Fast Geostatistical Stochastic Inversion in a Stratigraphic GridI. Escobar∗, P. Williamson and A. Cherrett - Total UKP.M. Doyen, R. Bornard, R. Moyen and T. Crozat - Compagnie Generale de Geophysique

SUMMARY

We have developed an efficient stochastic AVA inversion tech-nique that works directly in a fine-scale stratigraphic grid, andis conditioned by well data and multiple seismic angle stacks.We use a Bayesian framework and a linearized, weak con-trast approximation of the Zoeppritz equation to construct ajoint log-Gaussian posterior distribution for P- and S-waveimpedances. We apply a Sequential Gaussian Simulation al-gorithm to sample the posterior PDF. We perform a trace-by-trace decomposition of the global posterior into local poste-rior distributions, conditioned by previously simulated traces.Trace-by-trace sampling of the local PDFs generates multiple,high-resolution realizations of the elastic properties. The newsequential algorithm has been implemented to take full advan-tage of parallel architectures and scales approximately linearlywith the number of CPUs. The technique has been success-fully tested using real data and a large layered model contain-ing more than 30×106 grid-cells.

INTRODUCTION

In order to overcome the band-limited nature of determinis-tic inversion methods (Oldenburg et al. (1983)), stochastic ap-proaches have been proposed as a way of generating multi-ple high-frequency realizations of elastic properties from seis-mic data (Francis (2002)). By construction, the realizationsare constrained to reproduce the observed seismic data, withinsome noise-dependent tolerance limits and to honour condi-tioning well data. The multiplicity of models reflects the in-herent non-uniqueness of the inverse problem.

Haas and Dubrule (1994) have developed a rejection-basedstochastic method using trace-by-trace Sequential GaussianSimulation (SGS), where a simulated acoustic impedance traceis accepted only if the corresponding match between syntheticand real seismic is good. More recently, Debeye et al. (1996)and Contreras et al. (2005) have extended this work using sim-ulated annealing and MCMC techniques, respectively. Thesemethods, which apply sample-by-sample model perturbations,remain very computer-intensive. Buland et al. (2003) havedeveloped an elegant Bayesian linearized AVA inversion ap-proach that works in the Fourier domain and allows fast calcu-lation and sampling of a joint Gaussian posterior distributionfor the elastic parameters. However, their approach assumesthat the data and model are sampled regularly in time and thatspatial covariance functions and noise covariance are strictlystationary.

To facilitate integration with reservoir modelling workflows,the Geostatistical Stochastic Inversion (GeoSI) presented hereoperates in a stratigraphic grid defined in the time domain, withhorizontal sampling fixed by the seismic bin size and verti-cal columns of cells with variable thickness, typically muchsmaller than the seismic resolution.

Following the work of Buland and Omre (2003), we use aBayesian framework and a linearized AVA model to calculatea log-Gaussian joint posterior distribution for the elastic pa-rameters. We use an SGS procedure to decompose the jointposterior into a series of local posteriors for each column ofcells that are sampled sequentially to generate multiple real-izations of elastic properties. Our fast, layer-based stochasticinversion procedure supports non-stationary spatial statisticsvia vertically or laterally variable variogram models, incorpo-rated in the prior model. Non-stationary noise statistics canalso be handled by inputting trace-dependent S/N information.

METHODOLOGY

The proposed GeoSI algorithm relies on a linear forwardmodel linking elastic parameters of to seismic reflectivi-ties. When considering for example the acoustic and shearimpedancesIP andIS, the angle-variant reflectivity for a singleinterface can be approximated by (Fatti et al. (1994)):

rθ≈ 1

2cos2 θ∆ ln(IP)−4

IS2

¯IP2 sin2

θ∆ ln(IS) (1)

In a stratigraphic grid, it is straightforward to extend this equa-tion to a full vertical columni of layers. If we denote bymithe discrete vector containing the logarithms of theIP andIS,the reflectivity series can be written as:

r i,θ ≈ A i,θ mi (2)

whereA i,θ differences ln(IP) and ln(IS) and multiplies themwith the coefficients of equation (1). Strictly speaking, thisequation is linear in log-impedances only if the squaredimpedance ratios from equation (1) are assumed to be known.Following Buland and Omre (2003), we express the inverseproblem in a Bayesian framework with Gaussian PDFs. If theseismic noise is assumed to be uncorrelated across traces andangles, the likelihood of seismic datas can be written as:

p(s|m) ∝

exp

(−1

2 ∑θ

∑i(si,θ −Gi,θ mi)

TC−1s i,θ (si,θ −Gi,θ mi)

)(3)

whereCs i,θ are the noise covariance matrices. The rectangu-lar matricesGi,θ group togetherA i,θ from equation (2) withthe wavelet convolution and transform a model trace with ir-regular layers into a seismic trace with regular samples. On theother hand, considering theIP andIS as a log-Gaussian randomfield, a Gaussian prior PDF form can be defined as:

p(m) ∝ exp

(−1

2(m−µm)TC−1

m (m−µm))

(4)

Fast Geostatistical Stochastic Inversion in a Stratigraphic Grid

Figure 1: Sections showing invertedIP andIS. Top panels show the mean impedance of 25 realizations. Bottom panels show the prior mean models.MeasuredIP at two wells is shown as black curve. Time axis is in milliseconds, impedances are in g/cc× m/s.

(a) Near angle (10 degrees) (b) Far angle (30 degrees)

Figure 2: QC sections across the center of the area. Top: real seismic data. Middle: difference between real and synthetic seismic data using meanof 25 realizations. Bottom: standard deviation of 25 realizations. Measured P-wave impedance at two wells is shown as black curve. Time axis isin milliseconds.

whereµm andCm are respectively the prior mean and covari-ance matrix, which are not required to be stationary.Cm istypically modelled using vertical and lateral variograms andcross-correlations. The Gaussian posterior PDFp(m|s) is con-structed by combining equations (3) and (4). When well mea-surements are available, they can easily be integrated to the

posterior PDF with their uncertainties as an additional likeli-hood term. We have developed an SGS-type algorithm to sam-ple the posterior PDF: it is possible to perform a trace-by-tracedecomposition of the global PDF into a number of approxi-mate local PDFs, which are conditional to previously visitedtraces. These local Gaussian PDFs can be sampled easily and

Fast Geostatistical Stochastic Inversion in a Stratigraphic Grid

multiple realizations can therefore be efficiently generated.

As most of the computations can be shared between realiza-tions, extra simulations can be generated at little computationalexpense. The algorithm has been implemented to take full ad-vantage of parallel architectures because a significant part ofthe computation does not depend on the previously generatedtraces. Use of a local trace search neighbourhood allows avery efficient parallelisation of the algorithm with near-linearscalability with the number of CPUs.

The restrictive assumption that the impedance ratios in equa-tion (1) are known can be overcome by a post-processing step.The consequence of using erroneous impedance ratios is thatthere is usually a mismatch between input seismic and synthet-ics computed from the stochastically simulated impedance ra-tios. We have implemented post-inversion corrections, whichare applied independently to each trace and each realization.Small impedance perturbations are calculated so that the cor-rected simulated reflectivities match the reflectivities com-puted using the initial fixed impedance ratios. These pertur-bations give the correct fit to the seismic data, at the expenseof a slight degradation of the realization geostatistics.

TEST ON REAL DATA

The GeoSI method was tested using data from a deep-waterfield. The reservoir is a faulted anticline comprising turbiditechannels and lobe complexes. The main challenge in field de-velopment has been the assessment of lateral and vertical con-nectivity between channels, which is a hard task given the rel-atively low resolution of the seismic in the area.

The data include two angle stacks (10 and 30 degrees), dipolesonic and density logs at 4 wells and 3 seismic horizons in-terpreted over the 8.1km× 8.7km project area. Zero-phasewavelets were estimated independently for the two partialstacks using a simultaneous multi-well extraction procedure.The near angle wavelet has a dominant frequency around25Hz. Although similar in shape, the extracted far anglewavelet is consistent with a reduction of about 10% in fre-quency content compared to the near angle data.

S/N map estimation was performed separately on near and farangle cubes using a standard coherence-type analysis, result-ing in average S/N values of 6 and 4 respectively. Very lowvalues of S/N were observed in the Southwest region, partlydue to strong migration artifacts. These maps were used todefine trace-by-trace seismic noise statistics in the GeoSI in-version.

A stratigraphic grid framework was constructed in time byproportional interpolation from the three interpreted horizonsto define micro-layers in each macro-interval with maximumtime-thickness of 2 msec. The inverted volume correspondsto 325×350 seismic traces at 25 m CDP-spacing and spans a400 msec time window. This results in a layered model con-taining about 35×106 cells. It took 70 hours to generate 25realizations of bothIP and IS on a dual-Opteron 64-bit com-puter. Figure 1 is a vertical section showing the prior modelfor IP andIS, together with the corresponding inversion results

obtained by averaging the 25 realizations.

Prior mean models forIP and IS were obtained by 3-D krig-ing interpolation of the well log data after Backus averagingin the layered framework. Spatially constant relative stan-dard deviations of about 10% were used in the prior modelfor both elastic attributes. A prior linear correlation betweenIP and IS log residuals of about 0.7 was estimated from thewell data and used as a constraint in the inversion. A ver-tical variogram model with exponential structure and correla-tion length of about 20 ms was estimated by statistical analysisof the log residuals after blocking onto the grid and subtract-ing the prior model values. Lateral variograms were estimatedby cross-correlation between the near and far seismic data af-ter resampling into the stratigraphic grid. The analysis wasaveraged over several micro-layers in each interval. Similarisotropic exponential models were fitted to both macro-layerswith a range of 200 meters.

Synthetic and residual seismic cubes were generated using themean from the 25 realizations (Figure 2). Good agreement isobtained for both angle stacks with RMS errors below 10% and15% for the near and far angle stacks respectively. A better re-construction is achieved for the near angle stack because of itshigher signal-to-noise ratio. Some remaining signal is visiblein the far angle residuals, probably due to a wavelet scalingproblem and/or non-stationarity. The impact of well condi-tioning is demonstrated by the reduced variance of simulatedimpedance values in the neighbourhood of the well shown bythe black vertical line on the figure.

A blind well test was performed to assess the quality of theinversion results. Figure 3 shows a comparison of measuredand predictedIP andIS and confirms the good level of matchachieved with GeoSI. Individual realizations were also inter-

Figure 3: Comparison of measured (green) and inverted (black)impedance values at blind well with predicted confidence margins (2.5standard deviation) in light blue.

Fast Geostatistical Stochastic Inversion in a Stratigraphic Grid

Figure 4: 3D view of a single high-resolution realization of inverted impedances clearly showing one of the main channel system in the lowerreservoir section. Left: colourmap is between 3000 (black) and 9000 (red) g/cc× m/s. Right: colourmap is between 2000 (black) and 5000 (red)g/cc× m/s

preted and used to evaluate uncertainty in the spatial distribu-tion of the channelized sand deposits. Figure 4 shows 3D vi-sualizations of one high-resolution realization for both P-waveand S-wave impedance.

CONCLUSIONS

We have illustrated a new Bayesian linearized stochastic in-version approach that operates directly in a fine-scale strati-graphic grid, simultaneously inverts multiple angle stacks, canbe conditioned to well data and controls spatial continuity vialayer-dependent variogram models. Use of a linearized for-ward model allowing analytical calculation of the joint pos-terior distribution, combined with a trace-by-trace sequentialsimulation strategy leads to an efficient parallel inversion im-plementation with fast turnaround time, even for large data vol-umes. Sampling from the joint posterior distribution deliversmultiple, high-resolution images of elastic attributes in the lay-ered framework. The multiple realizations ofIP andIS can beused for uncertainty analysis and cascaded stochastic simula-tion of petrophysical reservoir properties.

ACKNOWLEDGMENTS

The authors would like to acknowledge TOTAL E&P UK PLCfor permission to publish this paper.

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