An ensemble-based method for seismic inversion to estimate elastic attributes is considered, namely the iterative ensemble Kalman smoother. The main focus of this work is the challenge associated with ensemble-based inversion of seismic waveform data. The amount of seismic data is large and, depending on ensemble size, it cannot be processed in a single batch. Instead a solution strategy of partitioning the data recordings in time windows and processing these sequentially is suggested. This work demonstrates how this partitioning can be done adaptively, with a focus on reliable and efficient estimation. The adaptivity relies on an analysis of the update direction used in the iterative procedure, and an interpretation of contributions from prior and likelihood to this update. The idea is that these must balance; if the prior dominates, the estimation process is inefficient while the estimation is likely to overfit and diverge if data dominates. Two approaches to meet this balance are formulated and evaluated. One is based on an interpretation of eigenvalue distributions and how this enters and affects weighting of prior and likelihood contributions. The other is based on balancing the norm magnitude of prior and likelihood vector components in the update. Only the latter is found to sufficiently regularize the data window. Although no guarantees for avoiding ensemble divergence are provided in the paper, the results of the adaptive procedure indicate that robust estimation performance can be achieved for ensemble-based inversion of seismic waveform data.

Seismic waveform inversion is a nontrivial optimization task, which is often complicated by the nonlinear relationship between the elastic attributes of interest and the large amount of data obtained in seismic experiments. Quantifying the solution uncertainty can be even more challenging, and it requires considering the problem in a probabilistic setting. Consequently, the seismic inverse problem is placed in a Bayesian framework, using a sequential filtering approach to invert for the elastic parameters. The method uses an iterative ensemble smoother to estimate the subsurface parameters, and from the ensemble, a notion of estimation uncertainty is readily available. The ensemble implicitly linearizes the relation between the parameters and the observed waveform data; hence, it requires no tangent linear model. The approach is based on sequential conditioning on partitions of the whole data record (1) to regularize the inversion path and effectively drive the estimation process in a top-down manner and (2) to circumvent a consequence of the ensemble reduced rank approximation. The method is exemplified on a synthetic case, inverting for elastic parameters in a 1D medium using a seismic shot record. Our results indicate that the iterative ensemble method is applicable to seismic waveform inversion and that the ensemble representation indeed indicates estimation uncertainty.