The recommended algorithm for the correction of higher order data errors in severe storms doppler lidar signal processing is discussed

Lee, R. W.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19830005434 2020-03-21T05:21:24+00:00Z
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Prepared for the George C. Marshall Space
Flight Center, National Aerommtics and
Space Administration, 8nnwille, Alabama.
October 1, 1982
R. W. Lee tC	 4 A
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LA S'S E,N L40 RESE A ^(F
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MANTON CALIFORNIA 96459
YASA-CR-169557 1 SEVERE STORES WPPLIR	 H83-0705(
LIDAH SIGNAL PROCESSING Final Report
(Lassen Research) 4 p HC A02/HF A01
CSCL 04B	 ODClasG3/47 01421
I. Introduction
Contract NASS-34768 called for the development of algorithms for
correcting certain measurement artifacts present in the measurements made
during the 1981 flight program. This problem was addressed in Technical
Report no. 1, issued in April 1982, and subsequent progress reports. The
purpose of this supplement is to summarize the recommended algorithm for the
correction of higher-order data errors.
II. The source of higher-order errors
As noted in Technical Report no. 1, there are several sources of error
in addition to the drift-angle error :rested in that report. These include
the non-uniform attitude-measurement delay (due to the non-uniform scanning
intervals), possible laser drift, and laser scanning above or below the
horizontal. Due to their random natures these errors cannot be corrected by
deterministic means.
III. Suggested correction algorithm
Fortunately these higher-order errors are essentially independent from
scan to scan. As such the error spectrum they produce is white, and error
reduction can be obtained by recourse to spectral filtering.
Where the errors are constant with range, as in the case of most of the
instrumental errors, a continuity argument can be used to justify a simple
correction algorithm. Consider a time series of mean radial-velocity
measurements. Each item in the time series is an average ove. about 10 km of
range. The mean-square difference between adjacent items in the series will
arise from that portion of the two-dimensional velocity spectrum which is not
averaged out by the 10-km one-dimensional integration. That is, the major
part of the variance of the true time series will be due to spatial
variations in the wind field with scale sizes on the Girder of several km. Due
to the integration the spectrum of this true time series will be concentrated
at low frequencies.
The observed time series will contain in addition variance due to the
errors mentioned above. Since the errors are essentially independent from
-1-
scan to scan, the additional variance in the spectrum will be white. Since
these two contr:'wtions to the spectrum differ greatly in spectral form,
filtering techniques can substantially reduce the white component. A
transversal low-pass filter is used to estimate the true component of the
time series. The passband is chosen to be sufficiently broad to pass all
significant frequency components present in the integrated velocity
measurements. This is easily accomplished since the frequency range of the
true time series has been greatly restricted by integration along the lidar
beam.
Many types of filters could be used to implement this algorithm. The
following suggestion makes use of a least-squares fit of a cubic function to
the data, to allow for slow changes in the mean radial velocity. Forward and
aft scan data sets are considered separately. Radial velocities are averaged
over the longest range interval over which the measurements are highly
reliable (by signal-to-noise ratio criteria). Each point in the resulting
time series is treated independently: a correction for :hat point is derived
from adjacent uncorrected averages, and that correction is applied to
measurements at each range FAte.
The corrected mean radial velocity for a given point can be derived as a
simple weighted mean of the mean apparent velocities in the neighborhood of
that point. Using an odd number of points (2M+1) centered on the point in
question, let the mean radial velocities be Vi, i--M to M. For cubic or
quadratic approximation (they give the same result for symmetrical
filtering), the corrected mean velocity V is
sum(Vi)*sum(i**4)-sum(i*i)*sum(i*i*Vi)
V = -------------------------------------- ,
sum(1)*sum(i**4)-sum(i*i)*sum(i*i)
where the sums are over the full range of i. For 5 points (M =2), the
coefficients of Vi are -.0857, +.3429, +.4857, +.3429, and -.0857. For 7
points they are -.0952, +.1429, +.2857, +.3333, +.2857, +.1429, and -.0952.
The weighting is symmetrical about V0, the point to be corrected. The
difference between V and VO is used as a correction factor for all range
gates in that data scan.
-2-
Note that this algorithm cannot correct errors which vary with range,
such as those produced by non-horizontal scans or laser drift. These errors
are not amenable to correction. If possible they must be recognized and
edited out of the data prior to the application of any smoothing or oth.r
corrections.
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Severe storms Doppler lidar signal processing

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