Inverse modelling for surface methane flux estimation with 4DVar: impact of a computationally efficient representation of a non-diagonal B-matrix in INVICAT v4

Bannister, Ross Noel; Wilson, Chris

Prior information is essential to most inverse problems and the surface flux estimation problem is no exception. The uncertainties of the prior fields, and their inter-correlations, should ideally be reflected in the a-priori error covariance matrix, often called B. The B-matrix, is however, difficult to quantify partly because it is typically a large matrix and partly because its numerical values are unknown.

We present a highly efficient method of representing the B-matrix to represent prior errors in the initial concentration and in the time sequence of surface fluxes for the 4DVar-based inverse modelling system (INVICAT) used to estimate the surface fluxes of methane. Our formulation is based on a spectral formulation of the square-root of B, which we believe has not been used in any such inverse modelling system before. It allows horizontal and vertical error correlations of the initial concentration, and horizontal and temporal error correlations of the flux to be represented. We provide full mathematical details. Our scheme allows the various correlation components to be switched on/off and for the respective length and timescales to be set in a way that is much more computationally efficient than representing such a B-matrix explicitly.

We test 14 configurations of the B-matrix (including the diagonal configuration) in a 100 day test assimilation of surface flask measurements of methane. We measure the performance of each by comparing the analysis to unassimilated observations held back for evaluation purposes. We find that the diagonal configuration is amongst the poorest performing choices of B. The best performing choice uses the spectral method. It does not include correlations for the initial concentration field, but does account for spatio-temporal correlations for the fluxes. These have the form of a SOAR (second order auto-regressive) function with a correlation length-scale of 600 km and a timescale of 3 months. Our results demonstrate the effectiveness of our method, which is applicable to very high resolution inverse modelling systems. We propose that potential biases in the prior initial condition field may be the reason for the poor performance when correlations in the prior initial concentration field are used.

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Bannister, Ross Noel / Wilson, Chris: Inverse modelling for surface methane flux estimation with 4DVar: impact of a computationally efficient representation of a non-diagonal B-matrix in INVICAT v4. 2024. Copernicus Publications.

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