Statistically self-similar (SSS) processes can be used to describe a variety of physical phenomena, yet modeling these phenomena has proved challenging. Most of the proposed models for SSS and approximately SSS processes have power spectra that behave as 1/f/sup /spl gamma//, such as fractional Brownian motion (fBm), fractionally differenced noise, and wavelet-based syntheses. The most flexible framework is perhaps that based on wavelets, which provides a powerful tool for the synthesis and estimation of 1/f processes, but assumes a particular distribution of the measurements. An alternative framework is the class of multiresolution processes proposed by Chou et al. (1994), which has already been shown to be useful for the identification of the parameters of fBm. These multiresolution processes are defined by an autoregression in scale that makes them naturally suited to the representation of SSS (and approximately SSS) phenomena, both stationary and nonstationary. Also, this multiresolution framework is accompanied by an efficient estimator, likelihood calculator, and conditional simulator that make no assumptions about the distribution of the measurements. We show how to use the multiscale framework to represent SSS (or approximately SSS) processes such as fBm and fractionally differenced Gaussian noise. The multiscale models are realized by using canonical correlations (CC) and by exploiting the self-similarity and possible stationarity or stationary increments of the desired process. A number of examples are provided to demonstrate the utility of the multiscale framework in simulating and estimating SSS processes.

This paper covers the design of multiscale stochastic models that can be used to fuse measurements of a random field or random process provided at multiple resolutions. Such sensor fusion problems arise in a variety of contexts, including many problems in remote sensing and geophysics. An example, which is used in this paper as a vehicle to illustrate our methodology, is the estimation of variations in hydraulic conductivity as required for the characterization of groundwater flow. Such a problem is typical in that the phenomenon to be estimated cannot be measured at fine scales throughout the region of interest, but instead must be inferred from a combination of measurements of very different types, including point measurements of hydraulic conductivity at irregular collections of points and indirect measurements that provide only coarse and nonlocal information about the conductivity field. Fusion of such disparate and irregular measurement sets is a challenging problem, especially when one includes the objective of producing, in addition to estimates, statistics characterizing the errors in those estimates. In this paper, we show how modeling a random field at multiple resolutions allows for the natural fusion (or assimilation) of measurements that provide information of different types and at different resolutions. The key to our approach is to take advantage of the fast multiscale estimation algorithms that efficiently produce both estimates and error variances even for very large problems. The major innovation required in our case, however, is to extend the modeling of random fields within this framework to accommodate multiresolution measurements. In particular to take advantage of the fast algorithms that the models admit, we must be able to model each nonlocal measurement as the measurement of a single variable of the multiresolution model at some appropriate resolution and scale. We describe how this can be done and illustrate its effectiveness for an ill-posed inverse problem in groundwater hydrology.

Heterogeneities of hydraulic conductivity across multiple spatial scales can significantly affect the flow of groundwater, yet the sparsity of subsurface observations does not allow accurate reconstruction at all such scales. Instead, a common approach is to estimate an effective parameter which predicts large scale variations in flow. We build a new framework upon maximum a posteriori inversions to directly address the problems of scale. This framework consists of (1) a fractal prior model for conductivity which is an autoregressive (AR) process evolving from coarse to fine scale, and (2) a measurement model in which each observation is well-approximated by a linear functional of the AR process at some scale. This framework efficiently incorporates multiple measurement sources, which consist of point measurements of hydraulic conductivity and a piezometric head. The multi-resolution framework produces a conductivity estimate with spatially varying resolution tailored to the measurement sampling geometry.

Obtaining a map of the spatial distribution of some physical property of the Earth subsurface from sparse data is a fundamental problem in aquifer and petroleum reservoir characterization. This problem can be stated in terms of an inverse problem as the search for a map that satisfies certain constraints, i.e., measurements and information on the spatial statistics of the property. We present the results of numerical experiments applying the method of projection onto convex sets, or POCS, that functions both as an estimation procedure, where the objective is to obtain a unique guess of the unknown spatial distribution given some constraints, and as a simulation procedure, where the objective is to obtain a set of nonunique reconstructions that are consistent with some measurements and have given spatial statistics. Numerical experiments that use point measurements at wellbores and volume integral measurements show that incorporating data from the region of the reservoir between wellbores allows better localization of large-scale heterogeneities in the reservoir volume.>