Stabilization of the Euler deconvolution algorithm by means of a two steps regularization approach

Transcription

1 Stabilization of the Euler deconvolution algorithm by means of a two steps regularization approach R. Pašteka ( 1,2 ), D. Kušnirák ( 1 ), H.-J. Götze ( 2 ) ( 1 )Department of Applied Geophysics, Comenius University, Bratislava, Slovakia ( 2 )Institute of Geosciences, Dept. of Geophysics, Christian-Albrechts University, Kiel, Germany Summary Euler deconvolution is one of the best developed methods for semi-automated interpretation of potential fields in the last two decades. This paper describes a two steps procedure based on: the stabilization of the input derivatives by regularization (1.step) and involving the regularized solution of the linear equation system (included in a 2.step). This can significantly improve the Euler deconvolution (ED) method which leads to better interpretations of potential fields. In this study the two steps stabilization approach will be applied to a synthetic model and a real data set which stems from an archaeological microgravity survey. These two case studies were calculated by the classical ED and the new derivative ED methods (DED). The latter is a modification of the classical approach and bases on initial vertical derivative evaluation for input data and it results in a direct estimate of structural index values. Presented new two steps procedure provides better estimations of Euler depths and eliminates significantly false solutions. Introduction Since long, Euler deconvolution is one of the best developed methods and thoroughly tried and tested for any semi-automated interpretation of potential fields (e.g. Reid 1995, Reid 2007). The reason is that it bases on a clearly formulated theoretical background (the Euler s homogenity theorem), is relatively simple and ensures good accessibility to professional and/or freely available interpretation software. On the other hand this interpretation method is very sensitive to noise and errors in interpreted data sets. Anyway, it is very effective in the interpretation of well shaped and sampled anomalies, which are caused by isolated geological structures and near-surface objects (e.g.: Bilgili et al. 2009, Pašteka et al. 2009). The deconvolution of noisy data often results in a large amount of erroneous and defocused solutions. In the past many techniques have been published to improve and select satisfying solutions (or reject bad ones). The majority of them is focused on clustering algorithms and/or fulfilling some of selected statistical criteria (standard deviation and/or condition number of the solution). In some cases these methods are very helpful, however, we think that such improvements are included to late in the numerical process. Therefore, stabilization of solutions should start much earlier. Toward this end we are suggesting a two folded methodological procedure: a) stabilized evaluation of the input data derivatives (Pašteka et al. 2009) and b) stabilized solution of the linear equation system (by means of the Levenberg-Marquardt method). This second regularization step is based on an introduction of a smoothing term in the equation matrix system, controlled by a dumping (regularization) parameter. The search for the optimum value of this parameter remains still an open problem. At this stage we are experimenting with clustering principle - well behaved and focused depth-solution clusters are accepted. Here we will present the results and our experience with the two steps stabilization of the Euler method. In the first example we get use of synthetic gravity data along a profile with additional noise. The classical (standard) Euler Deconvolution will be applied with a prescribed structural index). In the second one real world grided microgravity data were interpreted. They stem from a survey, which was

2 conducted in a baroque church in Western Slovakia and was focused on the detection of medieval crypts. In this example we tested the derivative Euler Deconvolution which included the direct estimation of structural index values. Both results show that the new two steps procedure provides two kinds of results (a) more focused estimates of Euler depths and (b) significant reductions of false solutions. Methodology The "two steps stabilization" procedure is based on: a) a Fourier domain low-pass filtering of the evaluated higher derivatives, entering into the Euler deconvolution method by means of regularization (Pašteka et al. 2009) and b) use of the Levenberg Marquardt (LM-) method (e.g. Aster et al. 2005) which enables a stable solution of the linear equation system. This numerical method modifies the classical Gauss Newton method for the solution of equation systems by means of the term λi (where I is the identity matrix and λ the regularization parameter), entering into the main matrix of the equation system, which forms then a nonsingular matrix. The problem which is related to the Levenberg Marquardt method is determination of the optimum λ value. We do not have a realistic solution of this problem at the moment. Application of the L-curve (Hansen, 1994) did not lead satisfying results because the best λ values did not necessarily point to the important breaking-point of the L-curve. The optimum λ was estimated by means of a clustering principle which will be described below. In Fig 1. we present results which were obtained by the classical Euler Deconvolution method with a prescribed structural index (Thompson, 1982). As mentioned earlier we used synthetic gravity data. They are calculated along a profile over a 2D horizontal cylinder (SI = 1) in the depth of 20 m. White noise was added with the range of 5% of the main anomaly amplitude. The influence of the added noise is crucial and without any smoothing or regularizing of the orthogonal x- and z-derivatives, all obtained solutions are obviously wrong (Fig.1a, black crosses). By means of regularization of derivatives (Pašteka et al. 2009) the quality increased (Fig. 1a, blue crosses are building a cluster close to the location of the center of the cylinder). However, there still exists a larger amount of false solutions. Only the application of the second regularization step the LM-method excludes these false solutions (Fig. 1b), but we realize that the solution clustering strongly depends of the selection of the regularization parameter λ (best solutions: Fig. 1b, red crosses, λ=10-6 ). To select an optimum value for λ, we used a "clustering principle": Well developed clusters are selected which corresponds with the most important anomalous features of the field. Simultaneously a large amount of irregularly distributed false solution points have to been eliminated. For very low λ values (e.g ) solutions are almost identical with that for λ = 0 (not shown here). For higher values (e.g and larger) solutions are defocused and moving upwards, forming an artificial and false cluster at z = 0 m; it is well developed - shown in Fig. 1b, by violet crosses, λ = However, the influence of regularized derivatives is more important if integrated in the method of the derivative Euler Deconvolution (DED) (Hsu 2002, Fedi and Florio 2002). This modification is based on an initial evaluation of vertical derivative and subsequent application of the standard Euler deconvolution methodology. This initial evaluation of vertical derivative minimize and remove the role of the background term and the structural index value can be estimated directly from the equation system together with the coordinates of the source. In Fig.2 we present the results of the DED method, applied on a selected part of residual Bouguer anomaly field (correction density: 1.80 t m -3 ), obtained in the framework of a microgravity and GPR survey which aimed in the detection of medieval crypts in the St. Nicholas baroque church in Trnava, Western Slovakia (Pašteka et al. 2007). In this survey, seven unknown crypts have been detected by the above mentioned methods and finally verified by video-inspection with a mini-camera. The selected part of the residual Bouguer anomaly field contains two small negative anomalies (min.: approx. 45 μgal), caused by two crypts with almost identical depth position and volume. In this case

3 the solutions without regularized derivatives are defocused and too shallow (Fig. 2a and 2d; the approximate cross-section of the crypt based on GPR interpretation is plotted in blue lines). Fig.1 Results of classical Euler Deconvolution with a prescribed structural index value (SI = 1), obtained from the synthetic gravity of a 2D horizontal cylinder (depth 20 m) with added synthetic white noise; a) the results with and without regularized derivatives, b) the results with regularized derivatives and Levenberg-Marquardt method (for various values of damping parameter ). Fig.2 Results of the DED method, which includes a direct estimation of structural index values. They are obtained from data interpretation of a microgravity survey. In the upper row residual Bouguer anomaly maps are displayed together with DED solutions (colored dots); in the lower row

4 depth sections from a GPR survey are shown in combination with DED solutions; a) & d) present results without regularized derivatives, and = 0; b) & e) portray results which were calculated by regularized derivatives and = 0; finally c) & f) shows the results with regularized derivatives and = 10-5 (the optimum value, selected by the clustering principle) In the case of regularized derivatives no bigger differences can be detected in the map between the results which based on a processing with and without the LM-method (Fig. 2b and c). On contrary to Fig. 2b only a small artificial cluster with coordinates 85.8m, 86.3m disappeared from the map (Fig. 2c). However, in the xz-projections (Fig. 3e, f) it is shown that the amount of wrong solutions with SI = 1 is much lower. Actually this SI value is not real for such isometric anomalies, because it is more reliable for contact structures. In the case without LM-introduction the number of these solutions achieves a relative level of 65% (λ = 0) and with it 49% (λ = 10-5 ), if they are compared to the total number of all solutions. Clusters with SI=1 are similar and show the approx. depth of the crypt center. Conclusions Based on our numerical experiments and results we described the role of regularized derivatives (1. stabilization step) as an important item and the introduction of the Levenberg-Marquardt (2. stabilization step) as an item with minor importance. Without stabilization of the derivatives we could not obtain any reliable Euler solutions in a great variety of synthetic and real data studies. The LM-method can help to cancel defocused and erroneous solutions, but it strongly depends on the correct selection of the optimum value of the dumping (regularization) parameter λ. It is important to mention that in the case of noisy input data, final clustering or statistical selection of received solutions can not basically affect the received shallower depths (resulting from the noise influence). However, the presented "two step stabilization" procedure results in Euler solution points which are located closer to the real positions of the bodies. Again, the key role plays the search for the optimum regularization parameters. In the case of the regularized derivatives, the concept of C-norm functions analysis (Pašteka et al. 2009) led to acceptable results in many cases. The L-curve concept (Hansen, 1994) included in the LM-method did not result in satisfying solutions till now the most acceptable solutions (clustered around a real source position) was not found close to the important breaking-point of the L-curve. This task/problem remains still open, in the actual version of the presented procedure a rather time-consuming trial and error method is applied for searching the optimum λ-parameter based on the clustering principle. Development and test of the presented "two-steps stabilization procedure" for improvement of Euler Deconvolution have been done under the framework of bilateral cooperation between the geophysical departments of Comenius University in Bratislava and Christian-Albrechts University in Kiel. The Matlab programs REGDER and REGDED have been completed by RP in the framework of his Mercator Professor ship in Kiel which was financially supported by the Deutsche Forschungsgemeinschaft, DFG (INST 5659/1-1) project. References Aster,R., Borchers, B., Thurber, C., [2005] Parameter Estimation and Inverse Problems. Elsevier Press. Bilgili F., Götze H.-J., Pašteka R., Schmidt S., Hackney R. [2009] Intrusion versus Inversion - a 3D Density Model of the Southern Rim of the Northwest German Basin. International Journal of Earth Sciences, Vol. 98, Nr. 3, Fedi, M., Florio, G., [2002] Euler Deconvolution with no a priori definition of Structural Index. Geoph. Research Abstracts, vol. 4. Europ. Geophys. Soc. 27th General Assembly, Nice, april,

5 Hansen, P. C., [1994] Regularization tools. A MatLab Package for Analysis and Solution of Discrete Ill-posed Problems. Numerical Algorithms 6, Hsu S., [2002] Imaging magnetic sources using Euler s equation. Geophysical Prospecting, 50, Pašteka R., Terray M., Hajach M., Pašiaková, M., [2007] Microgravity measurements and GPR technique in the search for medieval crypts: a case study from the St. Nicholas church in Trnava, SW Slovakia. Proc. of the Archaeol. Prosp. 7th conference, Nitra, Štúdijné zvesti Vol. 41, Pašteka R., Richter F.P., Karcol R., Brazda K., Hajach M. [2009] Regularized derivatives of potential fields and their role in semi-automated interpretation methods. Geophys. Prospecting, Vol. 57, Nr. 4, Reid, A. B. [1995] Euler Deconvolution: Past, Present and Future. A Review. SEG, Exp. Abstr., Reid, A. [2007] Semi-Automated Methods of Potential Field Interpretation Innovations, and Recent and Future Developments. Expaned Abstracts, EGM2007 Intern. Workshop., 4 p.