Its been shown that the statistical properties of a reflected seismic wavefield can be related to the statistical properties of the crust. In this paper, we report on a method to invert a reflected seismic wavefield for vertical stochastic parameters. The method is founded on a deconvolution, thresholding, and numerical integration procedure to estimate the Earth's bi-modal velocity perturbation. We then fit a von Kármán autocorrelation function to the autocorrelation of the estimated velocity perturbation and record the von Kármán parameters which result in the minimum misfit. Tests show that our scheme is successful at recovering the vertical characteristic length in synthetic seismic reflection data.
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 Geologic structures within the Earth's crust span at least seven orders of magnitude in scale. A deterministic mapping of such a wide distribution of scale lengths is often impossible and thus, geologists often rely on statistical descriptions in order to quantify structural elements of interest. A useful form of stochastic representation of the crystalline crust is the von Kármán autocorrelation model (see Goff and Jordan  for details), which is parameterized by a dimensional characteristic length, the Hurst exponent, and an RMS velocity fluctuation. The success of this stochastic approach to characterizing the crust is evidenced by the similarity of synthetic seismograms to recorded field data [e.g., Levander et al., 1994; Holliger and Levander, 1992, 1993], and the explanation of scattered seismic phases (e.g., Pn phases) as arising from this type of lower crustal structure [Nielsen et al., 2003].
 Several workers have made progress on the problem of estimating the horizontal stochastic parameters of a stochastic velocity model from seismic reflection data [Pullammanappallil et al., 1997; Hurich and Kocurko, 1999; Bean et al., 1999]. However, we know of no attempts at estimating the vertical stochastic parameters from data. In this paper, we present a simple method to estimate the vertical characteristic length parameter from seismic reflection data. We present results that indicate that estimating the vertical characteristic scale from data produced by bi-modal velocity models is feasible.
2. Estimating the Vertical Characteristic Length
 The method we present is founded on three assumptions. First, we assume that the Earth's stochastic velocity distribution is bi-modal. Although a bi-modal velocity distribution is not geologically realistic everywhere, it faithfully represents a number of lower crustal exposures. Secondly, we assume that the autocorrelation of the velocity field can be modeled by a von Kármán autocorrelation function. These first two assumptions are corroborated by field data of various exposed crustal sections [e.g., Holliger and Levander, 1992, 1993]. Finally, we assume that the seismic wavefield is only weakly scattered. A singly scattered wavefield is a primaries-only wavefield, commonly known as a primary reflection section. While not entirely accurate, the single scattering approximation is a common assumption for most seismic imaging schemes and one which we argue is valid for the scattering regimes in which we are concerned (see Flatte et al.  for more details on scattering regimes).
 We model seismic reflection data d(t) as a convolution of a source wavelet w(t) with the Earth's reflectivity r(t), which we can relate to the velocity structure by
where is the background velocity field and δv(t) is the (small amplitude) stochastic velocity perturbation [Pullammanappallil et al., 1997]. We can model the stochastic portion of the velocity structure for some parts of the Earth's crystalline crust statistically via a von Kármán autocorrelation function which is parameterized by a directional characteristic length a and the Hurst exponent ν which quantifies roughness [Goff and Jordan, 1988]. In the interest of brevity, we refer the reader to Goff and Jordan  and Holliger and Levander  for the specifics of von Kármán modeling. Note that we assume a bi-modal velocity distribution, and therefore νd ≈ 0.5 νc where νd and νc denote the Hurst exponent for discrete and continuous velocity models, respectively [Goff et al., 1994].
 The first step in our approach is to convert our data to depth according to the background velocity field and then apply a long-window (≈1.0 sec) automatic gain control (AGC) in order to normalize the trace amplitudes. We then employ spiking deconvolution [Ziolkowski, 1984; Yilmaz, 1987] to obtain an estimate of the reflectivity (z) by removing the filtering effects of the band-limited source wavelet. We then apply a thresholding operator to the amplitudes of (z). The thresholding operation effectively ignores small amplitude multiple and sidelobe energy, both of which we assume to have lower amplitudes than primary arrivals. The thresholding operation takes the form
where RT is some arbitrary fraction of the maximum amplitude of the reflectivity estimated by the spiking deconvolution, (z) (typically, fifty five to sixty five percent). An additional reason to impose this thresholding condition is to force bi-modality to the recovered impedance model. Finally, to obtain an estimate of the stochastic velocity function, we numerically integrate (z);
where (z) is our estimate of the stochastic velocity perturbation model such that (z) ≈ δv(z).
 To estimate the von Kármán parameters, we compute the autocorrelation of (z) in the z direction and then fit this to a theoretical von Kármán autocorrelation function, where az and ν are free parameters. Where the RMS misfit between the data and the theoretical von Kármán is a minimum, we record az and ν. However, during our numerical tests, we observed that the parameter ν is poorly constrained by our method. Letting ν vary as a free parameter produced estimates that were generally overestimated, an effect also noted by others when inverting for horizontal statistics [e.g., Pullammanappallil, 1997]. However, we observed that allowing ν to vary did not significantly alter our estimates of az when we kept ν fixed.
 Because there exists a trade-off between ν and az in the von Kármán formulation, we would expect that the estimated values of az would be effected by erroneous values of ν. However, when we allowed ν to vary between geologically realistic values (typically 0.1 to 0.5) the variation of the estimated value of az was only a few percent. Therefore, in order to limit the number of free parameters in the inversion, we fixed ν to 0.3, as the parameter az appears to be a much more robust parameter.
3. Numerical Tests and Results
 In the first test (Figure 1) we demonstrate the feasibility of estimating an impedance model from seismic data, from which az is in turn recovered. For this proof-of-concept test, we generated a 1-D, bi-modal, stochastic velocity model with az = 350 m and ν = 0.3. We then generated a synthetic trace by convolving a zero phase Ricker wavelet (fc = 10 Hz) [Ricker, 1940, 1953] with the reflectivity of the velocity model. We then used the method outlined above to recover (z) from the synthetic data.
 For more realistic tests, we employ a finite difference solution to the acoustic wave equation (second order time, fourth order space) to generate synthetic data through a stochastic velocity model (Figure 2). The model had a total size of 30 km wide by 20 km deep, with a grid spacing of 25 m. The top 5 km of the model was a homogeneous 6 km/sec. The stochastic portion of the model (5 km–20 km depth) was characterized by a bi-modal (velocities of 6.0 and 6.3 km/sec), discrete, von Kármán model with stochastic parameters of ax = 1000 m, az = 350 m, and ν = 0.3.
 For the first test using finite difference data, we computed the response of a single shot, which was located on the surface of the model at x = 15 km. The source was a zero phase Ricker wavelet (fc ≈ 10 Hz) and its reflections were recorded at the surface by 160 receivers located 25 m apart, in a split-spread configuration. For this simulation, we estimated az for multiple data windows which were 5 km in depth and 500 m in width (Figure 3, Table 1). Specifically, we applied the deconvolution/thresholding/integrating procedure to the entire data set, and then for each data window we compute an average vertical autocorrelation function by averaging all of the vertical autocorrelation functions within that data window. We then fit that resulting averaged autocorrelation function to a theoretical von Kármán function, to obtain an averaged estimate of az for that given data window. The window is then shifted in location 250 m horizontally and/or 1 km vertically and the procedure is repeated.
 For the single-shot simulation, we observed that when we use the known wavelet to deconvolve the seismic data, the resulting average estimates of az are higher than when we estimate the source wavelet directly from the data (Figure 3, Table 1). We suspect that the reason for this is that the bandwidth of the estimated source wavelet is not as broad as that of the original wavelet, leading to deconvolution errors. Presumably, when only a portion of the bandwidth of the source wavelet is recovered for use in the deconvolution operator, or when the data are contaminated by uncorrelated noise, the estimate of the reflectivity is degraded, leading to less robust estimates of az.
 In the last synthetic test, we simulate a full 2-D seismic survey through the same model as described for the previous test, in order to test our ability to recover az with a typical stacked data set (Figure 4). The survey consisted of 200 shots, with a shot increment of 100 m, with a ±4 km offset range (all the offsets were used in the final stack). The source was a zero phase Ricker wavelet (fc ≈ 20 Hz). After processing (AGC, dip-moveout, NMO, stack, poststack migration) the data were deconvolved and thresholded at sixty percent of the maximum amplitude and then inverted for az, with a fixed value of ν = 0.3. To compute az, we used the same sliding window procedure as with the single-shot test, except that the window size was 5 km square and moved in steps of 1 km horizontally and vertically. There are three key observations: 1) the recovered values of az closely match that of the input model, 2) the estimated value of az for stacked and imaged data isn't as variable as for the single shot test, and 3) when we use the known wavelet for the deconvolution the recovered values of az are not as variable as when we estimate the wavelet directly from the data.
4. Discussion and Summary
 In order for the method that we present here to work effectively, we introduce two conditions. The first is inherent in the algorithm: that is, the assumption of a bi-modal velocity model. For data reflected from a poly-modal, or a continuous velocity field, the method presented here is not appropriate. The second condition that we impose is more subtle and not as severe. Namely, we fix the parameter ν, as we found that estimating this parameter with our method was highly error-prone. Regardless, we found that at the highest and lowest ranges of ν, the magnitude of the estimated value of az was not effected by more than a few percent. It was for this reason that we simply fixed the parameter ν in the inversion. For both of the inherent limitations in our method, it would be most helpful to have a-priori information available from, for example, core samples or outcrop information. In such a case where such information is available, the assumption of bi-modality and/or a fixed estimate of the parameter ν can be made with more confidence in regards to extrapolating these assumptions to a given crustal data set.
 By enacting the thresholding/integration procedure, we artificially broaden the spectrum of the deconvolved trace by ignoring small amplitude side-lobe energy. The result is that we artificially introduce high and low frequencies to the data. However, our method of fitting the autocorrelation function to a theoretical von Kármán function is most sensitive to the corner wavenumber between the semi-white portion of the autocorrelation function and its roll-off portion. Because the corner wavenumber tends to be in the middle of the autocorrelation function's power spectrum we argue that artificially broadening the power spectrum of the data will not have a major impact on the estimate of az. However, the artificial broadening of the data's power spectrum is likely the reason for the erroneous estimates of the parameter ν, which is quite sensitive to the higher wavenumber spectral components.
 An important aspect of this analysis method is its dependence on the thresholding value. Because the data has an AGC operator applied to it (and is thus effectively normalized) increasing RT to a value greater than about sixty-five percent will effectively ignore small-amplitude/high-frequency fluctuations in . Thus, for excessively high values of RT, the estimated value of az becomes erroneously large. The inverse is true of excessively small values of RT; that is, estimated values of az become erroneously small. Although the effect of the magnitude of RT on the final az estimate is rather non-linear, there appears to be a flat spot in the estimate of az as a function of RT for the range of RT between fifty-five and sixty-five percent. Thus we claim that the estimated value of az at these thresholding values are relatively robust.
 For the full survey data, we observe that the estimated value of az increases with depth, likely a result of the illuminating wavelet changing shape (i.e., phase) with depth. Because we estimate the wavelet by taking the spectrum of the entire trace, we are obtaining an average wavelet for the entire trace. This neglects the (likely) changing wavelet shape as it propagates through the velocity model. To test this hypothesis more thoroughly would involve wavelet estimation occurring in discrete depth windows, and is beyond the scope of this paper.
 Additionally, we note that the tests presented here are for noise free data. Its not clear how noise, the noise spectrum, or the signal-to-noise ratio would effect the estimate of az. However, given that its well known that colored noise can degrade the results of deconvolution, its expected that noise would degrade the accuracy of the az estimates. Indeed, the effect of noise is the subject of future research.
 We thank William W. Symes for interesting ideas and thoughtful reviews of this manuscript. We graciously acknowledge the anonymous reviewers of this manuscript. This work was supported by NSF EAR-0222270.