Physics of the Earth and Planetary Interiors 124 (2001) 33–43
Optimizing model parameterization in 2D linearized seismic traveltime tomography Zoltán Wéber∗ Seismological Observatory of the Hungarian Academy of Sciences, H-1112 Budapest, Meredek u. 18, Hungary Received 14 February 2000; received in revised form 8 October 2000; accepted 26 January 2001
Abstract In seismic traveltime tomography, a set of linearized equations is solved for the unknown slowness perturbations. The matrix of this set of equations (the tomographic matrix) is usually ill-conditioned, because the model space contains more details than can be resolved using the available data. The condition of the tomographic matrix can be characterized by its singular value spectrum: the larger the normalized singular values and the greater the rank of the matrix, the smaller the null space and the better the condition of the inversion problem. The structure of the tomographic matrix depends on the source–receiver configuration and the model parameterization. Since the source–receiver geometry is often fixed (e.g. in earthquake tomography), conditioning can only be influenced through the model parameterization, i.e. the structure of the model space. In this paper, we demonstrate a method for finding an optimal, irregular triangular cell parameterization that best suits the raypath geometry. We define a practical cost function, whose minimization is equivalent to the minimization of the null space of the tomographic matrix. Since the cost function depends on the model discretization in a highly nonlinear manner, a simulated annealing algorithm is used to find the optimal parameterization. We show that the cost value for the optimal triangulated model is about two times smaller than that for the regular gridded model with the same dimension, resulting in more accurate and reliable inversion results. The method is demonstrated through some cross–borehole tomographic examples with given acquisition geometries. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Seismic tomography; Model parameterization; Global optimization; Simulated annealing; Delaunay triangulation
1. Introduction A linearized seismic traveltime tomographic inverse problem can be defined as that of solving the matrix equation Gm = t
(1)
where m ∈ Rn denotes the vector of the unknown slowness perturbations, t ∈ Rm denotes the vector ∗ Fax: +36-1-2482301. E-mail address:
[email protected] (Z. W´eber).
of data misfit (i.e. the difference between the measured and calculated traveltimes), G ∈ Rm×n is the “tomographic matrix” and n < m. The structure of the tomographic matrix G depends on the source–receiver configuration and the model parameterization. Since the source–receiver geometry is often fixed (e.g. in earthquake tomography), model parameterization plays an important role in determining the properties of the matrix G in Eq. (1). Most often in traveltime inversion, the Earth is modelled by regular grid of cells, where the local slowness is assumed to be constant. The element Gij of the tomographic matrix G is then the length of the ith ray
0031-9201/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 ( 0 1 ) 0 0 1 8 5 - 6
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Z. W´eber / Physics of the Earth and Planetary Interiors 124 (2001) 33–43
in the jth cell. Because of the regular parameterization not taking into account the raypath geometry, cells not hit by any rays can often be found and it is also often the case that some rows of G are linearly dependent. This means that G may be rank-deficient and has a null space. As a consequence, an infinite number of solutions can be obtained by adding any linear combination of vectors in the null space to the solution of Eq. (1). The least squares solution of the ill-posed system (1) is very sensitive to data errors, because the small eigenvalues of G cause strong fluctuations in the solution (van der Sluis and van der Vorst, 1987; Farra and Madariaga, 1988). One way of suppressing these undesirable effects is to use the damped least squares approach (Marquardt, 1963; van der Sluis and van der Vorst, 1987; Phillips and Fehler, 1991). Damping reduces the magnitude of the solution and makes the tomographic equations stable, but at the same time, it distorts the results (Michelena, 1993; Vesnaver, 1996). Tinti and Ugolini (1990) found that there are circumstances in which instabilities of Eq. (1) may be mitigated by reducing the number of seismic raypaths taken into account for the inversion. In this way, they can eliminate some linearly dependent rows from G resulting in more stable tomographic equations. Unfortunately, this method throws away some expensively acquired information, which is usually not desirable. Another way to improve the stability of Eq. (1) may be changing the structure of the tomographic matrix G by using a different model parameterization. With different parameterization, we introduce different physical information in the inversion method leading to different results, i.e. with the same data set, it is possible to obtain different results just because different parameterizations have been used (Trampert and Leveque, 1990). However, the goal is to obtain a reconstructed model that is independent of the parameterization. A best way to overcome this problem is to find a parameterization dependent only on the raypath geometry. In this way, information based only on the measured data is introduced in the inversion problem. Michelini (1995) introduces an adaptive-grid formalism, in which traveltime data are simultaneously inverted for both velocity and position of the grid points of the model discretization mesh. This method
seeks to improve the model parameterization, and its results are found to be superior to those from rectangular grids. Michelena and Harris (1991) propose another model parameterization consisting in constant regions along the beampaths called natural pixels. The discretization along the beampaths comes from the fact that they are the regions sampled with each measurement in traveltime tomography. This type of parameterization depends only on the particular data set to be inverted assuming no prior information about the model. As Vesnaver (1995, 1996) points out, there are two situations giving rise to the null space of the tomographic equations: some cells not being hit by any rays and some rays having rows in G that are linearly dependent. Vesnaver tries to reduce the null space by merging cells, shifting cell boundaries and splitting cells into two or more pieces. This procedure is carried out interactively until an irregular parameterization with small or even zero null space is achieved. Curtis and Snieder (1997) try to find the optimal irregular parameterization that gives the best inverse problem conditioning by maximizing the normalized trace of G TG . However, the method does not penalize zero eigenvalues explicitly, so it is unlikely to be always capable of eliminating the null space. Moreover, Curtis and Snieder (1997) do not compare the inverse problem conditioning for the optimized model and for the conventional regular grid parameterization. They also do not illustrate how the accuracy and reliability of the tomographic inversion results change as the model discretization changes from the conventional rectangular grid to the optimized parameterization. In this paper, we propose a method for finding an optimal, irregular triangular cell parameterization for a given raypath geometry by trying to minimize the null space of system (1). Since this is a highly nonlinear problem, a simulated annealing algorithm is used to find the optimal parameterization. Our method is similar to that of Curtis and Snieder (1997), but differs from it in some important aspects. We show that the condition of the tomographic matrix corresponding to the parameterization determined by optimization is much better than that corresponding to a regular rectangular grid parameterization with the same dimension. The method is demonstrated through some cross–borehole tomographic examples with given acquisition geometries.
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2. The method The singular value decomposition (SVD) of G ∈ Rm×n can be written as VT G = U ΣV
(2)
where U ∈ Rm×m is an orthonormal matrix of eigenvectors that span the data space, V ∈ Rn×n , an orthonormal matrix of eigenvectors that span the model space, and Σ ∈ Rm×n , a sub-diagonal matrix whose elements are the singular values of G (Aki and Richards, 1980). The columns of U are the eigenvectors of GG T and the columns of V are the eigenvectors of G TG , where G T denotes the transpose of the matrix Without loss of generality, the singular values are assumed to be arranged in decreasing order. When a singular value is zero, the corresponding singular vector in data space cannot be mapped into the model space or vice versa. Data vectors or model vectors with zero singular value belong to the null space of system (1) and cannot be resolved. When a singular value is not zero but is small relative to the largest one, the contribution of the corresponding eigenvectors to the solution must be regularized, otherwise the solution may be unstable. So, we define the quasi-null space corresponding to those singular values that are smaller than a predefined threshold chosen according to the largest singular value (Vesnaver, 1996). Our aim is to minimize this quasi-null space of G . Eq. (1) is usually solved in the sense of the L2 norm. The solution can be expressed as m ˆ = G −g t where G−g is the generalized inverse of G. Using the SVD, the generalized inverse can be written as G −g = V p Σp−1U Tp
(3)
where p is the number of the singular values outside of the quasi-null space, i.e. the effective rank of G . In Eq. (3), U p and V p consist of the first p columns of U and V , respectively, and Σp (a p × p matrix) has the corresponding singular values. The greater the rank and the larger the singular values of G , the more stable the inversion problem and the more independent pieces of information may be
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gained from the data. Now, we can define a measure of stability to be maximized as the sum of the normalized singular values and the rank (p) of G . Since the singular values of G are the positive square roots of the eigenvalues of G TG , the stability index of G can be defined as n
SG =
1 λi + p λ1
(4)
i=1
where λi is the ith eigenvalue of G TG . In Eq. (4), the first eigenvalue λ1 can be calculated quickly using the power method (Ralston and Rabinowitz, 1978), while the sum of the eigenvalues is equal to the trace of G TG . Unfortunately, determining the rank would require the whole SVD of G , which is a computationally expensive task, given the size of the matrix. However, p can be estimated by p, ˆ the number of model cells in which ray coverage is greater than a predefined threshold. (The ray coverage of a given cell is defined here as the total length of ray segments intersecting that cell.) This threshold should be chosen according to the average ray coverage. Although high ray coverage is not a sufficient criterion to eliminate the null space, it is a necessary criterion for a well-determined inverse problem. Thus, instead of maximizing the stability index in Eq. (4), the cost function G) = E(G
nλ1 + qˆ G TG ) trace(G
(5)
is minimized, where n is the number of the model cells and qˆ = n − pˆ denotes the estimated number of singular values in the quasi-null space. The cost funcG) has the minimum of 1, and any reduction tion E(G G) results in a better conditioned G with smaller in E(G quasi-null space. As Vesnaver (1995, 1996) points out, there are two situations giving rise to the null space of the tomographic equations: some cells not being hit by any rays and some rows in G that are linearly dependent. Linear dependence is mainly caused by regular discretization of the model, so it can be reduced by irregular model parameterization. Another source of zero eigenvalues is the inadequate ray coverage, which can also be reduced by appropriate model discretization. Moreover, the additional term of qˆ in Eq. (5) penalizes
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Fig. 1. Optimizing model parameterization for a simple cross–borehole raypath geometry. (a) Raypath geometry with rectangular grid parameterization (cost value = 9.28). (b) Raypath geometry with optimized triangular cell parameterization (cost value = 4.47). (c) Normalized eigenvalue spectra of G TG for the two types of parameterization. For the gridded model G TG has four zero eigenvalues while for the optimized triangulated model, it has full rank.
this type of zero eigenvalues explicitly, so if ray coverage becomes too small in a model cell, the cost value becomes greater. It is this additional term that G) from the cost function proposed distinguishes E(G by Curtis and Snieder (1997). Although high ray coverage cannot guarantee empty null space, the inclusion of qˆ in Eq. (5) has advantageous effects on the cost function. For many poorly conditioned model configurations, for which qˆ > 0, the cost value is much greater than that defined by
Curtis and Snieder (1997), and the surroundings of the global minimum probably become a deeper and narrower depression. And the more pronounced the global minimum is, the more easily it can be found, because the convergence of any optimization procedure depends on the shape of the cost function to be minimized. For example, the first term in Eq. (5) may decrease even if the null space of the tomographic matrix increases, since the effect of a small eigenvalue becoming zero can easily be compensated by increasing
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some greater eigenvalues at the same time. However, if the newly arising zero eigenvalue is associated with G) is very likely to increase due small ray coverage, E(G to the additional term of q. ˆ Since any procedure used to minimize Eq. (5) is eager to get rid of model configurations with large cost value, the new definition of G) proposed here can improve the convergence of E(G any optimization method towards the desired global minimum.
3. Parameterization of the model The cost function E in Eq. (5) depends on the structure of the tomographic matrix G . As a consequence, E is influenced by both the source–receiver configuration and the model parameterization. Since the source–receiver geometry is often fixed (e.g. in earthquake tomography), we minimize the cost function by finding an optimal model parameterization.
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The best parameterization is expected to consist of irregularly sized cells, so we use Delaunay triangular cells to describe the model. Delaunay triangulation is a procedure to generate a unique set of triangles from arbitrarily distributed nodes in two-dimensions (2D) (Delaunay, 1934; Sambridge et al., 1995). Each triangle has the property that the circle passing through its vertices does not contain other nodes. The freedom allowed by the Delaunay triangulation to build triangles from arbitrary nodal distribution in 2D makes them ideal for the basis of parameterization. Since the size of the Delaunay triangles is determined by the density of the nodal distribution, it is possible to build 2D models with extremely large variation in cell sizes and complex geometry. Fine detail can be imposed anywhere by increasing the local density of the nodes and letting the Delaunay triangulation produce the cells. The concepts of Delaunay triangulation can be generalized easily to higher dimensions allowing our proposed method to be used in three-dimensional
Fig. 2. Spatial distributions of the velocity discrepancy for the (a) gridded and (b) triangulated models shown in Fig. 1. The reference homogeneous velocity was 11% in excess of the true field in this experiment. There are only two triangles with velocity discrepancy values greater than 0.004%.
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tomography as well (Delaunay tetrahedra) (Sambridge et al., 1995). The above mentioned properties of the Delaunay triangulation allow us to find the optimal model parameterization by an automatic procedure. Since the parameterization depends on the nodal configuration, we seek the best model by searching for the optimal nodal distribution inside the model. We construct two sets of nodes. One of them contains the nodes defined by the source and receiver positions and these are kept fixed. The nodes of the other set are placed inside the model area and their positions are varied until the minimum of the cost function E is found. The coordinates of these internal nodes can be varied freely and continuously, while Curtis and Snieder (1997) try to find the optimal nodal distribution by selecting a subset from a number of fixed nodes. Therefore, our method gives us greater freedom to find the optimal node positions. The cost function to be minimized depends on the node positions in a highly non-linear manner, so a global optimization procedure must be used to
find the optimal parameterization. Since the work of Kirkpatrick et al. (1983), the simulated annealing method has been used in many parameter optimization problems (van Laarhoven and Aarts, 1987; Press et al., 1992). The usefulness of this method, when applied to global optimization problems in geophysics, has already been demonstrated by many authors (Rothman, 1985, 1986; Landa et al., 1989; Mosegaard and Vestergaard, 1991; Sen and Stoffa, 1991; Vasudevan et al., 1991; Vestergaard and Mosegaard, 1991; Pullammanappallil and Louie, 1994; Wéber, 2000). A simulated annealing procedure similar to that described in Wéber (2000) is used here for finding the optimal model parameterization that best suits the raypath geometry.
4. Numerical experiments We illustrate the proposed procedure by finding the optimal triangular cell parameterization for a
Fig. 3. Standard deviation (in arbitrary units) of the model parameters for the (a) gridded and (b) triangulated models shown in Fig. 1.
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cross–borehole tomographic inversion. The source– receiver geometry is similar to that used by Curtis and Snieder (1997) and is shown in Fig. 1. The ray coverage threshold used to estimate qˆ in Eq. (5) is chosen to be equal to one tenth of the average ray coverage (sum of ray lengths divided by the number of cells).
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The cross–borehole raypath geometry with rectangular grid parameterization and with the optimized triangular cell parameterization is illustrated in Fig. 1a and b, respectively. Fig. 1c compares the normalized eigenvalue spectra of G TG for the two cell types. Although the gridded model consists of 40 cells, while
Fig. 4. Optimizing model parameterization for a real experiment in a coal mine. (a) Raypath geometry and regular grid parameterization (cost value = 24.49). (b) The optimized triangular cell parameterization (cost value = 13.49). (c) Normalized eigenvalue spectra of GT G for the two types of parameterization. For the gridded model, G TG has nine zero eigenvalues while for the optimized triangulated model, it has full rank.
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the triangulated one contains only 39 triangles, the presented results can be readily compared. The normalized eigenvalues for the optimized triangulated model are considerably larger than those for the gridded one. The grid parameterization has resulted in four zero eigenvalues with cost value of 9.28, while for the triangulated model, the tomographic matrix has full rank and the cost value is 4.47. The distribution of the triangular cells in Fig. 1b is correlated with the raypath distribution: the greater the ray and angular coverage, the smaller the triangles. The shape and orientation of the triangles also depend on the ray distribution. In the upper part of Fig. 1b, for example, where sub-horizontal raypaths dominate, the triangles have horizontally elongated shape according to the well-known fact that sub-horizontal rays
produce little horizontal resolution. In other words, the optimized model with its triangles of different sizes and shapes reflects the local resolution of the inversion. Since the source–receiver geometry in Fig. 1 is practically identical to that used by Curtis and Snieder (1997), we have the opportunity to compare the two results. First of all, the cost value achieved by using the improved cost function defined in Eq. (5) is smaller than that produced by the method of Curtis and Snieder (1997): 4.47 versus 5.3. Secondly, in the upper part of Fig. 1b, the triangle distribution is approximately symmetric, according to the almost symmetric raypath distribution in this area. The final parameterization obtained by Curtis and Snieder (1997) does not show this symmetry at all. These results suggest that the model optimization technique proposed in this paper is
Fig. 5. Spatial distributions of the velocity discrepancy for the (a) gridded and (b) triangulated models shown in Fig. 4. The rectangular cell at the bottom-left corner of the model area is hit by no rays. The reference homogeneous velocity was 11% in excess of the true field in this experiment. There are only two small triangles with velocity discrepancy values greater than 0.01%.
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superior to that described by Curtis and Snieder (1997). In order to further investigate the advantages of the inversion with optimized model parameterization, it is useful to introduce some indicators. One of these is the relative discrepancy vv between the true velocities v , which are assumed to be known in our simulations, and the computed field vˆ vi = 100 ×
|vi − vˆi | vi
Fig. 2a and b show the distributions of the velocity discrepancy for the gridded and triangulated models, respectively. The reference homogeneous velocity was 11% in excess of the true field in this experiment. For the gridded model, the solution has been
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achieved by p = 35, while for the optimized model SVD truncation was not necessary. The images of the discrepancies between the true and the calculated velocity values show that the inversion produces more accurate results with the optimized parameterization. The explicit computation of the generalized inverse G −g allows us to deduce the variance–covariance matrix C m . If C t is the covariance matrix for the data, the covariance matrix for the model parameter vector may be easily obtained with the aid of the expression G−g ]T C m = G −g C t [G The standard deviation of the model parameter in the C m )i,i . ith cell is σi = (C In the following, the data covariance matrix is assumed to be the identity matrix in arbitrary units.
Fig. 6. Standard deviation (in arbitrary units) of the model parameters for the (a) gridded and (b) triangulated models shown in Fig. 4. The rectangular cell at the bottom-left corner of the model area is hit by no rays. For the gridded model, standard deviation is greater than 2 near the sources and receivers.
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The units are not important, because the major concern here is not the absolute magnitude of the standard deviations σ i , but rather their distribution in the cells and how they change as the model parameterization changes from the conventional rectangular grid to the optimized triangular configuration. Therefore, in this paper, the corresponding figures do not show the units of the standard deviation. Fig. 3a and b illustrate the standard deviation distributions for the gridded and triangulated models, respectively. Comparing these images, one can easily verify that σ is much smaller for the triangulated model than for the gridded one and that its distribution is more homogeneous. The proposed method has also been tested on a real source–receiver configuration used in a Hungarian coal mine. The raypath geometry and regular-grid parameterization are shown in Fig. 4a. Since the source–receiver positions do not coincide with the model boundaries, they do not define the convex hull of the model space correctly. To overcome this problem, the projections of the source–receiver coordinates onto the model boundaries are defined as fixed nodes instead of the source–receiver points themselves. The resulted optimized triangular cell parameterization is depicted in Fig. 4b. The ray coverage threshold used to estimate qˆ in Eq. (5) is again chosen to be equal to one-tenth of the average ray coverage. Both the gridded and the triangulated models contain 224 cells. The normalized eigenvalue spectrum of GT G for the optimized triangulated model is again superior to that of the regular gridded model (Fig. 4c). The grid parameterization has resulted in nine zero eigenvalues with cost value of 24.49, while for the triangulated model, the tomographic matrix has full rank and the cost value is 13.49. Fig. 5a and b illustrate the spatial distributions of the velocity discrepancy vv for the gridded and triangulated models, respectively. The reference homogeneous velocity was 11% in excess of the true field in this experiment. For the gridded model, the solution has been achieved by p = 215, while for the optimized model, SVD truncation was not necessary. The images of vv again show that the inversion produces more accurate results with the optimized parameterization. The same conclusion can be drawn from Fig. 6 which compares the standard deviation distributions for the two types of parameterization.
5. Conclusions We have defined a cost function in Eq. (5), whose minimization is equivalent to minimizing the null space of the tomographic matrix G defined in Eq. (1). We have also presented a method with which the cost function can be calculated efficiently for any type of model parameterization and source–receiver configuration. This efficiency allows us to use a global optimization method, such as simulated annealing, to find the global minimum of E. Using global optimization procedure is necessary because the cost function depends on the model discretization in a highly nonlinear manner. The cost value for the model with optimal Delaunay triangular cell parameterization is about two times smaller than that for the regular gridded model with the same dimension, resulting in more accurate and reliable inversion results. Since the structure of the tomographic matrix G and, as a consequence, the cost function E, depend on both the source–receiver configuration and the model parameterization, the proposed method may be used not only to optimize the model discretization but to optimize the source–receiver geometry for a given model parameterization as well (Curtis, 1999a, b). Its application is not even restricted to seismic tomography. Inversion problems arising from geomagnetic, electric or gravitational data interpretation may also be made better conditioned by means of the described optimization method. Acknowledgements The reported investigation was financially supported by the Hungarian Scientific Research Fund (No. F019277). The author is also grateful to the Department of Mining Geophysics, ELGI for providing the field data in the study. I would like to thank two anonymous reviewers for making various suggestions which have helped me in improving this paper. GMT (Wessel and Smith, 1998) was used for plotting the figures. References Aki, K., Richards, P., 1980. Quantitative Seismology: Theory and Methods. Freeman, San Francisco, CA.
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