Objectives of the Algorithm:
The primary objective of 3a-26 is to compute the rain rate statistics over 5 degree (latitude) x 5 degree (longitude) x 1 month space-time regions. The output products include the estimated values of the probability distribution function of the space-time rain rates at 4 'levels' (2 km, 4 km, 6 km and path-averaged) and the mean, standard deviation, and probability of rain derived from these distributions. Three different rain rate estimates are used for the high resolution rain rate inputs to the algorithm: the standard Z-R (or 0th-order estimate having no attenuation correction), the Hitschfeld-Bordan (H-B), and the rain rates taken from 2a-25. (Fits based on the high resolution inputs from the surface reference technique are output to the diagnostic file for evaluation).
This algorithm is based
on a statistical procedure. Although the radar team believes that
a statistical method of this type should be implemented for TRMM, the method
is
relatively new and
the testing has been carried out only on simulated data and on preliminary
TRMM data. Caution on the use of the results is well warranted.
Description of the HDF output variables from 3a-26 can be found in Volume 4 - levels 2 and 3 file specifications provided by the TRMM Data and Information System (TSDIS). The document is available at: http://tsdis02.nascom.nasa.gov/tsdis/Documents/ICSVol4.pdf.
Description of the Method:
Algorithm 3a-26 serves as an alternative way of estimating the space-time rain statistics. The idea behind the method is that because of attenuation at high rain rates and low signal to noise ratios at light rain rates, there will usually exist an intermediate region over which the rain rate estimates are most accurate. Using only these estimates and an assumption as to the form of the probability distribution function (log-normal), the parameters of the distribution can be found by minimizing the rms difference between the hypothetical distribution and the values of the distribution obtained directly from the measurements. Once the distribution is estimated, the mean and standard deviation of the distribution can be calculated [Refs. 1-2].
Useful by-products
from the calculation of the probability distribution of rain rates are
the fractional areas above (or below) particular rain rate thresholds.
These data can be used as inputs to the area-time integral (ATI) methods
that have been proposed [Refs. 3-5]. Although the data can be used
to implement the ATI method, the method used in 3a-26 is not itself an
ATI method.
The behavior of the
estimates depends strongly on the magnitude and type of threshold as well
as the method that is used to determine the high resolution rain rates.
There are 3 methods
that are used to determine the high resolution rain rates: the Z-R
(0th order without attenuation correction), the Hitschfeld-Bordan (H-B),
and the hybrid method of 2a-25. A fourth method, based solely upon
the surface reference method, is implemented in the code but the results
are output only to a diagnostic file for evaluation. For the 3 estimates
of rain rate (Z-R, HB and 2a-25), Q (or zeta as defined in 2a-25)
is used as the threshold parameter. What this means is that if the
threshold is set to a particular value, Q*, then if the measured value
of Q is less than Q*, the corresponding rain rate is accepted - that is,
it is used to update the distribution function of rain rates. On
the other hand, if Q exceeds Q* the corresponding rain rate estimate is
rejected - that is, it is not used to update the distribution function.
As the threshold value, Q, is increased a larger percentage of the rain
rates will be accepted. The converse holds so that as Q is decreased
a smaller percentage of the rain rates will be used in estimating the distribution
function. It should be noted that Q is a proxy for the attenuation
and usually assumes a value between 0 and 1.
If the Z-R method of estimating the high resolution rain rates is considered, the corresponding output files include the rain rate distribution function, zeroOrderpDf, and the mean, standard deviation, and probability of rain derived from the distribution, zeroOrderFit, for 6 different values of the Q threshold. The six values of Q are: 0.1, 0.2, 0.3, 0.5, 0.75 and 0.9999. Which set of values corresponding to which threshold should be used? Simulations suggest that if the total number of rain points is on the order of 500 to 1000, the best accuracy is usually obtained by using a threshold value of 0.3. This corresponds to the 3rd array element so that the monthly mean rain rate (using the Z-R method) over the 5 x 5 degree box (lat, long) at height level, ih, is given by:
mean = zeroOrderFit(lat, long, 1, ih, iq = 3)
The standard deviation and probability of rain are given by:
std dev
= zeroOrderFit(lat, long, 2, ih, iq = 3)
Pr (Rain)
= zeroOrderFit(lat, long, 3, ih, iq = 3)
Simulations indicate
that for a large number of rain points (N > 5000), the use of smaller threshold
values (Q = 0.1 or 0.2) leads to better estimates of the mean space-time
rain rate. In the case of Q = 0.2 we have:
mean
= zeroOrderFit(lat, long, 1, ih, iq = 2)
std dev
= zeroOrderFit(lat, long, 2, ih, iq = 2)
Pr (Rain)
= zeroOrderFit(lat, long, 3, ih, iq = 2)
A useful set for comparison is the choice: Q = 0.999 (array element 6). In this case nearly all of the Z-R rain rate estimates are accepted so that the method reduces to fitting almost all the Z-R derived rain rates to a log-normal distribution:
mean
= zeroOrderFit(lat, long, 1, ih, iq = 6)
std dev
= zeroOrderFit(lat, long, 2, ih, iq = 6)
Pr (Rain)
= zeroOrderFit(lat, long, 3, ih, iq = 6)
The estimate of the mean as determined from the zeroOrderFit HDF output variable should be considered the primary output of the algorithm. Since Q = 0.3 is considered, nominally, as the optimum choice of threshold, the variable, rainMeanTH, has been defined to store these values. In particular:
rainMeanTH(lat,long,ih) = zeroOrderFit(lat,long,ih,1,3)
The accuracy of the
results at other Q thresholds and the statistics derived from the Hitschfeld-Bordan
(hbFit) and rain rates from 2a-25 (fit2A25) will be evaluated as additional
data from the TRMM radar become available.
Relationship of
3a-26 outputs to those of 3a-25:
In comparing the statistics from 3a-25 and 3a-26 there are 2 differences between these data sets that should be kept in mind. The first is that the statistics produced from 3a-25 are conditioned either on the presence of rain or on the presence of a particular type of rain (stratiform or convective). For the 3a-26 products the means and standard deviations derived from the zeroOrderFit, hbFit and fit2A25 arrays are unconditioned - that is, the statistics include both rain and no-rain events. The second difference is that the set of heights for the 3a-26 products is a subset of the heights used for the (low resolution) products of 3a-25.
For the 3a-26 products,
the height levels relative to the ellipsoid are:
hlevel
height above ellipsoid
____________________________________________
1
2 km
2
4 km
3
6 km
4
path-average
For 3a-25 products, the height levels relative to the ellipsoid are:
hlevel height above ellipsoid
______________________________________________
1
2 km
2
4 km
3
6 km
4
10 km
5
15 km
6
path-average
In an earlier versions
of the program, the height levels were defined relative to the local surface.
In the latest versions of 3a-25 and 3a-26 all heights are measured relative
to the earth's ellipsoid.
As an example, assume
that the monthly rain accumulations, MRA (millimeters/month), are to be
computed over the 5 degree x 5 degree latitude-longitude box specified
by
(lat, long) for the
rain rates measured at a height level given by hlevel.
From 3a-25,
the mean rain rate (mm/hr), conditioned on rain being present at height
level, ih, is given by:
rainMean1(lat, long, ih).
To convert this to an unconditioned mean rain rate the quantity is first multiplied by the probability of rain. This can be approximated by the ratio of the number of rain counts (rainPix1(lat,long,ih)) to the total number of observations over the month: ttlPix1(lat, long). To convert this to a monthly accumulation, the unconditioned rain rate is multiplied by the number of hours in a (30 day) month, 720, so that the MRA (mm/month) as derived from the 3a-25 products, is:
MRA(3a-25) = rainMean1(lat,long,ih)*PrRain(lat,long,ih)*720
where
PrRain(lat,long,ih) = rainPix1(lat,long,ih)/ttlPix1(lat,long)
From the 3a-26 products, the MRA (mm/month), using the zeroth-order estimate (Z-R) is:
MRA(3a-26) = zeroOrderFit(lat,long,ih,1,iqthres)*720
For the 3rd threshold, Q = 0.3, the MRA is
MRA(3a-26) = zeroOrderFit(lat,long,ih,1,3)*720
or, equivalently,
MRA(3a-26) = rainMeanTH(lat,long,ih)*720
Processing Procedure:
The basic steps in
the procedure are (first 4 are similar to 3a-25 algorithm):
i. read in data (scan by scan) from 2a-21, 2a-23, 2a-25 and 1c-21
ii. adjust the range gate numbering conventions so that Zm, Zt and R are aligned properly
iii. find the coarse boxes to which the 49 IFOVs belong (coarse resolution boxes are 5 degree x 5 degree latitude-longitude boxes)
iv. resample Zm, Zt and R from the range direction onto the vertical
v. update the estimated probability distribution function for the various rain rate methods at each 5 x 5 degree box at the various heights, and for threshold values.
vi. if the granule crosses the month boundary, do a nonlinear least squares fit to the distributions determined in step 5, assuming a log-normal distribution; from the fitting parameters, calculate the mean, std deviation and probability of rain for each distribution.
vii. Re- initialize the intermediate file
Comments and Issues:
i. Presently, the height
levels are being defined relative to the ellipsoid and not the local surface.
ii. Resampling of
the radar data from the range direction to the vertical is done differently
in 3a-25 and 3a-26. In 3a-25, the estimate of the reflectivity factor
at a particular height is done by a gaussian weighting of the range gates
that intersect that height. 3a-26 uses only a single value
of Z and R - that gate, the center of which, intersects the height of interest.
iii. The Z-R or 0th
method refers to the zeroth order solution of the reflectivity factor from
the basic weather radar equation. In this approximation, no compensation
is made for attenuation so the reflectivity factor is directly proportional
to the measured radar return power. This approximate reflectivity
factor is sometimes called the apparent or measured reflectivity factor.
In converting any estimate of the reflectivity factor, Z(est), to rain
rate, R, the power-law approximation is used:
R = a * Z(est) ** b
where a and b are obtained from 2a-25.
The HB or Hitschfeld-Bordan solution to the reflectivity factor, Z, is obtained by using an specific attenuation-reflectivity factor (k-Z) relationship and then solving the weather radar equation for Z.
Description of the
rain rates from 2a-25 is given in the
documentation for
this algorithm.
References:
[1] Meneghini, R.,
1998, J. Appl. Meteor., 37, 924-938.
[2] Meneghini, R.,
and J. Jones, 1993, J. Appl. Meteor., 32, 386-398.
[3] Short, D.A., K.
Shimizu, B. Kedem, 1993, J. Appl. Meteor., 32, 182-192.
[4] Kedem, B., L.S.
Chiu, G.R. North, 1990, J. Geophys. Res., 96, 1965-1972.
[5] Atlas, D., D.
Rosenfeld, D.A. Short, 1990, J. Geophys. Res., 95, 2153-2160.