Phys. Chem. Earth (B), Vol. 25, No. 1012, pp, 12771280.2000
0 2000 Elsevier Science Ltd
Pergamon
All rights reserved 14641909/00/$  see front matter
PII: S14641909(00)001933
Observational
Error Covariance
Matrices
for Radar Data Assimilation
R. J. Keeler and S. M. Ellis NCAR Atmospheric
Technology
Division,
P.O. Box 3000, Boulder,
Colorado
80307, U.S.A.
Received 26 June 2000; accepted 7 July 2000
Abstract.
what the modelers need from the radar observations. Similarly, we attempt to define for the modelers the origins of radar observation statistical errors and artifacts of these observations that cause the measurements to have errors and be correlated in time and space. Sun and Crook (1994) recently described the Variational Doppler Radar Analysis System (VDRAS) technique as an extension of their adjoint data assimilation system. This system requires, as do other data assimilation schemes, knowledge of the error covariance matrix of the radar observations  both for reflectivity and radial velocity. For simplicity, these errors are frequently modeled as independent and identically distributed (iid) equal errors, whereas in reality the errors for all the radar resolution volumes may be vastly different. We investigate a well known technique for determining the observational instrumentation error of each radar measurement. We require knowledge only of the standard “base data”, the reflectivity (Z), radial velocity (V), and velocity spectrum width (W) estimates, which are routinely available from NEXRAD system WSR88D radars and most research radars.
Optimal assimilation of meteorological radar data into numerical models requires knowledge of the observation error covariance matrix, i.e., the error variance magnitude at each data point and its correlation to adjacent data points. We use knowledge of basic reflectivity, radial velocity and spectrum width measurements obtainable from most weather radars to determine the instrumentation error component of the complete observation error covariance matrix. Specifically, the technique will be used to ingest radar data into mesoscale numerical weather prediction models. We perform an experimental validation of the predicted errors from the Memphis, Tennessee NEXRAD (WSR88D) data. 0 2000 Elsevier Science Ltd. All rights reserved.
1.0 Introduction. Assimilation of meteorological radar observations using 4 dimensional variational (4DVAR) techniques, including the adjoint technique, is presently an active research topic because the technique is considered mature and several countries have installed national weather radar networks. Fitting a numerical model that satisfies atmospheric fluid dynamic constraints to a sequence of weather radar observations has been demonstrated to provide real time vector wind and potential temperature analyses in the dry boundary layer (Sun et al 1991). Data assimilation techniques are not familiar to the vast majority of the observational radar meteorological community. Likewise, weather radar data interpretation and assessment is not a familiar concept to an equally vast This majority of the numerical modeling community. treatise is a coarse attempt to explain, in familiar terms,
2.0 Data assimilation and the error covariance matrix. Variational data assimilation utilizes constrained optimization techniques to ingest observational data while simultaneously minimizing some defined cost function. The goal is accurately analyzing the initial conditions for the prediction of a future state of some multidimensional process, in this case, the state of the atmosphere. The mathematical technique is similar to least mean square estimation and adaptive filtering of stochastic processes that are familiar in estimation theory and state space analyses. The total cost function of a typical assimilation system describes the discrepancy between the model variables and the observations. The observation error Jobs is only one component of the error budget in the total cost function. Here we discuss only the observation error in the data assimilation scheme.
Correspondence
to: R. Jeffrey Keeler,
[email protected], Tel: 303497203 I, Fax: 3034972044 1277
1278
R. J. Keeler and S. M. Ellis: Error Covariance Matrices for Radar Data Assimilation
The departure of the atmospheric numerical model variables from the related radar observations is given in Sun and Crook (1999) as
Jobs= c [Hxk
 ykl*
0
’
[H
xk
 ykl
,
(1)
where the summation is over the time index k, xk is the model state vector (e.g., u, v, w), yk is the radar observation vector (e.g., Z, V, W), H is the observation operator and 0 is the observation error covariance matrix. Since x and y are typically different variables stated in different coordinate systems, the observation operator H represents the functional relationship y = f (x) and the transformation between the Cartesian model state variables and the polar coordinate observations. The observation error covariance matrix must include the error resulting from the conversion of the observed quantities to the model state variables, for example converting reflectivity to rain water content. We do not examine the conversion error in this study. The error covariance matrices typically are quite large. For optimization in polar coordinates, the number of radar resolution volumes is of order 1000 range gates x 500 azimuths x 20 elevations = 10 million elements. However, by transforming the polar measurements into a 3 km (horizontal) x 333 m (vertical) Cartesian grid, for example, the size is a more manageable 100 x 100 x 40 = 400,000 resolution elements covering a 300 km region 13 km in altitude. Tests have shown only negligible loss in accuracy of the resulting velocity fields (Sun and Crook 1999). The full observation error is further composed of two primary variance components  an instrumentation error (O(l) and a representativeness error (0,‘). The instrumentation error is the statistical error in making an accurate measurement of the actual mean value that is influenced by the fluctuating nature of the distributed weather targets and the power signal to noise ratio (SNR) of the measurement. Furthermore, contamination caused by sidelobe chitter from the ground, birds, and other undesired returns introduce an additional error that may be included with Oi‘. The representativeness error is the statistical error of a measurement that is assumed to include the entire variance of the underlying physical process. Real measurement systems sample the physical process (the signal) at temporal rates and spatial separations that do not include the entire measurement variance. This noninclusion of all the real variance is the primary contributor to the 0,’ (Daley 1991). That is, the observations represent the actual physical process except for the specified representativeness error. We may say that the (assumed) unbiased observations measure the mean of the physical process and have a total observation error that includes both the representativeness error and the instrumentation error. Quantitative expressions for radar reflectivity error have been determined by Marshall and Hitschfeld (1953) and for velocity and width errors by Doviak and Zrnic (1993). Keeler and Passarelli (1990) summarize these expressions
as a function of the signal strength and the number of independent samples in the radar dwell time, or beam acquisition time. More formally, we can express these instrumentation errors in terms of the signal to noise ratio (SNR), the number of samples (M), and the velocity spectrum width (W) of the data set. We focus our following discussion on the instrumentation error and only briefly comment on the error of representativeness.
2.1 Reflectivity
error
variance.
To initially simplify, let us assume that all measurements are made with high SNR. For NFXRAD and virtually all research radars, this means that we use echoes having reflectivity greater than 10 dBZ for ranges < SO km. This subset assures SNR > 10 dB and includes most clear air values and certainly all precipitation values. Weaker reflectivity near the radar also satisfies this high SNR assumption. For example, at ranges c 25 km a 4 dBZ echo will have SNR = 10 dB, etc. Thus, the high SNR assumption is not limiting and it simplifies the resulting reflectivity variance estimate for the discussion. The process can easily be extended to lower SNR values. Keeler and Passarelli (1990) give an expression for echo received power P, (or reflectivity) error using a square law receiver (such as used in NEXRAD) as: Var (P,) = P,’ / Mi + P,’ / M
(2)
where P, is the weather echo or signal power and P, is the noise power. Mi is the number of independent signal samples and M = Td / T, is the total number of samples in the dwell time  also the number of independent noise samples. Td is the dwell time and T, is the sample spacing. For the high SNR assumption, we may neglect the noise term and get Var (PJ = P: / Mi .
(3)
Thus, we see that the reflectivity instrumentation variance is simply related to the squared received power divided by the number of independent samples Mi during the dwell time. We can estimate this number from the spectrum width value W (in m/s) in the base data for that range gate and the dwell time Td (in seconds): Mi=4dItWTd/h.
(4)
Mi is a number like 410 for clear air and nonconvective situations and for typical scan and sampling rates. Also, averaging over spatial volumes using multiple range gates and beams will further reduce the variance of the reflectivity estimate. Thus, the reflectivity instrumentation error of observation is readily computed by making the rarely satisfied assumption that there are no artifacts to
R. J. Keeler and S. M. Ellis: Error Covariance Matrices for Radar Data Assimilation contaminate
be determined
The error of representativeness independently.
must
the data.
1279
for obtaining O,‘, the simplest being the velocity (spectrum) width of the observed data at each range bin.
2.2 Velocity error variance.
2.3 Correlation between measurements.
For the velocity error variance expression, we again assume high SNR to simplify the resulting estimate. Doviak and Zrnic (1993), also reproduced in Keeler and Passarelli (1990), show the following well established curves for the radial velocity error variance as function of SNR and spectrum width:
It is fair to assume that the instrumentation errors at contiguous range gates are independent at ranges greater than twice the pulse length. However, the receiver matched filtering process causes a strong correlation between these range bins when the data are sampled at more closely spaced range increments. Typically, the measurement correlation of adjacent range gates sized equal to the pulse length will be about 0.5 and the correlation will be zero at all other (larger) range gate spacing. In azimuth and elevation, the correlation in adjacent data samples (space and time) is determined by the antenna pattern and the scan parameters. The NEXRAD radars scan only in azimuth at predetermined elevation angles and rates. Therefore, it is easy to estimate a beamtobeam spatial correlation simply by measuring the average value of the azimuth 2way antenna pattern at one beam spacing (typically one beam width) away from the beam center. Typically, this value is 6 dB down  a good approximation is 0.25. Beams spaced 2 beam widths are much less correlated since the dwell times are in the sidelobe region that is typically more than 40 dB down ( 10m4)from the main lobe  zero is a good assumption. In elevation, the angles are separated by one beam width to several beam widths and the spatial correlation also changes accordingly in the range 0.25 to zero. Furthermore, there are usually several seconds to a few minutes between adjacent sweeps at the same azimuth, so time decorrelation reduces these values even more. One must use caution with the small correlation values approximated as zero. If the correlation is of order 40 dB ( 10e4) but a strong reflectivity difference between the two measurement volumes exists, say 60 dB ( 106), then we observe 20 dB leakage of the stronger signal when the main beam probes into weaker reflectivity regions. Does this factor enter the covariance matrix or not‘? It is a contamination, not an error correlation. If we approximate 10e4 as 0, then we falsely assume that no correlation exists, when in fact, a contamination is readily observable. We may assume the instrumentation errors have a normal distribution. The wellknown “law of large numbers” applies because usually several independent estimates are made and averaged together during the measurement interval, or “beam dwell time”.
SNR
 5dB
/
/
/
y’
"0
0.1
/
/ / /
Contiguous 
0.2

Independent
Pairs Pairs
0.3
0.4
NORMALIZEDSPECTRUM WIDTHrvn Fig. 1. Expected velocity error as function of normalized W and SNR. The width and error are normalized to the Nyquist interval 2v,. M is the number of sample pairs. Small circles represent simulation values. (From Doviak and Zrnic, 1993)
For the high SNR assumption, we obtain the velocity error from the general expression with the weak constraints given in Doviak and Zrnic (1993), Var(V)=hW/[847tMT,].
(5)
Thus, the instrumentation variance of the radial velocity estimate depends on the estimated width (W) in m/s, the number of pulse pairs (M) and the sample spacing (T,) in seconds. Standard velocity errors of 0.2  4.0 m/s are common. Generally, the error of representativeness is larger than the instrumentation error, but it is not clear that it may be computed so easily. Various estimates have been proposed
3.0 NEXRAD data analysis. We have taken a stratiform precipitation case from the WSR88D NEXRAD radar in Memphis, TN to demonstrate
R. J. Keeler and S. M. Ellis: Error Covariance
1280
and test the observation error due to instrumentation uncertainties. Inphase and quadrature Archive 1 data samples were taken in a high reflectivity (e.g., high SNR) ,region that had low velocity shear. The elevation angle was 1.5 degrees and no clutter filtering was performed so ground clutter contamination is likely present in the data. Instrumentation error histograms of the expected (Eq. 5) velocity errors (standard deviations) and the measured velocity error using MATLAB routines are shown in Fig. 2. We readily note that the measured errors are much larger than those expected from an uncontaminated data source. The ground clutter and other artifacts must be accommodated to provide a realistic instrumentation error variance.
Matrices
for Radar Data Assimilation
data can be properly ingested in 4 dimensional variational data assimilation systems. It is equally important to recognize that the representativeness error variance must also be included in the observation error covariance matrix. Additional research must be done to define a quantitative expression or, at least, a better approximation to the error of representativeness. Finally, clutter filtering radar data taken near the ground allows observation of weather that would otherwise be obscured or strongly biased by much stronger ground echo. However, we have ignored the reflectivity and velocity errors associated with clutter filtering. The bias in these errors can be minimized, but the random error variance will increase wherever the clutter filters are applied. Fortunately, we think these errors can be specified as an additional instrumentation component in the error covariance matrices. An important conclusion is that for a properly calibrated radar, having high data quality, the instrumentation error is small compared to representativeness error, artifacts in the data, low data quality, and clutter filter effects. Additional research is currently underway to quantitatively characterize the total radar measurement error covariance.
Acknowledgements.
1
1.5
?.
Radial Velocity (m/s) Fig. 2. Histograms of expected velocity error (left, oTp = .034 m/s) and measured velocity error (right, ovrnea = 0.63 m/s). Shear and variability within analysis region gives mean velocity variability of 0.5 m/s.
The authors acknowledge the assistance given by Jenny Sun, Ron Errico and Jay Millerall of whom have much greater data assimilation expertise than the authors and who provide a continuing collaboration for a greater understanding amongst us all. We also wish to thank Drs. Andrew Crook, and Tammy Weckwerth for their valuable comments as well as Mr. Benjamin Vinson for help with formatting.
References. Daley, Roger 1991: Atmospheric Duta Analysis, Cambridge Press, England, 457 pp.
4.0 Discussion. Given the above approximations, we may generate the instrumentation related component of the total observation error for the reflectivity and velocity error covariance matrices. Also, the error due to artifacts in the data must be counted and more analyses must be performed to realistically estimate this component. Additional research is required to obtain more complete error covariance matrices in order to account for lower data quality, representativeness errors and errors associated with ground clutter filters. In this study we have concentrated only on data with high SNR, moderate velocity spectrum width, and generally high data quality. It turns out that the variance of velocity and reflectivity estimates increases in regions of low SNR, low data quality or large spectrum width. It is important to quantify these relationships and include them in the error covariance matrices so that radar
University
Doviak, R.J. and D. S. Zmic, 1993: Doppler radar and weather observarions, 2” Ed., Academic Press, San Diego, 562 pp. Keeler, R.J. and R.E. Passarelli, 1990: Signal processing for atmospheric radars, Chap. 20 in Atlas (Ed.), Radar in Meteorology, AMS Press, Boston, 199229. Marshall, J.S. and W. Hitschfeld, 1953: Interpretation of the fluctuating echo from randomly distributed scatterers, Part 1, Can. J. Physics, 3 1, 962994. Sun, J., D.W. Flicker and D.K. Lilly, 1991: Recovery of three dimensional wind and temperature fields from single Doppler radar data, J. Amos. Sri., 48,876980. Sun, J. and N.A. Crook, 1994: Wind and thermodynamic retrieval from single Doppler measurements of a gust front observed during Phoenix II. Mon. Weu. Rev., 122, 10751091. Sun, J. and N.A. Crook, 1999: Real time boundary layer wind and temperature analysis using WSR88D observations. Preprints
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