We propose a new deterministic approach for remote sensing retrieval, called modified total least squares (MTLS), built upon the total least squares (TLS) technique. MTLS implicitly determines the optimal regularization strength to be applied to the normal equation first-order Newtonian retrieval using all of the noise terms embedded in the residual vector. The TLS technique does not include any constraint to prevent noise enhancement in the state space parameters from the existing noise in measurement space for an inversion with an ill-conditioned Jacobian. To stabilize the noise propagation into parameter space, we introduce an additional empirically derived regularization proportional to the logarithm of the condition number of the Jacobian and inversely proportional to the L2-norm of the residual vector. The derivation, operational advantages and use of the MTLS method are demonstrated by retrieving sea surface temperature from GOES-13 satellite measurements. An analytic equation is derived for the total retrieval error, and is shown to agree well with the observed error. This can also serve as a quality indicator for pixel-level retrievals. We also introduce additional tests from the MTLS solutions to identify contaminated pixels due to residual clouds, error in the water vapor profile and aerosols. Comparison of the performances of our new and other methods, namely, optimal estimation and regression-based retrieval, is performed to understand the relative prospects and problems associated with these methods. This was done using operational match-ups for 42 months of data, and demonstrates a relatively superior temporally consistent performance of the MTLS technique.
- Condition number of matrix
- Ill-conditioned inverse methods
- Satellite remote sensing
- Sea surface temperature (SST)
- Total error
- Total least squares (TLS)
We have demonstrated in this work the advantage of the MTLS, which is the family of the deterministic inverse methods, for producing SST retrievals compared with other prevailing methods. In addition, it is noteworthy that MTLS does not require additional error information, e.g., well-specified errors in observational and a priori information. This may provide a significant advantage for climate-based applications where retrievals should be as independent of external error sources as possible. The MTLS retrieval is improved by using the newer version of CRTM, which implies that more accurate forward models and ancillary data can further reduce the remaining MTLS error. This package can also calculate a metric relating to the total retrieval error and automatic QI at individual pixel level. Apart from the QI, MTLS is also capable of identifying the most difficult retrievals due to cloud contamination or high WV profile error. The sensitivity analysis confirms that MTLS solution is independent of a priori/IG error. The data driven dynamic regularization property of MTLS regularizes solutions toward the IG when the problem is either highly ill-conditioned or has high observation error or both to keep the solution below the a priori error. It is found that OEM retrieval, at least as implemented for this problem, is worse than the LS solution, and sometimes worse than the a priori error, irrespective of the version of CRTM. OEM is the most popular choice for physically based operational retrievals due to the assumption that a priori based constraining of an ill-posed inversion should still yield reasonable reasonable results under conditions where there may be unaccounted for parameters or unforeseen errors, as may be the case in real-world retrieval problems. However, these results suggest that this view may be based more on perception of idealized Bayesian statistics rather than comparative scientific study with respect to alternative methods. This study has also demonstrated that the sensitivity of OEM retrievals under practical circumstances renders it more vulnerable to noise than MTLS retrievals. Even by employing dynamic error covariance matrices, OEM is unable to produce the best retrieval for a fairly linear and moderately ill-conditioned problem of SST retrieval. Moreover, the estimation of error of the errors, which is a prerequisite for OEM, is rather difficult in practice, which perhaps explains why OEM results do not match the expectation from the theory of adding to/constraining by a priori knowledge. To date, operational SST retrievals are dominated by regression (REGB), which highly simplifies RT physics. Mostly, it does produce reasonable results (SD) due to the fact that the global variance of SST fields itself is not very high (e.g., compared with gaseous distributions) and the atmospheric attenuation for 3.9-μm channel is rather low, but such methods are still subject to biases on a spatial and temporal basis, with seasonal variations, and has no inherent means of correcting for them. This derivation of MTLS is based on linear algebra. However, this paper illustrates that a deterministic classical mathematics approach can produce better retrievals for real-world RT problems compared with more recent probability-based mathematics that solve ill-posed problems using covariance matrices. The MTLS retrievals outperform the OEM retrievals due to the fact that the regularization in MTLS is data driven. As opposed to OEM that uses regularization from user-defined a priori knowledge of measurement error and forward model error, as well as a priori knowledge error of the retrieved target parameter. A reliable estimation of both the errors in an operational environment is very difficult due to the highly dynamic atmosphere, fast forward model error, including NCEP data, as well as error in the measurements. An alternate effort toward error estimation using simulation minus observation (S-O) bias correction leads to further ambiguities and may potentially mislead our fundamental science understanding. With the advent of newer sensors with improved multispectral capabilities (e.g., the Visible and Infrared Imaging Radiometer Suite and the future Advanced Baseline Imager), employing a deterministic physical method for simultaneous retrieval of SST and WV (critical for weather and climate studies), such as the MTLS package, has the potential to provide substantial improvements in the use of satellite data and derived products.
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