Mathematical Geology, Vol. 31, No. 6, 1999
Improvement of Fourier-Based Unconditional and Conditional Simulations for Band Limited Fractal (von Ka´rma´n) Statistical Models1 John A. Goff 2 and James W. Jennings, Jr.3 We evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation. The method is applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization (the von Ka´rma´n model). The simple Fourier unconditional simulation is conducted by Fourier transform of the amplitude spectrum model, sampled on a discrete grid, multiplied by a random phase spectrum. Although computationally efficient, this method failed to adequately match the intended statistical model at small scales because of sinc-function convolution. Attempts to alleviate this problem through the ‘‘covariance’’ method (computing the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function) created artifacts and spurious high wavenumber content. A modified Fourier method, consisting of pre-aliasing the wavenumber spectrum, satisfactorily remedies sinc smoothing. Conditional simulations using Fourier-based methods require several processing stages, including a smooth interpolation of the differential between conditioning data and an unconditional simulation. Although kriging is the ideal method for this step, it can take prohibitively long where the number of conditions is large. Here we develop a fast, approximate kriging methodology, consisting of coarse kriging followed by faster methods of interpolation. Though less accurate than full kriging, this fast kriging does not produce visually evident artifacts or adversely affect the a posteriori statistics of the Fourier conditional simulation. KEY WORDS: simulation, conditional simulation, fourier methods, band-limited fractal, variogram, fast kriging.
INTRODUCTION Field simulation techniques enjoy a diverse and substantial body of literature, based primarily in the oil and mineral exploration sciences. Fourier1
Received 15 May 1998; accepted 18 August 1998. University of Texas Institute for Geophysics, 4412 Spicewood Springs Rd., Bldg. 600, Austin, Texas 78759. e-mail:
[email protected] 3 University of Texas Bureau of Economic Geology, Austin, Texas 78759. e-mail: jenningsj@ mail.utexas.edu 2
627 0882-8121/99/0800-0627$16.00/1 1999 International Association for Mathematical Geology
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based methods for simulation of random fields are attractive owing to their efficiency of operation, but problematic because of difficulties associated with aliasing and other artifacts at high wavenumbers. Fourier-based methods can also incorporate conditioning information, but in a separate step requiring use of kriging interpolation. Where the number of conditioning data points is large, kriging interpolation can severely curtail the efficiency of this operation. In this paper we attempt to improve Fourier-based methods for unconditional and conditional simulation. Our primary goal is to establish the most efficient and accurate simulation method under circumstances that we expect to find in real-world applications. Such circumstances may include large grids, large numbers of conditioning criteria in awkward geometries, and anisotropic, multiscale statistical models that allow for neither a maximum correlation distance nor a spectral density that decays to zero prior to the Nyquist wavenumber. Efficiency is important for large grids or large numbers of conditioning criteria, especially in computational applications where many conditional simulations must be generated in order to empirically define the error space of predicted results (such as may be necessary for Monte Carlo, simulated annealing, or other numerical inversion methods) (e.g., King and others, 1995; Sen and Stoffa, 1995). Accuracy of simulation methodologies can be severely taxed by multiscale statistical models, which void common simplifying assumptions used to increase efficiency. We use an anisotropic, upper-band limited fractal model (the ‘‘von Ka´rma´n,’’ or ‘‘K-Bessel’’ model) to characterize the random heterogeneity of the simulated fields (e.g., Goff and Jordan, 1988). The von Ka´rma´n model has been successfully applied to a wide variety of geophysical fields, including turbulence variations (von Ka´rma´n, 1948), seafloor morphology (Goff and Jordan, 1988), ice draft morphology (Goff, 1995), and most widely, in crustal heterogeneity (e.g., Tatarski, 1961; Wu and Aki, 1985; Frankel and Clayton, 1986; Gee and Jordan, 1988; Fisk, Charrette, and McCartor, 1992; Holliger, Levander and Goff, 1993; Levander and others, 1994; Holliger and Levander, 1994; Goff and Levander, 1996; Holliger, 1996). Although relatively simple in its parameterization, this model is extremely diverse, encompassing a spectrum of random field properties—e.g., from very rough to very smooth, from pure fractal to pure white noise, from purely isotropic to highly anisotropic. The oft-used exponential model is a special case of the von Ka´rma´n model. Hence, we anticipate that the work presented here will have wide applicability. Although for example purposes we shall focus on a single parameter set, our methods have been tested against a wide variety of realizations of the von Ka´rma´n model. For this experiment we also assume a Gaussian probability distribution. However, application to continuous-distribution non-Gaussian fields can be accomplished through normal-score transformation (e.g.,
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Deutsch and Journel, 1992; Christakos, 1992), provided the transformed field is reasonably multi-Gaussian as well. This paper begins with a brief review of the standard Fourier method for generating unconditioned and conditioned simulations and the anisotropic von Ka´rma´n statistical model. We then present unconditional simulations, discuss shortcomings, and establish methods for optimization of accuracy and efficiency. Finally, we generate conditional simulations based on Fourier methods, and present a means of increasing efficiency of this procedure without undue sacrifice of accuracy through use of a ‘‘fast’’ kriging technique.
BACKGROUND Fourier-based methods for unconditioned and conditioned simulations are by no means new. A wide body of literature exists on the subject. For a comprehensive treatment we refer the reader to Christakos (1992). Here we shall summarize only briefly. In the following, we assume a grid geometry for field Z(x) where the position vector x is discretized on an m by n grid. The discrete Fourier transform-based methods require the grid to be uniform and rectangular, but not necessarily isotropic; the x and y grid spacings need not be the same. If prior conditions exist, we specify them ˆ (xi ), i 僆 1, 2, . . . , N. by Z
Fourier Transform Simulation An unconditioned Gaussian-distributed simulation Zu(x), with a second-order statistical model expressed as the amplitude spectrum A(k), can be generated by discrete Fourier transform (DFT) via: Zu(x) ⫽ DFT兵A(k) exp[i2앟(k)]其
(1)
where the wavenumber vector k is also discretized on an m by n grid, and , the phase value, is a uniformly distributed random number sampled on [0, 1). Hermetian symmetry is enforced on the product of amplitude and phase spectra (Bracewell, 1978). We refer to this straightforward Fourier simulation as the ‘‘spectrum’’ method. An example of a Fourier simulation, here for an anisotropic, pure fractal model, is presented in Figure 1. For spectral amplitude functions, such as the fractal model, which do not decay to zero, the simulation can be significantly affected by ‘‘sinc’’ smoothing; because the true spectral function being transformed in Eq. (1) consists of A(k) multiplied by a ‘‘box’’ function (1 inside ⫾Nyquist
Figure 1. A, Unconditional Fourier simulation generated using the simple spectral method [Eq. (1)]. B, Unconditional Fourier simulation generated using the ‘‘covariance’’ methodology as described in the text. The statistical model and random phase spectra are identical in each case. The model is specified by parameters H ⫽ 1, kn ⫽ 2.5 (length⫺1), ks ⫽ 0.5 (length⫺1), and v ⫽ 0.4 (fractal dimension D ⫽ 2.6). The scale parameters are small enough that this model is equivalent to a ‘‘pure fractal’’ model; in particular, the covariance function does not decay to near zero within the size of the grid, especially in the horizontal direction of this anisotropic model. Under such circumstances, the covariance methodology of Fourier simulation creates significant artifacts in the realization, seen in B as vertical striping.
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wavenumber, 0 elsewhere), the simulation actually consists of a field with correct spatial statistics convolved with the Fourier transform of the box function: the sinc function (Bracewell, 1978). Hence the simulation is smoother than it ought to be. One method often considered to circumvent sinc smoothing begins with the covariance model rather than the spectral model (e.g., PardoIgu´zquiza and Chica-Olmo, 1993, 1994). This algorithm, which we refer to as the ‘‘covariance’’ method, proceeds as follows: (1) specify the covariance model function on the discrete spatial grid, (2) compute the DFT to generate an aliased power spectrum on the discrete wavenumber grid, and then (3) take the square root to compute the aliased amplitude spectrum and use this in place of A(k) in Equation (1). Theoretically this should produce a simulation with the desired spatial statistics, i.e., the appropriate amount of aliasing in the wavenumber domain. However, the DFT of the discrete covariance field generates negative values. The algorithm can proceed simply by zeroing-out the negative values, but the resulting simulation can exhibit significant artifacts if the covariance function does not decay to zero within the size of the simulation grid (Fig. 1B). In the Unconditional Fourier Simulation section we will describe our successful attempts to alleviate sinc smoothing by ‘‘pre-aliasing’’ the amplitude spectrum. Conditional Fourier simulations (e.g., Journel and Huijbregtts, 1978; Fouquet, 1994) typically require kriging interpolations of the conditions as well as of an unconditional simulation sampled at the same locations. Additional efficiency can be garnered by combining these into a single interpolation of a difference field. Hence we utilize the following algorithm: Step 1: Compute unconditional simulation Zu(x) via the Fourier method. Step 2: Sample Zu(x) at locations xi (i 僆 1, 2, . . . , N), and compute ˆ (xi ) ⫺ Zu(xi ). ˆ (xi ) ⫽ Z differences ⌬Z ˆ (xi ) 씮 ⌬ZI (x). Here we note that Step 3: Perform interpolation of ⌬Z ˆ (xi ), the subtraction of two independent the covariance of ⌬Z random fields of identical covariance Czz( ), is simply 2Czz( ). Step 4: Conditional simulation Zc(x) ⫽ Zu(x) ⫹ ⌬ZI (x). In practice, kriging, while the ideal solution in step 3, can be computationally expensive where large numbers of conditioning criteria exist. In the Conditional Fourier Simulation section we shall discuss a computationally efficient, multistep algorithm for approximating the kriging interpolation. The Statistical Model The anisotropic von Ka´rma´n model (e.g., Goff and Jordan, 1988) is a characterization of the second-order (2-point) statistical properties of a
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spatial field. It provides a simple parameterization of basic physical properties observed in many natural systems: RMS variation, characteristic horizontal scales, orientation of dominant strike, and fractal dimension. The current work focuses on two dimensional fields, but neither the statistical model nor the simulation methodologies described above preclude extension to three dimensions. The von Ka´rma´n statistical model can be described as a ‘‘band limited’’ fractal, in that fields corresponding to this model exhibit fractal behavior at scales below a specified outer scale. Limiting the fractal behavior in this way allows for a finite variance to be defined. In the space domain, the von Ka´rma´n model is expressed through the covariance Czz( ) which, for a zero-mean, homogeneous field Z(x), is defined by Czz( ) ⫽ E[Z(x)Z(x ⫹ )]
(2)
where E[ ] is the expectation function. The anisotropic von Ka´rma´n covariance model is expressed as (Goff and Jordan, 1988) Czz(r( )) ⫽ H 2
21⫺v r( )vKv(r( )), ⌫(v)
0 ⱕ r( ) ⱕ 앝, v 僆 [0, 1]
(3)
where Kv is the modified Bessel function of order v, ⌫ is the gamma function, and H is the rms variation of Z. Azimuthal variation is determined within the lag function r( ), which in two dimensions can be expressed through the dimensionless ellipsoidal norm r( ) ⫽ [TQ ]1/2, Q ⫽ k 2neˆneˆnT ⫹ k 2s eˆseˆsT
(4)
where k 2n and k 2s , with dimensions (length)⫺2, are the ordered eigenvalues (k 2n ⱖ k 2s ) of the matrix Q, and eˆn and eˆs are its normalized eigenvectors. Because the eigenvectors are orthogonal, they depend on only one orientation parameter, which we choose to be the azimuth s of the eˆs eigenvector. Hence, five parameters determine the model: the rms H, horizontal scale parameters kn and ks , azimuth s , and the order parameter v, which is related to the fractal dimension for a two-dimensional field by D ⫽ 3 ⫺ v (Goff and Jordan, 1988). Estimation of these parameters from data sets can be accomplished routinely through inversion of the sample covariance function (e.g., Goff and Jordan, 1988; Goff, 1995; Goff and Levander, 1996). In the limit of infinite grid size and infinitesimal grid spacing, the power spectral form Pzz of the von Ka´rma´n model is expressed as a function of wavenumber vector k by
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Pzz(k) ⫽ 4앟vH 2兩Q兩⫺1/2[u 2(k) ⫹ 1]⫺(v⫹1) u(k) ⫽ [kTQ⫺1k]1/2
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(5) (6)
where Q is the same as in Eq. (4). The amplitude spectrum A(k) is computed simply by the square root of P(k). The nondimensional parameters of this model chosen for the synthetic experiment are H ⫽ 1.0 (z units), kn ⫽ 25 (length⫺1), ks ⫽ 5 (length⫺1), and v ⫽ 0.4 (D ⫽ 2.6 for a two-dimensional field), with the eˆs vector oriented along the row (horizontal) direction of the grid. Note that these scale parameters are an order of magnitude larger than those used for Figure 1, which is essentially a ‘‘pure fractal’’ model; for this test case we wish to adequately sample the correlation length in the column direction. In all the examples presented below, we simulate a field sampled on a 1001 by 1001 grid with dimensionless grid spacing of ⌬x ⫽ 0.001 in both the row and column directions.
UNCONDITIONAL FOURIER SIMULATION An unconditional Fourier simulation generated via the spectrum method is presented in Figure 2. The 1001 by 1001 grid is sampled from a 1024 by 1024 realization; cropping the edges effectively randomizes the amplitude of the sample power spectrum. Using a public domain Singleton FFT FORTRAN algorithm on a sun Ultra 1 computer, the run time was 21 s. A common assumption regarding the Fourier method is that it guarantees that the simulated field conforms to the desired second order statistical properties. Indeed, if we generate an uncropped Fourier simulation via the spectrum method [Equation (1)] and then estimate its two-dimensional power spectrum through (1) forward Fourier transformation and (2) computing the square of the spectral amplitude, the result conforms exactly to the functional form P(k) used to construct the simulation in the first place. However, as discussed above, in reality the space domain image is affected by sinc smoothing, which decreases power at high wavenumbers. Sinc smoothing in the space domain image is unavoidable when inverse transforming from a fractal spectrum, which has no upper wavenumber limit to nonzero spectral density. To ascertain the conformance of the simulation to the desired statistics, we compute the sample variogram in both the row and column directions and compare these to the respective one-dimensional model functions. The variogram 웂zz( ) is defined as the variance of the differences of two points separated by lag , and is related to the covariance by
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Figure 2. Unconditional Fourier simulation generated by simple Fourier transform of the von Ka´rma´n amplitude spectrum sampled on a discrete grid and multiplied by a random phase spectrum. Model parameters are given in the text.
웂zz(x) ⫽ 2(1 ⫺ Czz(x))
(7)
(note that the oft-used ‘‘semivariogram’’ is one half the full variogram). Sample variograms for row and column directions for the unconditional Fourier simulation (Fig. 2), averaged over all available rows and columns, respectively, are compared with model variograms in Figure 3. There are two primary factors in judging the variogram comparison. Over larger scales we check to see if the sample variogram increases at the same rate as the model variogram, and in cases where the characteristic scale is observable (such as for the column direction), we check to see if the sample variogram ‘‘sills’’ at approximately the same value as the model function. The error in a variogram estimated from a finite sample increases with increasing lag, and it is normal for the estimate to fluctuate about its expected value.
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Figure 3. Comparison of model (dashed lines) and sample variograms (solid) for A, row, and B, column directions of the unconditional Fourier simulation presented in Figure 2. The sample variograms for row and column directions represent the averages of variograms computed for all individual rows and columns, respectively.
Variogram error is also highly correlated (Priestly, 1981), so it is also normal for such fluctuations to be smooth in character. With these considerations in mind, the sample variograms in Figure 3 adequately match the model variogram at larger scales.
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The second primary factor in assessing the a posteriori statistical character of the simulation is the behavior of the sample variogram at small lag, especially right at the origin. In particular, we check to see if the slope of the sample variogram at zero lag, a primary indicator of the fractal dimension, matches that of the model variogram. Unlike the larger-scale portion of the variogram, there is very little error in the smallest lag values, especially where all row and all column variograms are averaged over a large grid. For this test case, we find no discernible variation in the sample variogram at the smallest scales from one realization to the next. The deleterious effects of sinc smoothing of the Fourier simulation are clearly evident at these smaller scales (see insets of Figure 3), where the sample variogram begins at a more gradual slope than the model variogram. This is a consistent feature for all Fourier simulations examined. The discrepancy is more noticeable on averaged one-dimensional row and column power spectra (not shown for brevity): the difference with the model power spectrum at the highest wavenumbers is as large as one order of magnitude (factoring in anticipated aliasing).
Corrections for Sinc Smoothing There are three possible routes to alleviating the effects of sinc smoothing. One is the ‘‘covariance’’ methodology (Pardo-Igu´zquiza and ChicaOlmo, 1993, 1994), described above in the Background section, which is expected to produce the appropriate amount of aliasing in the wavenumber domain prior to transformation to the spatial domain. This method works well (i.e., visually artifact free) only if the covariance function decays to near zero within the spatial constraints of the grid (Fig. 1B, which is generated from an essentially ‘‘pure fractal’’ model presents a counter example). However, for the parameter set we have chosen for our test case, the covariance function does decay reasonably close to zero within the grid dimension, enabling us to assess the accuracy of the covariance methodology in its applicable domain. Figure 4 presents an unconditional covariance method simulation using the same parameters and random phase spectrum (to enhance the visual comparison) as used for the spectrum method Fourier simulation (Fig. 2). This simulation was completed in 43 s on a Sun Ultra 1 computer. Visually we can identify no artifacts in the covariance method simulation, and the overall character is indistinguishable from that of the simple Fourier simulation. Differences are notable, however, in comparison of the sample and expected variograms (Fig. 5). Here we find that the covariance method appears to have overcorrected the sinc smoothing effect, yielding a sample variogram that is steeper at the origin than expected;
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Figure 4. Unconditional Fourier simulation generated by the ‘‘covariance’’ methodology. Here the amplitude spectrum is estimated by taking the square root of the discrete Fourier transform of the covariance function (zeroing out any negative values prior to taking square root). Model parameters are given in the text. Random phase spectrum is same as Figure 2.
i.e., there is more variability at the finest scales than the model calls for. This effect is more pronounced in the row direction. A second means of counteracting sinc smoothing is to increase the Nyquist wavenumber by decreasing ⌬x of the simulated grid, and then subsample at the desired grid spacing; this, in essence, will decrease the width of the sinc function being convolved with the spatial realization. We have experimented with simulation grids as large as 8008 by 2002 (it is most important to decrease grid spacing in the eˆn direction). While the a posteriori statistics of such fields subsampled to 1001 by 1001 represent a significant improvement relative to Figure 3, it is not a satisfactory solution due to the great decrease in computational efficiency required to generate simulations on grids much larger than required.
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Figure 5. Comparison of model (dashed lines) and sample variograms (solid) for A, row, and B, column directions of the unconditional Fourier simulation (‘‘covariance’’ method) presented in Figure 4. The sample variograms for row and column directions represent the averages of variograms computed for all individual rows and columns, respectively.
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The third possible cure to sinc smoothing is to ‘‘pre-alias’’ the amplitude spectrum directly in the wavenumber domain by folding in contributions from higher wavenumbers than the Nyquist wavenumber. In our first attempt at pre-aliasing, we perform additive folding on the power spectrum in the row, column, and diagonal directions, and then take the square root of the sum to compute the aliased amplitude spectrum. This procedure reduced sinc smoothing, but did not satisfactorily eliminate it (as evidenced by analysis of a posteriori statistics). Subsequent experimentation established that a simple additive constant to the amplitude spectrum (i.e., a white noise process), in addition to the pre-aliasing described above, was sufficient to remedy sinc smoothing almost entirely. This constant Ca , is predominantly dependent on only two factors: (1) the ratio between scale parameters in the row and column direction, kr and kc , respectively, and (2) the value of the amplitude spectrum at the Nyquist wavenumber along the row axis, A(uN , 0), if kr ⬍ kc , or along the column axis, A(0, vN ), if kc ⬍ kr . The row and column scale parameters, derived from Equation (4), are given by kr ⫽ 兹(kn cos(s ))2 ⫹ (ks sin(s ))2 kc ⫽ 兹(kn sin(s ))2 ⫹ (ks cos(s ))2
(8)
For the example considered here, s ⫽ 90⬚, so kr ⫽ ks and kc ⫽ kn . Our formulation for Ca , derived from trial-and-error experimentation over a wide range of von Ka´rma´n fields, is as follows: Ca ⬵
冦
0.25(kc /kr )A(uN , 0), kr ⬍ kc 0.25(kr /kc)A(0, vN ),
kc ⬍ kr
(9)
An example of a Fourier unconditional simulation using the prealiasing method described above is presented in Figure 6. The run time was 27 s on a Sun Ultra 1 computer. The simulation looks visually comparable to the Fourier simulation of Figure 3 (both simulations used identical random phase spectra to enhance the visual comparison), while its a posteriori variogram statistics (Fig. 7) are superior to those of the simple Fourier simulation (Fig. 3) and the covariance method Fourier simulation (Fig. 5) at small lag. The pre-aliasing procedure has been tested over a wide variety of parameters and grid spacings for the von Ka´rma´n model, including ‘‘pure fractal’’ configurations (grid size small enough that the corner wavenumbers are not evident, as in Fig. 1), and with different azimuths s , aspect ratios kn /ks , and fractal dimensions D. In all cases the pre-aliasing procedure worked exceptionally well, and we are confident that it is robust, at least
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Figure 6. Unconditional Fourier simulation generated after ‘‘pre-aliasing’’ the amplitude spectrum as described in text. Model parameters are given in text. Random phase spectrum is same as in Figure 2.
with respect to the von Ka´rma´n model (in more ‘‘well-behaved’’ models the procedure is unnecessary). In a final test of the a posteriori statistics of the unconditional modified Fourier simulation, we generated a suite of 20 simulations and computed the mean and standard deviation of the sample variance. The computed mean was 0.97; a slightly lower value than 1.0 as expected since we are not completely sampling the full 2-D covariance over the sample grid; in any field with nonzero correlation distance, variance increases with sample size, asymptotically approaching the input variance (essentially matching it when enough correlation distances are sampled). The standard deviation was computed to be 0.014, a value that is nearly an order of magnitude less than the expected value of 앑0.11 (the 1- error on the variance is estimated as a byproduct of the two-dimensional covariance inversion procedure of
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Figure 7. Comparison of model (dashed lines) and sample variograms (solid) for A, row, and B, column directions of the unconditional Fourier simulation (pre-aliasing method) presented in Figure 6. The sample variograms for row and column directions represent the averages of variograms computed for all individual rows and columns, respectively.
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Goff and Jordan, 1988). This illustrates one minor drawback of the Fourier method: by explicitly specifying the amplitude spectrum, the variance of the field (the area under the amplitude spectrum) is not subject to random variation. What little variation we do have is a result of cropping the edges. However, this is not a serious complication; if an appropriate random variance is required, one may simply modify H in Eq. (5) to be a random variable of appropriate mean and variance (e.g., Gutjahr, Bullard, and Hatch, 1997).
THE CONDITIONAL FOURIER SIMULATION We take as our conditioning criteria the leftmost and rightmost columns of the pre-aliasing method unconditional Fourier simulation presented in Figure 6, which we consider our most accurate realization of the model statistics. We shall refer to this as the ‘‘master’’ grid; in this way we know a priori what ‘‘reality’’ looks like, and thus we have a means of visually checking the performance of the conditional simulation. The conditioning geometry is intended to simulate a situation in which two wells are drilled and logged, and we wish to realistically model, say, the porosity structure of the intervening region. However, the approach described below is general and can easily accommodate other conditioning geometries. The critical step in the conditional Fourier simulation algorithm is to generate a kriged interpolation from the differential between the conditioning data and an unconditional simulation (step 3 as given above in the Background section). With the large number of conditioning information (2002 points), a complete kriging would require both an inversion of a matrix numbering 2002 by 2002 points and a very large multiloop summation at each interpolated grid point. This computation is not practical; we estimate a run time for this procedure of over 20 days on a Sun Ultra 1 computer. However, while kriging is the ideal solution for this step in the algorithm, a full kriging is much more rigorous than necessary. In regions far from data conditions, a coarse sampling of the conditions are sufficient to estimate the conditional expectation, whereas in regions near data conditions, a sampling of the nearest conditions is sufficient. Futhermore, kriging is a smooth interpolator, especially so far from any data constraints. As long as kriging is used to provide the general framework for the interpolation, we anticipate that less rigorous interpolation methods can be used to fill in the majority of the grid. We therefore propose an approximate kriging methodology, which we term ‘‘fast’’ kriging, whereby we first perform kriging on a subset of the full grid, and then use these points and the original data conditions as a new set of conditions for a simpler and faster
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smooth interpolation method; in essence to interpolate the coarse kriging interpolation. A final, local kriging step completes the interpolation in the vicinity of the original data conditions. For this implementation we choose as our Kriging subset a coarse grid sampled at a spacing of Ns ⌬x (the value of Ns will depend on efficiency requirements). This effectively divides the full grid into approximately n/Ns ⫻ m/Ns ‘‘cells’’ (Fig. 8). A subset of the original data conditions are chosen as constraints for the coarse kriging. For this implementation we chose every Nsth point of the conditioning columns. We follow the coarse
Figure 8. Schematic representation of the ‘‘fast’’ kriging approach. First, kriging is performed on a coarse grid subset of the full grid, which divides the grid into cells. Second, cells that do not contain or abut data conditions are bilinearly interpolated, using the kriged cell corners as constraints. Finally, the remaining cells are interpolated by kriging of both the original data conditions and the edges of adjacent cells interpolated earlier in the sequence.
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kriging with a two-step procedure for fast interpolation. First, in cells that do not contain or abut data conditions, we perform a bilinear interpolation (Press and others, 1986) of the four kriged corners of the cell (Fig. 8). Although this procedure produces minor ‘‘kinks’’ along cell edges that eventually get factored into the conditional simulation (step 4 of the conditional simulation algorithm), the a posteriori statistics for this example as demonstrated below do not appear to be adversely affected. However, for very low fractal dimension fields, a smoother interpolator such as a bicubic spline (Press and others, 1986) may be more desirable for this step. Over the remaining cells, i.e., those containing or bounded by data conditioning points, we perform a kriging interpolation using as constraints both original data conditions and edges of any adjacent cells interpolated earlier in the sequence (Fig. 8). Figure 9 displays a fast kriging interpolation of the conditioning col-
Figure 9. ‘‘Fast’’ kriging interpolation of the leftmost and rightmost columns of the unconditional modified Fourier simulation shown in Figure 6. See text for details.
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umns (the left and right edges of Figure 6). (This grid was not actually used in the conditioning algorithm; we show it here so that the results of the fast kriging algorithm can be compared with the conditioning data. Rather, a fast kriged difference grid was used as described in the Background section.) Here we use Ns ⫽ 10, dividing the grid into 10000 cells of equal size (note that, for visualization purposes, the schematic presentation in Figure 8 is for Ns ⫽ 100). The bilinear interpolation was then computed within the innermost 9800 cells. Finally, along the leftmost and rightmost 100 cells, we applied the local kriging as described above. Figure 10 displays our example of a conditional Fourier simulation, using the pre-aliased Fourier simulation for step 1 and the fast kriging interpolation for step 3. The run time for the full procedure was 113 s on a Sun Ultra 1 computer. Visually this simulation compares well to the master grid (Fig. 6); the left and right edges are identical to those of the
Figure 10. Conditional Fourier simulation (pre-aliasing method), using as conditions the leftmost and rightmost columns of the unconditional modified Fourier simulation shown in Figure 6.
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master grid, the overall character is similar, and there are no visually evident artifacts. Comparison between model and sample variograms (Fig. 11) indicate no degradation in the a posteriori statistical character of the Fourier conditional simulation relative to the master grid. We surmise, therefore, that our fast approximation to kriging (note that there are certainly other possible methods that can be invoked) did not have a significant detrimental affect on the appearance or statistical accuracy of the conditional Fourier simulation. As with the pre-alias method unconditional Fourier simulation, we generated 20 conditional simulations, all using the same conditions, to examine the distribution of the variance. The mean and RMS of the sample variance for these samples were 0.96 and 0.09, respectively. Both values are close to expectation, despite the fact that the unconditional Fourier simulations had an RMS variance of only 0.014. Hence, a fortunate if unintended side benefit of the conditioning algorithm appears to be the better randomization of the sample variance.
CONCLUSIONS In this paper we have evaluated the performance and statistical accuracy of a common method for unconditional and conditional simulation: discrete Fourier transformation of the wavenumber spectrum. Although we restricted ourselves in this application to two-dimensional fields with a Gaussian probability distribution, application to three-dimensional and non-Gaussian fields is straightforward, but requires calibration of prealiasing. We used an anisotropic von Ka´rma´n statistical model, which is an upper band limited fractal characterization. Although it has limited parameterization, this model covers a large range of random field behavior, and has found wide applicability in geophysics. The von Ka´rma´n model, as with all fractal models, is not ‘‘well behaved’’ in that the covariance does not decay to zero within the space of the grid, and the power spectrum does not decay to zero prior to the Nyquist wavenumber. The simple unconditional Fourier simulation fails to adequately match the intended statistical model at high wavenumbers due to sinc smoothing (convolution with the Fourier transform of the ‘‘box’’ function, which limits the spectrum). One possible remedy for sinc smoothing is the ‘‘covariance’’ method of Fourier simulation; i.e., estimating the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function (zeroing out any negative values prior to taking the square root). However, where the covariance function does not decay to near zero within the space of the sample grid, serious artifacts can result. Even where artifacts
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Figure 11. Comparison of model (dashed lines) and sample variograms (solid) for A, row, and B, column directions of the conditional Fourier simulation presented in Figure 9. The sample variograms for row and column directions represent the averages of variograms computed for all individual rows and columns, respectively.
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are not evident visually, the a posteriori statistics indicate greater variability at small scales than the model allows. We formulate a modified Fourier method, consisting of prealiasing the wavenumber spectrum, that satisfactorily remedies sinc smoothing. A critical step in the Fourier conditional simulation algorithm is to perform a smooth interpolation of the difference between conditioning data and an unconditional simulation. Kriging is the ideal solution for this step, but the kriging computation can be prohibitively time-consuming where there are abundant conditioning data. We have devised a computationally efficient approximation to kriging (termed ‘‘fast’’ kriging), a multistep procedure consisting, in essence, of simple interpolation of a coarse kriging interpolation. First, a kriging interpolation of the conditioning data is performed on a coarse grid, separating the full grid into as-yet uninterpolated cells. Next, over those cells that neither contain nor abut conditioning data, a bilinear interpolation of the kriged points at the four corners of the cell is performed. Finally, over the remaining cells, at local kriging is performed using proximal conditioning data and previously interpolated points (both coarse kriged and bilinearly interpolated). The resulting conditional simulation is visually artifact free, conforms properly to the conditioning data, and satisfactorily matches the desired statistical model.
ACKNOWLEDGMENTS We appreciate the constructive comments of two anonymous referees on an earlier draft. This work was supported by ONR grant N00014-95-10067 and AFOSR grant F49620-94-1-0100 through Rice University subcontract R11800-42300094. UTIG contribution number 1397.
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