A Statistical Description of Intense Rainfall
A E Freeny, J D Gabbe
Bell Labs, Holmdel
1969
Bell System Technical Journal
In 1968, a movie, Image of a Thunderstorm, was produced by Freeny and Gabbe modelling the intensity and movement
of a heavy rain shower using data collected from an 11 by 10 grid
of rain gauges, 0.8 miles apart at Crawford Hill, Holmdel, New Jersey in
July 1966.
The movie consisted of an 11 by 10 set of shaded squares where rain fall of
200mm/hr was represented as a white square and a black square indicated
no rain fall. The grey values between these two indicated the amount of rain fall
that was encountered at a specific time. The film showed 20 minutes
of rain fall in this early 4 minute computer generated film generated
via an SC-4020 microfilm recorder.
Each shaded square consisted of an 8 by 8 array of points with
0, 16, 32, 48, or 64 points being drawn to give the various grey scales.
This paper contains a statistical summary of the 14,000,000 measurements taken during 27 rainfalls
in a six-month period in 1967 from a 96-station, rapid-response rain qauge network spread over a
rectangular area 13 by 14 kilometers centered near Crawford Hill, New Jersey. The analysis emphasizes
rain rates greater than 50 millimeters per hour, which interfere with radio transmission in the
10 to 30 GHz frequency range.
Heavy rain rates are relatively rare events, come in irregular bursts, and do not appear amenable
to description by simple analytic distributions. This paper presents statistics concerning the behavior
of rain rates at a point in space, the relationship of rain rates separated in space or time, and the
relationship of average rain rates on pairs of paths in various configurations.
I. INTRODUCTION
This paper presents some statistics from the rainfall data collected
on a rain gauge network during the period from June 1 to
November 30, 1967. The network consists of 96 gauges spaced
approximately 1.3 km apart on a rectangular grid centered near
Crawford Hill, Holmdel, New Jersey. The design of the rain gauges and
the equipment for recording data from the network are described
elsewhere. [1] [2]
In communications, interest in rain-rate data arises from the
relationship of attenuation of radio signals in the 10 to 30 GHz
frequency range to the number and size of raindrops present in the
transmission path. The quantity of water that the signal penetrates
is directly related to the average rain rate on the path. Thus a major
direction of our analysis was toward a statistical description of the
behavior of rain rates at a point in space, the relationship of two
rain rates separated in space or in time, and the relationship of
average rain rates on pairs of paths in various configurations.
Knowledge about these relationships, particularly for rain rates
greater than 50 mm per hour, where substantial attenuation occurs [3],
is important for the design of microwave radio transmission systems.
A major characteristic of the rain-rate data taken in this experiment
is its extreme variability, which increases as the rain rate increases.
The separations between successive readings in time (10 s) and neighboring
readings in space (∼ 1.3 km) are too large to provide continuous
representation of rain-rate behavior through time and space. Because
the time series at the measuring stations cannot be considered even
piecewise stationary during intense rainfalls, the usual time series
techniques are not applicable. This leads to large numbers of descriptive
statistics rather than a concise representation of the characteristics
of rainfall.
A brief summary of the results and general conclusions given in
Section VIII is: On the basis of 14,000,000 measurements
obtained during 27 rainfalls that occurred in the Crawford Hill locale
during the 1967 recording season, the empirical probabilities of
observing point rain rates above 50, 100, 150, and 200 mm per hour are
found to be 4.3 × 10-4, 1.3 × 10-4,
4.2 × 10-5, and 1.0 × 10-5
respectively; the joint probability that the rain rate exceeds a given
value at both of two stations simultaneously decreases rapidly at short
distances as the separation between the stations increases, and goes
through a minimum at a separation of about 12 km; the joint probability
that the average rain rate on both of a pair of parallel paths exceeds a
given value decreases as the path length increases, and shows a minimum
for paths separated by 9 km; and the probability that the average rain
rate will exceed 150 mm per hour on a single path 6.5 km long is 200 times
greater than the probability that the average rain rate will
simultaneously exceed 150 mm per hour on both of two parallel paths
6.5 km long and 6.5 km apart.
Detailed descriptions of the components of the analysis are:
Section II: treatment of the data, selection of a
subset for analysis and procedures used for the detection and removal
of spurious data;
Section III: general characteristics of observed
rain, rainfall behavior at individual stations and in individual
rainfalls, and the partitioning of the data made necessary by the
variability of rain;
Section IV: statistics on point rain rates, both
for the complete subset of the data and under the condition that the
rain rate is greater than 50 mm per hour;
Section V: conditional and joint probabilities of
various events for pairs of stations at selected separations in space
or time, and the relationship of the probabilities with distance and
time;
Section VI: results of an analysis of average rain
rates on pairs of paths in various configurations;
Section VII: engineering calculations and
benchmarks for translating the relative probabilities calculated
from the selected sample to annual probabilities based on the
duration of the experiment, comparison of the results of various analyses
using these data, and presentation of a result from another body of
data.
II. TREATMENT OF THE DATA
Most bodies of data containing several million data points can be
separated into segments of primary and secondary interest in the context
of a particular analysis. Furthermore, raw data in such quantities
inevitably contains some fraction of specious readings. In this section
a brief description of the data collection system is followed by several
presentations of rain-rate data and an outline of the data selection and
screening procedures. The closing subsection contains remarks concerning
the data retained for analysis.
2.1 The Basic Data
The data were acquired with a network of rapid-response rain-rate
gauges most of which were mounted on telephone poles about 15 meters
above the ground and well clear of all obstructions [1] [2]. The
collecting surface of the gauge has an area of 478 cm2;
the response characteristics of the gauge are such that the rain rate
represents an average over less than one second. Although occasional
anomalies in the data appear to be traceable to the gauges, they seem to
give satisfactory readings for rain rates greater than about 10 mm per
hour.
The network consisted of 96 stations, about 85 percent of which were
operational at any given time. The area around Crawford Hill was divided
into squares 1.29 km on a side by grid lines oriented north-south and
east-west; one of the gauges was emplaced as close to the center of each
square as practicable. The actual positions of the devices are shown
on the map in Fig. 1. The data from the rain gauges,
in the form of oscillator frequencies, is telemetered via telephone lines
into the telephone central office serving the gauge location. At the
central offices, which act as collecting points, the readings are
commutated and forwarded to the Crawford Hill Laboratory, where the
frequencies are detected. These frequencies and some auxiliary data are
then automatically recorded on magnetic tape.
During periods of rain, the network is scanned once every ten
seconds to produce a scan of readings. The gauges are not read
simultaneously, but are scanned at the rate of ten gauges per second in a
sequence dictated by the telemetry arrangements and indicated by the
numbers in the lower left corners of the grid squares in Fig. 1.
The gauge at Crawford Hill (station 33) was read once per second
separately from, and in addition to, the regular scan. This high
frequency sample is recorded as part of the auxiliary data. The solid
line between the transmitter (T) and receiver (R) shows an experimental
microwave transmission path. A pair of the parallel paths (P11-P12, P21-P22)
and a pair of the adjoining paths (I1-IA-I2,) used in the path analysis
are indicated by dashed lines. During the time covered by this report,
the devices were not interchanged among grid locations, so that a station
number always refers to the same device. The telemetry system is exposed
to interfering signals which may produce spurious readings, creating some
of the data-screening problems discussed in Section 2.2.
The data recorded at Crawford Hill are transferred from the original
tapes onto tapes compatible with the computers used in the data analysis.
For a variety of reasons this process could not be carried out
successfully with about ten percent of the tapes; these data were lost.
The next major processing programs transformed the data from oscillator
frequencies to rain rates by applying calibration functions[4].
As noted in Ref. 4 these calibrations have been
optimized for the higher rain rates at the expense of some loss of
relative accuracy below 25 mm per hour. The programs also
(i) discarded time periods during which rain rates greater than 15 mm
per hour were recorded at fewer than four stations during each scan for
at least 20 consecutive minutes,
(ii) indicated some kinds of questionable data,
(iii) appended additional information, such as a list of stations known
to be inoperative,
(iv) organized the data into a format suitable for subsequent analysis,
and
(v) produced the isometric plots described in Section 2.2.
2.2 Data Selection and Screening
It is neither practical nor desirable to analyze all the data acquired
from the rain gauge network. This is partly because more than 70
percent of the data were taken when the rain was too light to seriously
affect microwave transmission in the frequency range of interest,
partly because a tiny but important fraction of the data are spurious,
and partly because the total number of data points is so large (over 14,000,000).
The first selection procedure operating on the data was natural
selection, that is, owing to various equipment failures we have no data
for a number. of rainfalls. The second selection took place during the
data processing where lengthy periods, during which virtually no rain
was falling, were discarded. At this point about 430 hours of processable
data remained. The procedures used to identify data of interest
from the available 430 hours and remove spurious values are based on
several graphical presentations of the data.
The isometric plots, which were produced for each rainfall in the
final stage of the data processing, are the most basic graphical presentation.
These plots give one a spatial and temporal appreciation of the
rainfall as a whole, in addition to displaying particular features of
the data in an illuminating manner.
Three hours of data from the rainfall of July 28, and one hour from
July 11, 1967 are displayed in Fig. 2a and b. Each solid
trace represents the rain-rate measurements from a single station, and is
constructed by connecting six-reading averages in Fig. 2a and two-reading
averages in Fig. 2b. The time in minutes is measured from the first
data set. The traces are arranged in a horizontal and vertical grid
which is isomorphic to the geographic grid in Fig. 1.
Each trace is labeled with the station number. No trace is plotted if the station
is known to be inoperative.
On the basis of the isometric plots, a sample of about 110 hours of
rain was chosen for analysis in the following somewhat subjective
but operationally convenient manner. An hour's worth of rain was
included or rejected as a unit, with the exception of one isolated
20-minute shower which was included. A unit during which any six-reading
average rain rate exceeded 50 mm per hour was included. A
unit during which no six-reading average exceeded 30 mm per hour was
excluded unless it fell between two units which were included on the
basis of the 50-mm-per-hour criterion. Units with occasional
six-reading averages greater than 30 mm per hour but less than 50 mm
per hour were included if there were other units on the same day
chosen on the basis of the 50-mm-per-hour criterion, and excluded
otherwise. Subject to these constraints, the beginning and end of each
time period was determined somewhat arbitrarily in terms of such
operational conveniences as accepting an entire rainfall or
eliminating a few bad magnetic tape records near the beginning or end
of a rainfall. For example the entire three-hour period shown in
Fig. 2a was accepted, and no effort was made to delete the first 20
minutes, during which there is little rain. On the whole, we were
generous in the inclusion of data. This selection procedure reduced the
data base to about 110 hours of interesting rain.
Attention was next directed to removing spurious values from the
data. It is clearly impractical to attempt a point-by-point examination
of so large a body of data (3.4 × 106 data points
remained after the selection process), so various anomalies in the data
were identified and studied with the aid of three additional
presentations of information on the distribution of rain rate.
The most primitive displays of distributional information consisted of
histograms giving the frequency of occurrence of rain rates for
each station for each rainfall. Figure 3 contains
four examples of the histograms from the rainfall of July 28, 1967,
comprising 3840 readings per station, or about 11 hours of rain. The
ordinate is the number of occurrences in 2 mm per hour intervals plotted
on a logarithmic scale so that the frequencies of occurrence of the
higher rain rates are visible in the diagram. Note the very substantial
differences in the frequency of rain rates greater than 50 mm per hour
at the four stations.
The frequency distributions were also displayed as probability plots
of the rain-rate observations (ordinate) against quantities of the
standard normal distribution (abscissa) [5].
(In this presentation, a normal distribution would plot as a straight
line with slope equal to the standard deviation.)
Figure 4 contains four examples of the
distributions of data pooled for an entire rainfall and two examples of
the distribution of rain rates from individual stations. Adjacent curves
are translated vertically by 100 mm per hour to increase the clarity of
the plot. Curves a and b are from the rainfall of July 28, 1967 pooled
over all operational stations. The data base for curve a
contains 314,021 observations; that of curve b is smaller by 22
observations identified as outliers. Curves c and d are from the rainfall
of July 11, 1967. Curve c is based on 285,098 observations pooled over
all stations, curve d on the 281,869 observations that remained after
deletion of the data from a malfunctioning station.
Curves e and f are based on the 3840-observation sets collected during
the rainfall of July 28, 1967 from stations 10 and 84, respectively.
The distributions are far from normal, having either an overwhelming
surfeit of low (less than 40 mm per hour) values or too long a
high-rain-rate tail, depending on the point of view. The presentation has
been retained because:
(i) of the large number of analytic distributions with fixed parameters
that we have tried, none provides a good fit to even a minority of the
empirical distributions;
(ii) the rain rates higher than 40 mm per hour, in
which we are particularly interested, are approximately linear on the
normal plots (implying that these observations may be considered to be
part of separate subpopulations); and
(iii) the normal plots are readily produced and provide a convenient
standard for comparisons among the empirical distributions.
Percentile information was also extracted from the frequency
distributions and used to make pseudogeographic percentile plots.
Figure 5 displays four of these. Each set of five dots connected by a
vertical line represents the five points of the empirical cumulative
rain-rate distribution corresponding to 99.9, 99.5, 99.0, 98.0, and 97.0
percent in descending vertical order. Each block along the abscissa
corresponds to a row of stations on the grid of Fig. 1.
Thus these plots display the geographic distribution of the
high-rain-rate tail of the rain-rate distribution.
The first step in the data screening process was to remove data from
malfunctioning stations. The term station covers all the equipment for
measuring, telemetering, and recording data associated with a single
rain gauge. Malfunctioning stations were identified by looking at the
isometric plots for stations behaving differently from the surrounding
stations, and at the pseudogeographic percentile plots for stations
having distributional tails that appeared peculiar in the context of
distributions of the surrounding stations. When a station was judged to
have been malfunctioning, it was treated as though it
had been inoperative throughout that rainfall; no more refined attempt
was made to separate the spurious from the valid data.
Figure 5b shows that the tail of the distribution
of data from station 67 during the rainfall of July 11, 1967, is much
longer than the tails of the distributions recorded by the other
stations. Part of the corresponding isometric plot, Fig. 6a,
shows a rain rate in excess of 400 mm per hour at station 67 starting
at minute 410 and lasting for nearly an hour, while the surrounding
stations show no more than a light drizzle. Such a rainfall event is
very unlikely in New Jersey and we discarded the data from station 67
for this storm. However the decision is not always so easy.
An earlier portion of the same storm is shown
enlarged (two-reading-averages, rather than six-reading averages)
in Fig. 2b. The behavior of station 67 after
minute 25 is peculiar, but not outstandingly so, and without the
subsequent evidence this data would have been retained. Another example
of anomalous data is the behavior of station 29 during the storm of
November 17, 1967. Figures 6b and 5d show this.
The high-rain-rate tail of the distribution from station 29 is
certainly very different from that of any other station. However, the
isometric plot shows that the burst of intense rainfall
(∼ 400 mm per hour) lasted less than five minutes.
As station 29 is on the edge of the network and there are only two
stations which can be classed as close neighbors, there is little
context in which to form a judgment of the validity of the data.
In this case the data were retained.
These data-screening decisions may have substantial effects on the
high-rain-rate tail of the distributions. Curves c and d in Fig. 4 are
probability plots showing the distribution of the data from the rainfall
of July 11, 1967 with and without the data from station 67,
respectively. The effect on the shape of the rain-rate distribution
for high rain rates is very evident. The probability of observing a
rain rate greater than 100 mm per hour is changed by a factor of about 3;
this factor increases rapidly at still higher rain rates (a factor of
more than ten at 200 mm per hour) . These results are typical for
malfunctioning stations. Substantially all the retained data for rain
rates greater than 350 mm per hour comes from station 29 during the
November 17 rainfall, so the shape of the distribution in this region
depends entirely upon this event.
After the malfunctioning stations had been disregarded, a number of
individual bad data points were removed from the retained stations.
Some of these data points are definitely not connected with the
rainfall measurements. For example, the source of the numerous spikes
at time 530 on Fig. 2a is a faulty magnetic tape
record. In other cases, such as the spikes at time 425 on station 36,
405 on station 45, and 370 and 435 on station 46, one cannot identify
the extraneous sources. Outliers have been eliminated by examining the
histograms and deleting any isolated points more than 50 mm per hour
above what would otherwise be the last point in the tail of the
frequency distribution. Such a point appears at 256 mm per hour
in Fig. 3b. This point is also shown as the highest
point on curve e in Fig. 4 (just under curve d) and
is obviously inconsistent with the rest of the distribution.
Although the consequences of discarding outliers are less drastic
than those of disregarding stations, the effect on the shape of the tail
of the distribution is significant. Curves a and b in Fig. 4
show two probability plots of the data from the rainfall of July 28, 1968
pooled over all the operating stations. Curve a includes the outliers;
curve b is without the 22 points identified as outliers by examining
the station histograms individually. The five highest readings were
among those judged outliers, and their deletion changes the shape of
the distributions for rain rates greater than 250 mm per hour
substantially. Extrapolations to rain rates above 300 mm per hour would
give very different results in the two cases. The fate of spurious data
in the more common rain rates is of less concern. They are overwhelmed
by the valid readings and have little effect on the observed
distribution.
In retrospect it is clear that the magnitude of the interference
problem was not fully appreciated by the designers of the telemetry
system. When one is interested in infrequent events
(a probability of 10-6 corresponds to 30 seconds per year)
cleanliness of the raw data is very important. We emphasize this point
for future experimenters.
2.3 The Retained Data
The selection and screening processes leave a body of
data (the retained data) containing 3,418,623 observations at
96 grid locations on about 110 hours of rainfall.
The observations are distributed roughly as in Table I.
TABLE I - DISTRIBUTION OF RAIN-RATEs IN THE RETAINED DATA
Rain Rate
(millimeters per hour) | min.
max. | 0
30 | 30
50 | 50
100 | 100
150 | 150
200 | 200
250 | 250
300 | 300
350 | 350
400 | 400
∞ |
|---|
| Number of observations | 3,280,000 | 90,600 | 35,300 | 9440 | 3600 | 948 | 124 | 32 | 12 | 12 |
|---|
| Percent of observations | 95.9 | 2.7 | 1.0 | 0.3 | 0.1 | 0.03 | 0.004 | 0.001 | 0.0003 | 0.003 |
|---|
These data were collected over 27 rainfalls of which nine may be
classed as heavy (that is, at least 12 gauges showed rain-rate readings
greater than 125 mm per hour). Parts a, c, and d of Fig. 5
show heavy rainfalls; part b shows a light rainfall.
The question of what population of rain rates the data represent
is more fully discussed in Section 3.6.
For the present, notice that the selection of the retained data is
operationally defined and does not lead to a sample whose relationship
to the sampled population may be precisely defined in statistical terms.
Moreover, the selection procedure severely decimates the low-rain-rate
part of the rainrate distribution. This is intentional because the
experiment is specifically oriented toward measurements of rain rates
high enough to interfere seriously with the transmission of microwave
radio signals, and the rain gauges are not designed to provide reliable
readings for rain rates below about 10 mm per hour (in fact, once wet
the rain gauges record some low rain rate for long periods of time) [6].
The retained data may be used to calculate empirical probabilities
regarding the occurrence of various events given the condition that the
rain rate at some station is greater than 50 mm per hour. Viewed this
way, the sample contains only 50,000 points, a number which must be
considered small in the context of the enormous variations observed in
rainfall. The top 1100 of these (rain rate greater than 200 mm per hour)
are in addition particularly subject to the screening problems discussed
in Section 2.2. Some evidence presented in
Section VII suggests that the distribution of rain
rates above 50 mm per hour derived from the retained data is not
atypical of summer and fall rains in littoral northern New Jersey.
Broader interpretations of the data should be viewed with appropriate
caution.
III. GENERAL CHARACTERISTICS OF OBSERVED RAIN
The outstanding characteristics of the rainfalls observed are the
variety of behavior, both within and among individual rainfalls, and
the extremely rapid fluctuations in the measured rain rates. First we
discuss a possible demographic model for rain rate and the poolings
of the data necessitated by the variability. Then we discuss sampling
rates at one station; typical spatial, temporal, and distributional
behavior; and a systematic difference among the observations from
different stations. Finally we consider the matter of the relationship
between this sample and the general population of rain rates.
3.1 A Demographic Model
It is convenient to have a demographic model in terms of which the
measurements can be discussed. The very elementary phenomenology offered
in the following paragraph is only a convenient hypothesis which
should not be regarded as being confirmed in any sense by the data,
some of which is discussed later in this section.
(Of course, the model would not be presented if it were contradicted.)
A large variety of populations of rain rates is produced by different,
very local meteorological conditions. Each population has a
characteristic distribution. A rain gauge records a sample of rain rates
from the populations associated with those rain clouds which happen to
pass over it. Thus a rain gauge observes a composite distribution which
is the result of a two-stage sampling process, the first stage of which
is the natural selection of the cloud, the second, the discrete
scanning interval. Many rain-rate populations are inextricably
intermingled as the data are recorded; indeed, as it may rain
simultaneously from several strata of clouds, even a single rain-rate
measurement may represent a mixture of populations.
Our extensive attempts to classify the heavy rain rates by
distributional characteristics have been defeated by the large
variety of distributions observed, and so subsets of the data have
been pooled in various ways in attempts to synthesize some typical
mixtures of populations. The unit of pooling is the data from one
station during one rainfall. Some pools are: the data from each station
pooled over all rainfalls for which the station was operational;
the data from each rainfall pooled over all operational stations;
the heavy-rainstorm pool, which contains all the data from the nine
heavy rainfalls; and the grand pool which contains all 3.4 × 106
retained observations.
Behavior at Stations
Figure 7 shows two time series obtained from
station 33 during the rainfall of July 25, 1967. The thin trace connects
successive readings from the every-one-second (fast) sample. The thick
trace represents the every-ten-second (slow) sample and is drafted as
if the rain rate remained at the sampled value for ten seconds.
(The equipment design prevents the every-ten-second and every-one-second
readings from ever being coincident in time.)
The plot shows two notable features: the rain rate changes very
rapidly from second to second, on occasion by a factor of three;
and the magnitude of the fluctuations varies with the rain rate.
These large, rapid fluctuations are characteristic of rain rates
measured on a short (less than one-second) time scale with a small-area
gauge [7]. It is clear that the every-ten-second
frequency of sampling the stations is not nearly fast enough to provide
an accurate time-series representation of the rain rate at any gauge,
but instead acts as a low-pass filter.
The rain-rate distributions of the slow and fast samples for an
hour of the rainfall of July 25 are plotted against quantiles of the
normal distribution in Fig. 8. Curve a, the slow
sample, is based on 360 observations; curve b, the fast sample, is based
on 3600 observations. The curves are nearly identical
below the +2 quantile, indicating that the slow sample may be regarded
as an unbiased sample of the fast sample and the distribution can be
expected to be the same for large enough samples. The last three points
in the upper tail of the slow sample affect the shape of the distribution
above the second quantile significantly. One of these points is shown at
time 370 seconds on Fig. 7. It is almost surely
spurious, as are the other two outliers; the distributional effect of
these three points should be discounted.
Another characteristic of rainfall, which is apparent in Fig. 2,
is that heavy rain comes in bursts. Although some of the showers last
for almost 30 minutes, longer showers usually contain a number of short
bursts. In general, these periods of rain cannot (at our sampling rate)
be considered piecewise-stationary time series. Our many attempts to
analyze periods during which the rain rate was greater than
35 mm per hour yielded results badly confounded by the nonstationarity
produced by occurrences of these bursts. Thus one cannot naively apply
time-series analysis techniques to the heavy rain rates; such common
statistics as correlation and autocorrelation coefficients,
and power spectra are not directly meaningful.
An Overview of the Grid
At about time 400 on July 28, 1967 it began to rain heavily on two
separate portions of the grid, the southeastern portion and the north
central portion (Fig. 2a). The southeastern rain
built up slowly (in general) for about half an hour, decreased abruptly,
and then resumed at lower intensities and intermittent intervals for
another two hours. The north central rain was heavy for about ten
minutes and was followed by 40 minutes of light rain. At about time 460,
very heavy rain started in the northwest portion of the grid and
continued for between 20 minutes and an hour before dying out.
The general behavior of this rainfall is typical. It rains heavily first
on one part of the grid then on another, the regions of heavy rainfall
are fairly local (often only a few stations register heavy rain), and
it seldom rains heavily for as long as an hour. The downpour is not
continuous but has substantial variations in intensity throughout its
lifetime.
In view of the rapid fluctuations of rain rate at the stations,
it is not surprising that the correlation among the fine structures
observed at different stations is poor. It would require an extraordinary
mechanism to synchronize one-second rain-rate fluctuations over an area
of several square kilometers. The temporal relationship among rain rates
at the different stations may be seen on Fig. 2b.
Even allowing for the fact that the stations are not sampled
simultaneously (see Section 2.1) and that there may
be a propagation lag, the structure of the rain-rate traces is not
especially similar from station to station.
Of the 27 rainfalls, that of July 11 is one of the best examples of
systematic motion of a rainstorm. The traces in Fig. 2b
behave as though a rain cloud roughly 3 km wide and many kilometers long,
with the long axis oriented in the WNW-ESE direction, moved NNE across
the grid with a velocity of about 15 km per hour.
3.4 Distributional Characteristics
In discussing the distributional characteristics of rain rates,
our attention is focused on rain rates greater than 50 mm per hour.
Over our sample this is the upper 1½ percent of the empirical distribution,
although for particular stations and rainfalls the amount of data
above 50 mm per hour varies from zero to five percent.
The purpose of this subsection is to demonstrate that there seem to
be many different kinds of rainfall, so the data contain observations
from many different populations with different rain-rate distributions
and mixture ratios. Many common analytic distributions (the normal,
log normal, gamma family, and so on) have been tried in attempts to
find a simple description of the observed composite distributions which
behaves reasonably in the tail region (rain rate ≥ 50 mm per hour);
but none have been found that can serve even approximately.
We conclude that the large sampling variation and the complexity
of the mixtures precludes obtaining reliable estimates of distributional
parameters of rain rates above 50 mm per hour from any small sample.
Some samples we consider small in this context are single stations for
a season and the entire network for a single intense rainfall. For still
higher rain rates the present overall sample may be inadequate (there
are only 1100 observations for rain rates above 200 mm per hour).
In the absence of a distribution on which to base precise estimates of
formal statistics, such as confidence limits, discussion of these matters
can best await the analysis of the 1968 data.
The remainder of this subsection indicates some of the distributional
variety observed. The arrival of heavy rainfall in bursts, as indicated
in Fig. 2, makes it obvious that the distribution of rain
rates is very different for different segments of the same rainfall measured at
the same station. The same holds true for the distributions observed at
different stations for the same rainfall. This may be seen from the two
probability plots for stations 10 and 84 (curves e and f in Fig. 4),
from the differences in the tails of the distributions shown in Fig. 5,
and the four histograms in Fig. 3. The rain-rate
distributions differ greatly among stations even for the rainfall of July 11,
whose active portion, shown in Fig. 2a, appears
deceptively similar for many of the traces. Apparently many populations of rain
rates coexist within rainstorms.
In an attempt to find a common mixture of populations, the stations
have been pooled within rainfalls. Curves b and d in Fig. 4
are examples of such pooled distributions, and the difference between them is
typical of what is observed. A glance at the percentile plots of
Fig. 5 indicates that it is unlikely that pooling across
these collections of stations can produce distributions with similar shapes
above 50 mm per hour. The next pooling is within stations, across rainfalls,
that is, at each station for the whole season. Naively this seems to be the
procedure most likely to produce a typical sample and thus similar
distributions. Cloudbursts, however, are limited in extent. Some of the
stations were never hit by cloudbursts, whereas others appear to be deluged
quite often. Figure 9a shows normal probability plots
of the rain rates from each of four stations pooled for the season,
and Fig. 9b shows the matching upper five percent
of the empirical cumulative distributions. Adjacent curves have been
translated vertically by 100 mm per hour. Curves aa and ba are for
station 29, ab and bb for station 1, ac and be for station 98,
and ad and bd for station 76. The data base for each station contains
about 39,000 observations. The distributions are seen to be quite different.
The overall pseudogeographic percentile plot, Fig. 10,
shows the upper tails of the distribution of rain rates from each station
pooled across all the rainfalls (for which the station was operational) in
the final sample. This plot covers the upper tail down to a rain rate
of 50 mm per hour for most of the stations. The figure indicates the
variety of distributions, and shows that the four samples selected for
Fig. 9 are not atypical. It also confirms the notion
that data from a single station over a season are not a sufficient sample
on which to base rainrate statistics for rain rates above 50 mm per hour.
3.5 Systematic Behavior
Figure 10 gives an idea of the areal distribution of
the intense rain rates during the 1967 season. It rained most intensely on the
northwestern part of the network and least intensely in the northeastern
and southerly central portions. Intense rain was also observed by the
stations in the southeastern section of the grid. A more detailed study
of the data reveal that station 1 always records more intense rain than most
of the network and station 76 always records rain of lower intensity than most
stations. Three possible explanations are confounded in the present data:
first, there may be a systematic geographic effect;
second, the effect may be inherent in the equipment of the
stations (the calibrations of the gauges may have shifted); and
third, one may be witnessing a sampling fluctuation. Action has been
initiated to determine whether the effect is connected with the equipment.
Analysis of data from 1968 may help to distinguish between the first and
third possibilities.
3.6 What Does the Sample Represent?
The sample contains virtually all of the available data for rain
rates greater than 50 mm per hour and about a quarter of the data
below 30 mm per hour. As such it can be expected to be representative
(with appropriate proportionality adjustments) of rainfall with rain
rates greater than 10 mm per hour within a small area of northern
littoral New Jersey over a relatively short period. Neither the year
nor the locality are known to be exceptional in any respect.
The distributional stability of the observations above 50 mm per
hour is poor, as has been indicated, which shows that the sample is
small in the context of the demographic composition. In Section VII,
the data are compared with the 1958 data taken at Island Beach, New
Jersey by Mueller and Sims and found to be reasonably similar [8].
Some informal indication of the dispersion of various statistics is given
in the following sections by noting the quartiles and interquartile ranges
of the distributions. (The quartiles are the 25 and 75 percent points
of the distribution; the interquartile range is the difference between
them.) While there is nothing to indicate that the results of this study
are in any way atypical of heavy rainfall in this part of New Jersey,
it is unlikely that they permit an accurate assessment of the extremes
of rainfall that may not infrequently be encountered.
Extension of these results outside the immediate locality will, of
course, be subject to even greater uncertainties.
IV. POINT-RAIN-RATE STATISTICS
4.1 The Grand Pool
Any interpretation of this section must take into account the fact
that the retained data include all of the available data for rain rates
greater than 50 mm per hour but only about 25 percent of the
available data for rain rates below 30 mm per hour.
Figure 11 is the grand histogram based on
the 3,418,623 observations (about 110 hours of rain) retained from 1967.
Log frequency of occurrence is plotted linearly on the ordinate in order to
accommodate the range of frequencies. The interval of the histogram is 2 mm
per hour. Although only about 25 percent of the data below 30 mm per
hour were selected, the relative frequencies in this range are unlikely
to have been substantially affected by this selection. However, the
general tendency of the rain gauges to give biased readings which
greatly overestimate the rain rates below about 15 mm per hour make
this histogram unsuitable for hydrological use.
The empirical cumulative distribution is shown in Fig. 12a,
and the upper four percent of the distribution is shown on an expanded scale
in Fig. 12b. The shape of these curves may be adjusted to
approximately represent the distribution of the entire 430 hours of rain
recorded by scaling the probabilities below 0.96 by the factor 0.99/0.96.
Above 0.96 the new probability, PN, would be given
approximately by the expression PN = 0.75 + 0.25 PB,
where PB is the probability on the plot.
The cumulative distribution points up the small fraction of the data
which lies above about 50 mm per hour. The empirical probabilities,
Fig. 12a, are transformed to quantiles of the standard
normal distribution and the rain rates as ordinates are plotted against the
quantiles in Fig. 12c.
(Only the data above the mean, quantile 0, rain rate about
8 mm per hour, are shown.) If the data were a random sample from a normal
distribution, the points would lie along a straight line. The
data are not even approximately normally distributed, which must of
course be the case as the frequency
distribution (Fig. 11) decreases monotonically from the
lowest interval.
As the distribution has such a long tail and is skew (no negative
rain rates have been recorded although all that water must have gotten
up there somehow), it seems reasonable to look at various longtailed
skew distributions such as the gamma, Wiebull, and extreme value.
These distributions yield probability plots which do have an appreciably
better appearance than the normal probability plot
of Fig. 12c; however, they all show substantial
deviations from linear behavior near rain rates of 100 mm per hour.
Efforts to improve the linearity by adjusting the proportion of the data
below rain rates of 30 mm per hour were not successful.
Since a distributional description in such circumscribed conditions is
of only very limited usefulness the details of these efforts are not
reported here. Typical of the results is Fig. 12d,
which is a plot of log rain rate vs. quantiles of the standard
normal distribution (that is, a probability plot of the log normal
distribution). As in Fig. 12c, only data above the
mean are shown. Those familiar with probability plots will recognize that
the inflection at the 2.5 quantile cannot be removed by adjusting the
fraction of data assumed to lie above 30 mm per hour [5].
In the present data-analytic situation, where there is no simple
analytic distributional description of the data, and indeed the data may
represent a drawing in unknown proportions from many populations,
it is not possible to compute formal confidence limits for the various
estimates of probabilities. An informal indication of the
dispersion in the data may be obtained by examining the behavior of the
distribution from individual stations, some of which are
displayed in Fig. 9. The behavior of all the stations
is summarized in histograms showing the frequency distribution of rain
rates corresponding to 95, 97, 98, 99, 99.5 and 99.9 percent for the data
from each station pooled over all the rainfalls. These histograms are
presented in Fig. 13. All 96 stations were used.
While 11 stations were operational only about half of the time and four
were operational less than one-third of the time, inspection of the
distributions from some of these stations indicates that an assumption
of random loss of observations is not unreasonable. Summary statistics
comprising the value of rain rate from the grand pool, together with
mean, upper, and lower 25 percent points (quartiles), and the
interquartile range (spread of the middle 50 percent) of the histograms,
are given in Table II.
TABLE II - STATISTICS ON THE DISTRIBUTION OF VARIOUS PERCENTILES FROM THE RAIN-RATE DISTRIBUTION
| Percent of rain rate distribution | Rain rates in mm per hour |
|---|
| Value from the grand pool | The 99 stations each station pooled for the season |
|---|
| Lower quartile | Upper quartile | Interquartile range | Mean |
|---|
| 95.0 | 28 | 17 | 29 | 12 | 25 |
|---|
| 97.0 | 34 | 22 | 38 | 16 | 32 |
|---|
| 98.0 | 41 | 27 | 47 | 20 | 41 |
|---|
| 99.0 | 62 | 37 | 75 | 38 | 59 |
|---|
| 99.5 | 91 | 50 | 109 | 50 | 81 |
|---|
| 99.9 | 162 | 79 | 160 | 81 | 127 |
|---|
The overall rain-rate percentiles are larger than the means
of the corresponding histograms. The amount by which they are larger increases
as the overall percent (and thus the rain rate) increases until the largest
overall percentile corresponds to the upper quartile of the histograms.
This indicates that most of the high-rain-rate data came from only a few
stations. The interquartile ranges are between 50 and 75 percent of the means
of the histograms and tend to increase with rain rate. The histograms of the 99th
and higher percentiles suggest a bimodal distribution, as if the data contained a
component of very high rain rates which (because it is infrequent) is never
observed at a large number of the stations. Thus Fig. 13
indicates a substantial sampling uncertainty above the 99th percentile;
this uncertainty would be much greater for data collected at only a few stations.
While it is possible to make histograms similar to those of Fig. 13
for other poolings of the data (say across stations for each rainfall)
this has not been done because the rain-rate percentiles are affected by
the dilution of the high-rain-rate data when periods of light rain are
included. As a result of the selection procedure (Section II)
this dilution varies greatly (by a factor of five) from rainfall to
rainfall. Thus percentiles for different rainfalls are not really
comparable, but percentiles from poolings across rainfalls are.
This remark does not apply to the data when the condition in
Section 4.2 is applied.
4.2 Conditioned Probabilities
All the probabilities in this section are based on the condition that
the rain rate is greater than 50 mm per hour. This simple condition
may be applied straight-forwardly to the point-rain-rate data because
in arriving at the point-rain-rate statistics each data point is
treated as independent, and no relationships in time or space are taken
into account. The 50-mm-per-hour condition is chosen because
our selection procedure accepts virtually all data meeting this criterion.
Thus at the cost of severely limiting the range of rain rates examined,
one may work with probabilities that are well-defined and independent
of the subjective criteria used to determine the beginning and end of
periods of interesting rain.
The frequency distribution of rain rates greater than 50 mm per hour
(in 2 mm per hour intervals) is given by the part of Fig. 11
which lies above this value. The corresponding empirical cumulative
distribution appears in Fig. 14, which shows the full
range from 0 to 1.0, as well as the upper tail on an expanded scale from
0.9 to 1.0. Once more there is no obvious overall correspondence between
the data and the more common distributions. We have however fitted the
function
N = ½ α exp(-βR), R ≥ 65mm/hr (1)
where N is the number of observations in a 1 mm per hour interval,
R is the rain rate, and α and β are fitted coefficients,
to the tail of the frequency distribution of Fig. 11.
The fitting procedure was weighted least squares, with an ad hoc
weighting that emphasizes the goodness
of fit for the higher rain rates. The estimates a and b of α
and β are 7754 and 0.02408, respectively. The properties of the model
are best displayed by the residuals, which are plotted against rain rate in
Fig. 15. The fit is seen to be moderately good for
rain rates greater than 200 mm per hour, and fair for rain rates between
65 and 200 mm per hour. Below 65 mm per hour the function and the
observations diverge rapidly.
The model provides a useful smoothing function between rain rates of 65
and 400 mm per hour, and also represents the best currently available
extrapolation of these data above 400 mm per hour. However, the systematic
behavior of the residuals shows that an exponential function is not really
a suitable distributional description of the data.
Figure 16 shows empirical cumulative distributions for rain rates
above 50 mm per hour for the same four stations whose unconditioned
cumulative distributions appear in Fig. 9.
This once more illustrates the large differences among stations.
Histograms of the frequency with which certain percentiles fall in
various rain-rate intervals are once more used to quantify the dispersion
in the data. Figure 17 contains the results obtained
when data are pooled across the season's rainfalls for each station;
Figure 18 contains analogous information for data
pooled across stations for individual rainfalls. The outliers on the
two 99 percentile histograms are both caused by the same event,
a very high rain rate at one station during one rainfall
(see Section 2.2). Smaller poolings generally contain
too few data points above 50 mm per hour to be informative.
The results are summarized in Table III.
TABLE III - STATISTICS ON THE DISTRIBUTION OF VARIOUS PERCENTILES FROM THE RAIN-RATE DISTRIBUTION
For rain rates greater than 50 mm per hour
| Percent of rain rate above 50mm per hour | Rain rates in mm per hour |
|---|
| Value from the grand pool | The 96 stations each pooled for the season |
|---|
| Lower quartile | Upper quartile | Interquartile range | Mean | Lower quartile |
|---|
| 50 | 74 | 65 | 78 | 13 | 73 | 64 |
|---|
| 75 | 106 | 80 | 112 | 32 | 95 | 81 |
|---|
| 90 | 148 | 97 | 152 | 55 | 125 | 100 |
|---|
| 95 | 174 | 110 | 167 | 57 | 142 | 108 |
|---|
| 99 | 222 | 132 | 204 | 72 | 174 | 141 |
|---|
Most notable in Table III is the great
similarity between the distributions of upper-tail percentiles for
the two poolings. There is no obvious reason why this should be so,
and the result may be peculiar to this set of data.
As in Table II, the interquartile range
increases with the mean (ranging in value from about 20 to about
40 percent of the mean) indicating the increasing uncertainty
associated with the small sample of high rain rates.
Above 90 percent, the value from the grand pool lies above all
other values in the same line of the table, showing once again
the influence of a few local cloudbursts.
V. SPATIAL AND TEMPORAL RELATIONSHIPS
Figure 2 shows the general space-time relationship
of rain rates:
When it is raining heavily at a station it is likely to be raining heavily
at nearby stations, and it is also likely to be raining heavily a short
time later. This section quantifies these general observations.
If the time series representing the rain-rate observations were
stationary the relationships would be given by the cross correlation as a
function of distance and the autocorrelation as a function of lag (time
difference). However as indicated in Section 3.1
the time series are not even piecewise stationary in the regions containing
the high rain rates. As the situation is more complex than that represented
by stationary time series, the statistical summary is somewhat less
succinct. The spatial relationships are described in terms of the
distribution of the rain rate at a point there given certain conditions of
rain at point here. The distance between here and there and the rain rate
at point here are varied as parameters. The temporal relationships are
described analogously in terms of distributions of the rain at time hence
given certain conditions of the rain at time now. The lag, that is, the
interval between now and hence, and the rain rate at
time now are varied as parameters.
5.1 Spatial Relationships
The frequency distributions underlying this presentation were collected
as follows. The readings from each scan of the network were
assumed to be simultaneous. The very rapid variations in rain rate
(Fig. 7) and lack of detailed correlation over
distances of the magnitude of the station separations (Section III)
provide some statistical justification for this treatment.
Each station (here) was then paired with every other
station (there), giving 962 pairs. Each pair was
classed as belonging to one of 238 possible histograms according to
the value of the rain rate here, which is sorted into 17 intervals,
and the nominal distance (stations are assumed to be at the center of the
grid square) between the stations, which is sorted into 14 separation
intervals. Results are labeled with the mean of the rain-rate interval,
and the mean of the nominal distances in the separation interval. For
the rain rate here the interval is increased as the rain rate
increases; for the separation, the interval is about 20 percent of the mean.
Once the histogram was selected, the frequency of the rain rate there
was accumulated in intervals of 5 mm per hour.
The 110 hours of rain data contain about 4 × 104 scans of
the network and each scan yields about 104 pairs giving a total
data base of 4 × 108 pairs. The data are accumulated
separately for each of the 27 rainfalls and then pooled for many
presentations.
In spite of the large number of pairs, there are relatively few pairs
at the high rain rates. Thus the high-rain-rate results show noticeable
sampling fluctuations and remain sensitive to the data-screening
procedures. Probabilities involving rain rates of less than 50 mm per
hour are distorted by the data-selection processes as already discussed.
Three typical histograms (from the set of 238) are shown in Fig. 19.
Figures 19a and b show the smooth behavior produced by the large
numbers of pairs at low and intermediate rain rates;
Figure 19c shows
the fluctuations that result from the relatively small numbers of pairs
at high rain rates. These fluctuations are reflected in occasional
irregular behavior seen in other displays in this section.
Three sets of empirical probabilities for the rainfall of July 25, 1967
are shown in Fig. 20. The curves give the probability
of the rain rate there being less than R mm per hour when the rain rate
here falls in the interval indicated on the curve.
Figure 20a shows that when it is raining lightly at
station here, it is unlikely to be raining heavily at
stations 1.3 km away. As the rain rate at station here increases
it becomes increasingly more likely to be raining heavily at nearby
stations.
Rain Rate Here (mm per hour)
| A | B | C | D | E | F | G |
|---|
| Min | 0 | 20 | 40 | 70 | 110 | 170 | 200 |
|---|
| Max | 10 | 30 | 50 | 90 | 140 | 200 | 240 |
|---|
The strength of this relationship decreases with increasing separation
until at about 8 km separation the probability of observing a rain rate
less than R mm per hour is largely independent of the rain rate observed
at the distant station; this is demonstrated by the contraction of curves
in Fig. 20b into a narrow band. As the separation
increases still further the curves reverse (that is, the probabilities
associated with curve 5 ± 5 mm per hour are generally lower than those
associated with curve 200 ± 20 mm per hour for the same R. This may be
seen on Fig. 20c which shows the results for stations
separated by 12 km. The reversal means that if it is raining heavily at
station here it is less likely to be raining heavily at a
station 12 km away than if it were raining lightly at station here.
This effect is noticeable at separations up to 16 km, but few pairs of
stations are so widely separated (the diagonal measurement of the network
is 18 km). This behavior suggests that regions of heavy rainfall have
a range of influence of about 16 km (twice the 8 km distance
corresponding to Fig. 22b) and that the centers
of these regions are separated by more than the dimension of the network,
that is, more than about 16 km.
The probabilities may also be plotted against distance,
with the rain-rate-here intervals as parameters and different values of R
appearing on different plots. Figures 21a shows the
behavior of the probability that the rain rate there is less
than 35 mm per hour (the threshold) as a function of distance for various
intervals of the rain rate here. The probability that the rain
rate there is less than 35 mm per hour increases monotonically
with distance when the rain rate here is greater than 30 mm per
hour. For rain rates here that are less
than 30 mm per hour, the probability for rain rate there being
less than 35 mm per hour goes through a minimum at a separation that
depends on the rain rate here. Figure 21a also
displays the crossover of the probability curves at a separation of
about 8 km. For smaller separation it is less probable for the rain rate
there to be below 35 mm per hour if it is raining heavily
here than if it is raining lightly here.
At separations exceeding 8 km, the inverse is true. Figures 21b and 21c
show the analogous plots for rain rates less than thresholds of 60 mm per
hour and 125 mm per hour, respectively. The pattern is the same, and if
70 and 125 mm per hour respectively are substituted for 30 mm per hour in
the previous discussion it remains applicable. In particular the
crossover still occurs at 8 km separation. As might be expected, the
probability of finding the rain rate there lower than the
threshold increases with increasing threshold.
For completeness we include four plots in Fig. 22,
which show the behavior with separation of the probability that the rain
rate there
is less than certain values for the rainfall of July 25, 1967.
Each plot is for a given interval of the rain rate here.
The behavior of the curves is generally smooth without any abrupt changes
of slope. The direction of curvature and position of the maxima and minima
depend on the particular interval of rain rate here displayed.
We are unable at this time to evaluate the significance of the fine
structure that appears occasionally.
Rain Rate Here (mm per hour)
| A | B | C | D | E | F | G | H |
|---|
| Min | 0 | 20 | 40 | 70 | 110 | 170 | 200 | 240 |
|---|
| Max | 10 | 30 | 50 | 90 | 140 | 200 | 240 | 280 |
|---|
The patterns of the other heavy rainfalls have been examined and
are similar to that of the July 25, 1967 rainfall. In particular the
range of influence of the rainfalls is approximately
16 to 20 km. When the data from all 27 rainfalls (or even for the
nine heavy rainfalls) are aggregated, the differences among the
rainfalls conceal the individual patterns, and the reversal of the
order of the curves that take place between Figs. 20a and 20c
disappears. However, the concept of a range of influence,
that is, a separation at which the rain rate there is relatively
independent of the rain rate here, is still valid for the
aggregate when the rain rate there is below
about 125 mm per hour. The range of influence is about 20 km.
Of particular engineering interest is the probability that the rain
rate is greater than a particular value at two stations
simultaneously. Results in this form are presented
in Fig. 23, which is a plot of the joint
probability that the rain rate is greater than R mm per hour at both
of two stations against the separation in km. These probabilities
change smoothly with distance, and have a minimum at a separation
of about 12 km for rain rates with sufficient data to establish the
shape of the curves. The joint probabilities were also computed for
each of the 27 rainfalls separately; the upper quartiles of the
27 rainfall distributions (a distribution for each separation) are
shown for minimum rain rates of 50 and 110 mm per hour by the crosses
in Figs. 23a and b, respectively. The lower
quartiles are indistinguishable from the abscissa on the scale of the
plots. The greater-than-50-mm-per-hour curve lies close to its upper
quartiles; the greater-than-110- mm-per-hour curve lies far above the
corresponding upper quartiles. These results again demonstrate the
sensitivity of the upper tail of the aggregated distribution to a
small number of cloudbursts. Although the plots for
individual rainfalls differ substantially in behavior and statistical
stability, about one third of the rainfalls display a minimum in the
joint probabilities at a separation of about 12 km.
Minimum Rain Rate (mm per hour)
| A | B | C | D | E | F | G | H | I | J |
|---|
| 30 | 40 | 50 | 70 | 90 | 110 | 140 | 170 | 200 | 240 |
Because of the bias introduced by the method of selecting the data,
the numerical values of joint probabilities for rain rates below 50 mm
per hour are distorted relative to those for rain rates above that value.
5.2 Temporal Behavior
The frequency distributions on which descriptions of the temporal
behavior are based are accumulated in a manner completely analogous
to that used to collect the spatial distributions. Instead of distance intervals,
time intervals (lags) are used, and each station is paired
only with itself. Thirteen lags were used and there are about 100
stations and 40,000 scans. This leads to 5 × 107
pairs which (as 17 intervals of rain rate now are used)
fall within 13 × 17 = 221 histograms. The value of the rain
rate hence is sorted into intervals of 5 mm per hour to
form the histograms. All the data presented in this section are
pooled across the 27 rainfalls.
The probabilty that the rain rate hence is less than
R mm per hour for various intervals of the rain rate now is given
by the curves in Fig. 24. While the curves shift
toward lower rain rates at the longer lags, the pattern remains
essentially the same with time. If it is raining heavily now
it is more likely to be raining heavily a short time hence
than if the rain now were light. Curves G and H, which
show rain rate now greater than 240 mm per hour are much
more variable than the others because of the small number of
observations in this range.
Rain Rate (mm per hour)
| A | B | C | D | E | F | G | H |
|---|
| Min | 0 | 20 | 40 | 70 | 110 | 170 | 240 | 290 |
|---|
| Max | 10 | 30 | 50 | 90 | 140 | 200 | 280 | 320 |
|---|
Figure 25 shows curves giving the empirical probability that the
rain rate hence will be less than the value indicated on the
curve versus lag for rain rates now in the ranges 20 to 30,
50 to 70, 90 to 110, and 200 to 240 mm per hour, respectively.
These plots show that the probability that the rain rate will fall at
or below its present value is about 0.7 (increasing to 0.9 at the
highest rain rate) and relatively independent of the lag
(the increase is about 0.1 over the 360 seconds).
The tendency of the curves to move in the direction of the line
corresponding to a probability of 0.8 is noticeable at the small lags.
This means that the probability, that the rain rates a short time
hence are very different from the current rain rate, is small.
The relationship has both short and long term components.
The short term component dies down within about five minutes, as
evidenced by the decrease in the slopes of the curves with increasing
lag, and is probably associated with the fine structure of the
precipitation. The long term component is present after five minutes
as evidenced by the different values for the same curves on the four
plots. This component is probably associated with some average property
of the local precipitation. Except for the highest rain rates
(Fig. 25d), the spacing between the curves after a
360-second lag is larger for the lower rain rate hence
limits (in spite of the fact that the rain rate interval is also
smaller for the lower rain rate hence limits) , indicating that the
probability density for the rain rate hence peaks at the low rain
rates after a lag of 360 seconds for values of the rain rate now
below about 200 mm per hour.
An interesting feature appears in Fig. 25d.
All the curves with rain
rate hence limits of less than 240 mm per hour show a dip at a
lag of 20 seconds, and the curve labeled 240 mm per hour shows a double
dip. This indicates a tendency for the heavy rain rates to have a fine
structure oscillation with a period of about 20 seconds. The conjecture
is confirmed by examination of plots (not presented here) for rain rate
now intervals with lower bounds greater than 200 mm per hour.
Another presentation, Fig. 26,
plots the empirical probability that the rain rate hence is less than
R mm per hour (the threshold) for various intervals of the rain rate
now as a parameter. In agreement with the previous plots, one notes
that the probability that the rain rate exceeds its present value
decreases as both lag and rain rate now increase and is
smaller than about 0.2. There is a tendency for the rain-rate intervals
below the threshold (for example, the 40 to 50 mm per hour curve
in Fig. 26b) to be slightly concave upward,
with a minimum at about 180 seconds. This might be associated with a
characteristic growth time for showers; but the averaged effect is very
weak.
Lastly, the empirical probability that the rain rate exceeds a given
value both now and after various lags is given in Fig. 27
for a number of rain rates. The probabilities are seen to decrease
slowly and monotonically with time up to lags of 360 seconds.
The crosses in Fig. 27 are analogous to those
in Fig. 23 and indicate the upper quartiles of the
distributions obtained (a distribution for each lag) when the joint
probabilities of minimum rain rates of 50 and 110 mm per hour are
calculated separately for each of the 27 rainfalls.
(The lower quartiles are too close to zero to show on the plots.)
That the joint probabilities in time are also very sensitive to small
numbers of heavy-rain-rate events is demonstrated by the fact that the
mean value is higher than the crosses for a large fraction of the data.
However the situation is less sensitive than in the case of joint
probabilities in space (Fig. 23).
Minimum Rain Rate (mm per hour)
| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O |
|---|
| 20 | 30 | 40 | 50 | 70 | 90 | 110 | 140 | 170 | 200 | 240 | 280 | 320 | 360 | 400 |
This presentation does not provide direct information on the
probability that the rain rate will remain above a certain value
for the time given by the abscissa. We have not attempted to provide
statistics on the length of time the rain rate exceeds given values
because (as shown in Fig. 7) the rain rate
changes rapidly within the ten second sampling interval and the
form of distribution of which the rain rates within the interval
are a sample is not known.
The remark on sampling bias at the end of Section 5.1
applies to the temporal as well as the spatial probabilities.
VI. PATH STATISTICS
The average rain rate along the path between a microwave transmitter
and receiver affects the amplitude of the received signal. This
section presents the probability that the average rain rate exceeds a
given value on a single path or on two paths simultaneously for a
variety of conditions. We calculate the probabilities directly from the
data without recourse to any of the statistics and intermediate
results previously presented. Because of the very large amount of
computation involved, it is essential to reduce the data base and
regularize the geometry as much as possible. We chose to examine only
the nine heavy rainfalls (see Section 2.3), and
from these we selected only those 6725 scans during which at least
one station showed a rain rate in excess of 50 mm per hour.
The geometry has been regularized by rectangularizing the network
to 11 east-west rows of ten stations each. The averages of the values
from the surrounding stations have been substituted for values missing
because stations were inoperative (or fictional, such as the northeast
corner) or where the values were judged to be outliers. Substituting for
missing values has only a small effect (statistically) on the
distribution of average rain rates but a large effect on the population
of paths in various categories. This matter has been examined in some
detail and, because of the configuration of heavy-rain-rate occurrences,
it has been concluded that omission of paths containing missing
observations introduces artifactual sampling variations that distort the
results of the path analysis very seriously in unpredictable ways, and
that this approach is unacceptable. The distortion introduced by using
averages of surrounding values to estimate missing observations is that
of generally lowering the probability of occurrence of high rain
rates. As noted in Section 7.3, the effect is most
severe on single station paths.
Omission of data from rows 1, 2, and 11 of the grid, in an attempt
to make the grid rectangular without filling in corners, seriously
biases the results by throwing out too much of the heavy-rain-rate data
and many of the long paths. As justified in Section V,
stations were treated as if they were located at the centers of their
grid squares and scanned simultaneously. The paths were taken as
starting and ending at stations; in computing the average rain rate the
values at the stations at the ends of the path were given the weight ½
and the values at the intermediate stations, the weight 1.
The diameter of a rain gauge is only 30 cm and the distance between
grid square centers is 1.3 km, so that much less than 1/4000 of the path
is sampled. As the rain rate has a great deal of fine structure and the
number of samples per path is small (less than 11), these averages are
themselves subject to very substantial fluctuations about the true
path average.
We considered many possible paths and pairs of paths through the
network. Because a single station is generally included in several paths
of the same length and orientation, the separate path averages are not
statistically independent. This factor has not been taken into account
explicitly in our analysis.
6.1 Parallel Paths
Parallel paths are characterized by three parameters: length,
separation and orientation. (To allow consistency of notation, length
zero indicates paths which include only a single station and separation
zero indicates a single path.) We consider four orientations:
two square, north-south, and east-west; and two diagonal,
northeast-southwest and northwest-southeast. A pair of parallel
north-south paths of length 5.2 km and separation 2.6 km is indicated
by the two dashed lines, P11-P12 and P21-P22 toward the western side of
the grid in Fig. 1.
The relative probability that the rain rate on both paths exceeds the
value on the abscissa is given by the ordinate for nine separations as
parameters and six lengths of path as parts of Fig. 28
for nine heavy rainfalls in 1967. (The abscissa is then the minimum
average rain rate for the pair.) The results for the north-south and
east-west paths have been pooled. The largest number of pairs of paths
entering into the data base for an individual curve is 1,338,275 (Fig. 28a curve B)
the smallest is 80,700 (12 per scan, Fig. 28f curve I).
Separation (km)
| A | B | C | D | E | F | G | H | I |
|---|
| 0 | 1.3 | 2.6 | 3.9 | 5.2 | 6.5 | 7.8 | 9.1 | 10.4 |
Except for a small amount of straggling in the lowest decade,
the curves are smooth and well behaved. In order to display the small
probabilities and high rain rates effectively, the logarithm of the
probability has been used as the ordinate, and rain rate has been
plotted linearly as the abscissa. The curves are generally quite
straight on this semilog plot, but tend to droop somewhat at the high rain
rates. This indicates that for purposes of extrapolation at the high
rain rates the customary log-probability versus log-rain-rate plot is
unsuitable and that improvement should be sought in the opposite
direction of scaling (for example, by raising the rain rate to a power
slightly greater than 1). For paths up to 10 km long, the joint probabilities
decrease with increasing separation until a minimum is reached at a
separation of about 9 km.
For larger separations the joint probability increases again. This
result was observed for single stations in Section V.
Comparison among the parts of Fig. 28 shows that
for fixed probabilities the minimum average rain rate decreases as the path
length increases at all separations. Thus the product of the minimum average
rain rate and the path length increases more slowly than the path length
itself. (This product is approximately proportional to the quantity of
water along the path.)
Results from the diagonal paths generally confirm the results from the
square paths. Pooling has not been carried out, however, because of the
differences in path lengths and separations, and the fact that the absence
of long diagonal paths at large separations tends to weight the stations
in the center of the network very heavily.
The curve for zero length and zero separation corresponds to single
stations, that is, point-rain-rate statistics. Similarly the curves for zero
length and various separations are subsets of the space pairs of
Section V. However, the data bases are not the same;
missing observations have been estimated for the subset of the data used
for the path analysis, making direct comparison of these results very
difficult. Section VII presents a rough comparison and a
discussion of the differences.
In order to minimize edge effects (the stations on the edges are included
in fewer paths, that is, less heavily weighted than the central section even
for the square paths) the maximum length of path considered should be short
compared with the dimensions of the network. This is certainly not the case
for the paths more than 5 km long. Furthermore, the results appear quite
sensitive to the particular patterns of rainfall observed. The relative
values, while subject to some distortion introduced by the estimation of
missing points, are probably fairly good; however, we do not believe that
the numerical values of the results in Fig. 28 should
be considered better than an order of magnitude for the high rain rates at
this stage of the analysis.
This conservative view is vindicated by the large dispersion among the
results for the four orientations considered individually.
For example, Fig. 29 shows the results for all parallel
paths about 7.5 km long and 3.7 km apart pooled over the nine heavy rainfalls
categorized by orientation. The orientation effect exceeds a factor
of 2 at a minimum average rain rate of 60 mm per hour and a factor of 10
above about 100 mm per hour. Part of the dispersion may be real, that is, a
result of systematic factors (such as the direction of the prevailing winds
or the location of local updrafts) and part may be a result of differences
in the population of stations which go to make up the different sets of
paths. The dispersion also indicates the additional caution with which
results from single lines of stations should be viewed.
6.2 Adjoining Paths
The three parameters characterizing adjoining paths (two paths
having an end point in common) are the length of the arms, the
angle between the arms, and the orientation of the apex. We consider
three angles, π/4 (acute), π/2 (right), and 3π/4 (obtuse)
and the eight orientations corresponding to rotations through an angle of π/4.
Right angles containing diagonal paths (D) are differentiated from right
angles containing square paths (R) to avoid confounding the different lengths
of arm, thus there is a total of four categories: acute (A), obtuse (O), D,
and R. Two paths with arms about 3 km long joined in an obtuse angle are
indicated by the two dashed lines I1-IA-I2 toward the eastern side of the
grid in Fig. 1.
The relative probability that the average rain rate on both paths
exceeds the value on the abscissa is given by the ordinate for the four
categories as parameters and six lengths of arm in Figs. 30a through f.
The data, which are from the nine heavy rainfalls in 1967, have been pooled
across orientations. The smallest and largest data bases for individual curves
contain 53,800 and 4,842,000 points, respectively. For the same length of
path, the results for adjoining paths lie between those for single
paths (zero separation) and those for two paths separated by 1.3 km; and show
the same decrease in probability with increasing length and rain rate.
A given rain rate is exceeded most frequently for the acute angles and
least frequently for obtuse angles. The factor between the two increases
with both path length and rain rate, and ranges from less than
two (Fig. 30a) to about ten (Fig. 30d).
Except for the acute angle, long paths cannot transit the center of the
network so the results for various path lengths are based on different
populations of stations, an effect which may introduce some bias.
Very long path pairs with abuse and diagonal angles cannot be formed in the
network, and are missing from Figs. 30e and f. Once
again the concatenation of uncertainties suggests that these results should
not be considered better than an order of magnitude in absolute values.
Figure 31 contains results for the four pairs of
adjoining paths of the square family (R) whose arms are 5.2 km long.
The paths whose apex angles point to the northwest yield probabilities
substantially higher than the other three orientations. This result is not
peculiar to this category of angle or length of path. Symmetry considerations
would lead one to expect the results for diagonally opposed
pairs (that is, NW, SE and NE, SW) to be the same. The symmetry expectation,
however, is quite generally violated. The explanation is that for a small
grid the population of path pairs is very different for the two sets of
pairs. In this case the SE, NE, and SW path pairs can never have their
apexes in the northwest corner of the grid where heavy rain rates are most
frequent. This sensitivity to sampling bias further confirms our reluctance
to regard the numerical probabilities derived from the results of this
section as better than an order of magnitude.
Some further work on statistical dependence, edge and other geometric
effects, and dispersion in the data is being considered.
VII. BENCHMARKS FOR ENGINBEERING APPLICATIONS
All the probabilities presented so far have been calculated with
respect to their own particular data bases. Thus the results do not
contain assumptions about the representative properties of the data or
subsets thereof, correction factors for missing data, or other adjustments
which may require future revision. In this section we supply, for those
who must make engineering calculations, factors which allow conversion
of the relative probabilities to annual probabilities. The annual
probabilities should be used only in the light of the various caveats
and discussions regarding representativity, precision, and accuracy
contained in this paper.
All benchmarks are computed at rain rates of 50 mm per hour. They
may be applied to higher rain rates directly, but can be applied to
lower rain rates only if they are scaled (see Section 4.1)
to compensate for the bias introduced by excluding low-rain-rate data
in the selection process.
7.1 A Point-Rain-Rate Benchmark
During the time covered by these data. the average probability of
observing a rain rate greater than 50 mm per hour was 4.3 × 10-4.
This number may be used to transform the scale of Fig. 14
as follows:
P(R) = 4.3 × 10-4 [1 - p(R)],
where P(R) is the annual probability that the rain rate exceeds R
and p(R) (the empirical probability that the rain rate is less than
R) is given by Fig. 14. This benchmark is based on
a nominal observation period of 158 × 104
scans (June 1, 1967 through November 30, 1967), an allowance of
15 percent for data known to have
been lost, and a correction of all the stations to 40,069 scans
(110 hours of rain) on the assumption that the loss of data is
random over the data base.
For convenience, some of the point-rain-rate data have been
replotted in Fig. 32a on the log-log scales often
used in engineering. The figure shows the Crawford Hill 1967 data and
the 1958 Island Beach, New Jersey, data of Mueller and Sims [8].
(The curve in Fig. 32a was derived from Muellers
and Sims Figs. 25 and 26 by D. C. Hogg.)
Rain rates greater than 100 mm per hour are six times more probable
in the Crawford Hill than the Island Beach measurements. However,
rain rates above 250 mm per hour are equally frequent indicating
somewhat different distributions of the rain rates measured.
Fig. 32a also shows the results from stations 1 and 76.
The spread between them is more than two orders of magnitude
and encompasses most of the stations in the network.
In view of this spread there is no reason to regard either
the Crawford Hill aggregate or the Island Beach results as atypical.
The bars in the figure are the interquartile ranges from the
distributions of the results from the 96 stations (taken from Table III).
The aggregate curve lies above the upper end of the bar when the
rain rate is higher than 150 mm per hour, emphasizing the importance
of rare events at these rain rates. If essential,
extrapolation of the Crawford Hill curve to higher rain rates
may best be accomplished by using the formula in Section 4.2
to construct the necessary ratios.
| Label | Separation (km) |
|---|
| A | (●) | 1.3 |
|---|
| B | (○) | 3.9 |
|---|
| C | (▲) | 6.5 |
|---|
| D | (△) | 9.0 |
|---|
7.2 A Spatial-Temporal Benchmark
For rain rates above 50 mm per hour the relative probability scales
of Fig. 23 and Fig. 27
may be converted to annual probabilities by multiplying
by the factor 3.0 × 10-2. This factor contains a
correction of about 15 percent for rainfalls known to have been missed.
7.3 Path Probabilities Benchmark
The path probabilities benchmark is calculated on the premise that
the 6725 scans analyzed contain all the pertinent data. The
consequences of this premise are further discussed in this section. The
probability scales of Figs. 28 through 31 should be
multiplied by the factor 5.0 × 10-3 to
convert to annual probabilities. This factor also contains the
correction (about 15 percent) for rainfalls known to have been missed.
Curve A of Fig. 28a, which represents single
stations, is replotted using this benchmark
on Fig. 32 as the thin solid line labeled
path analysis. For single stations the corrected probabilities
from the path analysis are about 25 percent below those obtained from the
pointrain-rate analysis for rain rates over 100 mm per hour and 35 percent
lower for rain rates near 50 mm per hour. This suggests that about
30 percent of the data above 50 mm per hour have been excluded
from the data base for the path analysis. However, this has been
confounded by the inclusion of estimates for missing observations in the
path analysis. We feel that most of the data above 50 mm per hour
were included, and the lower result for the path analysis is mainly
artifactual.
Relative joint probabilities of pairs of stations for four different
separations are taken from Fig. 28a, corrected to
annual probabilities, and replotted as the solid curves
in Fig. 32b, together with the corrected relative
probabilities for the same nominal separations (the points) taken
from Fig. 23. The geometry of the pooling in the two
sets of data differs for all except for curve A
of Fig. 32b. The path analysis included only
separations along the square grid, while the spatial analysis included
all stations (both square and diagonal) within appropriately separated
concentric circles. As previously noted, the data base for the path
analysis is a subset of that used for the joint probabilities, and the
probabilities in the path analysis have been lowered by the inclusion
of estimates for missing observations. Figure 32b
demonstrates the results; the probabilities from both
analyses are in good agreement for low rain rates and diverge as the
rain rate increases. The values from the spatial analysis
(the points in Fig. 32b) are clearly more reliable.
Results for high rain rates on paths consisting of single gauges are
most affected by the estimation of missing observations, so that
Fig. 32b presents the worst cases for the path
analysis. As agreement is good near 50 mm per hour and the missing-data
effect is more serious at the higher rain rates and less severe for the
longer paths, no single compensatory adjustment of the benchmark can be
made.
7.4 Seasonal Correction
Data concerning fading on the microwave transmission
path (T-R in Fig. 1) indicate that during 1967
most of the heaviest rain rates, which are usually generated by summer
thunderstorms, occurred between June 1 and
December 1 [6]. This seasonal effect decreases the
probability of observing the highest rain rates by a factor of two,
without materially affecting the probability of observing light rain.
Insufficient data are available to correct our distributions for the
seasonal variation. However, an approximate adjustment may be made by
multiplying the annual probabilities associated with rain rates greater
than about 100 mm per hour by the factor ⅔, if the probabilities
are to be applied on an annual basis. Correction for the seasonal effect
would improve some aspects of the comparison between the Island Beach
and Crawford Hill data (Fig. 32) and worsen others.
VIII. SUMMARY AND CONCLUSIONS
This article presents some statistical summaries of rainfall data
acquired on the Crawford Hill, New Jersey, rain gauge network between
June 1 and November 30, 1967. The emphasis of the analysis is on rain
rates greater than 50 mm per hour, which cause substantial attenuation
of electromagnetic waves in the 10 to 30 GHz frequency range. The objects
of analysis are: the behavior of rain rates at a point in space, the
relationship of two rain rates separated in space or in time, and the
relationships of average rain rates on pairs of paths in various
configurations.
The network consisted of 96 stations approximately uniformly
distributed over a rectangular area 13 by 14 km [6]. The gauges are small
in area (478 cm2), measure the rain rate averaged over
less than one seconds, and are read out in sequence every ten seconds [1].
About 430 hours of data were recorded; the 110 hours of greatest interest
have been examined in detail. The data are generally satisfactory for
rain rates greater than about 10 mm per hour, although some substantial
data-screening problems were encountered. There is no indication that
these data are atypical of rain rates in northern littoral New Jersey,
but they constitute a very small sample at the high rain rates; there is
only informal indication of the large range of variation to be expected.
Heavy rainfall is found to come in irregular bursts. The time series
containing these bursts are nonstationary (not even piecewise stationary
at the sampling rate used), and no simple description of the wide
variety of observed distributions of the rain rates has been found.
Several poolings (for example, one station for the season, all stations
for a rainfall) of the data were examined in an unsuccessful effort to
find subsets of the data that behaved consistently enough to be described
by a single distribution.
Figures 11 and 12 summarize the distribution of the
point-rain-rate data. The high-rain-rate tail
(i) is longer than would be expected on the basis of the normal
distribution,
(ii) can be approximated by an exponential function [equation (1)],
(iii) depends heavily on a small number of cloudburst-like events
(and thus shows a lot of scatter, see Fig. 13 and
Table II), and
(iv) is sensitive to the data-screening procedures used in the data
processing.
Restricting the data to rain rates greater than 50 mm per hour gives a
better-defined subset of the data than does the original data-selection
process and allows a further examination of the dispersion in the
data (Table III), but fails to reduce the
distributional variety observed.
Empirical probabilities for point rain rates above 50 mm per hour
are given in Table IV. These measurements are
somewhat higher (up to a factor of 6 at 50 mm per hour) for rain rates
below 250 mm per hour than the Island Beach, New Jersey measurements of
Mueller and Sims [8]. However, better agreement
would be obtained if a seasonal correction were applied.
TABLE IV - EMPIRICAL PROBABILITIES FOR POINT RAIN RATES
Data are for Crawford Hill network during 1867.
| R mm per hour | Probaboloty (rain rate > R) | Total time (rain rate > R) minutes per year |
|---|
| 50 | 4.3 × 10-4 | 230 |
| 100 | 1.3 × 10-4 | 70 |
|---|
| 150 | 4.2 × 10-5 | 22 |
|---|
| 290 | 1.0 × 10-5 | 5 |
|---|
When it is raining heavily at a station, nearby stations are also
likely to show heavy rain. More detailed examination of the spatial
characteristic of the data in terms of the joint probability of the rain
rate being greater than R mm per hour at each of two stations
simultaneously shows a rapid decrease of the joint probability with
increasing separation for short separations and a minimum in the joint
probability when the stations are separated by about 12 km
(Fig. 23). Other spatial relationships also change
qualitatively when the separations exceeds about 12 km. The temporal
behavior may be characterized by the behavior of the joint probability
of the rain rate at a station exceeding R mm per hour at both the
beginning and end of a given time period (Fig. 27).
This probability declines smoothly and relatively slowly as the interval
increases from 10 to 360 seconds.
Of particular engineering interest are the joint probabilities that
the average rain rate will simultaneously exceed a particular value
on all of a number of paths. Figure 28 summarizes
results of analysis of various sets of paths within the network for
parallel paths; Figure 30 summarizes the same
results for adjoining paths. Figure 29 and Figure 31
give some indication of the dispersion among subsets of the paths.
For parallel paths up to 10 km long, the joint probability is minimum
for paths separated by about 9 km. The joint probabilities for parallel
paths 6.5 km long (and ordinary probabilities for single paths) are
tabulated in Table V. As seen in Table V, for paths
of this length the joint probability may be more than 200 times lower
than the ordinary probability. This indicates the improved reliability
of microwave transmission that may be obtained with appropriate choice
of redundant paths. While the relative values of the probabilities
obtained in this analysis are likely to be reasonably good, for reasons
given in Section VI the numerical values should be
regarded as correct only within an order of magnitude. The joint
probabilities associated with average rain rates on adjoining paths are
somewhat less than the probabilities associated with single paths through
the network.
Section VII supplies benchmarks for converting the
relative probabilities presented in the body of the paper to the annual
probabilities needed for engineering calculations and discusses an
additional correction for seasonal effects. Annual probabilities for
point rain rates are plotted in Fig. 32a, and
results of various analyses are compared and found to be consistent.
TABLE V-JOINT PROBABILITY THAT THE AVERAGE RAIN RATE ON BOTH OF TWO PARALLEL PATHS EXCEED R
The paths are 6.5 km long. Probability is tabulated by the separation between paths. (Ordinary probability for the single paths.)
| R mm per hr / Separation km | 0 (Single path) | 1.3 | 3.9 | 6.5 | 9.1 |
|---|
| 50 | 2.5 × 10-4
(2 hr per yr) | 1.5 × 10-4
(75 min per yr) | 5 × 10-5
(25 min per yr) | 1.5 × 10-5
(7.5 min per yr) | 7.5 × 10-6
(4 min per yr) |
| 100 | 2.0 × 10-5
(10 min per yr) | 7.5 × 10-6
(4 min per yr) | 1.3 × 10-6
(30 sec per yr) | 1.0 × 10-7
(3 sec per yr) | < 10-8 |
|---|
| 150 | 1.5 × 10-6
(30 sec per yr) | 1.0 × 10-7
(3 sec per yr) | < 10-8 | < 10-8 | < 10-8 |
|---|
IX. ACKNOWLEDGMENT
The authors thank D C Hogg and R A Semplak for the data and for lively
discussions thereof; Mrs C L Clark and Mrs R J Hedin for their assistance
with data reduction and analysis.
REFERENCES
1. Semplak, R A, Gauge for Continuously Measuring Rate of Rainfall, Rev Sci Instruments, 87, No. 11 (November 1966), pp 1554-1558.
2. Semplak, R A, Keller, H E, A Dense Network for Rapid Measurement of Rainfall rate, BSTJ, this issue, pp 1745-1756.
3. Semplak, R A, Torrin, R H, Some Measurements of Attenuation by Rainfall at 18.5 GHz, BSTJ, this issue, pp 1767-1787.
4. Freeny, A E, Statistical Treatment of Rain Gauge Calibration Data, BSTJ, this issue, pp 1757-1766.
5. Wilk, M B, Gnanadesikan, R, Probability Plotting Methods for the Analysis of Data, Biometrika, 65, No. 1 (March 1968), pp 1-17.
6. Semplak, R A, unpublished work.
7. Hogg, D C, unpublished work.
8. Mueller, E A, Sims, A L, "Investigation of the Quantitative Determination of Point and Areal Precipitation by Radar Echo Measurements, Technical Report ECOM-00032-F, U. S. Army Electronic Command, Fort Monmouth, New Jersey, December 1966. (Contract DA-28-043 AMC- 00032(E) with the Illinois State Water Survey, University of Illinois, Urbana, Illinois.)
