Seismic tomography is a cornerstone of geophysics and has led to a number of important discoveries about the interior of the Earth. However, seismic tomography remains plagued by the large number of unknown parameters in most tomographic applications. This leads to the inverse problem being underdetermined and requiring significant non-geologically motivated smoothing in order to achieve unique answers. Although this solution is acceptable when using tomography as an explorative tool in discovery mode, it presents a significant problem to use of tomography in distinguishing between acceptable geological models or in estimating geologically relevant parameters since typically none of the geological models considered are fit by the tomographic results, even when uncertainties are accounted for. To address this challenge, when seismic tomography is to be used for geological model selection or parameter estimation purposes, we advocate that the tomography can be explicitly parametrized in terms of the geological models being tested instead of using more mathematically convenient formulations like voxels, splines or spherical harmonics. Our proposition has a number of technical difficulties associated with it, with some of the most important ones being the move from a linear to a non-linear inverse problem, the need to choose a geological parametrization that fits each specific problem and is commensurate with the expected data quality and structure, and the need to use a supporting framework to identify which model is preferred by the tomographic data. In this contribution, we introduce geological parametrization of tomography with a few simple synthetic examples applied to imaging sedimentary basins and subduction zones, and one real-world example of inferring basin and crustal properties across the continental United States. We explain the challenges in moving towards more realistic examples, and discuss the main technical difficulties and how they may be overcome. Although it may take a number of years for the scientific program suggested here to reach maturity, it is necessary to take steps in this direction if seismic tomography is to develop from a tool for discovering plausible structures to one in which distinct scientific inferences can be made regarding the presence or absence of structures and their physical characteristics.
- Award ID(s):
- 1848192
- NSF-PAR ID:
- 10157849
- Date Published:
- Journal Name:
- Geophysical Journal International
- Volume:
- 221
- Issue:
- 1
- ISSN:
- 0956-540X
- Page Range / eLocation ID:
- 334 to 350
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
SUMMARY -
more » « less
Abstract Recent advances in deep learning for neural networks with large numbers of parameters have been enabled by automatic differentiation, an algorithmic technique for calculating gradients of measures of model fit with respect to model parameters. Estimation of high‐dimensional parameter sets is an important problem within the hydrological sciences. Here, we demonstrate the effectiveness of gradient‐based estimation techniques for high‐dimensional inverse estimation problems using a conceptual rainfall‐runoff model. In particular, we compare the effectiveness of Hamiltonian Monte Carlo and automatic differentiation variational inference against two nongradient‐dependent methods, random walk Metropolis and differential evolution Metropolis. We show that the former two techniques exhibit superior performance for inverse estimation of daily rainfall values and are much more computationally efficient on larger data sets in an experiment with synthetic data. We also present a case study evaluating the effectiveness of automatic differentiation variational inference for inverse estimation over 25 years of daily precipitation conditional on streamflow observations at three catchments and show that it is scalable to very high dimensional parameter spaces. The presented results highlight the power of combining hydrological process‐based models with optimization techniques from deep learning for high‐dimensional estimation problems.
-
Abstract Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.more » « less
-
Estimating and quantifying uncertainty in unknown system parameters from limited data remains a challenging inverse problem in a variety of real-world applications. While many approaches focus on estimating constant parameters, a subset of these problems includes time-varying parameters with unknown evolution models that often cannot be directly observed. This work develops a systematic particle filtering approach that reframes the idea behind artificial parameter evolution to estimate time-varying parameters in nonstationary inverse problems arising from deterministic dynamical systems. Focusing on systems modeled by ordinary differential equations, we present two particle filter algorithms for time-varying parameter estimation: one that relies on a fixed value for the noise variance of a parameter random walk; another that employs online estimation of the parameter evolution noise variance along with the time-varying parameter of interest. Several computed examples demonstrate the capability of the proposed algorithms in estimating time-varying parameters with different underlying functional forms and different relationships with the system states (i.e. additive vs. multiplicative).more » « less
-
more » « less
Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers.
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/gji/ggz559
