A tour of optimisation methods for facies estimation in AVO seismic inversion using Markov Random Fields

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1 A tour of optimisation methods for facies estimation in AVO seismic inversion using Markov Random Fields James Gunning, CSIRO ESRE Michel Kemper, Denis Saussus, Ikon Science Adrian Pelham, Elaine Fitzgerald, Tullow Oil January 16, 2013 Abstract We overview some joint lithology-fluid-class/rock-properties inversion techniques for AVO/quant- -itative-interpretation work from true-amplitude imaged seismic data, using a Bayesian hierarchical model. We use a Markov random-field (MRF) model for facies labels, and a facies-conditional Normal model for elastic rock properties. The MRF forces spatial smoothness in categories, and forbids non-hydrodynamical fluid-class transitions. The rock properties model captures loading effects and mutual correlation between elastic variables within a facies. We study the optimisation problem rather than sampling. The optimisation is a mixed integer-continuous problem, provably NP-hard, but susceptible to good heuristics in certain regimes we discuss. A natural approach to the problem is the EM algorithm, which provides fast local solutions using loopy-belief propagation for the E-step and conjugate-gradient least-squares for the M-step. This yields soft/marginalised estimates of facies labels. Industry-standard Bayesian classifications are related to single cycles of this EM algorithm, but without spatial coupling. A comparable method to the EM algorithm is relaxed joint inversion with continuous facies indicator variables confined to a simplex. The absence of middle frequencies and the multimodality of the objective make fast gradient methods, including EM and relaxation, vulnerable to local minima. Two globalising approaches, deterministic annealing, and analytical reduction to integer-optimisation, are discussed. Introduction Inversion of reflection seismic data for rock properties is a notoriously non unique problem. All acquisition frameworks have bandwidth and physics sensitivity limitations, which imply the data has insufficient information to allow unambiguous inference of absolute rock properties. Standard imaging/migration workflows acknowledge this problem by generating bandlimited images of reflection coefficient only, and, where seismic amplitudes are expected to be useful in reservoir characterisation, aim to produce true amplitude volumes where the imaged amplitudes are well modelled by specified incidence reflectivity equations. This passes much of the non uniqueness problem on to downstream workflows, where quantitative interpretation or AVO teams can work on a simplified inversion problem with migrated seismic as 1D bandlimited data, and the 3 dimensional aspects of the wave propagation largely removed. In AVO inversions, additional information brought in to address non uniqueness often consists of low frequency trends for rock properties, extrapolated from well data into space. These extrapolated trends are usually pooled/weighted across rock types, and their contribution to AVO inversion is usually a convex stabilising term added to an AVO amplitude mismatch objective function, taken in either a Bayesian or regularising sense. Wavelets and noise levels are assumed known from well ties. The overall AVO inversion problem is then convex and efficiently solvable using standard gradient based techniques, characteristics which we think very attractive. A significant problem with this approach is that, far from

2 wells, the background trend information yields only very low frequency drift. After inversion, the middle frequencies get lost, we do not recover blocky looking models, and direct facies classification becomes unreliable. (Broadband seismic is obviously a measurement approach to this issue). Most convex stabilising terms, especially those of l 2 form, are hostile to blocky model restorations, and the latter is what we commonly like to see when the subsurface geology is expected to have distinct and extensive facies. An approach to this problem is to drop the convex, facies pooled stabilising apparatus, replacing this with facies dependent rock property distributions in a Bayesian framework, and attempting joint inversion for facies types and rock properties. The inversion problem becomes a nonconvex, mixed continuous/discrete one, and thus computationally very challenging. The first difficulty is in formulating a discrete prior distribution for the facies labels in space. Many techniques for stochastic facies simulation, such as multipoint or indicator models, generate this distribution implicitly, and the consequent lack of closed forms make these choices problematic in either an optimisation or MCMC framework. In grid free or off lattice approaches, stochastic geometries represented by marked point models do have closed forms, and can be very useful in appraisal/production phases, since the facies sequence and positions may be nearly known. For models on a lattice, Markov random fields (MRFs) are a workable choice, and have the advantage that spatial interactions operate only locally. Recent treatments of joint facies and elastic inversion using MRFs are Larsen et al. (2006), Ulvmoen and Omre (2010). These approaches hinge on linearisations about a particular agnostic background model, from which the continuous rock properties can be approximately marginalised out of the problem. This leaves an inversion problem for the posterior distribution of the discrete facies labels only, and the authors demonstrate how to sample this using MCMC. In the example problems they discuss, the facies uncertainty is noticeably small in the posterior samples. We think this implies that optimisation for the maximum aposteriori (MAP) state would therefore be very useful. In this paper we focus on the problem of joint MAP estimation of facies labels and rock properties, using a Bayesian model with MRFs for the facies labels, and conditional normal distributions for the rock properties which capture rock physics effects and loading. The full optimisation problem is without doubt NP hard, so we present several approaches spanning a range of rigour and CPU cost. We sketch some local approaches based on the expectation maximisation (EM) algorithm, including graph cutting and loopy belief propagation. Closely related to the EM algorithm are gradient methods based a relaxation of the discrete labelling space to indicator variables. A further layer of globalisation can be attempted by embedding the algorithms above inside a deterministic annealing loop (Rangarajan, 2000). Finally, we discuss some more exact methods based on integer quadratic programming over moving templates. Theory In rough notation, the model to be estimated is the model vector of elastic properties m = {v p, v s, ρ} together with facies labels F (F i [1...N f ]) at all the lattice sample locations i = 1...N p in the volume (x, y, t). The data y are angle stacks of true amplitude imaged seismic reflectivities. The prior distribution of rock properties is a facies conditional joint distribution, constructed from rock physics regression equations developed from regional well log analysis, which we write as P(m F). The facies labels are assigned a discrete prior distribution P(F) over the volume. Given the forward model f(m), and a suitable likelihood of the data given the model L(y m) = L(y f(m), the joint posterior distribution of the model m and F can be written P(m, F y) L(y m)p(m F)P(F). (1) The MRF distribution for the facies labels, set on a 3D stratigraphic grid with pairwise neighbours denoted ( j i), is of form log(p(f)) = i (e Fi + j i 2β (D i j) I(F i F j )). Parameters e Fi are for

3 proportions and β (D i j) for connectivity of labels in direction D i j, presumed known in this note. This allows control of facies transitions in all directions. The conditional prior for the rock properties P(m F) is a product of independent Normals of form P(m i ={v p,i,v s,i,ρ i }) N( m i (F i ),C p,i (F i )). For simplicity, we focus on the simplest case of a fully linearised likelihood (Fatti model reflectivity) with N(0,C d ) Gaussian noise model, where the likelihood is L(y m)=n(y Xm,C d ), and X a multi stack block matrix representing the matrix product of vertical differentiation, Fatti AVO coefficients, and wavelet convolution. Finding the joint MAP estimate of facies and properties then amounts to minimising the -ve log posterior of (1), which looks like E(m,F) = (y Xm) T Cd 1 +log C p (F) + i (y Xm)+(m m(f)) T C 1 (F)(m m(f)) (e Fi + A hierarchy of increasingly difficult optimisation strategies follow j i p 2β (D i j) I(F i F j )). (2) 1. One shot vanilla Bayesian classification (closely related to various industry standards). Amounts to fixing m(f) at some background/hybrid mean, (say m ), minimising (2) over m (F bits neglected), then fixing the resulting m, & minimising over F (with all β = 0) for most likely facies. Many approaches depend on the accuracy of marginal distributions estimated/approximated from this inversion, e.g. Ulvmoen and Omre (2010), and some compensations can be made (Buland et al., 2008). Usually sensitive to the initial hybrid. 2. Alternating (EM like) methods. (i) K means/graph cut: minimise (2) alternately in spaces of m (large scale least squares via conjugate gradient) andf (using graph cutting, possible since the objective is submodular (Boykov et al., 2001)). The MRF cleans up short spatial scale noise in the model considerably. CPU costs are a small multiple of classical least squares inversion. Usually rather sensitive to choice of initial hybrid. (ii) Full EM algorithm: here the E step estimates soft facies memberships by loopy belief propagation with current model m fixed. The M step to update m is least squares but with a prior formed by a current membership weighted average of the facies prior means and covariances. With membership step limiting this helps mitigate the sensitivity to initial guesses appreciably. CPU costs are O(10-100) times costs of classical inversion. 3. Relaxed membership joint optimisation. Here, the discrete variables F are relaxed to continuous indicators γ il [0,1] (labels l, voxel i), and the MRF is recast to a quadratic form in the γs yielding the same energy transitions. The optimisation is constrained to the simplex l γ il = 1. The objective is then positive definite in m, but negative definite in the γs, which is a quadratic programming problem known to be NP hard. Local optima can be found efficiently using projected gradient descent, and again, step limiting in the membership (γ) space has proved helpful. This method is about comparably effective (and expensive) to the full EM algorithm, but bypasses belief propagation. The initial guess sensitivity seems similar. 4. Deterministic annealing (DA) variations. In either of (2) or (3) from above, the nonconvexity of the objective makes for initial guess dependence. DA methods wrap the optimisation in an outer cooling loop, with a temperature parameter gradually evolving the objective from some initially convex hybrid to its final multimodal form. The MAP point is tracked from its initial unique minima as the cooling proceeds. Optimisation is gradient based and efficient for each temperature. Our DA peeling scheme consists in pulling the facies normal distributions slowly away from a common starting hybrid which is surely convex. 5. Moving template integer quadratic programming (IQP). Facies clusters are approximated, sharing a common (abundance weighted average) rock properties covariance. The m optimisation of (2)

4 is done analytically, yielding an IQP problem for the facies labels. For realistic signal to noise ratios, the IQP matrix is not even close to submodular, so efficient graph cutting techniques are inapplicable. For smallish moving spatial templates, this can be converted to a binary linear programming problem and fed to a commodity IP solver. This has no startup dependence (template optima are global), but computational demands are very fierce. Two Examples Problems with 2 or 3 facies clusters which align well in P impedance space, in a dominant lithology background, yield to most of the faster methods above, with some tuning required. Example 1, NW Australian shelf. Here, Fig. 1 shows facies inversions for a turbidite field with strong Gassmann oil response. Peak S/N ratio is perhaps 5 here. The insets d,e,f, (detail at a trace) shows EM can be mildly robust to starting guesses cf a global optima. Example 2, North Sea: when the facies clusters are not t(s) t t (a) near-stack seismic + well data 2400 (b) facies, EM algorithm 2400 (c) facies, IQP algorithm (d) trace properties, EM run1 (e) trace properties, EM run2 (f) trace properties, IQP run Figure 1: (a-c) Seismic and MAP facies estimates from EM and IQP algorithms. Single trace detail: (d,e,f) show facies clusters (brown=shale, blue=brine, orange=oil), P impedance contours (thin red), (d,e) hollow squares=1st sweep inversion from 2 different facies-proportioned starting hybrids, filled squares=final model (f)=direct MAP properties from IQP optimum aligned, images are noisier etc, inferences can be much more fragile to the starting/background hybrid model and/or convexity approximations in the faster methods; see Fig 2. Here, the S/N ratio is much worse, and classification is much more reliant on the model and optimisation algorithm. Cheap methods can work very well with excellent S/N. Our Monte Carlo studies of noise effects show that facies misclassification rates, even with perfect optimization, grow sharply for S/N< 4, so the hard regime, where expensive optimization is warranted, may not be large. Conclusions Joint MAP inversion for rock properties and facies is a hard problem. Fast, local optimisation methods based on the EM and related algorithms can be remarkably effective, but good starting guesses and S/N

5 (a) Basic EM (b) EM within DA (c) IQP Figure 2: More difficult, 4 stack, noisy data, non aligned facies; behaviour at a single trace. (a) Basic EM algorithm (b) DA algorithm with inner EM optimisation (c) IQP. The DA algorithm, of controllable CPU budget (perhaps 10-fold the cost of basic EM ), is not too far from very expensive IQP. are needed. More ambiguous rock physics mixtures will require expensive globalising methods, and here, DA looks an affordable alternative to fearsome integer methods. References Boykov, Y., Veksler, O. and Zabih, R. [2001] Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell., 23, , ISSN , doi: / Buland, A., Kolbjornsen, O., Hauge, R., Skjaeveland, O. and Duffaut, K. [2008] Bayesian lithology and fluid prediction from seismic prestack data. Geophysics, 73, 13 21, doi: / Larsen, A.L., Ulvmoen, M., Omre, H. and Buland, A. [2006] Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model. Geophysics, 71(5), R69 R78. Rangarajan, A. [2000] Self annealing and self annihilation: Unifying deterministic annealing and relaxation labeling. In Pattern Recognition, Ulvmoen, M. and Omre, H. [2010] Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 1 methodology. Geophysics, 75(2), R21 R35.