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Water Resources
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The Evaluation of Coherent Scatterng in Rain Radar
Jiro Aoyagi and Nobuhiko Kodaira
Remote Sensing Technology Centre of Japan
Uni Roppongi BLDG 7-15-17, Roppongi
Miato-ku, Tokyo 106 Japan
1. Introduction
The main characteristic of rain radar is to be able to measure rainfall intensity quantitatively. The radar equation (6) is used to extract the rainfall intensity from the radar received power. However, there exists one problem in which the rainfall intensity is underestimated by several dB comparing with the actual ground rainfall intensity. Therefore, it is usual to introduce a corection factor to the radar equation so as to reproduce the actual rainfall intensity.
In order to improve such a situation, the authors proposed to introduce the idea of cohererft se\cattering for the mechanism of rain scattering 1 . The idea of the mechanism has been previously confined to incoherent scattering. Incoherent scattering is applicable to the reflection mechanism among the different frequency components by which the Doppler spectrum of the rain echoes is consisted of. However, the authors pointed out that the mechanism of coherent scattering had been overlooked so far, which is never avoidable to describe the interaction among the received amplitudes due to many rain drops contributing to the same frequnecy component.
We reported that the power ratio of coherent scattering to incoherent scattering is 3.6 dB for Marshall-Palmer rain dropsize distribution, which is given as an exponential distribution, which is given as an exponential distribution. And its Z-R relation is 296R. where Z is radar reflectivity where Z is radar reflectivity factor and R is rainfall intensity. It was found that the falue of 3.6 dB acts so as to correct the discrepancy between the radar equation and the rain fall measuremnts.
This paper will make it clear the power ratio more systemstically with the uses of the model distributions and unified series of random numbers. Furthermore. It will be emphasized that the power ratio varies with the rainfall intensity, because the rain drop size distribution based on Webull function 7 is introduced.
This study was carried out as one of the improvement of the accuracy of the rainfall measure ment by the rain radar installed on TRMM (Tropical Rainfall Measuring Mission Satellite which will be launched in August 1997.
2. Data Processing
The radar received poswer due to incoherent scattering is given by the arithmetic sum of the power caused by each rain drop. Such mechanism is valid to deal with the power due to the amplitudes between the different Doppler freqency components. Meanwhile, since the power for coherent scattering is dealed with the amplitudes caued by the rain drops in the same frequency component, the resultant power is affeted due to the relative positioning of the rain drops in space expressed in terms of the phase.
Therefore, in this study, the non-uniform distribution of rain-drops in the phase domain is reproduced using random numbers. Action 32 phase elements are used, on which rain-drops condisting of various sizes are randomly arragned in the number of 4096 as mentioned below. Then the residual amplitude as the result of the vector sum of each amplitude among those phase elements will contribute to the power which is called coherent scattering.
It is already known frm our study that it is not enough to increase only the number of random numbers which correspond to that of rain drops on the phase elements to get the reliable value of the power fatio. If there is a number (m.n) of the random numbers, the values of the power ratio calculated using the number, n. are fluctuating each other. So. By repeating such a calculation for m times using different series of random numbers, we can get fmuch reliable result from the averaged data of m power ratio values, This scheme is similar to that of the averaging of the fluctuating amplitudes of rain echoes in order to get the radar mean received power.
In our calculations, 4096 was chosen for the number, n, and 256 for m by taking account of the ability of the computer used . In this case, the calculation time is beyond 2 hours. If the number becomes twice like 2 n and the number m is fixed then the calculation time will increase by four times.
3. Model Distributions
We assumed that the factor contributing to the power ratio is caused only by the rain drop-size distribution in space (2) Therefore, it will be studied in this section that how much the power rations will vary by various types of model dropare shown in Table 1 The model distributions used here are six types such as are of quadrant, exponential mode, right-half Gaussian mode, right-angled triantle, quadrant and rectangle which are monotonously decreasing functioons except the rectangle.
Table 1 : The relationship between the statistical index and the power ratio of coherent scattering to incoherent scattering for six model drop size distributions.
| No |
Mode |
Graph |
s/x |
 |
Power ratio (dB) |
| 1 |
Arc of quadrant |
 |
7.717 |
0.992 |
- 3.7 |
| 2 |
Exponential dist (N(x) = N0 exp (-x) |
 |
1.0 |
0.707 |
-3.3 |
| 3 |
Right half Gaussian dist |
 |
0.756 |
0.603 |
-3.8 |
| 4 |
Right angles triangle |
 |
0.707 |
0.577 |
-4.4 |
| 5 |
Quadrant |
 |
0.623 |
0.528 |
-5.2 |
| 6 |
Rectangle |
 |
0.577 |
0.5 |
-5.9 |
The value 0.01 was chosen for the increment , dx, of x shown in the row of the exponential distribution of the table. From the viewpoint of the limited number 4096 . if one trys to make it narrower the increment more. There is a risk not to be able to reproduce rightly the original distribution.
As shown in Table 1. Each model distribution has its inherent value of the statisticalindex given by the ratio of the standard deviation to the mean value or the square root of the effetive value, and the corresponding value of the power ratio.
In Figure 1. The relations between the power rations and the indexes for six model distributions are shown by blank dots. And it is quite interesting to see from the figure that the values of the power ratio can be expressed by a quadratic equation as a function of the statistical indexes. Therefore, if the statistical index or the form of the rain drop size distribution is assumed, the values of the power ratio cab be estimated or in reverse.
Figure 1 The Relationships of the power ratio of coherent scattering to incoherent scattering to the
statistical index. Black points show the model distributions. Circles show Weibull distributions.
For instance, if the rain drop-size distribution in given by the exponential mod, the power ratio will be given as 3.3l dB. It is also known that the maximum value of the power ratio is never beyond 3.1 dB Meanwhile, if the rain drop size distribution so as to yield the values below the statistical index for the quadrant model do not exist actually, the power rations less than about -5 dB will never occur in the radar rainfallmeasurement . It is necessary to take notice that the model distributions are monotonously decreasing functions in this case.
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