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2.4. Extinction by Condensed Waters
2.4. Extinction by Condensed Water* 2.4.1. Introduction
Individual hydrometeors such as rain, hail, and snow particles both absorb and scatter incident radiation. The interaction between a particle and the incident field may be computed using electromagnetic scattering theory. Exact solutions are available only for simple shapes and distributions of dielectric properties within the scatterer. Rain may be reasonably modeled with spheres having the dielectric properties of water. More exact models of rain that take into account the nonspherical nature of the raindrops have been tried’ but are not generally used because they require relatively large amounts of computer time and the differences between their results and results based on the exact solution for spherical particles are smaller than the statistical uncertainty of the results due to variations in the drop-size distributions. Approximate calculations of hail and snow scattering have also been made, but the problem of rain scattering has received most attention. From the point of view of both probability of occurrence and severity of effect, rain is most important. 2.4.2. Solution t o the Scattering Problem
The theory of scattering by a single lossy dielectric sphere of size comparable to a wavelength is based on a solution to the scattering problem.’ The solution which is attributed to Mie has been verified experimentally many times (see, for example, Gerhardt el u / . ) . The ~ use of the scattering properties of a single raindrop in the description of the effects of rain also requires a description of the drop-size distribution and an assumption about the statistics of drop location within a volume. The use of measured drop-size distributions and the assumption that the drops are distributed throughout the volume in accordance with a Poisson process allows one to compute the per unit volume extinction (attenuation) and scattering cross sections of
’ T. Oguchi, Radio Sci. 8, 31 (1973).
H . C. Van de Hulst, “Light Scattering by Small Particles.” Wiley, New York, 1957. J. R. Gerhardt, C. W. Tolbert, S. A. Brunstein, and W. W. Bahn, J . Mereorol. 18, 340 (1961).
* Chapters 2.4 and 2.5 are by
R. K. Crane. 177
2.
178
ATMOSPHERIC EFFECTS
rain.4 Computations of the extinction cross section per unit volume or the attentuation coefficient (specific attenuation) made using the average measured dropsize distribution reported by Laws and parson^,^ are given in Fig. 1. For comparison, the attenuation coefficients for liquid water clouds 6in./hr(152.4mm/hr), RAIN ( 2 = 4 . 6 ~ 1 0 ' ) 2.5gm/m3, ---Cumulus Congestus Cloud(Z= 6 . 9 ) 1 inJhr(25.4 mm/hr), Roin(2 3.9 x 10')
l / l O i n . / h r (2.5mm/hr), Rain(Z=l.6x1O3)
(z
0 . 2 ~ ~ 1 ~ 3= , 3.1 x lo-*) Fair Weather Cumulus Cloud
0
20
40
60
80
1
100
FREOUENCY (GHz)
FIG. 1. Attenuation coefficient as a function of frequency for liquid scatterers, rain and clouds. The coefficients were coniputed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. The quantity Z is a measure of the radar cross section per unit volume.
are also presented. The cloud computations were made in an identical manner to those for rain with the exception that the cloud particle-size distributions of Weickmann and aufm Kampe6 were used.
2.4.3. Effects of Drop-Size Distributions Measurements of raindrop-size distributions show large variations for the same location, rain type, and rain rate. These variations imply that the attenuation and scattering properties of rain will also vary for a given rain rate. Figure 2 shows for a frequency of 16 GHz a scatter diagram which presents the results of computations of the attenuation coefficient and rain rate for 4741 drop-size distributions taken from December 1960 to March R. K. Crane, M.L.T. Lincoln Lab., Lexington, Massachusetts, Tech. Rep. 426, ASTIA Doc. AD-647798 (October 1966). J. 0. Laws and D . A. Parsons, Trans. Amer. Geophys. Un. 24,452 (1943). H. K. Weickmann, and H. J . aufm Kampe, J . Mereorol. 10, 204 (1943).
2.4.
179
EXTINCTION BY CONDENSED WATER
F' C
/
5w
/
0 LL
w
0
u
0.1
2
0
&3 2
w I-
a c
-
/
RAIN RATE ( mm/ hr 1 I
I
1 111111
0.01
I
I
I IIOII
0.1
I
I
I 1 1 1 1 1 1
I
I
1
I I l l l U
10
RAIN RATE ( i n c h l h r l
FIG.2. Scatter diagram of attenuation versus rain rate for a frequency of 16.03 GHz. and drop temperature of 10°C. Regression curves shown are linear (L) and logarithmic (R),
1962 at Franklin, North Carolina.' The dashed lines indicate the maximum and minimum possible attenuation coefficients for a given rate if we assume the worst and best possible monodisperse drop-size distributions with the limitation that the drop size lie within the range of sizes attributable to rain. It is evident that, in natural rain, the fluctuations in attenuation due to variations in the drop-size distributions are significantly less than those theoretically possible. The root mean square (rms) variation in the data points about the linear least square fit or linear regression curve at a fixed rain rate is 28 %, and the rms variation about the logarithmic regression line is 22 %. Figure 2 shows that the linear and logarithmic lines are better fits at opposite ends of the rain-rate scale.
' E. A. Mueller and A. L. Sims, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071-RR3 (September 1967).
2.
180
ATMOSPHERIC EFFECTS
2.4.4. Models for Attenuation Computations
The statistical analysis of many drop-size distributions provides models for use in calculating attenuation effects. The models are easier to use than the standard table of computations made from averaged drop-size distributions and have the advantage of also providing an estimate of the possible variation in the estimated attenuation. Linear and logarithmic regression models for several frequencies are listed in Table I . The data for North TABLE I . Attenuation versus Rain-Rate Models Logarithmic model" A = uR4 Frequency (GHz) 2.8' 7.5 9.4 16.0 34.9 69.7b
U
0.000459 0.00459 0.00870 0.0374 0.225 0.729
P 0.954 1.06 1.10 1.10 1.05 0.893
Linear model" A = UR
rms error
(%I 28 30 22 I0
rms error U
0.00481 0.00932 0.0403 0.234
( %)
31 36 28 12
Models computed from 4741 data points with Franklin, North Carolina, drop-size measurements, R in millimeters per hour and A in decibels per kilometer. [See E. A. Mueller and A. L. Sims, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071 -RR3 (September 1967)]. Data based on 4590 drop-size measurements analyzed by E. A. Mueller and A. L. Sims [Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-0271-F (May 1967)l. (I
Carolina are reasonably representative. The results of the analysis of the North Carolina data do not differ significantly from computations made with drop-size data from other areas.839Using the models given in Table I we computed the attentuation coefficients for a 100 mm/hr (4 in./hr) rain rate given in Fig. 3 together with the values computed using the Laws and Parsons and Marshall and Palmer drop-size distributions." The scatter E. A. Mueller and A. L. Sinis, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071-F (May 1967). R. K . Crane, M.I.T. Lincoln Lab., Lexington, Massachusetts, Tech. Note 1968-33. ASTlA Doc. AD-678079 (September 1968). l o J. S. Marshall and W. McK. Palmer,J. Mefeorol. 5, 165 (1948).
:
100 I
-5
2.4.
I
181
EXTINCTION B Y CONDENSED WATER
I
1
I
10
011 0
I
10
I
20
1
I
30 40 FREQUENCY ( GHz 1
1
50
I
60
I
70
FIG. 3. Comparison of attenuation versus rain-rate models for a drop temperature of 10°C and a rate of 101.6 mniihr (4 in./hr). Coefficients were computed using the Marshal and Palmer drop-size distribution (the dashed curve) and the Laws and Parsons drop-size distribution (the solid curve). Logarithmic and linear regression of attenuation on rain rate are indicated by ( 0 ) and (O), respectively, where wings on the points represent rms fluctuations of data about the model curve.
2.
182
ATMOSPHERIC EFFECTS
diagrams for each of the other frequencies are similar to the one presented in Fig. 2. For frequencies below 35 GHz and at the 100 mm/hr (4-in./hr) rain rate, the logarithmic regression model provides a better fit to the data than either of the averaged drop-size distributions. At 35 GHz, the linear model provided a fit that was as good as either of the averaged distributions. Both the model computations and the computations based on averaged drop-size distributions provide an estimate of the attenuation coefficient. In general, however, it is not sufficient to compute the attenuation for a line-ofsight path through rain. Rain both scatters and absorbs the incident radiation. The scattered radiation may be singly or multiply scattered and eventually reach the receiving antenna. For incoherent measurement systems (radiometers) the radiative transfer equation governs propagation through the absorbing and scattering medium. The detailed mathematical analysis of this radiation-transfer problem has been summarized by Crane." We shall consider here only the single-scattering approximation. For this purpose let us define the single-scattering albedo as the ratio of energy scattered to total energy lost from the incident wave including both absorption and scattering. 2.4.5. Single-Scattering Albedo
The single-scattering albedo is readily computed for a given drop-size distribution using Mie t h e ~ r y The . ~ results obtained using the Laws and Parsons drop-size distribution are given in Fig. 4. For rain, the single4/40in /hr. SLEET i grn/rn3, HAIL
-
s
08-
w m
-
4
06-
fl
E5 In
04-
W
m
z
02-
0
20
40
60
80
400
FREQUENCY ( G H z )
FIG. 4. Single-scattering albedo versus frequency. Albedo values were computed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. [Note that 1/10 in./hr = 2.5 mni/hr, 1 in./hr = 25 mm/hr, and 6 in./hr = 150 rnmihr.1 R. K . Crane, Proc. IEEE 59, 173 (1971).
2.4.
EXTINCTION BY CONDENSED WATER
I83
scattering albedo is less the 0.6, and at frequencies below 10 GHz it is less than 0.1. At the lower frequencies, where the total attenuation is low, it is expected that the nonscattering-transmission equation will be sufficient. At higher frequencies, multiple scattering may be important. Using the criterion that the product of the single-scattering albedo and the optical depth should be less than unity, we can calculate the distance at which multiple scattering becomes important. The results of computations using the Laws and Parsons distribution are given in Fig. 5. The computations show that in rain and at frequencies above 60 GHz, multiple-scattering effects may be important.
FREQUENCY (GHz)
FIG.5. Mean distance to multiple scattering as a function of frequency. Distance values were computed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. [Note that 1/10 in./hr= 2.5 mm/hr, I in./hr=25 rnm/hr, and 6 in./hr = 150 mmihr.1
2.4.6. Multiple Scattering
Multiple scattering does not affect coherent transmission systems (interferometers) in the same way as incoherent systems. Except for scattering by drops spaced closer than a wavelength, the multiple-scattered signals have random phases when compared with the attenuated direct signal. Since the drops are separated by distances larger than a wavelength the effect of multiple scattering on the attenuation of the coherent signal is negligible.' The transmission equation for a single-scattering medium, therefore, applies
184
2.
ATMOSPHERIC EFFECTS
to coherent transmission systems for all combinations of optical depth and single-scatter albedo. The multiply scattered signal is, however, present in the forin of excess noise. The effect of multiple scattering on an incoherent transmission system is to increase the received signal when compared to that predicted on the basis of single scatter. The attenuation experienced by an incoherent transmission system, therefore, should be less than that for coherent systems. At present, neither adequate measurements nor an adequate theoretical treatment of multiple scattering effects have been made. The computations of attenuation using the calculated attenuation coefficients as reported in the literature are all made using the transmission equation for a single-scattering medium and strictly apply only to coherent transmission systems. No significant departures from estimates based on single-scattering theory have been reported. 2.4.7. Measured Attenuation
The first comprehensive tabulation of experimentally determined values of the attenuation coefficient versus rain rate for a large number of frequencies in the centimeter and millimeter wavelength bands was made by Ryde and Ryde.” Recently, Medhurstt3 corrected and extended the earlier work of Ryde and Ryde and compared the updated curves of attenuation versus rain-rate with the experimental data then available in the literature. A number of additional measurements were available for comparison with the calculated results. Comparison did not, however, validate the method. Medhurst observed a “ marked tendency for the observed attenuations to fall well above levels which, according to the theory, cannot be exceeded.” From this he concluded that “the applicability of the Mie theory to the practical rainfall situation cannot be said to be demonstrated.” Since the publication of Medhurst’s paper, several additional experiments have been performed and rep~rted.’~.’’ Except for the work of Semplak and Turrin,I6 each of the papers reported the same tendencies as Medhurst noted. The measurements just cited and those reviewed by Medhurst tend to be crude in the handling of meteorological data. Generally long paths were used with relatively few rain gauges. The only data that show good agreement between estimated and measured attenuation were published by Usikov l 2 J . W. Ryde and D. Ryde, General Electric Co., Res. Lab., Wernbley, England, Rep. 8516 (August 1944). l 3 R . G . Medhurst, IEEE Trans. Antennas Propagat. AP-13, 550 (1965). l 4 J. Bell, Proc. Inst. E k e . Eng. 114, 545 (1967). I s J. F. Roche, H . Lake, D. T. Worthington, C. K . H . Tsao, and J. T. deBettencourt, IEEE Trans. Antennas Propagat. AP-18, 452 (1970). l 6 R . A. Sernplak and R . H . Turrin, EellSysr. Tech. J . 48, 1767 (1969).
2.4.
EXTINCTION BY CONDENSED WATER
185
et al."
In this work an extremely short, 50 m path was used, and data were reported only when identical rain rates were observed at two gauges 30 m apart along the path. The tendency toward agreement in experiments with short paths and dense rain gauge networks together with differing amounts of disagreement in others suggests that the discrepancies are not due to the inapplicability of Mie theory as suggested by Medhurst but due to the inadequacy of meteorological data. More intensive radar observations' show that rain is composed of many small, relatively heavy showers imbedded in a latger area of light rain. A network of rain gauges spaced far apart could easily miss these localized showers. Radar measurements may be used to observe the spatial changes in rain intensity that correspond to the temporal changes in rain rate at the surface. A radar measurement of the structure of a New England summer shower obtained on July 28, 1967 is shown in Fig. 6. The radar used was the Contours 5 d 0 Apart
0
10
20
30
40
TRANSVERSE DISTANCE AT 150 KM (km)
FIG.6. Weather-radar observations of showery rain using the Millstone Hill L-band radar at 1.4" elevation and 275"-290" azimuth. Data was compiled on July 24, 1967.
Millstone Hill L-band radar which has approximately a 2-km3 resolution volume.' The radar map of rain intensity shows several small cells with widths the order of 5 km across and peak to minimum rain intensity distances the order of 3 km. "0.Ya. Usikov, V. L. German, and I . Kh. Vakser, Ukr. Fiz. Zh. (Ilkr. Phys. J . ) 6, 618 (1961).
Individual hydrometeors such as rain, hail, and snow particles both absorb and scatter incident radiation. The interaction between a particle and the incident field may be computed using electromagnetic scattering theory. Exact solutions are available only for simple shapes and distributions of dielectric properties within the scatterer. Rain may be reasonably modeled with spheres having the dielectric properties of water. More exact models of rain that take into account the nonspherical nature of the raindrops have been tried’ but are not generally used because they require relatively large amounts of computer time and the differences between their results and results based on the exact solution for spherical particles are smaller than the statistical uncertainty of the results due to variations in the drop-size distributions. Approximate calculations of hail and snow scattering have also been made, but the problem of rain scattering has received most attention. From the point of view of both probability of occurrence and severity of effect, rain is most important. 2.4.2. Solution t o the Scattering Problem
The theory of scattering by a single lossy dielectric sphere of size comparable to a wavelength is based on a solution to the scattering problem.’ The solution which is attributed to Mie has been verified experimentally many times (see, for example, Gerhardt el u / . ) . The ~ use of the scattering properties of a single raindrop in the description of the effects of rain also requires a description of the drop-size distribution and an assumption about the statistics of drop location within a volume. The use of measured drop-size distributions and the assumption that the drops are distributed throughout the volume in accordance with a Poisson process allows one to compute the per unit volume extinction (attenuation) and scattering cross sections of
’ T. Oguchi, Radio Sci. 8, 31 (1973).
H . C. Van de Hulst, “Light Scattering by Small Particles.” Wiley, New York, 1957. J. R. Gerhardt, C. W. Tolbert, S. A. Brunstein, and W. W. Bahn, J . Mereorol. 18, 340 (1961).
* Chapters 2.4 and 2.5 are by
R. K. Crane. 177
2.
178
ATMOSPHERIC EFFECTS
rain.4 Computations of the extinction cross section per unit volume or the attentuation coefficient (specific attenuation) made using the average measured dropsize distribution reported by Laws and parson^,^ are given in Fig. 1. For comparison, the attenuation coefficients for liquid water clouds 6in./hr(152.4mm/hr), RAIN ( 2 = 4 . 6 ~ 1 0 ' ) 2.5gm/m3, ---Cumulus Congestus Cloud(Z= 6 . 9 ) 1 inJhr(25.4 mm/hr), Roin(2 3.9 x 10')
l / l O i n . / h r (2.5mm/hr), Rain(Z=l.6x1O3)
(z
0 . 2 ~ ~ 1 ~ 3= , 3.1 x lo-*) Fair Weather Cumulus Cloud
0
20
40
60
80
1
100
FREOUENCY (GHz)
FIG. 1. Attenuation coefficient as a function of frequency for liquid scatterers, rain and clouds. The coefficients were coniputed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. The quantity Z is a measure of the radar cross section per unit volume.
are also presented. The cloud computations were made in an identical manner to those for rain with the exception that the cloud particle-size distributions of Weickmann and aufm Kampe6 were used.
2.4.3. Effects of Drop-Size Distributions Measurements of raindrop-size distributions show large variations for the same location, rain type, and rain rate. These variations imply that the attenuation and scattering properties of rain will also vary for a given rain rate. Figure 2 shows for a frequency of 16 GHz a scatter diagram which presents the results of computations of the attenuation coefficient and rain rate for 4741 drop-size distributions taken from December 1960 to March R. K. Crane, M.L.T. Lincoln Lab., Lexington, Massachusetts, Tech. Rep. 426, ASTIA Doc. AD-647798 (October 1966). J. 0. Laws and D . A. Parsons, Trans. Amer. Geophys. Un. 24,452 (1943). H. K. Weickmann, and H. J . aufm Kampe, J . Mereorol. 10, 204 (1943).
2.4.
179
EXTINCTION BY CONDENSED WATER
F' C
/
5w
/
0 LL
w
0
u
0.1
2
0
&3 2
w I-
a c
-
/
RAIN RATE ( mm/ hr 1 I
I
1 111111
0.01
I
I
I IIOII
0.1
I
I
I 1 1 1 1 1 1
I
I
1
I I l l l U
10
RAIN RATE ( i n c h l h r l
FIG.2. Scatter diagram of attenuation versus rain rate for a frequency of 16.03 GHz. and drop temperature of 10°C. Regression curves shown are linear (L) and logarithmic (R),
1962 at Franklin, North Carolina.' The dashed lines indicate the maximum and minimum possible attenuation coefficients for a given rate if we assume the worst and best possible monodisperse drop-size distributions with the limitation that the drop size lie within the range of sizes attributable to rain. It is evident that, in natural rain, the fluctuations in attenuation due to variations in the drop-size distributions are significantly less than those theoretically possible. The root mean square (rms) variation in the data points about the linear least square fit or linear regression curve at a fixed rain rate is 28 %, and the rms variation about the logarithmic regression line is 22 %. Figure 2 shows that the linear and logarithmic lines are better fits at opposite ends of the rain-rate scale.
' E. A. Mueller and A. L. Sims, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071-RR3 (September 1967).
2.
180
ATMOSPHERIC EFFECTS
2.4.4. Models for Attenuation Computations
The statistical analysis of many drop-size distributions provides models for use in calculating attenuation effects. The models are easier to use than the standard table of computations made from averaged drop-size distributions and have the advantage of also providing an estimate of the possible variation in the estimated attenuation. Linear and logarithmic regression models for several frequencies are listed in Table I . The data for North TABLE I . Attenuation versus Rain-Rate Models Logarithmic model" A = uR4 Frequency (GHz) 2.8' 7.5 9.4 16.0 34.9 69.7b
U
0.000459 0.00459 0.00870 0.0374 0.225 0.729
P 0.954 1.06 1.10 1.10 1.05 0.893
Linear model" A = UR
rms error
(%I 28 30 22 I0
rms error U
0.00481 0.00932 0.0403 0.234
( %)
31 36 28 12
Models computed from 4741 data points with Franklin, North Carolina, drop-size measurements, R in millimeters per hour and A in decibels per kilometer. [See E. A. Mueller and A. L. Sims, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071 -RR3 (September 1967)]. Data based on 4590 drop-size measurements analyzed by E. A. Mueller and A. L. Sims [Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-0271-F (May 1967)l. (I
Carolina are reasonably representative. The results of the analysis of the North Carolina data do not differ significantly from computations made with drop-size data from other areas.839Using the models given in Table I we computed the attentuation coefficients for a 100 mm/hr (4 in./hr) rain rate given in Fig. 3 together with the values computed using the Laws and Parsons and Marshall and Palmer drop-size distributions." The scatter E. A. Mueller and A. L. Sinis, Illinois State Water Survey, Urbana, Illinois, Tech. Rep. TR-ECOM-02071-F (May 1967). R. K . Crane, M.I.T. Lincoln Lab., Lexington, Massachusetts, Tech. Note 1968-33. ASTlA Doc. AD-678079 (September 1968). l o J. S. Marshall and W. McK. Palmer,J. Mefeorol. 5, 165 (1948).
:
100 I
-5
2.4.
I
181
EXTINCTION B Y CONDENSED WATER
I
1
I
10
011 0
I
10
I
20
1
I
30 40 FREQUENCY ( GHz 1
1
50
I
60
I
70
FIG. 3. Comparison of attenuation versus rain-rate models for a drop temperature of 10°C and a rate of 101.6 mniihr (4 in./hr). Coefficients were computed using the Marshal and Palmer drop-size distribution (the dashed curve) and the Laws and Parsons drop-size distribution (the solid curve). Logarithmic and linear regression of attenuation on rain rate are indicated by ( 0 ) and (O), respectively, where wings on the points represent rms fluctuations of data about the model curve.
2.
182
ATMOSPHERIC EFFECTS
diagrams for each of the other frequencies are similar to the one presented in Fig. 2. For frequencies below 35 GHz and at the 100 mm/hr (4-in./hr) rain rate, the logarithmic regression model provides a better fit to the data than either of the averaged drop-size distributions. At 35 GHz, the linear model provided a fit that was as good as either of the averaged distributions. Both the model computations and the computations based on averaged drop-size distributions provide an estimate of the attenuation coefficient. In general, however, it is not sufficient to compute the attenuation for a line-ofsight path through rain. Rain both scatters and absorbs the incident radiation. The scattered radiation may be singly or multiply scattered and eventually reach the receiving antenna. For incoherent measurement systems (radiometers) the radiative transfer equation governs propagation through the absorbing and scattering medium. The detailed mathematical analysis of this radiation-transfer problem has been summarized by Crane." We shall consider here only the single-scattering approximation. For this purpose let us define the single-scattering albedo as the ratio of energy scattered to total energy lost from the incident wave including both absorption and scattering. 2.4.5. Single-Scattering Albedo
The single-scattering albedo is readily computed for a given drop-size distribution using Mie t h e ~ r y The . ~ results obtained using the Laws and Parsons drop-size distribution are given in Fig. 4. For rain, the single4/40in /hr. SLEET i grn/rn3, HAIL
-
s
08-
w m
-
4
06-
fl
E5 In
04-
W
m
z
02-
0
20
40
60
80
400
FREQUENCY ( G H z )
FIG. 4. Single-scattering albedo versus frequency. Albedo values were computed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. [Note that 1/10 in./hr = 2.5 mni/hr, 1 in./hr = 25 mm/hr, and 6 in./hr = 150 rnmihr.1 R. K . Crane, Proc. IEEE 59, 173 (1971).
2.4.
EXTINCTION BY CONDENSED WATER
I83
scattering albedo is less the 0.6, and at frequencies below 10 GHz it is less than 0.1. At the lower frequencies, where the total attenuation is low, it is expected that the nonscattering-transmission equation will be sufficient. At higher frequencies, multiple scattering may be important. Using the criterion that the product of the single-scattering albedo and the optical depth should be less than unity, we can calculate the distance at which multiple scattering becomes important. The results of computations using the Laws and Parsons distribution are given in Fig. 5. The computations show that in rain and at frequencies above 60 GHz, multiple-scattering effects may be important.
FREQUENCY (GHz)
FIG.5. Mean distance to multiple scattering as a function of frequency. Distance values were computed using the Laws and Parsons drop-size distribution with a drop temperature of 0°C. [Note that 1/10 in./hr= 2.5 mm/hr, I in./hr=25 rnm/hr, and 6 in./hr = 150 mmihr.1
2.4.6. Multiple Scattering
Multiple scattering does not affect coherent transmission systems (interferometers) in the same way as incoherent systems. Except for scattering by drops spaced closer than a wavelength, the multiple-scattered signals have random phases when compared with the attenuated direct signal. Since the drops are separated by distances larger than a wavelength the effect of multiple scattering on the attenuation of the coherent signal is negligible.' The transmission equation for a single-scattering medium, therefore, applies
184
2.
ATMOSPHERIC EFFECTS
to coherent transmission systems for all combinations of optical depth and single-scatter albedo. The multiply scattered signal is, however, present in the forin of excess noise. The effect of multiple scattering on an incoherent transmission system is to increase the received signal when compared to that predicted on the basis of single scatter. The attenuation experienced by an incoherent transmission system, therefore, should be less than that for coherent systems. At present, neither adequate measurements nor an adequate theoretical treatment of multiple scattering effects have been made. The computations of attenuation using the calculated attenuation coefficients as reported in the literature are all made using the transmission equation for a single-scattering medium and strictly apply only to coherent transmission systems. No significant departures from estimates based on single-scattering theory have been reported. 2.4.7. Measured Attenuation
The first comprehensive tabulation of experimentally determined values of the attenuation coefficient versus rain rate for a large number of frequencies in the centimeter and millimeter wavelength bands was made by Ryde and Ryde.” Recently, Medhurstt3 corrected and extended the earlier work of Ryde and Ryde and compared the updated curves of attenuation versus rain-rate with the experimental data then available in the literature. A number of additional measurements were available for comparison with the calculated results. Comparison did not, however, validate the method. Medhurst observed a “ marked tendency for the observed attenuations to fall well above levels which, according to the theory, cannot be exceeded.” From this he concluded that “the applicability of the Mie theory to the practical rainfall situation cannot be said to be demonstrated.” Since the publication of Medhurst’s paper, several additional experiments have been performed and rep~rted.’~.’’ Except for the work of Semplak and Turrin,I6 each of the papers reported the same tendencies as Medhurst noted. The measurements just cited and those reviewed by Medhurst tend to be crude in the handling of meteorological data. Generally long paths were used with relatively few rain gauges. The only data that show good agreement between estimated and measured attenuation were published by Usikov l 2 J . W. Ryde and D. Ryde, General Electric Co., Res. Lab., Wernbley, England, Rep. 8516 (August 1944). l 3 R . G . Medhurst, IEEE Trans. Antennas Propagat. AP-13, 550 (1965). l 4 J. Bell, Proc. Inst. E k e . Eng. 114, 545 (1967). I s J. F. Roche, H . Lake, D. T. Worthington, C. K . H . Tsao, and J. T. deBettencourt, IEEE Trans. Antennas Propagat. AP-18, 452 (1970). l 6 R . A. Sernplak and R . H . Turrin, EellSysr. Tech. J . 48, 1767 (1969).
2.4.
EXTINCTION BY CONDENSED WATER
185
et al."
In this work an extremely short, 50 m path was used, and data were reported only when identical rain rates were observed at two gauges 30 m apart along the path. The tendency toward agreement in experiments with short paths and dense rain gauge networks together with differing amounts of disagreement in others suggests that the discrepancies are not due to the inapplicability of Mie theory as suggested by Medhurst but due to the inadequacy of meteorological data. More intensive radar observations' show that rain is composed of many small, relatively heavy showers imbedded in a latger area of light rain. A network of rain gauges spaced far apart could easily miss these localized showers. Radar measurements may be used to observe the spatial changes in rain intensity that correspond to the temporal changes in rain rate at the surface. A radar measurement of the structure of a New England summer shower obtained on July 28, 1967 is shown in Fig. 6. The radar used was the Contours 5 d 0 Apart
0
10
20
30
40
TRANSVERSE DISTANCE AT 150 KM (km)
FIG.6. Weather-radar observations of showery rain using the Millstone Hill L-band radar at 1.4" elevation and 275"-290" azimuth. Data was compiled on July 24, 1967.
Millstone Hill L-band radar which has approximately a 2-km3 resolution volume.' The radar map of rain intensity shows several small cells with widths the order of 5 km across and peak to minimum rain intensity distances the order of 3 km. "0.Ya. Usikov, V. L. German, and I . Kh. Vakser, Ukr. Fiz. Zh. (Ilkr. Phys. J . ) 6, 618 (1961).