. J., 1988, Gravity inversion of
an interface above which the density contrast varies exponentially with depth: Geophysics, 53(6), 837– 845. Chenot, D., and Debeglia, N., 1990, Three-dimensional gravity or magnetic constrained depth inversion with lateral and vertical variation of contrast: Geophysics, 55(3), 327–335. Commer, M., 2011, Three-dimensional gravity modeling and focusing inversion using rectangular meshes: Geophysical Prospecting, 59, 966–979. Cordell, L., and Henderson, R. G., 1968, Iterative threedimensional solution of gravity anomaly data using a digital computer: Geophysics, 33(4), 596–601. Guillen, A,, and Menichetti, V., 1984, Gravity and magnetic inversion with minimization of a specific functional: Geophysics, 49(8), 1354–1360. Guo, L. H., Meng, X. H., and Shi, L., 2009, 3D correlation imaging for gravity and gravity gradiometry data: Chinese J. Geophys. (in Chinese), 52(4), 1092–1106. Hamarik, U., Palm, R., and Raus, T., 2008, Extrapolation of Tikhonov and Lavrentiev regularization methods: Journal of Physics: Conference Series, 135, 1–8. Hamarik, U., Palm, R., and Raus, T., 2007, Use of extrapolation in regularization methods: Journal of Inverse and Ill-Posed Problems, 15, 277–294. Joerg L., and Heinrich V., 2013, Large-scale Tikhonov regularization of total least squares: Journal of Computational and Applied Mathematics, 238, 95–108 Last, B. J., and Kubik, K.,1983, Compact gravity inversion: Geophysics, 48(6), 713–721. Lawson, C. J., and Hanson, R. J., 1974, Solving least squares problems: Prentice-Hall Inc. Li, Y. G., and Oldenburg, D. W., 2003, Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method: Geophysical Journal International, 152, 251–265. Li, Y. G., and Oldenburg, D. W., 1996, 3-D inversion of magnetic data: Geophysics, 61(2), 394–408. Li, Y. G., and Oldenburg, D. W., 1998, 3-D inversion of gravity data: Geophysics, 63(1), 109–119. Liu, Y. P., Wang, Z. W., Du, X. J., et al., 2013, 3D constrained inversion of gravity data based on Extrapolation Tikhonov regularization algorithm: Chinese J. Geophys. (in Chinese), 56(5), 1650-1659. Medeiros, W. O., and Silva, J. B. C., 1996, Geophysical inversion using approximate equality constraints: Geophysics, 61(6), 1678–1688. Meng, X. H., Liu, G. F., and Chen, Z. X., et al., 2012, 3-D gravity and magnetic inversion for physical properties based on residual anomaly correlation: 147
Extrapolated Tikhonov method and inversion Chinese J. Geophys. (in Chinses), 55(1), 304–309. Pedersen, L. B., 1977, Interpretation of potential field data, a generalized inverse approach: Geophys. Pros ., 25, 199–230. Silva, J. B. C., and Barbosa, V., 2006, Interactive gravity inversion: Geophysics, 71, J1–J9. Silvia, G., and Paolo, N., 2014, Automatic parameter setting for Arnoldi–Tikhonov methods: Journal of Computational and Applied Mathematics, 256, 180– 195 Tikhonov, A. N., 1963, Regularization of ill-posed problems: Doklady Akad. Nauk, SSSR, 153, 49–52. Tikhonov, A. N., and Arsenin, V. Y., 1977, Solution of ill-posed problems: V. H. Winston &Sons. Yao, C. L., Zheng, Y. M., and Zhang, Y. W., 2007, 3D gravity and magnetic inversion for physical properties using stochastic subspaces: Chinese J. Geophys. (in Chinese), 50(5), 1576–1583. Yao, C. L., Hao, T. Y., and Guan, Z. N., et al, 2003, High-speed computation and effective storage in 3-D gravity and magnetic inversion based on genetic algorithms: Chinese J. Geophys. (in Chinses), 46(2),
148
252–258. Zhdanov, M., 1988, Integral Transforms in Geophysics: Springer Verlag. Zheng H. L., 2005, Gravity field and gravity exploration: Geological Publishing House, BeiJing.
Wang Zhu-Wen received his PhD from the Department of Applied Geophysics at China University of Geosciences (Beijing) in 1994, M.Sc. in Structure Geology from East China College of Geology in 1990 and B.Sc. in Exploration Geophysics from East China College of Geology in 1987. He is currently a Professor at Jilin University. His interests are new methods and techniques in Geophysical Well Logging, Reservoir Formation Evaluation, Applied Geophysics, Nuclear Geophysics, and Radiation and Environmental Evaluation.
