Fractal Inversion

Chapter 3

Fractal Inversion 3.1. INTRODUCTION Inversion of seismic data plays a vital role in reservoir characterization. Highresolution inversion methods provide models with higher resolution than those built from the conventional stacked seismic traces (e.g., Rowbotham et al., 2003; Sen, 2006). Such models are good for reservoir characterization and model building. Edited well logs provide the most accurate and the best vertical resolution of geophysical information of a subsurface reservoir. Though vertical resolution of well logs is very good, horizontal resolution is poor because of sparse availability and the small extent of lateral depth of investigation. One approach to obtain a high-resolution three-dimensional (3D) description of a reservoir is to use geostatistical interpolation, extrapolation, or simulation (Hass and Dubrule, 1994) using available well-log data at a few sparse locations. Accuracy of such models is dictated by the number and spatial distribution of the wells over a reservoir. Therefore, with seismic data being the most continuous information available (although at a lower vertical resolution), a stochastic inversion of seismic data that integrates seismic and well-log data can add great value in reservoir characterization (Francis, 2006a) as it combines seismic data with well logs and uses vertical resolution from well logs and good horizontal resolution from seismic data into the estimated model. The fusion of seismic and well-log data is possible using seismic inversion, which converts seismic information into petrophysical properties, such as acoustic impedance and shear impedance (e.g., Dimri, 1992; Russell and Hampson, 2006; Srivastava and Sen, 2010; Vedanti and Sen, 2009). Merging seismic data directly with the log data is difficult because they have a different range of scale/frequency of measurement compared to well logs and also their recording is basically in a different domain, viz., seismic data in the time domain, whereas logs are recorded with depth. Seismic inversion allows estimation of several attributes, viz., acoustic impedance, density, Poisson ratio, net pay thickness, and porosity (Francis, 2006a,b; Torres-Verdin et al., 1999). Due to the limited frequency band of the seismic data, the inversion estimates are significantly below the resolution

Handbook of Geophysical Exploration: Seismic Exploration, Vol. 41. DOI: 10.1016/B978-0-08-045158-9.00003-8 # 2012 Elsevier Ltd. All rights reserved.

47

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Fractal Models in Exploration Geophysics

desired by reservoir engineers. Regularization methods are employed in deterministic inversion algorithms (Vedanti et al., 2005) to address some of these issues, but even with regularization such as sparsity constraints, blockiness, and maximum entropy constraints, inversion results do not match the frequency band of the well logs very well (Oldenburg et al., 1983; Rietsch, 1988; Russell and Hampson, 1991). The very low- and high-frequency components of the model reside in the null space, which remains unconstrained by the input of seismic data. One approach to overcome the resolution problem of deterministic estimates is to include the high- and low-frequency information from the well log in an inversion algorithm. The inversion methods that combine probabilistic broadband information are often termed stochastic methods. In a stochastic method, several model estimates are derived, all of which represent possible models of the subsurface. As low- and high-frequency components of the model space are not constrained by the seismic data, stochastic methods try to achieve the best possible model of the model null space. A starting random initial model that contains all the frequencies can leave unwanted high and low frequency information not constrained by the seismic data in the model space. We have applied an alternative approach to generate a realistic initial model from a fractional Gaussian distribution derived from well logs. This approach provides a natural way of generating suitable initial models that is used in the optimization module. The probability density function (pdf) of the fractional Gaussian noise (fGn) is a generalization of the Gaussian noise and represents variations in subsurface properties recorded in well logs better than any other distribution (Hardy, 1992; Hewett, 1986; Painter et al., 1995). The fGn is characterized by the power law (scaling) behavior of its power spectrum, variogram, and covariance with the Hurst coefficient as an exponent. Unlike a Gaussian distribution that is described by mean and variance only, a fractional Gaussian pdf is described by three parameters: mean, variance, and Hurst coefficient, which we estimate from available well logs. The initial model generated in this way facilitates consistent high-resolution estimates of the model parameters in the same frequency range similar to the given well log. Thus, such an initial model reduces the probability of having spurious high-frequency estimates beyond the frequency band of the known log data. We have developed a stochastic inversion method that makes use of the initial model generated by the fractional Gaussian process (fGp) in very fast simulated annealing (VFSA), one of the global optimization methods. Thus, our approach constrains the solution to honor well-log statistics that matches the seismic response. Unlike deterministic methods, our method does not require the addition of a low-frequency model in the inversion result as it is derived from the fractal-based initial model itself in our algorithm. The application of our method is demonstrated on a real two-dimensional (2D) seismic data for estimation of P- and S-impedances.

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Fractal Inversion

49

3.2. SEISMIC INVERSION Prestack seismic inversion uses amplitude as a function of offset (angle) to estimate simultaneously the P-impedance, S-impedance, and density at each common midpoint (CMP) location. It is usually termed simultaneous inversion as it inverts for several parameters simultaneously using several common angle traces. A typical approach is to generate normal-move-out corrected gathers, which are converted from offset to angle domain using sonic log or root mean square (RMS) velocities. The data are converted to angle domain as it is convenient for the formulation of the forward problem. A basic flowchart describing fractal-based stochastic inversion algorithm is shown in Figure 3.1. The inversion algorithm needs the following steps to be performed: (1). Background model generation: We follow the standard procedure of interpolation and extrapolation of well logs through the entire seismic line or volume using interpreter-picked horizons. Thus, at each CMP location we have a interpolated well log. Each interpolated broadband log is used to compute statistics to be used in the inversion. Further, a low-frequency model is derived using logs that are filtered to a low frequency (typically below 8 Hz) and a user-supplied bound is computed around these smooth bounds. The optimization algorithm is restricted to search within these bounds. (2). Wavelet estimation: Wavelets at each angle either from a well log or using seismic data by statistical method are derived at the well locations. As shown in the flowchart (Figure 3.1), a realization of zp, zs and density is drawn from a fGp and is used as a starting model. Synthetic angle gathers are computed by convolving reflectivity derived from this model and the wavelets. They are compared against the recorded data and updated using an update rule in VFSA until convergence is reached. Different steps of our algorithm are described in detail below.

3.2.1

Forward Modeling

In a seismic experiment for an incident plane P wave, various phases of useful waves are generated, for instance, reflected and refracted P and S waves. The amplitude of various phases can be computed using the Zoeppritz equation (Zoeppritz, 1919). Linear approximation of the Zoeppritz equation (Aki and Richards, 1980; Bortfeld, 1961) provides a simple expression for PP reflection coefficient (Rpp) by assuming isotropic, homogeneous, and elastic layers with welded contact and small contrast in material properties. This computes amplitude as a function of angle of incidence, which forms the basis of amplitude variation with offset (AVO) and amplitude variation with angle. Here, we use linear approximation of the Zoeppritz equation rearranged by Fatti et al. (1994) in terms of zero-offset

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Fractal Models in Exploration Geophysics

Use well log and seismic data to perform well tie at each well location and extract a wavelet

Pick horizons in seismic data (Normal move out (NMO)-corrected angle gathers) in the zone of interest

Interpolate the well logs (Zp, Zs, density) between the picked horizons corresponding to each Common mid point (CMP) gather

Pick an interpolated log, compute the mean, variance, and Hurst coefficient. Generate fractal-based initial models for Zp, Zs, and density

Select a CMP gather, run the Very fast simulated annealing (VFSA) algorithm, which uses fractal-based initial model for forward modeling

Inverted acoustic and shear impedance

FIGURE 3.1 Flowchart showing the steps of a stochastic algorithm.

P-wave reflectivity (Rp), zero-offset S-wave reflectivity (Rs), and density (r) for total representation of an elastic earth. The equation is given as:

"   # " #  2
 Vs 2 2 1 2 Vs 2 2 Rpp ðyÞ ¼ 1 þ tan y Rp  8 sin y Rs  tan y  2 sin y Rd ; ð3:1Þ Vp Vp 2

where

Chapter

3

Fractal Inversion

      1 DVp Dr 1 DVs Dr Dr ; Rs ¼ ; and Rd ¼ : Rp ¼ þ þ 2 Vp r 2 Vs r r

51

ð3:2Þ

In the above equations, Vp and DVp are the P-wave velocity and change in P-wave velocity across an interface, respectively; Vs and DVs are the S-wave velocity and change in S-wave velocity, respectively; r and Dr are the density and change in density across an interface, respectively; y is the angle of incidence. To invert for P-impedance (Zp ¼ rVp) and S-impedance (Zs ¼ rVs), we combine the approach given by Ma (2002) and Russell and Hampson (2006). This leads to the following set of equations, which allow to invert for Zp and Zs: Lp ðiÞ ¼

 1
1 lnZp ði þ 1Þ  lnZp ðiÞ ; Ls ðiÞ ¼ ½ lnZs ði þ 1Þ  lnZs ðiÞ; 2 2

ð3:3Þ

where ln represents the natural logarithm and Lp and Ls are the logarithmic differences of the P- and S-impedance values. According to the logarithmic approximation of reflectivity (assuming reflectivity  0.5), Lp and Ls represent P- and S-wave reflectivity, respectively. Now, replacing the Rp, Rs in Equation (3.1) by Lp, Ls, respectively, and Vs/Vp ratio by Zs/Zp, we get the final expression only in terms of Zp, Zs, and r parameters, given by: "   # " #  2
 Zs 2 2 1 2 Zs 2 2 sin y Ls  tan y  2 sin y Rd : Rpp ðyÞ ¼ 1 þ tan y Lp  8 Zp Zp 2 ð3:4Þ Equation (3.4) is an approximation of the Zoeppritz equation and valid only within small angles. It is also assumed that the variation in Vp and Vs is not very large in successive layers. Further, this approximation assumes a horizontal layer earth model. Convolution of Rpp (y) with angle-dependent wavelets produces the required synthetic seismograms.

3.3. GENERATION OF FRACTAL-BASED INITIAL MODEL Analysis of the well logs shows that the power spectrum, variogram, and covariance of the well logs follow power law (scaling) behavior with a scaling exponent defined in terms of a so-called Hurst coefficient (Dimri, 2000, 2005; Emanual et al., 1987; Hardy, 1992; Hewett, 1986). Similarly, a reflectivity sequence also follows the fractal behavior unlike the common assumption of whiteness (Saggaf and Robinson, 2000; Saggaf and Toksoz, 1999; Todoeschuck and Jensen, 1988; Toverud et al., 2001). Fractal characteristics of well-log data and the seismic reflectivity sequence motivated us to generate a trace-by-trace one-dimensional (1D) realization of the model parameters as an input to our

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stochastic inversion algorithm. Synthesis of fractal time series is discussed in detail, for instance, in Davies and Harte (1987), Peitgen and Saupe (1988), Percival and Walden (1993), and Caccia et al. (1997). We closely follow the method of Caccia et al. (1997) to generate exact fGn for 1D time series using fGp. The fGp algorithm to generate fGn is already given in detail in Chapter 1 (Section 1.9).

3.3.1 Computation of Hurst Coefficient One of the important parameters used to generate a fractal-based signal is the Hurst coefficient. There are several methods available for estimation of the Hurst coefficient (Caccia et al., 1997; Chamoli et al., 2007; Turcotte, 1997). It is known that if the input signal has a sufficient number of samples, then rescaled range (R/S) analysis works well. In this paper, we have used R/S analysis to compute the Hurst coefficient. Hurst et al. (1965) found empirically that many data sets satisfy in nature the power law relation:  H R N ¼ ; ð3:5Þ S 2 where H is known as the Hurst coefficient, N is the number of data points, R and S stand for the range and standard deviation obtained from the R/S analysis. The R/S analysis is easily extended to discrete time series. The running sum of the series relative to its mean is: yn ¼

n X

ðyi  yn Þ; n ¼ 2; N  1:

ð3:6Þ

i¼1

The running sum of the series is computed for several segments of the given time series relative to its mean. The running sum range is defined by: RN ¼ ðyn Þ max  ðyn Þ min ;

ð3:7Þ

with standard deviation of the same denoted by: SN ¼ sN :

ð3:8Þ

The plot of log(R/S) versus log(N/2) is a straight line whose slope gives value of the Hurst coefficient H. To assess the uncertainty in estimation of the Hurst coefficient, 100 realizations of a synthetic fGn series with H ¼ 0.85 were generated using Equations (1.18–1.21, Chapter 1). Further, using Equations (3.5)–(3.8), the Hurst coefficient of these realizations is estimated (Figure 3.2). The value of the Hurst coefficient used to generate the 100 realizations of fGn is plotted with a continuous line (H ¼ 0.85) and estimates of the H are

Chapter

3

53

Fractal Inversion

1 0.95

Value of H

0.9 0.85 0.8 0.75 0.7 0.65

0

10

20

30

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Number of runs FIGURE 3.2 Uncertainty analysis for estimation of the Hurst coefficient. An fGn series with H ¼ 0.85 was generated 100 times and the estimated Hurst coefficient is shown with dots for each run.

shown by the dots. It was demonstrated by Caccia et al. (1997) that R/S analysis underestimates the value of H for time series of H > 0.7 and overestimates the value of H for time series of H < 0.7. As we are interested in a persistent time series (H > 0.5), the uncertainty in H estimates using R/S analysis is not a big problem. We find that in our tests even if H ¼ 0.55, the estimated values are always > 0.5; thus, it does not change the nature of our time series. Usually, well logs have H values in the range of 0.75–0.99.

3.4. VERY FAST SIMULATED ANNEALING VFSA is a modified form of simulated annealing (SA)—a global optimization method used in many geophysical parameter estimation problems. SA is a concept borrowed from solid state physics, where it was used as a sampling method (Metropolis et al., 1953). In this study, we have used VFSA as an optimization module where we draw a prior realization from a fractal-based technique. To make the discussion self-contained, we outline here, in brief, the concepts behind VFSA optimization. Detailed description of VFSA and its applications to exploration geophysics can be found in Ingber (1989) and Sen and Stoffa (1991, 1995). SA is a process in which a melt of solid material is cooled slowly to form a crystal. Formation of a crystal occurs at the lowest energy state, which corresponds to the global minimum in the optimization scheme. The analogy between the terms used in SA of solids with those used in optimization methods follows. Solid particles in melt state are analogous to the set of model parameters in an inverse problem. The temperature in the case of annealing of solids

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Fractal Models in Exploration Geophysics

corresponds to a control parameter that plays a key role in determining probability of acceptance/rejection of the model parameters in an inversion algorithm. The energy function is analogous to the objective function in an optimization problem, and finally, the crystallization energy of solid material refers to the global minimum error in an inversion procedure.

3.4.1 Examples The application of the fractal-based stochastic inversion is shown using both synthetic and real data. Also, comparison of results obtained from deterministic and stochastic inversion is shown. We test our algorithm on a known data set given as a demonstration data set in Hampson and Russell software (CGGVeritas). The data set consists of super gathers available in the AVO module. For inversion analysis, we used an AVO well log consisting of density, P-wave and S-wave velocity logs available in the same module. An angle gather is created first by making a super gather and using the AVO well log. The velocity information to generate an angle gather is used from the available P-wave curve in the AVO well log. Figure 3.3 shows some of the angle gathers used in our prestack inversion analysis, two picked horizons are also shown with the black lines. The well logs used in this analysis for generation of the background model are shown in Figure 3.4. Further, we extracted angle-dependent wavelets; however, because variations in wavelets are not significant, the average of wavelets over the angle range of 0–10 is assumed as a near-angle wavelet and the average of wavelets over 11–21 is taken as a far-angle wavelet. Thus, in this Xline 70 Angle 0 3 6 9 12 15 18 21 24 2730

71 0 3 6 9 12 15 18 21 24 2730

AVO-well

72 0 3 6 9 12 15 18 21 2427 30

550

Time (ms)

600 Super-gather horizon-1

650

Super-gather horizon-2 700

FIGURE 3.3 Prestack seismic angle gathers near the AVO-well location. The data consist of 1 inline and 131 cross lines. The whole volume has been used in prestack inversion analysis within the time window shown as horizon-1 and horizon-2.

(a)

3

VP (m/s)

2300 2400 2500 2600 2700 2800 2900 560

TWT (ms)

55

Fractal Inversion

(b)

VS (m/s) 900 1000 1100 1200 1300 1400 1500

(c) 2.1

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Chapter

FIGURE 3.4 Checkshot-corrected (a) P wave, (b) S wave, and (c) density log used in prestack inversion.

study, we use two angle-dependent wavelets corresponding to near angle (0–10 ) and far angle (11–21 ).

3.5. STOCHASTIC INVERSION In the case of prestack inversion, three initial models corresponding to P-impedance, S-impedance, and density are generated using the fractal-based method. Further, these initial models are used to generate angle-dependent reflectivity using a formula (Fatti et al., 1994; Ma, 2002; Russell and Hampson, 2006) described in Section 3.2.1 (Equation 3.4). Only the near-angle wavelet is used in our code instead of two wavelets used in the deterministic case. The functional form of the objective function, which is being minimized in the VFSA modeling module, is: P jdobs  dcal j jm  mcal j js  scal j  þ b  obs  P P þ a  obs E¼2 s2  s2  jdobs þ dcal j þ jdobs  dcal j m2obs  m2cal  obs cal jHobs  Hcal j ; ð3:9Þ þ g  2 2  Hobs  Hcal where dobs, mobs, sobs, and Hobs are observed seismic data, mean, standard deviation, and Hurst coefficient of the available impedance log, respectively, and dcal, mcal, scal, and Hcal are calculated seismic data, mean, standard deviation, and Hurst coefficient of the impedance log being used in the inversion algorithm, respectively. The a, b, g are weighting factors, || represents absolute values (L1 norm), and the sum is over the number of data points. Like any other optimization problem, the weighting factors a, b, and g are obtained by trial and error. In our application, we used a ¼ 0.01, b ¼ 0.025, g ¼ 0.01.

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Fractal Models in Exploration Geophysics

3.5.1 Synthetic Example To test the efficacy of the algorithm, a synthetic study is done using the available well log shown in Figure 3.4. Synthetic convolutional model angle gathers (Figure 3.5a) are generated using Fatti et al. (1994). The inversion of (a)

(b) Trace number 1

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FIGURE 3.5 (a) Synthetic and (b) inverted seismic data; (c) P- and (d) S-impedance estimate at the well location using synthetic data (no random noise added).

Chapter

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57

synthetic seismic data is carried out using the fractal-based stochastic method. The objective function described in Equation (3.9) is used for the model optimization. Our method does provide a high-resolution model estimate (Figure 3.5c,d). Further, to test the effect of noise on the inversion algorithm, random noise is added to the synthetic data and inversion is performed. The inversion results are shown in Figure 3.6. The data match (Figure 3.6a) is not very good, but the estimated acoustic impedance model (Figure 3.6c,d) is in good agreement with the real acoustic impedance log.

3.5.2

Field Data Example

Acoustic impedance models computed by the fractal-based inversion algorithm are shown in Figure 3.7a,b at the well location. Multiple realizations of P- and S-impedances are also shown in Figure 3.8a,b, which corresponds to the result at the well location; these can be used to assess uncertainty in our results from stochastic inversion. It is evident from Figure 3.7 that the results from the stochastic inversion scheme have high resolution because the model estimates from stochastic method demonstrate a better match with the well log. The stochastic inversion result for the entire line is shown in Figure 3.9a,b, and the combined plot of observed versus best-fit seismic data is shown in Figure 3.10a,b corresponding to the near-angle, mid-angle, and far-angle gathers. The results from deterministic and stochastic inversion are compared in Figures 3.7a,b and 3.9a,b.

3.6. DETERMINISTIC INVERSION We used the model-based constrained inversion algorithm available in the STRATA module of Hampson-Russell software (CGGVeritas) for prestack seismic inversion to compare with the results of the fractal-based inversion. The algorithm uses angle-dependent reflectivity for the forward model, which is then convolved with multiple angle-dependent wavelets extracted from seismic data and well-log. Three low-frequency a priori trends are used corresponding to each model parameter (P-impedance, S-impedance, and density), which are derived from the interpolation and filtering of the available well logs using a 10-Hz low-pass filter. This comprises the low-frequency models that attempt to constrain the inversion results to predict a consistent earth model (Pendrel, 2006; Sen, 2006). Inversion analysis at the well location was performed to test the optimum parameters for stochastic inversion (fractal-based VFSA) and verify the inversion results. The model estimates obtained by both the fractal-based inversion method and the model-based deterministic method at the well location are shown in Figure 3.7. Next, both inversion algorithms were applied to invert an entire 2D line with 131 CMP gathers using the same parameters as established at the well location. The comparison of the model estimate for the entire line is given in Figure 3.9. The best-fit seismic data and residual using deterministic inversion

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Fractal Models in Exploration Geophysics

(a) 550

1

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FIGURE 3.6 (a) Synthetic seismic data with random noise; (b) inverted seismic data; (c) P- and (d) S-impedance estimate at the well location using noisy synthetic data.

are shown in Figure 3.11. Inversion for density is not done because the angle information available in the data is not sufficient to obtain reliable estimates of the density.

Chapter

(a)

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Fractal Inversion

(b)

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FIGURE 3.7 (a) Comparison of observed P-impedance (black) with deterministic (red) and stochastic inversion (blue) model estimates. (b) Comparison of observed S-impedance (black) with deterministic (red) and stochastic inversion (blue) model estimates. Inversion window 600–780 ms.

3.7. ANALYSIS OF RESULTS In stochastic inversion, several realizations of the model estimate are generated to appraise the sensitivity of the solution. It is obvious from Figure 3.7a,b that stochastically inverted P- and S-impedance values are in close agreement with the measured values of these parameters from the well log. Further, we computed standard deviation using all the realizations and found that the maximum standard deviation is 15%. Qualitative analysis shows a high degree of confidence in realizations for both P- and S-impedance. Figure 3.9 shows comparison of results obtained from the deterministic and fractal-based stochastic inversion algorithms. We note that the stochastic inversion results show higher resolution compared to the deterministic results, which show an average estimate of the model parameters. Moreover, the vertical resolution in stochastic inversion results is consistent

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Fractal Models in Exploration Geophysics

(a)

(b)

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FIGURE 3.8 (a) Multiple realizations of Zp at the well location; (b) realizations of Zs at the well location, thick black line shows the mean of realizations.

with the well-log measurements and helps to delineate thin beds. The amplitude anomalies on a seismic section around 640 ms (Figure 3.3), known as bright spots, are associated with gas sand (Russell and Hampson, 2006). This amplitude anomaly corresponds to a classic class 3 AVO anomaly characterized by low-impedance sand having fairly large reflectivity at all offsets and no polarity change (Rutherford and Williams, 1989), showing low impedance at the gas sand zone (Figure 3.9), which is to be expected. However, there are also low-impedance zones elsewhere, probably due to wet sand. The same anomaly is smeared in the deterministic results (Figure 3.9—right panel) and poorly resolved. The vertical resolution is greatly enhanced in stochastic inversion results compared to those in deterministic inversion.

Chapter

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Fractal Inversion

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FIGURE 3.9 (a) Comparison of inverted Zp using stochastic (left) and deterministic (right) inversion. (b) Comparison of inverted Zs using stochastic (left) and deterministic (right) inversion.

3.8. CONCLUSION The initial model generated by the fractal method using available well logs helps to improve the resolution of the model space and avoids unwanted frequencies that could creep in the model if a white noise were taken as an initial

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Fractal Models in Exploration Geophysics

(a)

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FIGURE 3.10 (a) Observed and (b) inverted seismic data using stochastic inversion. Traces correspond to 3 angle at each CMP location (near-angle gather). (c) Observed and (d) inverted seismic data using stochastic inversion. Traces correspond to 21 angle at each CMP location (farangle gather).

model. The fractal-based model generation can be much more useful in local optimization methods, which needs to be tested. The method can be extended to generate prior models in other geophysical problems where model (source) follows fractal behavior. The synthetic and field examples presented in this study substantiate the usefulness of the fractal-based stochastic inversion

Chapter

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Xline 70 Angle 0 3 6 9 12 15 18 21 24 27 30

71 0 3 6 9 12 15 18 21 24 27 30

AVO-well

72 0 3 6 9 12 15 18 21 24 27 30

550

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FIGURE 3.11 Data match using a deterministic algorithm. The upper panel shows synthetic seismic data obtained by the best-fit model and the lower panel shows the residual between observed seismic data (Figure 3.3) and synthetic seismic data.

method for high-resolution model estimates. Forward seismic modeling in this study is based on the Fatti’s approximation of the Zoeppritz equation, which is used for small reflectivity ( 0.5) and isotropic medium.