A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing

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.