A Method for Correlated
Noise Modeling in Satellite
Radiance Simulation
Thomas J. Kleespies
David Crosby
NOAA/NESDIS
Thomas.J.Kleespies@noaa.gov
1.
Introduction
This work is motivated by an
Observing Systems Simulation Experiment in which NESDIS is collaborating with
NCEP/EMC and the NASA Data Assimilation Office. The immediate goal is to determine the incremental improvement on
the forecast of a Doppler Wind Lidar instrument over present day observing
systems. In order for the current
satellite instruments to be accurately simulated, it is necessary to model
instrument noise, both independent and correlated. This paper presents such a method, and notes a high level of
correlation in the inter channel noise in one of the High Resolution Infrared
Sounder (HIRS) instruments.
2.
Methodology
A number of standard methods
of modeling correlated noise are available.
We chose one such method presented in Searle (1982). Let S be the sample covariance matrix from
the measured noise, then the vector of simulated correlated noise is given by
1)
where
G is the eigenvectors of S
is a matrix whose
diagonal is composed of the square root of the eigenvalues of S
X is a vector of gaussian
random numbers with mean =0.0, variance=1.0
The sample of the measured
noise is from HIRS space calibration observations for an entire day. An example of inter channel noise for NOAA
11 is given in figure 1. The gaussian
random numbers are computed by (Box and Muller, 1958),
2)
where U1 and U2
are numbers drawn from a uniform random distribution. We used the FORTRAN RAN function for this purpose. This approach was applied to a realization
of size 100,000 and the inter channel correlation coefficients were
computed. Results for selected channel
pairs are presented in Table 1.
Figure 1. HIRS 11 Space
calibration radiances, Ch 10 vs Ch 15, n=16032, r=.492
units are mw /(cm2
sr cm-1)
Table 1. Correlation coefficient for selected
channels of NOAA 11 HIRS observed and simulated space calibration observations.
3.
Conclusions
We have presented a method
for simulating correlated noise and applied it to radiances from the NOAA 11
HIRS instruments. This method
accurately reproduces the observed correlation structure. The NOAA 11 HIRS exhibits inter channel
noise correlations that approach 0.5 .
This fact has implications for the use of off diagonal elements in the
observational error covariance matrix used in variational data assimilation.
4. References
Box, G. E. P. and M. E.
Muller, 1958: A note on the generation of random normal deviates. Annals of
Mathematical Statistics, 29, pp 610-611.
Searle, S. R., 1982: Matrix
Algebra Useful for Statistics, Ch. 13, Wiley&Sons.