ABSTRACT:
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.
KEYWORDS:
1. Condition number of matrix
2. Ill-conditioned inverse methods
3. Regularization
4. Satellite remote sensing
5. Sea surface temperature (SST)
6. Total error
7. Total least squares (TLS)
SOFTWARE: MATLAB/SIMULINK
CONCLUSION:
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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