Abstract
Abstract. Addressing a variety of questions within Earth science disciplines entails the inference of the spatiotemporal distribution of parameters of interest based on observations of related quantities. Such estimation problems often represent inverse problems that are formulated as linear optimization problems. Computational limitations arise when the number of observations and/or the size of the discretized state space becomes large, especially if the inverse problem is formulated in a probabilistic framework and therefore aims to assess the uncertainty associated with the estimates. This work proposes two approaches to lower the computational costs and memory requirements for large linear space–time inverse problems, taking the Bayesian approach for estimating carbon dioxide (CO2) emissions and uptake (a.k.a. fluxes) as a prototypical example. The first algorithm can be used to efficiently multiply two matrices, as long as one can be expressed as a Kronecker product of two smaller matrices, a condition that is typical when multiplying a sensitivity matrix by a covariance matrix in the solution of inverse problems. The second algorithm can be used to compute a posteriori uncertainties directly at aggregated spatiotemporal scales, which are the scales of most interest in many inverse problems. Both algorithms have significantly lower memory requirements and computational complexity relative to direct computation of the same quantities (O(n2.5) vs. O(n3)). For an examined benchmark problem, the two algorithms yielded massive savings in floating point operations relative to direct computation of the same quantities. Sample computer codes are provided for assessing the computational and memory efficiency of the proposed algorithms for matrices of different dimensions.
Highlights
The CryosphereAddressing a variety of questions within Earth science disciplines including environmental science, hydrology, geology, geophysics, and biogeochemistry entails the inference of the spatiotemporal distribution of parameters of interest based on observations of related quantities
Computational limitations arise when the number of observations n and/or the size of the discretized state space m becomes large, especially if the inverse problem is formulated in a probabilistic framework and aims to assess the uncertainty associated with the estimates
We propose a computational approach for evaluating HQHT + R −1 (HQ), and by extension HQHT for very large inverse problems, for the case where the covariance matrix Q can be expressed as a Kronecker product of two smaller matrices
Summary
Addressing a variety of questions within Earth science disciplines including environmental science, hydrology, geology, geophysics, and biogeochemistry entails the inference of the spatiotemporal distribution of parameters of interest based on observations of related quantities Such estimation problems often represent inverse problems, with examples including the estimation of hydraulic conductivity or other aspects of subsurface structure using geophysical (e.g., Aster et al, 2013) or hydrologic (e.g., Hyndman et al, 2007) observations, the identification of environmental contaminant sources using downstream concentrations (e.g., Atmadja and Bagtzoglou, 2001; Liu and Zhai, 2007; Michalak and Kitanidis, 2003; Zhang and Chen, 2007), the characterization of atmospheric and oceanic processes (Bennett, 2002), and the quantification of budgets of atmospheric trace gases using atmospheric observations (e.g., Ciais et al, 2011; Enting, 2002). We use the study of Gourdji et al (2012) as a computational benchmark
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