1 Stochastic vs Deterministic Pre-stack Inversion Methods Brian Russell
2 Introduction Seismic reservoir analysis techniques utilize the fact that seismic amplitudes contain information about the geological properties of the reservoir. The mathematics behind this observation was developed in the early 1900s, but its application to exploration seismic data did not start until the 1970s. We classify these methods into two categories: methods that analyze only the amplitudes, and methods that invert the amplitudes to reservoir properties. Newer methods analyze pre-stack data, where the analysis of the amplitudes without inversion is called Amplitude versus Offset, or AVO. Pre-stack inversion has many forms, where the major division is between deterministic and stochastic, or geostatistical, methods. In this talk I will discuss these methods and look at their assumptions and limitations.
3 A suggested workflow Well Log Data Seismic Data V P, ρ V P, V S, ρ Modeling for V S Build rock physics model Post-stack only Post-stack inversion Gathers, only offsets AVO & Prestack inversion Gathers with azimuths AVAz / Fracture Identification Integrate using multivariate or Bayesian statistics
5 The basic model for inversion The zero offset, or stacked, seismic trace can be modeled as the convolution of the acoustic impedance (AI) reflectivity with the wavelet. As shown in the next slide, this is the basis for post-stack inversion. Acoustic Impedance AI = ρvp Reflectivity AI R AI = 2AI Wavelet W Seismic W S = W * R AI
6 Post-stack inversion Post-stack seismic Inversion, developed in the 1970s, reverses the forward modeling procedure, allowing us to derive the impedance from the reflectivity: Inverse Wavelet 6 Impedance Reflectivity Seismic
7 Qualitative AVO In the 1980s, geophysicists observed that the amplitudes in a seismic gather could be written in linearized form using the amplitude versus offset (AVO) equation, a reformulation of the Aki-Richards linearized solution to the Zoeppritz equations: Note that this has added two extra terms to the zero-offset case, a gradient term G and a curvature term C, often referred to as A, B and C, where the term A is called the intercept. This formed the basis to what I refer to as qualitative AVO. where :, sin tan sin ) ( θ θ θ θ C G R R AI P + + =. 2 and, 2 4 2, P P P S S S P S P P P P AI V V C V V V V V V V V G V V R = = + = ρ ρ ρ ρ
8 Intercept and gradient analysis The amplitudes are extracted at all times, two of which are shown here: Offset or Angle θ +R AI +G sin 2 θ Time The AVO equation predicts a linear relationship between these amplitudes and sin 2 θ. Regression curves are calculated to give R AI and G values for each time sample. -R AI - G
9 Using the angle gathers for inversion Fatti et al. (1994) re-formulated this equation to show that the pre-stack seismic data is a function of the acoustic impedance reflectivity (R AI ), shear impedance reflectivity (R SI ) and density reflectivity (R D ) term: R P( θ ) = arai + brsi + crd, where R AI AI VS ρ SI =, RSI = + =, SI = ρvs, RD 2AI 2V 2ρ 2SI S = ρ, 2ρ a = 1+ tan 2 θ, b V = 8 V S P 2 sin 2 θ, and c = V 4 V S P 2 sin 2 θ tan 2 θ.
10 Independent pre-stack inversion Independent pre-stack inversion is implemented by first extracting the reflectivity components, and then inverting them separately. To estimate the reflectivities, the amplitudes at each time t in an N- trace angle gather are picked as shown here, to give R P (θ 1 ) R P (θ N ): We can then solve for the reflectivities at each time sample using least-squares inversion. Finally, these estimates are inverted using a post-stack type scheme. Time (ms) t R R R AI SI D Reflectivities Angle 1 N Generalized inverse weight = matrix 1 R R P P ( θ 1 ) ( θ ) N Observations
11 Simultaneous Pre-stack Seismic inversion Pre-stack inversion is also based on an extension of the Fatti formulation of the Aki-Richards equation: S( θ ) = c1w ( θ ) DLP + c2w ( θ ) DLS + c3w ( θ ) DLD, where : S(θ) = seismic trace at angleθ, L, L, L = logarithms of Z, Z W(θ) = P S, and ρ, the extracted wavelet at angleθ, and D is the derivative operation. D P P As in our discussion of AVO and independent inversion, this can again be set up as a least-squares problem: model parameters = generalized inverse x observations As we discussed earlier, there are two main types of pre-stack inversion, deterministic and stochastic.
12 Deterministic vs Stochastic Inversion First of all, let us define the fundamental difference between deterministic and stochastic inversion: In deterministic inversion we produce what we consider to be a single best solution. In stochastic inversion we produce many possible solutions, all plausible, which average to the deterministic solution. The advantage of deterministic inversion is that we get the best leastsquares solution to our problem. The advantages of stochastic inversion are its higher frequency nature and the calculation of uncertainty.
13 Deterministic pre-stack inversion example On the next slide, I will show an example of deterministic pre-stack inversion. A Gulf Coast dataset (shown on the left of the slide) was inverted for P- impedance, S-impedance and density (which are shown on the right). The inverted volumes were transformed to Vshale, porosity and Sw (also shown on the right of the slide). Our assumption is that each inverted or transformed result is the correct answer. However, this will not allow us to obtain uncertainty estimates from of the rock properties.
16 Deterministic pre-stack inversion example Seismic Amplitude Map Inverted Inverted Acoustic Inverted Shear Impedance Density Impedance
17 Deterministic pre-stack inversion example Seismic Amplitude Map Derived V shale Map Inverted Inverted Acoustic Inverted Derived Shear Impedance Density Vshale Impedance
18 Deterministic pre-stack inversion example Seismic Amplitude Map Derived V shale Map Inverted Inverted Acoustic Inverted Derived Shear Impedance Density Porosity Vshale Impedance
19 Deterministic pre-stack inversion example Seismic Amplitude Map Derived V shale Map Inverted Inverted Acoustic Inverted Derived Shear Impedance Density Porosity Vshale Sw Impedance
20 Stochastic inversion In stochastic inversion, the least-squares inversion method is extended by formulating the problem using a Gaussian or Log Gaussian posterior probability density function, or pdf (Tarantola, 1987). This allows us to sample various scenarios from the pdf using the Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC) approach. The earliest approach to stochastic inversion was by Haas and Dubrule, 1994, in which Sequential Gaussian Simulation (SGS) is used. Buland and Omre (2003) developed a fast approach to stochastic linearized inversion which utilized a Gaussian pdf. The GeoSI method that I will discuss today combines both a Gaussian pdf and the SGS approach (Doyen, Williamson et al., 2007) My colleague Ali Tehrani discussed the Jason StatMod approach yesterday.
21 Geostatistical inversion (Haas and Dubrulle) Actual seismic trace Adapted from Dubrule, 2003 (x,y) Best simulated synthetic trace wavelet AI simulations Populate model with AI data at wells Define a random path through all (x,y) trace locations At each trace location perform a local optimization Generate a large number of trial AI sequences using SGS with spatial and vertical variograms. Compute reflectivity series and convolve with extracted wavelet. Compute misfit against observed seismic. Retain best matching AI (ρ >0.8). Go to next trace
22 Variogram models Vertical (temporal) variogram Horizontal variogram map showing anisotropy Anisotropic variograms in principal directions Here are the variograms computed by Haas and Dubrulle (1994), showing the vertical, or temporal change, and the horizontal change including anisotropy.
23 Bayesian stochastic inversion Although geostatistical stochastic inversion produces reasonable results, it has two limitations: It is quite slow. It has difficulty in converging to an answer. Buland and Omre (2003) introduced a new type of stochastic inversion which was based on multivariate Bayesian statistics. To illustrate the concept of Bayesian statistics, I will first consider the standard least-squares regression problem. We will then look at the general theory proposed by Buland and Omre. We will then extend this method by combining it with SGS.
24 Least-squares regression Consider a regression fit to 16 measured porosity values (φ i ) plotted against seismic impedance (z i ), shown by the red line in the plot. This can be written: φi = a + bz i The regression line is the least-squares fit between porosity and impedance and is considered the right answer, even though only one point falls on it.
25 Statistical interpretation In the statistical interpretation of this crossplot, each variable (porosity and impedance) is given as a Gaussian probability distribution function (pdf) defined by its mean (µ) and variance (σ). p(φ) µ φ σ φ p(z) σ z Joint pdf p(φ,z) µ z The joint pdf p(φ,z) is the probability of φ and z occurring, and is defined by the variances and means, as well as the covariance between φ and z.
26 Bayesian regression Bayesian statistics tells us that the conditional probability of φ given z, or the posterior, equals the joint probability divided by the probability of z, or the prior. p( φ z) = p( φ, z) p( z) σ φ µ φ z conditional pdf p(φ z) σ φ z The conditional mean µ φ z is the least-squares fit, and the conditional variance σ φ z gives us the scatter in this fit. Note it is narrower than p(φ).
27 Bayesian stochastic inversion Generalizing the previous example to inversion, Buland and Omre (2003) showed that: where: µ C m d m and : µ = model T 1 1 = C ( G C d + C µ ) m d m d d m m = conditional mean, C C m d = ( G covariance, T C 1 d G + C 1 m = conditional This equation reduces to the least-squares solution if we assume that µ m = 0, and C d = σ d2 I: T 2 1 mˆ = µ m d = ( G G + σ dcm ) d µ ) = data m 1 = model covariance mean (prior), 1 G covariance. T, d
28 GeoSI The GeoSI method, as implemented by CGG and ported to the Hampson- Russell suite of software, involves the following steps: Build a stratigraphic grid using horizons, well logs and layer-based kriging. Bring in partial angle stacks and wavelets. Compute the Bayesian posterior distribution by combining the model, seismic data and well logs. Create multiple P and S-impedance realizations using the SGS technique. Compute the mean and standard deviations from the impedance realizations. These steps are shown diagrammatically on the next two slides.
29 Building the stratigraphic grid Stratigraphic grid Ip Horizons in time Layer-based Kriging Is Stratigraphic layering style Low-pass filtering Well logs in time (Vp, Vs, Density) Low-frequency prior model in stratigraphic grid R. Moyen and J. Frelet
30 Stochastic Inversion Workflow Partial angle stack seismic cubes n Ip-Is realisations Well logs (Vp, Vs, Density) Well uncertainty Bayesian stochastic inversion time AI Ip-Is prior mean & standard deviation in stratigraphic grid Horizontal & vertical variograms R. Moyen and J. Frelet Posterior mean & standard deviation Ip-Is
31 Bandwidth components For all inversion methods, the prior model is constructed by interpolating filtered logs, and controls low frequencies. For both deterministic and stochastic inversion, the seismic amplitudes control intermediate frequencies within the seismic bandwidth. In stochastic inversion, the vertical variogram model controls the high frequencies. Prior model Seismic Variogram model Power Spectrum Frequency (Hz) Adapted from Moyen and Frelet
32 Offshore West Africa example Elastic inversion (Ip-Is) 3 seismic angle stacks ,000 traces Time window of 200 ms 132 layers in grid 500 realisations (59 Gb total) 3 wells with Vp, Vs and density logs Computations on standard workstation Courtesy of R. Moyen and J. Frelet
33 Ip-Is Prior Model P Impedance 200 ms m/s x g/cm 3 S Impedance Courtesy of R. Moyen and J. Frelet m/s x g/cm 3
34 Ip-Is Realisations P Impedance 200 ms 5200 m/s x g/cm S Impedance Courtesy of R. Moyen and J. Frelet 2200 m/s x g/cm3 4200
35 Ip Posterior Mean and Standard Deviation P Impedance mean 200 ms m/s x g/cm 3 P Impedance std. dev. Courtesy of R. Moyen and J. Frelet σ Ip (km/s. g/cm 3 )
36 Vp/Vs Mean vs Realisations 1.5 Vp/Vs 2.5 Mean Realisation Posterior mean Sand/shale Cutoff Realisation
37 Inversion Results Vp/Vs Ratio Mean of 500 realisations Vp/Vs Courtesy of R. Moyen and J. Frelet
38 Using the realizations One of the key questions about stochastic inversion is: what do we do with all the realizations? In other words, wouldn t a single answer (i.e. the deterministic solution) be better? The answer is that with multiple realizations we can generate a number of new results, such as: Seismic lithology prediction. Facies classification. Volumetric uncertainty analysis. Petrophysical property analysis. These concepts are illustrated in the next few slides.
39 Stochastic Lithology Prediction N sand / shale simulations Is Ip Sand probability cube N realisations of Ip, Is Histogram of sand volume Courtesy of R. Moyen and J. Frelet
40 Facies Discrimination 0.5 Poisson s ratio Ip/Is Courtesy of R. Moyen and J. Frelet VSH P Impedance
41 Individual Realisations Courtesy of R. Moyen and J. Frelet
42 Histogram of sand volume For each realization, we can compute the sand volume from the number of cells with sand. This can then be arranged in histogram format, and the probability percentiles can be computed. A percentile is computed from the total area under the probability curve. Note that the percentile maps do not indicate a higher probability of sand, only where the map falls within the distribution. As shown by an earlier slide, the percentile values will in general be larger than mean computation. These concepts are illustrated in the next few slides.
43 Histogram of Sand Volume from Realizations 40 P50 Number of realisations P10 P90 0 Sand volume P10 P50 P90 Ranked lithology simulations
44 Sand Volume from Realisations and Mean 40 P50 Number of realisations Sand volume from inversion mean P10 P90 0 Sand volume
45 Connected Sand Geo-bodies Geobodies connected to at least one well Courtesy of R. Moyen and J. Frelet Color-code: geobody volume (only largest are displayed)
46 Facies Probability from Stochastic Inversion Sand probability Large volume but small probability Courtesy of R. Moyen and J. Frelet Smaller volume but high probability
47 Stochastic Petrophysical Modelling Multiple I p & I s models Multiple V sh and Φ models Φ I p Statistical petro-elastic calibration Courtesy of R. Moyen and J. Frelet
48 Geostatistical inversion vs other modelling techniques Geostatistical reservoir modeling Interpolate between the wells Plausible details Accurate near wells Not elsewhere Deterministic seismic inversion Optimize P-Impedance to minimize synthetic-to-seismic misfit Accurate within seismic bandwidth Unrealistically smooth Only one possibility StatMod/GeoSI geostatistical seismic inversion Subsumes geostatistical modeling and deterministic inversion Does both, simultaneously and in a statistically rigorous way Multiple plausible realizations at high detail (e.g. 1ms 25m) Yet also coherent interpretations of the seismic up to the km scale
49 Conclusions Stochastic inversion is a natural extension of deterministic inversion (mean of realizations deterministic inversion) But it can provide extra information, such as: Lithology probability Facies distribution Volumetrics Petrophysical parameters Our case study focussed on a channel sand play from West Africa.
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