Statistics of rainfall over paths from 1 to 50 km

Atmospheric Enuironmenr Vol. 12, pp. 2333-2342. 0 Pergamon Press Ltd.1978. Printed in Great Britain.

STATISTICS

000..6981/78/1201-2333

502.00/o

OF RAINFALL OVER PATHS FROM 1 TO 50 km G. DRUFUCA Politecnico di Milano, Italy and R. R. ROGERS McGill University, Canada

(First received 2 November 1977 and in final form 9 May 1978) Abstract-Ten years of data from the tipping-bucket raingauge of the McGill Observatory, Montreal, were analyzed using the synthetic storm technique to generate statistics of rain along paths ranging from 1 to 50 km in length. For each rain event and for every path length considered, the mean and mean-square rain rate were calculated. These were then employed to obtain the standard deviation and the uniforn&y of the rain for each event and every path. Probability distributions of these quantities were compiled and their dependence on path length noted. The tendency for uniform structureis strong on short paths, but decreases with increasing path length. Probability distributions of rain rate were calculated for paths of various lengths, conditioned by the path-average rate. These distributions can be used in applications in which there is a nonlinear dependence on rainfall rate, such as electromagnetic interactions with rain and the scavenging of pollutants. For paths up to 10 km in length, the modal value of rain rate was found to be equal to the pathaverage rate for all combinations of path length and average rate. With increasing path length, a second mode corresponding to zero rain rate or very weak rain becomes increasingly important and eventually dominant. Anuroximatelv half of the total variance of snatial fluctuations in rain rate is accounted for by scales shorter than 8 km. _

2. METHOD OF ANALYSIS

1. INTRODUCTION

Rainfall and its time and space variability have a crucial bearing on such diverse fields as agriculture,

hydrology, radiowave propagation, and the scavenging of atmospheric pollutants. Statistics on short-term and fine-scale rain variability have been derived from raingauge networks (Huff and Shipp, 1969; Freeny and Gabbe, 1969; Sims and Jones, 1975) from singleraingauge records (Briggs, 1968 ; Drufuca and Zawadzki, 1973, and from weather radar data (Austin and Houze, 1972; Katz, 1976; Drufuca, 1977), to cite only a few references. The spatial variability is especially important in hydrology, in which the classical problem is to estimate the area-integrated rainfall from a few point measurements, and in microwave propagation through the atmosphere, for which the path-integrated rain rate is closely related to attenuation. This paper presents statistics of rain rate on straightline paths of various lengths. Results are given which illustrate, from several points of view, the effect of spatial averaging on the pattern of rain rate. Being directly related to the scales of variability in rain, these results indicate which of the scales are most important. Estimates are presented of the probability distribution of rain intensity along a specified path, given only the mean rain rate. The data employed in the analysis are 10 years of tipping-bucket raingauge records (1961-1970) from the McGill Observatory. The same data base was used in an earlier study (Drufuca and Zawadzki, 1975), where it is described in detail.

The raw data consisted of records of rain rate vs time at a point, with a resolution of approx 1 min. During the ten-year sample period, a total of 527 events were recorded, as rain, storms, and showers moved over the site of the gauge. These were converted artificially to profiles of rain rate vs distance by the “synthetic storm” technique, in which time is transformed to distance for each record by applying the translation speed of the rain pattern over the gauge. The speed used was that of the wind at 7OOmb, which is the effective “steering level” for rain systems at Montreal and othei mid-latitude localities. The records thus transformed give rain rate as a function of distance along the direction of motion of the rain patterns, with a spatial resolution which equals 800m on the average, because the translation speed averages about 50 km h-‘, or 0.8 km min-‘. This procedure ofcreating synthetic profiles ofram rate has been employed to generate rain attentuation statistics by Drufuca (1974), Bertok et al. (1977), and Watson et al. (1977). It was also used by Drufuca and Zawadzki (1975) in their study of the joint statistics of rain at two points as a function of point separation distance. Although for any particular rain event the corresponding synthetic profile might give only a poor description of actual rain structure, evidence is good that the statistical properties of rain derived from a large sample of profiles agree closely with actual rain statistics. The apphcations cited have shown that reliable results are obtained if the time lag does not exceed approx 40 min. For the Montreal data, this corresponds to a distance of about 30 km. In the present study, the profiles of rain rate vs distance are visualized as moving over a line segment of a specified length. In l-minute steps, each profile moves across the line from the beginning to the end of the rain event, according to the speed of motion of the pattern. At each step, the profile is used to give the linear extent of rain in increments of 1 mm h- I. Data are thereby accumulated on the rain rate along paths of

2333

2334

G. DRIJFUCA and R. R. ROGERS

Table 1.

various lengths ranging from 1 to 50 km. At every l-mm time step and for each path length considered, the following statistics were computed : the pathaverage rain rate R, the standard deviation ua of rain rate on the path, and the quantity U which we call the uniformity of rain rate on the path. Over a path of length I, these quantities are defined by

s

O
.X+1

R(ybJy

(1)

x

s=-

1

X+1 R’(y)dy

R classes

R values (mm h-t)

32 128 256 512

(2)

1s I

CT: = R2 - (R)’

(3)

u = (R)@.

(4)

In (1) and (2), x denotes the end position of the path on the rainfall profile R(y). This position is shifted each minute according to the speed of motion of the rain pattern. Depending on its duration and the path length considered, each rain event provides a number of samples of each of these quantities. The samples from all 527 events were combined to give ensemble statistics on R, ox, and U. These are described in the next section.

3.

PATH-INTEGRATED STATISTICS

Figure 1 shows the cumulative probability distribution of i? for the various path lengths considered. Owing to the analysis procedure, these are conditional probabilities, given that rain is occurring somewhere on the path. To accommodate the wide range of rain rates encountered, R is measured in terms of class interval on a logarithmic scale. The relation between rain rate and class interval is listed in Table 1. The figure indicates, for example, that the conditional

-2_

IO -

b”

-

a Id37

-4_ IO -

Zkm

lo-5123456789 CLASSES

J

10-5U

I OF R

Fig. 1. Cumulative probability distributions of the l-mitt, path-averaged rainfall rate, provided that rain is occurring somewhere on the path. (For the rain rate intervals corresponding to classes of l?, see Table 1.)

2

3

4

5

CLASSES

6

7

8

9

OF UR

Fig. 2. Cumulative probability distributions of the standard deviation of l-min rainfall rate along the path, provided that rain is occurring somewhere on the path. (The values of us for the different classes are the same as given for R in Table 1.)

Rainfall statistics over paths from 1 to 50 km probability that i? > 4 mm h- ’ (lower limit of Class 3) on a 50-km path, given that there is some rain on the path, equals 9 x 10m2. As the path length becomes shorter the probability of any given k? increases. Thus, 4 mm h- 1 is exceeded twice as frequently on a l-km path as on the 50-km path. A similar general behaviour has been found by Sims and Jones (1975) with lines of recording gauges in Florida and Illinois. Cumulative distributions of the standard deviation are plotted in Fig. 2 for the various paths. The class intervals of ba have the same relation to ffR (in mm h-i) as shown in Table 1. On the 50 km path, for example, oR > 2 mm h-r (lower limit of class 2) for 3.6% of the time that rain occurs. For standard deviations lying within Class 5 or weaker values, the frequency of occurrence increases as the path is shortened. The highest values of OR, however, are observed on 5 and lo-km paths. The rapid decrease in probability from Class 1 to Class 2 signifies, for all path lengths, a preponderance of (TRvalues in Class 1. This implies a tendency for weak spatial variability. For each path length, several ensemble average quantities were calculated. In particular, we determined the ensemble averages of the (path-integrated) mean rain rate, the squared-mean rain rate, and the variance of rain rate. These are denoted, respectively, by (8), ((R)‘), and (a;), where the bent brackets denote ensemble averages, and all three quantities depend on the path length. Figure 3 is a plot of (a;) and oi as functions of the path length, where ai, the variance of the mean rain rate, is defined by aa = ((k?)2) - (8)2. These quantities and their dependence on the averaging distance 1 indicate the relative importance of different scales of variability in the rain pattern. u$, the variance of the mean, reflects the contributions which, roughly speaking, are larger than the path length 1in size. Thus cri decreases with increasing path length as fewer and fewer irregularities remain with scales large enough to be detected after the smoothing over distance 1. (a;), however, the ensemble average of the variance, is sensitive primarily to scales smaller than 1, and hence increases with path

2335

length. These functions are therefore complementary to one another. For unbiased sampling of a stationary, ergodic process, it is well known that ai+ (ai) = const., independent of averaging distance (e.g. Rogers and Tripp, 1964). The sum is not constant for these data because the method of analysis introduces a bias. For any path length 1, data are accumulated only so long as there is rain somewhere on the path. Suppose, for example, a particular rain event has extent L. Data points will then be collected along a total distance approximately equal to L + 21.The resulting sample is therefore smaller for short paths than for long ones, and it includes fewer points lying outside the rain. This exclusion ofzeros with decreasing 1introduces a bias in both u6 and (ui) and prevents their sum from being constant. Even so, the interpretation of these quantities in terms of scales of irregularity is still approximately valid. Figure 3 indicates, accordingly, that half the spectral energy of the fluctuations in rain rate arises from scales shorter than 8 km, the intersection point of the curves. From its defining equation (4), the uniformity parameter may be shown to be related to the mean rain rate and standard deviation by U = l/[l + ($#)‘].

(5)

Thus U is a function only of the coefficient of variations, uR/i?. Obviously U is limited to values between 0 and 1. If the rain is uniform along the path, U= 1. If the rain is limited to only one resolvable location on the path, U+O. Figure 4 shows the probability density function of U for the various path lengths. For short paths, the curves have a strong mode at 0.9~ U < 1, which corresponds to uniform rain along the path. This is explained in part by the fact that the averaging distance is not much longer than the effective resolution distance of the data. The magnitude of the mode in this interval diminishes steadily with increasing path length, and eventually a new mode emerges for U ,
Fig, 3. The average variance of rain rate (CT:) and the variance of the mean rain rate c$. as functions of path length.

G. DRWCA and R. R. ROGBRS

2336

ing class of R, and in fact the median lies within the class for all intervals of R. This correspondence between the median and the class of R holds for ill paths up to 10 km in length. Entries are also givenfor other quantiles and for the maximum R for any combination of R and L.Again for the 10-km path, it is seen that for the mean rain rate between 4 and 8 mm h- ‘, the I-min rain rate exceeds 32 mm h-i: for 1% of the observations and 50 mm h-r for 0.1% of the observations. The maximum rate observed for these conditions was 152mm h-i. Figure 5 shows the conditional probability densities for the case I = 5 km. Curves are entered for each class of R and for the overall (unconditional} probabihty density of R along 5-km paths. As an example, for 8
4. CONDITIONAL

RAIN

PROBABILITIES RATE

l **..OVERALL

OF

In several applications it would be useful to know the probability distribution of rain rate on a path, when given only the path-average rate R. For example, although the attenuation of microwaves by rain is approximately proportional to the path-average rain rate, other effects of rain, such as depolarization and scatter int~feren~, have a strong nonlinear dependence on rain rate and cannot be estimated from R alone. They require knowledge of the distribution of rain rate along the path, at least in a statistical sense. Another example of a nonlinear interaction with’rain is the scavenging of dusts and gaseous pollutants (Stern et al., 1973), for which knowledge of the p~~~abilit~ distribution of R would therefore be helpful. We have computed the conditional probability distributions of 1-min rainfall rate over paths of various lengths, given 8. Results are summarized in Table 2. The entries for a lO-km path show, for example, that the median rain rate is 0.78 mm h- i for cases in which R is in Ciass 1, less than or equal to 2 mm h-i. The median increases steadily with increas-

1

*

l

* D

-5 IO

+

‘,

: 1

,

2

3

,

,

4 5 CLASSES

0

,

6 OF

,

,,

7 R

6

,

9

Fig. 5. Conditional probability densities of I-min r&i rate along a S-km path, for given values of the path-average rain rate. See Table 1 for classes of R and E.

Rainfallstatistics over paths from 1 to 50 km

2331

Table 2. Quantilesof cumulativeconditional probabilitiesof R given R K (mmh-I)

O-2

2-4

4-8

8-16

16-32

32-64

64-128

> 128

12 17

24 36

46 62

84 110

155 >300

l=lkm median 10%

0.87 2

ol$ n$murn

43 45

; 91

12 18 66

47 31 152

106 49 152

151 80 182

>300 137 >300

>300 >300

/=2km median 10% 1% 0.1% maximum

0.85 2 3 5 60

2.5 3.5 6 15 91

5 1.5 16 24 73

12 16 37 55 152

22 36 62 107 152

45 65 113 155 >300

80 116 >300 >300 >300

147 242 >300 >300 >300

0.83 7.8 4 8 60

2.4 4.5 8 17 91

5 8.5 20 40 106

10 22 47 15 152

20 45 85 135 220

40 80 149 >300 1300

15 110 224 >300 >300

152 >300 >300 >300 >300

0.78 1.8 3 5 121

2.5 5 14 32 106

5 10 32 50 152

16 26 55 107 s-300

29 52 125 >300 >300

37 92 145 >300 ,300

75 >300 >300 >300 >300

144 >300 >300 >300 2300

0.73 2.5 17 5

2.3 6 38 17

5 12 75 35

6 30 182 76

10 60 222 110

35 92 230 135

54 142 >300

_ _ _

2.5 3.5

5 7.5

I=5km median 10% 1% 0.1% maximum

I = 1Okm median 10% 1% 0.1% maximum I=20km median 10% O.$ 0 maximum

121

I=5Okm median 10% 1% 0.1% maximum

0.65 2 8 31 165

Fig.

135

165

>300

>300

>300

>300

_

1.5 7 26 52 >300

4 12.5 55 150 >300

5 32 105 220 >300

6 65 110 195 >300

30 95 152 >300 >300

_ _ _

_ _

6, where the conditional probabilities for 4
he considered the conditional probability distribution of 1-min rain rate over the network, given the networkaverage rain rate. He found that the conditional density functions have a dominant mode at the same class of rain rate as that of the area-average rain rate. Moreover, Rancourt’s density functions for the 16-mi2 area are very similar to the ones calculated here for a 5 km path. A direct comparison is shown in Fig. 7 of Rancourtls results and ours. For simplicity only two classes of R are shown, but these are representative of the agreement for all classes. Though it may only be a coincidence, this apparent equivalence between rain statistics over an area and over a path is of considerable interest. It suggests first, and somewhat surprisingly, that smoothing over a S-km path has about the same effect as smoothing over a 16-mi* area. The crucial factor in such averaging is, roughly speaking,

G. DR~FU~~ and R. R. ROGERS

2338

t

---

5km

---

l6mi

2

16<1i:32

10-33 123456769 CLASSES

OF

R

Fig. 7. Conditional probability densities of l-mm rain rate along a 5-km path given the path-average rate (heavy lines) compared with those on a 16-mi’ area given the area-average rain rate (thin lines).

CLASSES

OF

R

Fig. 6. Conditional probability densities of 1-min rain rate along paths of different length, given that the path-average

rain rate is between 4 and 8 mm h- l. the extent of the smoothing filter compared to the size of the region within which the rain is self correlated. It might, therefore, have been expected that a path somewhat longer than 5 km would have been equivalent to 16mi2. Secondly, the apparent equivalence is related to the problem of estimating area rainfall statistics from time statistics. The data employed in our study were actually time records, and for the 5-km path the corresponding time average is approx 6 mm. Therefore it appears that conditional probabilities of 1-min rain rate, relative to a 6-min average, are approximately the same as those of 1-min, point rain rate, relative to a 16-m? average. This can be interpreted as a symptom of ergodicity in the rainfall process, supporting other evidence for ergodicity that has been previously reported. Bussey (1950) found, for instance, a statistical equivalence between l-hour time averaged and 50-km path averaged rain rate. Drufuca (1977) compared cumulative probability distributions of point-rain rate based on (1) ten years of records from a single station, and (2) a few days of rain observations at a large number of locations in the same region. He found that the cumulative distributions were quite similar. The rain process is not truly ergodic because of climatic and topographic effects which, through strong

annual and geographical variability, introduce nonstationarity and inhomogeneity. Nevertheless the symptoms of ergodicity, which occasionally appear, are useful in statistical work by making it possible to build up a large sample of data by combining observations on the spatial structure of rain with those on its temporal structure. 5. CONCLUSIONS

The results of this study may be summarized by the following conclusions : (1) There is a tendency for uniformity in rain over short distances, as indicated by the fact that the modal value of R (in terms of rain rate class) corresponds to that of R for path lengths up to 10 km. (2) Even so, there is evidence that half of the observed variance in the spatial structure of rain is accounted for by variability in scales shorter than 8 km. (3) An apparent equivalence was found between conditional probabilities over a 5-km path and a 16mi’ network. (4) The observed dependence of the uniformity parameter on path length can be explained by a simple rain model. (5) There are further indications of ergodicity in the rain process.

Acknowledgements- The authors are grateful to E. Torlaschi for carrying out the calculations and to K. Rancourt for

2339

Rainfall statistics over paths from 1 to 50 km making his data available for comparison. The collaboration was made possible by a NATO travel grant. Partial support for data analysis and for the preparation of this paper was provided by the National Research Council of Canada and by Centro Naxionale delle Ricerche of Italy.

APPENDIX 1. A SIMPLE RAIN MODEL It is assumed that all rainfall profiles have the same rectangular shape with length Land uniform intensity c. It is straightforward to calculate the uniformity parameter V as these profiles move over a path of length 1. The p~~~lity density of V, from simple geometrical considerations, turns out to be, for L> 1, L-l -s(u-1), L+1

21

REFERENCES Austin P. M. and Houxe R. A. (1972) Analysis of the structure of precipitation patterns in New England. J. appf. Met. 11, 926-935.

Bertok E., De Renxis G. and Drufuca G. (1977) Estimate of attenuation due to rain at 11 GH from raingauge data. Proc. VRSI Commission F symp, La Baule, France. Briggs J. (1%8) Estimating the duration of high intensity rainfall. Met. Msg. 97, 289-293. Bussey H. E. (1950) Microwave attenuation statistics estimated from rainfall and water vapor. Proc. IRE 38, 781-784. Drufuca G. (1974) Rain attenuation statistics for frequencies above 10GHz from raingauge observations. J. Reck. atmos. 8, 399-411. Drufuca G. and Zawadxki I. I. (1975) Statistics of raingauge data. J. appl. Met. 14, 1419-1429. Drufuca G. (1977) Radar-derived statistics on the structure of precipitation patterns. J. uppf. Met. 16, 1029-1035. Freeney A. E. and Gabbe J. D. (1969) A statistical description of intense rainfall. Bell Syst. Tech. J. 48, 1789-1852. Huff F. A. and Shipp W. L. (1969) Spatial correlations of storm, monthly and seasonal precipitations. .I. appl. Met. 8, 542-550. Katz I. (1976) A raincell model. Preprints 17th Radar Met. Conf., Am. Met. Sot., Boston, 442-447. Rancourt K. L. (1977) A statistical study of rain rates in a raingauge network. Unpublish~ M.Sc. thesis, Dept. of Meteorology, McGill University, Montreal. Rogers R. R. and Tripp B. R. (1964) Some radar measurements of turbulence in snow. J. appl. Met. 3,603-610. Sims A. L. and Jones D. M. A. (1965) Frequencies of shortperiod rainfall rates along lines. J. uppl. Met. 14,1970-74. Stem C. A., Wohlers H. C., Boubel R. W. and Lowry W. P. (1973) PM~~ntals of Air Poli~~n. pp. 263-266. Academic Press, N.Y. Watson P. A., Papaioannou G. and Neves J. C. (1977) Attenuation and cross-polarization measurements at 36GHz on a terrestrial path. Proc. URSI Commission F symp., La Baule, France.

---+ L+1

P(V) =

O
otherwise

4

and for L d 1,

p(V)=

i

21 i-L -+-SW-l), l+L I+L

I

OGU
otherwise.

0,

The distribution thus consists of a uniform term plus a delta function, resembling the observed density functions in Fig. 4. The intensity c does not appear in the results. From (6) and (7), the expected value of U and the variance of U are found to be L

41

=

Vp(U)dU = J.+I --4)

d; = (V’) - (V)2 =

I

1 = 1+x

m V$(V)dV s --m

(8) _ ._?_.._ (1 f x)2

x(2 - x) 06x61

3(1+

=

I

(91 ,,

2x - 1

X>,l

3x2(1 + x)2 ’

where x = l/L. Figures 8 and 9 compare these analytical results, for L = 27 km, with the observed values of
p(R)=

-L +

Ic

L-l +-s(R-C), L+l

o
otherwise.

Fig. 8. The ensemble average (V) of the uniformity vs path length as observed (heavy line) and as calculated from the storm model for various values of L.

(10)

G. DRUFWA and R. R. RCIGERS

2340

I O.I-

-

OBSERVED

-

CALCULATED

0/

4

50

20

IO

5

2

I

(km)

Fig. 9. The ensemble variance tri of the uniformity versus path length as observed and as calculated from the storm model with L-27 km.

For L< I,

p(R)=

the same combination grossly underestimates the measured values of the variance. Consequently, the simple rectangular model is inadequate except for describing the uniformity.

T-1 +!&@-CL,/)

1 L+lc

4

I+L

(11)

OtlEWiSe.

The mean and standard deviation of fi, in terms of x= I/L, are then given by

CR)=&

uk =

1

APPENDIX 2 TABULATION OF CONDITIONAL PROBABILITIES

(12)

0< x < 1

3(1 + x) ’ x(22x- -x)cZ 1

x,

(13)

1.

3x2(1 + x)2 ’

It was found that (12) tits the observed values of(R) almost perfectly for L = 65 km and c = 3.4mmh-‘, while (13) for

It is essential in some applications to have nmrmrical values of the conditional probability of point rain rate, given the path-average rain rate. These are given in Tables 3-8 for lengths of 1,2,5,10,20 and 50 km. Point rates and averages areclassi6edonthesamelogarithmicscaleupto512mmh-1. Each entry corresponds to the fraction of the total sample which falls within the indicated class of R. For instance, in Table 3, which is for a l-km path, for 4-z Rd8mm h-r, 0.91 of the total contributions were from rain rates between 4 and 8 mm h-r, 1.9 x lo-’ from rates between 8 and 16mm h-r and 6.4 x lo-’ from rates between 16 and 32 mm h-l.

Table 3. Conditional probability density of R given i?for a 1 km path # (mm h-r)

o-2

2-4

4-8

8-16

16-32

32-64

64-128

>128

R (mmh-‘)

o-2 2-4 4-8 8-16 16-32 32-64 64-128 128-256 256-512

0.988 1.1 (-2) 5.3 (-4) 4.5 (-5) 2.4 (-6) 1.3 (-6) -

-

4.9 (-2) 0.93 2.0 (-2) 6.3 (-4) 9.9 (-5) 4.1 (-6)

1.6 (-2) 5.7 (-2) 0.91 1.9 (-2) 6.5 (-4) 1.7 (-4) 3.9 (-6)

-

2.0 (-2) 2.7 (-2) 0.11 0.81 2.5 (-2) 3.3 (-3) 1.6 (-4) 1.2 (-5) _

1.9 (-2) 1.6 (-2) 3.9 (-2) 0.12 0.74 6.4 (-2) 3.2 (-3) 4.7 (-4) -

7.4 (-3) 2.0 (-3) 1.0 (-2) 2.5 (-2) 4.7 (-2) 0.88 2.8 (-2) 1.5 (-3) -

3.4 (-3) 1.1 (-2) 7.6 (-4) 3.8 (-3) 1.4 (-2) 5.6 (-2) 0.88 3.2 (-2) 1.1 (-3)

4.0 (-2) 1.3 (-2, 6.2 (-2) 1.1 (-2) 2.6 (-2) 0.70 0.15

Rainfall statistics over paths from I to 50 km

2341

Table 4. Conditionai probability density of R given R for a 2 km path

R (mmh-‘)

o-2

2-4

4-8

16-32

8-16

32-64

64-128

> 128

R (mm h-l)

o-2 2-4 4-8 8--16 16-32 32-64 64-128 128-256 >256

0.97 2.4 (-2) 1.5 (-3) 7.8 (-5) 5.9 (-6) 1.9 (-6) -

0.11 0.85 4.0 (-2) 1.87 (-3) 1.5 (-4) 8.2 (-6) 4.1 (-6) -

3.8 (-2) 0.12 0.80 4.1 (-2) 1.4 (-3) 2.4 (-4) 5.3 (-6) -

4.7 (-2) 5.9 (-2) 0.20 0.62 6.8 (-2) 7.6 (-3) 2.9 (-4) 1.1 (-5) _

0.11

3.3 (-2) 8.9 (-2) 0.17 0.53 0.14 5.1 (-3) 7.1 (-4) -

3.4 (-2) 1.4 (-2) 2.3 (-2) 6.8 (-2) 8.6 (-2) 0.68 9.7 (-2) 6.2 (-3) 4.5 (-4)

1.2 (-2) 2.7 (-2) 5.6 (-3) 2.2 (-2) 3.9 (-2) 0.11 0.71 5.7 (-2) 1.0 (-2)

1 1.6 (-;) 7.9 (-2) 0.12 6.3 (-2) 0.65 7.2 (-2)

Table 5. Conditional probability density of R given li for a 5 km path

w(mmh-‘j

o-2

2-4

4-8

8-16

16-32

32-64

64-128

> 128

R (mmh-‘)

o-2 2-4 4-8 8-16 16-32 32-64 64-128 128-256 > 256

0.95 :; (-3) 3.9 f-4) 2.9 ( - 5) 5.2 (-6)

0.23 0.68 8.4 (-2) 6.6 (-3) 3.75 (-4) 7.4 (-5) 7.3 (-6)

9.8 (-2) 0.21 0.60 8.2 (-2j 9.2 (-3) 1.7 (-3) 3.0 (-5)

0.10 1.1 (-2) 0.24 0.40 0.12 3.0 (-2) 1.3 (-3) 9.0 (-5) _

8.4 (- 2) 5.7 (-2) 0.14 0.20 0.33 0.17 2.5 (-2) 1.3 (-3)

0.12 5.5 (-2) 5.4 (-2) 9.5 (-2) 0.13 0.43 0.15 2.2 (-2) 4.2 ( - 3)

2.5 (-2) 1.4(-2) 8.2 (-3) 4.6 (-2) 4.0 (-2) 0.19 0.63 4.2(-2) 8.6 (-4)

0.20

0.60 0.20

-

Table 6. Conditional probability density of R given iii for a 10 km path R (mm h-‘)

o-2

2-4

4-8

8-16

16-32

32-64

64-128

> 128

R (mm h-l)

o-2 2-4 4-8 8-16 16-32 32-64 64-128 128-256 >256

0.92 7.1 (-2) 7.8 (-3) 9.1 (-4) 5.8 (-5) 1.5 (-5) 6.3 (-7) _

0.345 0.51 0.13 1.6 (-2) 2.4 (-3) 4.3 (-4) 1.4 (-5)

0.18 0.23 0.46 9.6 (-2) 1.9 (-2) 5.8 (-3) 2.1 (-4) 9.4 (-6)

0.17 0.14 0.24 0.28 0.12 4.0 (-2) 5.4 ( - 3) 2.5 (-4) 7.6 (-6)

0.15 0.10 0.14 0.18 0.19 0.18 3.9 (-2) 8.0 (-3) 1.9 (-3)

0.11 5.4 (-2) 6.7 (-2) 0.11 0.11 0.30 0.23 1.4 (-2) 8.8 (-5)

9.3 (-3) 1.3 (-2) 1.3 (-2) 2.6 (-2) 8.8 (-2) 0.31 0.50 0.17 3.6 (-3)

0.10 0.20 5.0 (-2) 0.51 0.13

Table 7. Conditional probability density of R given R for a 20 km path R (mm

h-‘)

o-2

2-4

4-8

8-16

16-32

32-64

64- 128

R (mmh-I)

o-2 2-4 4-8 8-16 16-32 32-64 64-128 128-256 >256

0.90

0.46

8.1 (-2) 1.2 (-2) 2.4 (-3) 4.6 (-4) 9.0 (-5) 2.4’(-6)

0.36 0.15 2.3 (-2) 5.2 (-3) 1.7 (-3) 7.7 (-5) 1.2 (-6)

0.28 0.23 0.34 0.10 3.3 (-2) 7.0 (-2) 1.1 (-3) 4.1 (-5)

0.29 0.14 0.21 0.19 9.9 (-2) 5.9 (-2) 9.4 (-3) 2.8 (-3) 8.0 (-4)

0.27 0.10 0.12 0.15 0.12 0.15 7.2 (-2) 5.9 (-3) 4.4 (-6)

0.16 4.3 (-2) 6.7 (-2) 8.7 (-2) 0.12 0.26 0.26 1.0 (-2) 3.9 (-4)

2.7 (-2) 7.2 (-2) 4.4 (-2) 8.0 (-2) 0.15 0.14 0.36 0.10 2.6 (-2)

> 128

2342

G. DR~FLJCAand R. R. ROGEFS Table 8. Conditional probability density of R given ri for a 50 km path

d (mm h-‘)

o-2

2-4

4-8

8-16

16-32

32-64

64-128

R (mm h-‘) o-2 2-4 4-8 8-16 16-32 32-64 64-128 128-256 > 256

0.90 1.2 (-2) 1.9 (-2) 4.8 (-3) 1.1 (-3) 1.7 (-4) 4.5 (-5) 5.5 (-7)

0.58 0.24 0.14 3.1 (-2) 1.2 (-2) 3.5 (-3) 3.9 (-4) 1.4 (-5) 1.0 (-6)

0.43 0.20 0.23 8.0 (-2) 3.0 (-2) 2.3 (-2) 2.4 (-3) 1.2 (-3) 2.8 (-4)

0.40 0.13 0.16 0.14 9.6 (-2) 5.7 (-2) 3.3 (-2) 1.6 (-3)

0.37 8.7 (-2) 0.12 0.11 8.5 (-2) 0.73 9.8 (-2) 2.4 (-3) 2.1 (-5)

0.19 6.3 (-2) 7.9 (-2) 9.8 (-2) 0.13 0.26 0.15 2.7 (-2) 6.3 (-3)

_ _ _ _ _ _ _ _ _

> 128