A008 THE PROAILITY PERTURATION METHOD AN ALTERNATIVE TO A TRADITIONAL AYESIAN APPROAH FOR SOLVING INVERSE PROLEMS Jef AERS Stanford University, Petroleum Engineering, Stanford A 94305-2220 USA Abstract ayesian inverse theory provides a framework for solving inverse problems that are non-unique, e.g. history matching. The ayesian approach relies on the fact that the conditional probability of the model parameters given the data (the posterior) is proportional to the likelihood of observing the data and a prior belief expressed as a prior distribution of the model parameters. In case the prior is not Gaussian and the relation between data and parameters (forward model) is strongly non-linear, one has to resort to iterative samplers, often Markov chain Monte arlo methods, for generating samples that fit the data and reflect the prior model statistics. While theoretically sound, such methods can be slow to converge, and are often impractical when the forward model is PU demanding. In this paper we propose a new sampling method that allows to sample from a variety of priors and condition model parameters to a variety of data types. The method does not rely on the traditional ayesian decomposition of posterior into likelihood and prior, instead it uses socalled pre-posterior distributions, i.e. the probability of the model parameters given some subset of the data. The use of pre-posterior allows to decompose the data into so-called, easy data (or linear data) and difficult data (or non-linear data). The method relies on fast non-iterative sequential simulation to generate model realizations, instead of an iterative sampling. The difficult data is matched by a perturbing an initial realization using a perturbation mechanism termed probability perturbation. A simple example is used to illustrate the theoretical development, while a reservoir study is provided in an accompanying paper. Introduction Inverse problems are ubiquitous in the Petroleum Sciences. Sets of measurements d are used to determine the spatial distribution of a physical attribute, described mathematically by a model with a set of parameters m. In most applications, the measurement d is too sparse to determine uniquely the underlying subsurface phenomenon or the model m. For example, a well-test pressure (d) does not uniquely determine subsurface permeability at every location (m) unless the reservoir is purely homogeneous (single m). A set of geophysical measurements such as seismic data does not uniquely determine the subsurface porosity distribution. Most inverse problems are therefore fundamentally underdetermined (ill-posed), with many alternative solutions fitting the same data d. For this reason, a stochastic approach to solving the inverse problem is taken. Instead of determining a single solution, one generates a set of solutions distributed according to a probability distribution function. More specifically, in a ayesian context one aims at sampling such solutions from the posterior density distribution of the model parameters m given the data d, 9th European onference on the Mathematics of Oil Recovery annes, France, 30 August - 2 September 2004
2 ( ) ( ) ( ) f d m f f m m d () f ( d) The prior density f(m) describes the dependency between the model parameters and therefore constrains the set of possible inverse solutions. In a spatial context such dependency refers to the spatial structure of m, be it a covariance, oolean or training image-defined dependency. The likelihood density f(d m) models the stochastic relationship between the observed data d and each particular model m retained. This likelihood would account for model and measurement errors. Void of any such errors, the data and model parameters are related through a forward model g, d g( m ) (2) The density f(d) depends on the density f(m) and the forward model g, but its specification can often be avoided in most samplers of f(m d). The aim of ayesian inverse methods is to obtain samples of the posterior density f(m d). Other than the case where g is linear, such sampling will need to be iterative. Popular sampling methods are reection sampling and the Metropolis sampler (Omre and Telmeland, 996; Hegstad and Omre, 996). oth are Markov chain Monte arlo (McM) samplers that avoid the specification of f(d) but are iterative in nature in order to obtain a single sample m (l) of f(m d). Generating multiple (conditioned to d) samples m (l), l,,l in this manner quantifies the uncertainty modeled in f(m d). The problem with such approaches is that they either rely on simple probability models such as multi-gaussian distributions (Tarantola, 989) or require a very large number of forward model evaluations, hence are impractical when g is a flow simulation model. In this paper, we also treat the more general case where two types of dataset d and d 2 are available. In such case the ayesian decomposition of () is written as f( d, d2 m) f( m) f ( m d, d2) f ( d, d2) (3) f( d m) f( d2 m) f( m) ; f ( d, d2) where the latter expression relies on a conditional independence hypothesis between the two data types. We will consider the data d as the easy data, i.e. data than can be conditioned to within a more classical geostatistical context. d could consist of a set of point measurements (termed hard data ) combined with secondary information (termed soft data ) whose relationship to the model m is linear or pseudo-linear in nature. The data d 2 is termed the difficult data, such as production data, exhibiting a complex multi-point and non-linear relationship with the model m. In this paper we present a practical method, termed probability perturbation, that addresses the above mentioned limitations of traditional ayesian approaches. The method relies on: () The use of sequential simulation (Deutsch and Journel, 998) to sample the prior and posterior distributions. Sequential simulation is not iterative, hence more PU efficient than iterative McM samplers. (2) The use of so-called pre-posterior distributions f( m d) and f( m d 2), instead of likelihoods. We will show that this leads to an efficient sampling method, primarily by avoiding the calculation of f ( d, d 2). The purpose of this paper is to present the theory behind the probability perturbation method. For a practical reservoir example, we refer to an accompanying paper by Hoffman and aers (2004)
3 Methodology Sampling the prior Practical inverse methods should be able to incorporate a diversity of prior models, not limited to Gaussian ones. To emphasize the non-gaussianity of the proposed method, we consider a binary random function, modeling the absence or presence of a geological event, if the "event" occurs at u I( u ) 0 else where event could represent any spatially distributed, whether continuous or categorical phenomenon. The random function is discretized on a grid composed of a finite set of N grid node locations u i (x i,y i,z i ). The parameters of the inverse problem are given by the unknown indicator variables at each grid node location: m {( I u), I( u2), K, I( u N )} The grid need not be rectangular. The prior distribution is then simply the oint distribution of all indicator random variables at all grid node locations f( m) Prob{ I( u), I( u2), K, I( u N ) } A fast and general method for sampling a prior distribution is sequential simulation. Unlike McM methods, sequential simulation is non-iterative and is completed after a single pass over all grid node locations. Sequential simulation relies on the following decomposition of the prior: f( m) Prob{ I( u) } Prob{ I( u2) i( u)} K (4) Prob{ I( u ) i( u ), K, i( u )} N N This decomposition has led to a series of practical algorithms that can sample a wide variety of prior models (e.g. Deutsch and Journel, 998; Strebelle, 2002) ayesian inversion Priors need to be conditioned to data d and d 2, resulting in a posterior distribution of the model parameters given the data, as formulated in ayes rule () or (3). In determining the posterior, we will follow the same philosophy as in determining the prior, i.e. rely on a sequential decomposition into conditionals. We consider the case where the data d constitutes direct observations (hard data) of the model parameters at a set of n < N spatial locations (hard data), in the binary random function case, d {( i u α), α, K, n} d 2 contains any type of data that has a non-linear relationship with the model parameters d2 g( m) g( I( u), I( u2), K, I( u N )) (5) with g is the forward model. We will assume for now that there are no data and model errors. One is interested in generating samples from a posterior distribution: f( m d, d ) Prob{ I( u ), I( u ), K, I( u ) { i( u ), α, K, n}, d } 2 2 N α 2 To sample such posterior, we do not follow the traditional ayesian route of likelihood and prior. Instead, we rely on the a sequential decomposition similar as in the prior case: f( m d, d2) Prob{ I( u) { i( uα), α, K, n}, d2} Prob{ I( u2) i( u),{ i( uα), α, K, n}, d2} K (6) Prob{ I( u ) i( u ), K, i( u ),{ i( u ), α, K, n}, d } N N α 2 9th European onference on the Mathematics of Oil Recovery annes, France, 30 August - 2 September 2004
4 Sampling from the complex (oint) posterior f is equivalent to sequential sampling from a series of univariate conditionals of the type: Prob{ I( u ) i( u), K, i( u ),{ i( uα), α, K, n}, d2} Prob( A, ) with A { I( u ) } { i( u), K, i( u ),{ i( uα), α, K, n}} d2 where we have introduced a simpler notation A (unknown model parameter), data (easy data) and data (difficult data) to make further development clear. Since it is difficult to state the conditionals Prob( A, ) explicitly, we decompose it further into two pre-posterior distributions, Prob( A ) (pre-posterior to knowing ) and Prob( A ) (pre-posterior to knowing ), using Journel s (Journel, 2002) combination method: τ τ2 x b c Prob( A, ) with where: + x a a a (7) -Prob( A ) -Prob( A ) -Prob( A) b, c, a Prob( A ) Prob( A ) Prob( A ) In case, τ τ 2, Journel shows that Eq. (6) is equivalent (up to a simple standardization) to the hypothesis of conditional independence of Eq. (2). Assuming the prior Prob( A ) is known, the problem of stating the conditional Prob( A, ) is now decomposed into a problem of stating the pre-posteriors Prob( A ) and Prob( A ). Working with pre-posteriors will lead to an approach that is different from a classical ayesian inversion which would involve the likelihoods Prob( A) and Prob( A ). This difference may seem subtle but has important practical consequences. Stating pre-posteriors, instead of likelihoods, allows using (noniterative) sequential simulation via Eqns. (6) and (7), instead of (iterative) McM. The τ-values in Eq. (7) allow the user to model explicitly the dependency between the -data and -data. The τ-values can be interpreted as weights given to each data type (see Journel, 2002). Assuming a conditional independence (τ τ 2 ) results in a very particular dependency model that may often not reflect the actual dependency between and data. In the context of sequential simulation, the pre-posterior Prob( A ) is simply the conditional distribution of any previously simulated nodes i( u), K, i( u ) and the hard data i( u α), α, K, n. The remaining pre-posterior Prob( A ) cannot be directly estimated due to the complex nonlinear relationship (forward model) between A and. Instead, in the next section, we propose an iterative sampling method for determining this probability, termed probability perturbation.
5 Sampling Prob( A, ) : the probability perturbation method Given a random seed s, and given the pre-posterior Prob( A ) at all grid nodes u, a ( ) ( ) ( ) ( ) realization i s { s ( ), s ( 2), K, s i u i u i ( u N)} can be drawn by ways of sequential simulation. The subscript in i ( s) emphasizes that i ( s) is conditioned to data d (-data) only, not yet to data d 2 (-data), this would require somehow the use of the other pre-posterior, Prob( A ). To generate a realization i (l) matching both d and d 2, each grid node should be sequentially sampled from the oint distribution Prob( A, ). However, the oint distribution is unknown, since Prob( A ) is not known. The probability perturbation method performs a stochastic search for those probabilities Prob( A ) (at all N grid nodes) that achieve a match to the d 2 data. The key idea is to search for all N probabilities Prob( A ), such that, after combining Prob( A ) with Prob( A ) into Prob( A, ) using Eq. (7), a realization i (l) sequentially drawn from Prob( A, ),, K, N, matches the data d 2. Searching for all N probabilities Prob( A ) directly is impossible, particularly since N can be large. Therefore we rely on a perturbation parameterization of all N Prob( A ) using a single parameter as follows: ( ) Prob( A ) Prob( ( ) ) ( ) s I u r i ( u ) + rp( A),, K, N (8) where r is a parameter between [0,], not dependent on u. This independence on u is will be relaxed later. Given Eq. (5), Prob( A ) can be calculated for a given value of r and for a given initial realization ( s i ) ( u ). The importance of Eq. (8) is that it translates the search for N probabilities Prob( A ) into an optimization problem of a single parameter r as follows: hoose a random seed s Generate an initial realization i ( s) using the data d and random seed s Iterate: Until the data d 2 is matched, do the following:. hoose another random seed s 2. Determine a realization i ( s ) that matched better data d 2 as follows: From Prob( A ) in Eq. (8), Prob( A, ) can be determined using Eq. (7). This allows generating a realization ( s ) ( s ) ( s ) ( s ) i { i ( u ), i ( u ), K, i ( u )} drawn by sequentially sampling from r 2 N Prob( A, ),, K, N. This realization is dependent on the value of r, as well as a random seed s. To find an optimal r, hence per Eq. (8) best Prob( A ), the following obective function is formulated ( s ) Or (, s) d2 g( i r ) (9) which measures the distance between the data d 2 and the forward model evaluated on the realization ( s i ). The obective function (9) depends on r r 9th European onference on the Mathematics of Oil Recovery annes, France, 30 August - 2 September 2004
6 and a fixed random seed s. A simple one-dimensional optimization can be performed to find the value of r that best matches the data d 2. Once an optimal value r opt is found, generate a realization ( s i ) opt to be used r in Eq. (8) during the next iteration step. The two limit values r 0 and r clarify the choice of the parameterization in Eq. (8) ( ) () In case r 0, then Prob( A ) s i ( u ), hence per Eq (7) ( ) Prob( A, ) i s ( u ). Regardless of the random seeds s and s, any realization ( s ) ( s) r 0 i i. In other words, r 0 entails no perturbation of i ( s). (2) In case r, then Prob( A ) P( A), a simple calculation using Eq. (7) shows that Prob( A, ) PA ( ). Since the seed s is different from the seed s, the ( s ) ( ) realization s ir i is equiprobable with i ( s). In other words, r entails a maximum perturbation within the prior model constraints. A value r in between (0,) will therefore generate a perturbation ( s i ) r between some initial realization i ( s) and another equiprobable realization i ( s ) both conditioned to the data d and each following the prior model statistics. The optimization of r in (9), has to be repeated for multiple random seeds s since a single opt optimization of Or (, s ) with fixed random seed is likely not to reach satisfactory match to the data d 2. A simple illustrative example A simple example is presented to clarify the approach and illustrate various properties of the method in finding inverse solutions. The model consists of a grid with three nodes, u, u 2 and u 3. Each node can be either black, I(u) or white, I(u)0. The model m is therefore simply m {( I u), I( u2), I( u 3)} The spatial dependency of this simple D model is described by a training image shown in Figure. One can extract, by scanning the training image with a 3 template, the prior distribution (f(m)) of the model m, as shown in Figure. To test the probability perturbation method we consider two data: the first data is a point measurement (-data, or easy data ) namely, i(u 2 ), the second one is I( u) + I( u2) + I( u 3) 2 (-data or difficult data ). The problem posed is: What is Pr( I( u) i( u2), I( u) + I( u2) + I( u 3) 2)? Or in simpler notation, what is Pr( A, )? We consider three ways to solve this problem:. alculate the true posterior Pr( A, ) directly by simple elimination of those prior models that do not match the and data. 2. alculate Pr( A, ) by first calculating Pr( A ) and Pr( A ) based on the prior, then using Journel s Eq. (7) under conditional independence, namely τ, τ 2. 3. alculate Pr( A, ) by running a Monte arlo experiment using the probability perturbation method (PPM). On each realization drawn from the prior model, the PPM using the algorithm described above is applied. From the set of matched realizations, Pr( A, ) is calculated by checking how many times i(u ).
7 The solutions are as follows:. Directly: Pr( A, ) 3 2. Using Journel s Equation 5 3 5 6 3 Pr( A) a, Pr( A) b, Pr( A ) c 8 5 5 4 3 2 3 Applying Eq. (7) with τ τ 2: x Pr( A, ) 3 5 3. Using Monte arlo on the PPM method, Pr( A, ) 0.39 The following observations can be made: omparing the results of [.] and [2.], it is clear that the assumption of conditional independence is not valid for this case. A simple calculation shows that using τ 2.59 and τ 2 in Eq. (7) would provide an exact approximation of the true posterior. This indicates that the single hard conditioning data should receive substantially more weight than assumed under conditional independence. The relative differences, between P(A), P(A ), P(A ) and P(A,) indicate that P(A ) carries more relevance in determining Prob(A,) then Prob(A ). In other words, the single hard () data carries more relevant information about the unknown A than the sum-equals-two () information. The difference of results between methods [2.] and [3.] can be explained as follows: in the PPM method, perturbations are created by means of sequential simulation using Eqns. (6), (7) and (8). In sequential simulation, the -data (hard data at u 2 and any previously simulated values) changes at every node visited and depend on the order of visited nodes (u first, then u 3 or vice versa). In method [2.] the -data is fixed and consists of the single conditioning value at u 2. While the PPM relies on the same assumption of conditional independence, the result is much closer to the true posterior probability. This result was confirmed by applying a variety of other training images, i.e. other prior distributions. Method [2.] produces consistently a considerable overestimation of Prob(A,), the PPM is off by a few percentages only. The reason for the latter observation can be explained by means of Eq (8). In this equation, the pre-posterior Prob( I( u ) ) is a function of the data through the parameter r, and, a function of an initial realization {i(u ),i(u 2 ),i(u 3 )}. This initial realization depends itself on the pre-posterior Prob( I( u ) ) with depending on the random path. Hence, Eq. (8) forces an explicit dependency between the Prob( I( u ) ) and Prob( I( u ) ) prior to combining both into Prob( I( u ), ) using a conditional independence hypothesis (Eq. (7) with τ, τ 2 ). At least from this simple example, one can conclude that the sequential decomposition of the posterior into pre-posteriors has robustified the estimate of the true posterior under the conditional independence hypothesis. onclusion 9th European onference on the Mathematics of Oil Recovery annes, France, 30 August - 2 September 2004
8 The probability perturbation method shares two important properties of other inverse algorithms: () Inverse solutions generated using the probability perturbation method honor the prior statistics. Whatever value of r or random seed s, all model realizations are drawn using a sequential sampling method that depends on Prob( A ) (depending on prior statistics). The probability perturbation is essentially a search method for finding those prior model realizations that honor the data d 2. (2) The probability perturbation method can cover the entire space of the prior model realizations. At each step of the outer loop, the random seed is changed, hence, if infinite computing time were available, the method would cover the entire space of possible realizations, ust like a reection sampler would. References aers, J., (2003). History matching under a training image-based geological model constraint. SPE Journal, SPE # 7476, p. 28-226 Deutsch,.V. and Journel, A.G. (998). GSLI: the geostatistical software library. Oxford University Press. Hegstad,.K. and Omre, H. (996). Uncertainty assessment in history matching and forecasting. In Proceeding of the Fifth International Geostatistics ongress, ed. E.Y aafi and N.A. Schofield, Wollongong Australia, v., p. 585-596. Hoffman,.T. and aers, J., History matching with the regional probability perturbation method applications to a North Sea reservoir In: Proceedings to the EMOR IX, annes, Aug 29 - Sept 2, 2004. Journel, A.G. (2002). ombining knowledge from diverse data sources: an alternative to traditional data independence hypothesis. Math. Geol., v34 573-596. Omre, H. and Telmeland, H. Petroleum Geostatistics. In Proceeding of the Fifth International Geostatistics ongress, ed. E.Y aafi and N.A. Schofield, Wollongong Australia, v., p. 4-52. Strebelle, S., (2002). onditional simulation of complex geological structures using multiple-point geostatistics. Math. Geol., v34, p-22. Tarantola, A. (989). Inverse problem theory. Kluwer Academic Publisher, Dordrecht Training image Prior probabilities of the following realizations: 8 5 6 4 6 4 Figure : prior probabilities extracted from the training image