Reduced-cost construction of Jacobian matrices for high-resolution inversions of satellite observations of atmospheric composition

Nesser, Hannah; Jacob, Daniel J.; Maasakkers, Joannes D.; Scarpelli, Tia R.; Sulprizio, Melissa P.; Zhang, Yuzhong; Rycroft, Chris H.

doi:https://doi.org/10.5194/amt-14-5521-2021

Articles | Volume 14, issue 8

https://doi.org/10.5194/amt-14-5521-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/amt-14-5521-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 14, issue 8

Research article

|

12 Aug 2021

Research article |

| 12 Aug 2021

Reduced-cost construction of Jacobian matrices for high-resolution inversions of satellite observations of atmospheric composition

Hannah Nesser, Daniel J. Jacob, Joannes D. Maasakkers, Tia R. Scarpelli, Melissa P. Sulprizio, Yuzhong Zhang, and Chris H. Rycroft

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Final revised paper (published on 12 Aug 2021)
Preprint (discussion started on 16 Nov 2020)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Review of "Reduced-Cost Construction of Jacobian Matrices for High-Resolution Inversions of Satellite Observations of Atmospheric Composition" by Nasser et al.', Marc Bocquet, 15 Feb 2021
- AC1: 'Reply on RC1', Hannah Nesser, 04 Jun 2021
RC2: 'Review for paper', Anonymous Referee #2, 10 May 2021
- AC2: 'Reply on RC2', Hannah Nesser, 04 Jun 2021

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Hannah Nesser on behalf of the Authors (04 Jun 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (04 Jun 2021) by Helen Worden

RR by Anonymous Referee #2 (08 Jul 2021)

Suggestions for revision or reasons for rejection

Based on the changes made to the manuscript I recommend acceptance of the manuscript without any changes.

However, I want to direct authors attention to significant computational advancements that have already been made in inverse modeling in other fields especially hydrology and magnetic resonance imaging (not a list of all areas) that may help them in their future computational research. These include areas ranging from matrix multiplication based on hierarchical matrices, fast approximate calculation of covariance matrices, Jacobians, basis functions, Kalman filter-based methods and in case of MCMC based inversions Variational Approximation and Copulas. See for example (not a full list):
1. Scalable subsurface inverse modeling of huge data sets with an application to tracer concentration breakthrough data from magnetic resonance imaging.
2. Real-time data assimilation for large-scale systems: The spectral Kalman filter
3. Principal Component Geostatistical Approach for large-dimensional inverse problems (See section on Low Rank Approximation); Year 2013
4. Large-scale hydraulic tomography and joint inversion of head and tracer data using the Principal Component Geostatistical Approach; Year 2014. Nothing new in Geostatistical approach it is just a version of Bayesian inversion. Also note they suggest reduction before even attempting forward runs.
My last comment i.e.,
Hence, please explain or mathematically show how does the reduction in posterior variance translate from coarser resolution to finer resolution (not in terms of R i.e., correlation).
• You can mathematically bound your solution to show what happens to posterior variance when you change the resolution of the inverse problem. Look at Rao-Balckwell theorem as an example
Can an upper bound be found and does it have spatial structure i.e., what has happened to the error you obtained from the inversion (second part of equation 2)? Furthermore, what has happened to the trace of the averaging kernel. How has it distributed your trace at finer resolution?
• This question relates to the trace of the averaging kernel aka model resolution. What do you expect would happen to model resolution if you keep the parameters in prior covariance and model-data mismatch to be the same but change the spatial and temporal resolution of the problem? You can express it in terms of reduced chi-square statistic. For spatial structure again look at the averaging kernel matrix and compute changes in Frobenius norm.

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ED: Publish as is (08 Jul 2021) by Helen Worden

AR by Hannah Nesser on behalf of the Authors (09 Jul 2021)

Short summary

Analytical inversions of satellite observations of atmospheric composition can improve emissions estimates and quantify errors but are computationally expensive at high resolutions. We propose two methods to decrease this cost. The methods reproduce a high-resolution inversion at a quarter of the cost. The reduced-dimension method creates a multiscale grid. The reduced-rank method solves the inversion where information content is highest.