Efficient traveltime compression for 3D prestack Kirchhoff migration

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1 Geophysical Prospecting, 2011, 59, 1 9 doi: /j x Efficient traveltime compression for 3D prestack Kirchhoff migration Tariq Alkhalifah The Institute for Geophysical Research, King AbdulAziz City of Science and Technology (KACST), Riyadh, Saudi Arabia Received May 2009, revision accepted March 2010 ABSTRACT Kirchhoff 3D prestack migration, as part of its execution, usually requires repeated access to a large traveltime table data base. Access to this data base implies either a memory intensive or I/O bounded solution to the storage problem. Proper compression of the traveltime table allows efficient 3D prestack migration without relying on the usually slow access to the computer hard drive. Such compression also allows for faster access to desirable parts of the traveltime table. Compression is applied to the traveltime field for each source location on the surface on a regular grid using 3D Chebyshev polynomial or cosine transforms of the traveltime field represented in the spherical coordinates or the Celerity domain. We obtain practical compression levels up to and exceeding 20 to 1. In fact, because of the smaller size traveltime table, we obtain exceptional traveltime extraction speed during migration that exceeds conventional methods. Additional features of the compression include better interpolation of traveltime tables and more stable estimates of amplitudes from traveltime curvatures. Further compression is achieved using bit encoding, by representing compression parameters values with fewer bits. Key words: Compression, Migration, Traveltime. INTRODUCTION At the heart of any conventional implementation of Kirchhoff imaging is traveltime computation. Specifically, Kirchhoff migration requires either precomputing traveltime tables, or including traveltime calculation in its innermost computational loop. Traveltimes are often computed by solving the zero-order (high-frequency) approximation of the Wentzel, Kramers and Brillouin (WKB) expansion of the wave equation. This high-frequency approximation is more than sufficient for most practical imaging applications. It provides a traveltime equation known as the eikonal equation. The second term of the WKB expansion provides the far-field amplitude term, known as the transport equation. The prestack migration traveltime for 2D media is represented by a three-dimensional table with two of the dimensions corresponding to the 2D image field and the third tariq.alkhalifah@kaust.edu.sa dimension corresponding to possible source or receiver locations on the surface. This table is accessed for each input trace with a given source and receiver location to build the traveltime field necessary to smear the input trace over all possible contribution areas in the output image. The total traveltime field is calculated through the summation of the source and receiver traveltimes for each image point within the aperture of coverage (Alkhalifah 2003). The traveltime table for conventional 2D data does not usually exceed 2 GB. We can easily load such a table into the computer physical memory as we apply prestack migration. To implement prestack Kirchhoff migration in 3D, we need to compute a five-dimensional traveltime table. Three dimensions correspond to the image field and two additional dimensions correspond to the possible locations of the source or receiver on the surface. The size of this traveltime table can easily exceed hundreds of GB. Loading such a file into the computer memory for the application of prestack migration is impossible on many machines, resulting in slower implementations of migrations based on repeated C 2010 European Association of Geoscientists & Engineers 1

2 2 T. Alkhalifah access of a slow hard drive (Calandra et al. 2001; Alde et al. 2002). Solutions to the large traveltime table problem have rarely been discussed in literature. Cunha et al. (1995) suggested using a combination of bit-encoding and the third differences of traveltime to obtain reasonable traveltime compression. Their method excels in piece-wise constant velocity models where the increase in traveltime is well represented by first and second differences of traveltime and thus, the third differences have relatively small values easily describable by fewer bits. In this paper, we combine well-known tools of mathematical representation to develop a method for traveltime compression that is both efficient and reasonably accurate. I investigate the accuracy of the compression for practical traveltime fields. Traveltime compression presented here simplifies other critical operations, like data access and traveltime interpolation. The smoothness feature inherently embedded in the compression operation used here also provides an opportunity to compute stable amplitude fields from traveltime tables (but not necessary highly accurate ones). We investigate such features using the EAGE/SEG salt body model. THE TRAVELTIME FIELD At the heart of Kirchhoff prestack migration is traveltime calculation. We compute the traveltime field for each possible source or receiver location over a regular grid that covers the range of possible source and receiver locations of the acquisition (see Fig. 1). Bilinear or spline interpolation is used to evaluate the traveltime field at a certain source and receiver location that might not lay on this grid. Figure 1 A schematic plot depicting the process of traveltime calculation for 3D media where for each source location on the surface we compute the traveltime to all possible image point locations in the 3D image field domain. This results in a 5D traveltime table. Traveltimes are often calculated by solving the eikonal equation using finite difference methods, or indirectly using ray-based methods. Each method has its advantages and disadvantages. Among the finite difference method the key advantage for compression purposes is the general continuity (by virtue of causality) of the traveltime field and completeness of the solution space. Since the proposed compression approach strives on smoothness, the general smoothness of finite-difference traveltimes will help obtain higher degrees of compression of the traveltime without changing the traveltime. However, in ray-based methods in which discontinuities in the traveltime field are present the compression will smooth out the discontinuities. Specifically, we will use the fast marching method described by Alkhalifah and Fomel (2001) to solve the eikonal equation. The fast marching method is unconditionally stable and produces a complete solution that is continuous. SPHERICAL COORDINATES AND THE CELERITY DOMAIN Most transformation-based compression algorithms (especially those based on polynomial basis functions) excel with smooth data. The smoother the data field the greater its compressibility; the compression is achieved by dropping higher order terms of the polynomial representation, those coefficients that are associated with the more rigid parts of the field. To obtain optimal compression levels of the traveltime field we transform it to a more convenient (natural) coordinate system. For traveltimes describing a wavefield emanating from a point source, the wavefront has a general spherical shape with its centre given by the source location. As a result, representing such a traveltime in spherical coordinates results in a smoother traveltime field. In fact, for homogeneous media the traveltime field in spherical coordinates is constant in both angular directions and is linear in the radial direction (see Fig. 2). For media with more complicated velocity variations, the traveltime field in spherical coordinates will have higher order variations, yet is smoother (less curvature) overall than in Cartesian coordinates. The eikonal equation can also be described in spherical coordinates (r, θand φ, see Fig. 3), as follows: ( ) t r r 2 ( ) t 2 + θ ( ) 1 t 2 = s 2 (r,θ,φ), (1) r 2 sin 2 θ φ where s is the slowness. This equation in spherical coordinates is solved by the fast marching method of Alkhalifah and Fomel

3 Efficient traveltime compression 3 Figure 2 A schematic plot showing the traveltime in Cartesian coordinates (left) and Polar coordinates (right) for a homogeneous medium. Figure 3 A spherical coordinate system given by r, θ and φ. The source, s(x 0, y 0, z 0 ), is at the origin of the spherical coordinates where r = 0. The parameter r = (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 is the distance from the source to the point of interest along the wavefront. The parameter φ is the angle between the x-axis and the projection of r onto the x y plane. The parameter θ is the angle between the z-axis and r. (2001). The output of this traveltime calculation is a traveltime field in spherical coordinates. A more efficient alternative to spherical coordinates is the celerity domain, which has similar point source emanating properties but better transformation features. It was suggested originally by Pica (1997) as a tool for traveltime calculation. In fact, for homogeneous media, the celerity domain representation is given by one value, which is the value of the homogeneous velocity. In celerity, we represent the traveltime field by a parameter that has units of velocity and is given by the following relation: (x x0 ) c(x, y, z) = 2 + (y y 0 ) 2 + (z z 0 ) 2, (2) t(x, y, z) where t is the computed traveltime from the source (x 0, y 0, z 0 ) to a location (x, y, z). For homogeneous media, c is constant throughout and equals the original homogeneous velocity. As velocity varies, c varies in a somewhat velocity averaging sense given by the traveltime calculation (an integral of velocity). From equation (2), the traveltime can easily be extracted from the celerity representation within the prestack migration operation as (x x0 ) t(x, y, z) = 2 + (y y 0 ) 2 + (z z 0 ) 2. (3) c(x, y, z) The combination of the forward and inverse celerity transforms (equations (2) and (3)) provides the mechanism to represent the traveltime field in a compression-friendly domain in an efficient matter. In this domain, unlike in spherical coordinates, grid interpolation is not required in the transformation process. CHEBYSHEV POLYNOMIALS After establishing the domain of compression, we now choose the compression tool. Polynomial based transforms provide compressibility by exploiting the inherent smooth features of the traveltime field (integral of velocity) in its compression domain. A polynomial representation of functions with exceptional transform speed (fast Fourier transformation (FFT) speed) is Chebyshev polynomials (Gottlieb and Orszag 1977). The Chebyshev polynomial of degree n is denoted T n (x)and is given by T n (x) = cos(n arccos x). At first glance, this may look trigonometric, (similar to the Fourier cosine transform), however, the lowest order Chebyshev polynomials are given by T 0 (x) = 1, T 1 (x) = x,

4 4 T. Alkhalifah T 2 (x) = 2x 2 1, and the higher order polynomials by the following recurrence T n+1 = 2xT n (x) T n 1 (x) n 1. The Chebyshev polynomials are orthogonal in the interval [-1,1] over a weight (1 x 2 ) 1 2. Thus, we scale the actual range to fit this interval. Unlike cosine transforms, the second basis function of the Chebyshev polynomial is linear, which is convenient for traveltime field representation in homogeneous media. The coefficients of a Chebyshev polynomial representation decay faster than cosine transforms for smooth functions (Moin 2001) and since traveltimes and celerity velocities are calculated through integrating the medium velocity they are inherently smooth. In fact, Chebyshev polynomials can have exponential convergence in representing infinitely smooth functions (Press 2007). Chebyshev polynomial transformation has efficiency features similar to the Fourier-based cosine transform. The FFT speed for Chebyshev polynomials is obtained by interpolating nodes onto a sinusoidal grid and using proper scaling. Nevertheless, for highly complex media both Chebyshev polynomials and the cosine transform provide similar advantages. As suggested above, transform-based compression, like Chebyshev polynomials, perform better in continuous traveltime fields like those produced by the finite difference solution of the eikonal equation. Specifically, the continuous field allows us to avoid the artefacts associated Gibbs phenomenon that may occur at the discontinuities using truncated spectral representations, like those planned here. However, since these discontinuities represent artificial discontinuities due to choosing a certain arrival or branch (i.e., energetic arrival) from the ray-based traveltime field, the behavior of the traveltime field near them is not important and we can use this compression method even with ray-tracing traveltimes. resorting to bit-encoding (Cunha et al. 1995). Nevertheless, we use later bit encoding as a possible alternative or in addition to Chebyshev polynomial transformations for additional compression. Using the SEG/EAGE salt body model we compute the traveltime field via the fast marching method (Fig. 4). The complexity of the curvature of the contours increases below the salt body. A spherical coordinate transformation is then applied, followed by Chebyshev polynomial transformation of the traveltime field. Figure 5 shows the Chebyshev coefficients of significant value that are worth maintaining. There are clearly few compared to the size of the original field. If we keep only one tenth of the coefficients of the original field this gives a 10 to 1 compression. Using the saved coefficients to construct the traveltime field gives us the white contour curves in Fig. 4. They are not apparent because they predominantly coincide with the original traveltime field given in yellow. This implies that there is hardly any loss by excluding coefficients beyond the first one tenth of the smoothest coefficients. If we keep only one fiftieth of the significant coefficients (a fifty-to-one compression) we obtain the dark blue traveltime contours in Fig. 4; there are now differences in the contours, especially in areas of higher curvature. Specifically, the compression, in this case, tends to smooth the original traveltime field in areas of high curvature by keeping only the THE SPHERICAL ALGORITHM Now we combine the spherical coordinate implementation with the Chebyshev polynomial transformations to give coefficients that describe the traveltime field in a new domain. In this spherical domain, the resulting coefficients give a smooth representation of the traveltime field. The smoother the traveltime field, the faster the higher order coefficients disappear and thus the representation can be done with fewer coefficients; this is the essence of the lossy compression used here. Note, we are still using the same 32-bit representation before and after compression. Further compression is achievable by Figure 4 Contours of the traveltime field through the SEG/EAGE salt body model from a source on the surface located in the middle of the model. The green curves correspond to the original traveltime field of the model, the red curves (coincident with the green curve, and thus not observable) are the traveltime after a 10-1 compression and the blue curves correspond to the traveltime field after a 50-1 compression. The medium depicted by the plot ranges 13,500 m in the inline and crossline directions and 4100 m in depth here and throughout the paper.

5 Efficient traveltime compression 5 Figure 5 An isosurface plot of significant (in size) values of the Chebyshev coefficients representing the traveltime field through the SEG/EAGE salt body model from a source above the middle part of the model given in spherical coordinates. The field is represented by the same number of points as in the spherical space given by 61 samples for each of the F φ and F θ and 31 samples for F r. coefficients corresponding to the smoother Chebyshev functions. Representing the traveltime in spherical coordinates gives high compression levels with low errors. However, we can obtain even higher compression levels by using the more efficient and flexible celerity domain. THE CELERITY VERSION Though spherical coordinates are convenient to calculate waves emanating from a source, the interpolation required to represent traveltime in spherical coordinates induces accuracy and efficiency complications due to the uneven physical spacing. Such complications can be avoided using a celerity domain traveltime representation that has the curvature features of spherical coordinates but without the interpolation problems in the transform process. Combining the celerity representation with Chebyshev polynomial transformation provides a more accurate and efficient compression of the traveltime field, resulting in higher compression levels. Using this combination, we compress the traveltime field from the SEG/EAGE salt model and obtain coefficients of the Chebyshev polynomials of which we show only the significant values in Fig. 6. Clearly, they occupy a much smaller region than those associated with the spherical coordinates in Fig. 4. As a result we have an opportunity Figure 6 An isosurface plot of significant (in size) values the of the Chebyshev coefficients representing the traveltime field in celerity domain throughout the SEG/EAGE salt body model from a source above the middle part of the model given in celerity domain. The field is represented by the same number of points as in the physical space given by 68 samples for each of the F x and F y and 21 samples for F z. to obtain higher compression levels than those given by the spherical coordinates transform. Figure 7 shows a comparison between the original traveltime field and the celerity compressed field using a 10:1 Chebyshev polynomial compression level. Specifically, we keep only 10% of the coefficients corresponding to the smoothest functions of the transform. The two fields practically coincide and thus it is hard to distinguish between the two sets of curves. Specifically, there is hardly any loss of information with such a compression. On the other hand, a Figure 7 Contours of the traveltime field through the SEG/EAGE salt body model from a source on the surface located in the middle of the model. The yellow curves correspond to the original traveltime field or the model and the white curves (coincident with the yellow curve) are the traveltime after a 10:1 compression.

6 6 T. Alkhalifah celerity representation. In summary, the celerity representation is generally smoother with less curvature and thus less higher order variations. As a result, the combination of celerity domain representation and Chebyshev polynomial transformation provides an exceptional platform for traveltime compression. Such representation has additional features illustrated in the next three sections. THE INTERPOLATION ADVANTAGE Figure 8 Contours of the traveltime field through the SEG/EAGE salt body model from a source on the surface located in the middle of the model. The yellow curves correspond to the original traveltime field or the model and the white curves (coincident with the yellow curve) are the traveltime after a 125:1 compression. 125:1 compression level of the celerity transformed data results in observable differences in the traveltime field before and after compression (Fig. 8). Compression tends to smooth the traveltime field; however, in areas with low curvature the difference between the traveltime before and after compression is small even for such a high compression level. To explain the reason behind the high compression levels obtained in the celerity domain I compare, in Fig. 9, two isosurfaces: the yellow is the original traveltime field and the red is its celerity representation for the SEG/EAGE salt model. Clearly, the celerity representation is far smoother with less curvature than the traveltime field itself. This is further demonstrated in the contour curves comparison in Fig. 10 between the original traveltime field and its Using Chebyshev polynomial based compression, not only do we obtain the compression needed for prestack migration (the main focus of this paper) but we also obtain a platform for efficient and accurate traveltime interpolation. Since compression is applied to the traveltime field for each source location on a regular grid, interpolating for a point within such a grid is obtainable through bi-linear interpolation of the Chebyshev polynomial coefficients. Figure 11 demonstrates this concept as the traveltime transform coefficients from all four neighbouring points on the surface contribute to the desired location. This allows for efficient interpolation of fewer parameters taking advantage of the linear nature of the Chebyshev polynomial transforms. Interpolation of Chebyshev coefficients also results in more accurate construction of the traveltime field since such coefficients describe weights of continuous functions and thus, we can at least guarantee the continuity of the first and the second-order derivatives of the interpolated traveltime. This has an additional advantage for amplitude evaluation after traveltime extraction, as we see next. THE QUALITY ADVANTAGE Figure 9 Isosurfaces for the original traveltime field given in yellow (time = 3.5 s) and the celerity representation of it given in red. This is an extension to the interpolation features. First-order (and even higher order) finite difference traveltime calculation rarely provides traveltimes that are useful for amplitude calculation. Amplitude calculation, given for example by the transport equation, relies on second-order derivatives of traveltime with respect to position; a function that tends to be highly unstable and rarely usable for amplitudes when obtained using conventional finite difference solutions of the eikonal equation. Figure 12(a) shows a volume plot from the top of the curvature of the traveltime field for the SEG/EAGE salt model calculated using the second-order Fast marching method for solving the eikonal equation. The curvature is clearly not smooth enough to calculate the amplitudes needed for a Kirchhoff operator. Using such

7 Efficient traveltime compression 7 Figure 10 Contours of the original traveltime field given in dark blue and the celerity representation of it given in green. Also apparent is the salt body. THE SPEED ADVANTAGE Figure 11 A schematic plot depicting the process of traveltime interpolation for a source or receiver not laying exactly on the traveltime source grid. The blocks depicts a cube of the compression coefficients of the traveltime field. a function will certainly result in artefacts in the Kirchhoff implementation. However, the traveltime field after an 8:1 compression using the celerity domain to represent the traveltime field results in the traveltime curvature field shown in Fig. 12(b). This volume plot is much smoother than that shown in Fig. 12(a) for the original traveltime field. In fact, this field will result in fewer artefacts in describing an approximate amplitude distribution for the Kirchhoff operator. A key source for the smooth curvature is the function representation given by the cosine transforms as these functions are differentiable. One important consideration in the implementation of 3D prestack Kirchhoff migration is the speed of traveltime extraction. For each input trace we will need to extract the traveltime field for the source and receiver of that trace from the precomputed traveltime table. Generally, the source and receiver locations for the input trace will not lay on the regular grid of sources in the traveltime table. Thus, we will need to extract four neighboring sources for each source and receiver for a bi-linear interpolation and 16 for a cubic spline one. Of course, extraction speed is expected to increase for smaller size traveltime tables. Thus, for compressed tables the extraction speed exceeds that for uncompressed tables. Also, interpolating the compressed coefficients, due to their smaller range (compact), is cheaper than interpolating uncompressed traveltimes. The only additional cost added by the compression is the inverse Chebyshev polynomial and celerity transform. This cost is less than the other two factors given by the size of the traveltime field and the interpolation size. For the SEG/EAGE model, the cost of the extraction in Fig. 7 using the compression was half of that using conventional approaches. However, in many applications of 3D prestack migration, we combine the coarse grid traveltime field (to help with the memory limitation) with spline interpolation. Figure 13 (blue) shows the difference, given as an isosurface plot of 0.05 s, between the original traveltime and the one resulting from using half the number of traveltime samples in each direction, which results in an 8:1 compression of the traveltime table and cubic spline interpolation. Figure 13 (green) shows the difference between the original traveltime and the traveltime produced using Chebyshev

8 8 T. Alkhalifah Figure 12 A volume plot of the curvature of a) the traveltime field without any compression which is shown in Fig. 7 (yellow) and b) the traveltime field after celerity-based compression as that shown in Fig. 7 (white). Figure 13 A 3D plot displaying isosurfaces of errors in the traveltime larger than 0.05 s for the coarse grid plus spline interpolation at the image level (blue) and for Chebyshev plus celerity (green). A section of the contour plot of the traveltime for the two cases and the original traveltime is also plotted. It took about 1.4 seconds of CPU time on a PC Linux machine to extract the celerity-chebyshev traveltime. This is far less than the 5.6 s needed to obtain the coarser grid cubic spline interpolation result. Meanwhile, it takes only 2.25 s to extract the same traveltime over the coarse grid but with a linear interpolation. Thus, the cost of the spline interpolation relative to other extraction tasks is large. For the same case we applied a cubic spline source interpolation, instead of the image one used above, to the coarse grid traveltime. The errors are now less given by the blue isosurface of 0.05 s shown in Fig. 14. However, it took 6.2 s to extract this traveltime, which is even larger than the earlier spline interpolation. The combination of both spline interpolations in the program though should improve accuracy, which will result in a costly extraction not practical for imaging applications. In fact, one can easily envision the cost exceeding 20 s of extraction time. This time typically exceeds the cost of migration itself. compression and celerity domain with the same isosurface envelope of 0.05 s. The volume of the blue surface is larger than that of the green surface, which implies that errors are less for the Chebyshev plus celerity compression. The lower error is a result of the bi-linear source interpolation of the Chebyshev coefficients instead of the actual traveltime. The errors seem large at 0.05 s but when considering the dimensions of the problem with image grid spacing of 200 m and a source grid spacing of 500 m in x- and y-directions, the errors are reasonable. BIT ENCODING Further compression is attainable using bit encoding to represent numbers with fewer bits. Though bit encoding will not result in improved interpolation or amplitude calculation (features added by the use of Chebyshev polynomial or cosine compression), it will provide additional compression that might be needed at a low extraction cost. It will also, if applied to the Chebyshev polynomial (or cosine) coefficients, maintain all the advantages gained by the smoothing. The celerity representation of the traveltime field also provides a

9 Efficient traveltime compression 9 CONCLUSIONS Figure 14 A 3D plot displaying isosurfaces of errors in the traveltime larger than 0.05 s for the coarse grid plus spline interpolation for the source (blue) and for Chebyshev plus celerity (green). A section of the contour plot of the traveltime for the two cases and the original traveltime is also plotted. Using a combination of celerity representation of the traveltime field and Chebyshev polynomial (or cosine) based compression allows for high compression levels of traveltime from reasonably complex media with mild loss of information. We obtain reasonable compression levels that exceed 20:1 with minimal loss of accuracy, especially in areas of smooth and continuous traveltime fields. It also results in somewhat smoother traveltime (continuous first- and secondorder derivatives) fields beneficial for stable amplitude calculation. Furthermore, the reduction of the size of the traveltime table using the Chebyshev compression provides low extraction cost. Despite the complexity of the SEG/EAGE salt body model, this compression approach provided the desired compression levels and the additional features expected. ACKNOWLEDGEMENTS I thank KACST and KAUST for their financial and technical support. I also thank Bee Bednar for useful discussions as well as the anonymous reviewers, who have helped this paper immensely. REFERENCES Figure 15 Contours of the traveltime field though the SEG/EAGE salt body model from a source on the surface located in the middle of the model. The yellow curves correspond to the original traveltime field and the white curves (almost coincident with the yellow curve) are the traveltime after a 4:1 bit-encoding compression of the celerity transformed traveltime. useful platform for bit encoding. The celerity representation is more compact than the traveltime field; for example, the traveltime field in a homogeneous velocity medium is given by a single value in the celerity domain. Figure 15 shows a comparison between the traveltime field before and after compression but now with 4:1 bit encoding approach. There ise hardly any observable difference between the two fields. In fact, any difference in this case will have a random nature rather than a smoothing one as is the case with Chebyshev polynomials. Alde D., Fehler M., Hildebrand S., Huang L. and Sun H Determining the optimally smoothed slowness model for raytracing based migration using multiple-valued traveltime tables. 72 nd SEG meeting, Salt Lake City, Utah, USA, Expanded Abstract, Alkhalifah T. and Fomel S Implementing the fast marching eikonal solver: spherical versus cartesian coordinates. Geophysical Prospecting 63, Alkhalifah T Tau migration and velocity analysis: Theory and synthetic examples. Geophysics 68, Calandra H., Baina R., Hanitzsch C. and Rousseau J Improving 3D Kirchhoff prestack depth migration: Why not use regularization and multi-pathing? 71 st SEG meeting, San Antonio, Texas, USA, Expanded Abstracts, Cunha C.A., Pametta J., Romanelli A. and Pedrosa I Compression of traveltime tables for prestack depth migration. 65 th SEG meeting, Houston, Texas, USA, Expanded Abstracts, Gottlieb D. and Orszag S Numerical Analysis of Spectral Methods: Theory and Applications. SIAM. Moin P Fundamentals of Engineering Numerical Analysis. Cambridge University Press. Pica A Fast and accurate finite-difference solutions of the 3D eikonal equation parametrized in celerity. 67 th SEG meeting, Dallas, Texas, USA, Expanded Abstracts, Press W.H Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.