A simple method is proposed for the determination of effective rain cell dimensions and orientation. Two rain cell models were considered: the circular cell and the elliptical cell. In both models it is assumed that the rain rate is constant throughout the cell, that all cell locations are equally likely, and that the cell dimensions depend upon the rain rate. It is then shown that the effective cell dimensions and orientation may be deduced from rain rate statistics accumulated at two or three closely spaced sites, depending upon the model chosen. The results of this study alone with previous estimates of rain cell dimension indicate that the rain rate measurement points should be spaced on the order of one-half kilometer apart. The proposed method is simple and relatively inexpensive; thus, this approach is readily suited to the study of the dependence of rain cell size and orientation upon climatic region. The resulting rain cell characteristics are of direct value in the prediction of millimeter wavelength attenuation statistics on both single-terminal and diversity earth-space propagation paths as well as point-to-point terrestrial links

Hodge, D. B.

English

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A SIMPLE METHOD FOR THE DETERMINATION OF EFFECTIVE RAIN CELL
DIMENSIONS AND ORIENTATION
D. B, Hodge
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Department ©1 (Storied
s, Ohio 43212
TECHNICAL REPORT 2374-15
September 1973
Grant Number NGR 36-008-080
CASE FILE
COPY
National Aeronautics and Space Administration
Office of Grants and Research Contracts
Washington, D.C. 20546
https://ntrs.nasa.gov/search.jsp?R=19740002465 2020-03-23T14:10:48+00:00Z
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A SIMPLE METHOD FOR THE DETERMINATION OF EFFECTIVE RAIN CELL
DIMENSIONS AND ORIENTATION
D. B. Hodge
TECHNICAL REPORT 2374-15
September 1973
Grant Number NGR 36-008-080
National Aeronautics and Space Administration
Office of Grants and Research Contracts
Washington, D.C. 20546
ABSTRACT .
A simple method is proposed for the determination of effective rain
cell dimensions and orientation. Two rain cell models have been con-
sidered: the circular cell and the elliptical cell. In both models it
is assumed that the rain rate is constant throughout the cell, that all
cell locations are equally likely, and that the cell dimensions depend
upon the rain rate. It is then shown that the effective cell dimensions
and orientation may be deduced from rain rate statistics accumulated at
two or three closely spaced sites, depending upon the model chosen. The
results of this study along with previous estimates of rain cell dimen-
sions indicate that the rain rate measurement points should be spaced on
the order of one-half kilometer apart. The proposed method is simple
and relatively inexpensive; thus, this approach is readily suited to the
study of the dependence of rain cell size and orientation upon climatic
region. The resulting rain cell characteristics are of direct value in
the prediction of millimeter wavelength attenuation statistics on both
single-terminal and diversity earth-space propagation paths as well as
point-to-point terrestrial links.
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
II. THE RAIN CELL MODELS 2
III. CIRCULAR CELL MODEL . 3
IV. ELLIPTICAL CELL MODEL 6
V. SUMMARY 11
ACKNOWLEDGMENTS . .12
REFERENCES 13
m
I. INTRODUCTION . .
The development of wide-bandwidth communication systems has led to
an increased demand for utilization of the frequency spectrum above
10 GHz. Unfortunately, signals in this portion of the spectrum are
attenuated by liquid water; and, in fact, severe attenuation may be en-
countered when the signal passes through regions of intense rain rates
such as those associated with severe thunderstorms Ii.lj. Thus, the
overall communication system reliability may be ultimately determined by
the spatial extent of those regions of intense rain rate through which
the signal must pass and the probability of their occurrence.
A simple, cylindrical rain cell model was studied in an effort to
determine its utility in the prediction of attenuation statistics. Al-
though this model grossly oversimplifies the physical characteristics of
the actual rain cells, it was found to be quite useful for the prediction
of long term attenuation statistics within one or two decibels [2j. This
model has been generalized to an ellipsoidal shape in order to more
accurately reflect the effect of cell shape on space diversity communi-
cation systems where two spatially separated, parallel propagation paths
are used to improve the overall system reliability. In the course of
both of these model ing.efforts it has been found that little data re-
lating rain cell size, shape, orientation, and rain rate exists. Further,
the dependence of such data upon climatic region is virtually nonexistant.
The use of radar for the determination of these characteristics ;has
been proposed and is. being, pursued. However, this approach is expensive,
and it is unlikely that the instrumentation will be widely implemented
in the near future. Therefore, an alternative approach for the measure-
ment of these cell characteristics is proposed herein. This approach
would utilize either two or three rain rate gauges spaced relatively
closely, on the order of one-half kilometer, and a means of simultaneously
recording the measured rain rates. Thus, the instrumentation require-
ments would be minimal and the approach would be quite inexpensive to
implement.
The radar approach yields size, shape, and orientation information
concerning each rain cell observed. This information must then be pro-
cessed to obtain statistical cell properties. In contrast, the rain
gauge approach will not yield characteristics of individual cells but
will considerably simplify the data processing required to obtain the
desired effective cell characteristics. The radar approach can also
yield information concerning the vertical distribution of rain rates;
whereas, the rain gauge approach yields only surface rain rate statistics.
Initially, both techniques should be compared at one site in order to
determine the_.degree of agreement or disagreement.
II. THE RAIN CELL MODELS
The rain cell models to be considered were chosen because of their
tractability. The intent was to represent rain cells in a phenomenal
sense rather than to attempt to model individual rain cells. The first
model considered was simply circular cylinder having a finite height
above the earth's surface; the second model was an ellipsoid having two
axes on the surface of a flat earth. The first model has two geometrical
parameters, the diameter and the height, while the second model has four,
the length, width, height, and orientation angle. Since the rain gauge
method discussed herein deals only with surface rain rates, the vertical
distributions of rain rate will be ignored. Hence, in the following,
only the circular or elliptical surface cross sections of the cells will
be considered. These geometries are shown in Fig. 1 where the x- and y-
, coordinate axes lie in the plane of the earth's surface. The orienta-
tion of this coordinate system is rather arbitrary; but, for the sake of
(X,Y)
(X,Y)
CELL GEOMETRIES
Fig. T. The geometries of the circular and the
elliptical rain cell models.
completeness, let us define the positive y-axis such that it is oriented
in a northerly direction and the positive z-axis such that it is ori-
ented upward. In both cases the coordinates X,Y denote the position of
the center of the cell, and the remaining parameters are:
d .= the diameter of the circular cell model,
a,b = the major and minor axes of the elliptical cell model, and
<f> = the orientation angle of the elliptical cell model with
respect to the x-axis. '
The following basic assumptions are made in both cases:
(1) The rain rate is constant throughout each cell.
(2) In a statistical sense, there is a unique, single
, valued relationship between the cell rain rate
and the cell geometrical parameters.(3) All cell locations are equally likely.(4) Only one cell influences the rain rate measurements
at any instant of time.
Again, it must be stressed that the rain cells are to be modeled in a
phenomenal, rather tha'n physical, sense. Note, also, that assumption (2)
implies that d, a, b , ' a n d . < j > are all dependent upon the rain rate, R.
Finally, assumption (4) is in general too restrictive; however, in the
cases to be examined here, the gauge spacings will be small compared to
typical cell sizes. Therefore, since rain cells do not tend to occur
immediately adjacent to one another, it is not likely that two different
cells will simultaneously produce _rain at two closely spaced gauges.
III. CIRCULAR CELL MODEL
If a rain gauge is placed at the origin of the coordinate system, a
record of the rain rate, R, may be obtained as a function of time. The
percentage of time during which the rain rate is less than or equal to a
given value, RQ, may then be readily determined from such data as a func-
tion of R0. IT, then, the data gathering period is sufficiently long,
it will be assumed that this percentage distribution is equivalent to,
the rain rate probability distribution. This distribution will be rer
ferred to as the point rain rate distribution, Ppj.(R), where the rain
rate subscript has been dropped for simplicity. Since it has been
assumed that all rain cell locations are equally likely, it follows that
the point rain rate distribution does not depend upon the coordinate
location of the measurement point. The point rain rate probability
density function may then be defined in the usual manner,
(D P
d R
line of reasoning applies to all of the prob
not
The preceding pragmatic a-
bility density functions discussed in the following and, hence, will 
be repeated.
A cell probability density function, pcen(R), will be associated
with cells having rain rate R which are centered a t ' X , Y . Again, pcel l (R)
does not depend upon X or Y since the cells were assumed to be uniformly
distributed. It should be reiterated at this point that
tive cell diameter is assumed to be associated with
Consequently, a cell of rain rate R centered within
d about the gauge will produce a rain rate R in the
a unique effec-
each rain rate,
a circle of diameter
gauge, as shown in
Fig. 2. This area about the gauge within which the occurrence of rain
RAIN CELL
RAIN CELL
CENTER
AREA OF
INFLUENCE
GAUGE
CIRCULAR AREA OF INFLUENCE
Fig. 2. Area of influence.
cell center will produce rain in the gauge will be referred to as the area
of influence, Apj-(d). This area of influence depends upon the cell dia-
meter and, thus, implicitly upon the rain rate. In fact, it can be noted
that the size and shape of the cell and the area of influence are ident-
ical for any given rain rate. Now, since the cells are uniformly
distributed, it follows that the point rain rate and cell probability
density functions are related by
(2) ppt(R) = Apt(d) Pce l l(R)
or
(3)
If, how, a second gauge is placed a distance £ along the x-axis from
the first gauge, where a is less than the cell diameter, d, the geometry
shown in Fig. 3 results. For any given rain rate the areas of influence,
4
GAUGE I
APtc
2 GAUGE GEOMETRY
Fig. 3. Two gauge geometry.
t} and A . , are identical. The joint area of influence, A-01-_ t , is
at area in which the occurrence of a rain cell center will produce
rain at both gauges simultaneously. From geometrical considerations,
the joint area of influence is
d2 /i- *i - 1
J d2 2
The joint rain rate probability density function, p-0-jn j .(R,£) associated
with the two gauges may now be related to the cell probability density
function by
This joint rain rate probability density function is dependent upon only
one rain rate since the model considered permits the existence of only
one rain cell at any instant of time. Thus, the simultaneous occurrence
of dissimilar ,rain rates at the two gauges is not possible within the
confines of this model.
Substituting (4) into (5) yields
(6) sin-i d*
Lett ing
a n d e l iminat ing P R ) from E q . ( 3 ) a n d ( 5 ) gives
Ppt< R >
Here the ratio of the two rain rate probabili ty density functions is a
relatively simple function of the geometrical parameters i and d. This
functional relationship is shown in Fig. 4. Thus, if the two rain rate
probabi l i ty density functions are evaluated experimentally for a given
value of rain rate and, of course, the gauge spacing is known, the
effective rain cell diameter may be taken directly from Fig. 4. There-
fore, the effective rain cell diameter, d ( R ) , may be
 ;determined directly
and simply from rain rate measurements at two sites.
It is evident from Fig. 4 that the useful range measurement of y
extends from roughly 1 to 20, i.e., d ranges from £ to 2(k. The i m p l i -
cation here is that the gauge spacing should be small compared to the
cell diameters of interest. Previous estimates of cell diameters i n d i -
cate that they may range from 1 to 10 km with the smaller diameters being
associated with the more intense rain rates |3|. Consequently, one may
conclude that a gauge spacing of 0.5 km would be desirable for the im-
plementation of this approach. In i t i a l ly , this small spacing appears
to be contradictory with rain gauge spacings commonly used in the past.
However, these measurements have generally been concerned with
distr ibutions of point rain rates rather than the interpretation of the
data in terms of area! rain rate distr ibutions. This dist inct ion between
point and area! rain rate statistics has been pointed out by Epstein - | 4 | .
IV. ELLIPTICAL CELL MODEL
The treatment of the e l l ipt ical cell model follows that of the
 t
circular cell model with the exception of the increased number of degrees
of freedom. The area of inf luence associated with a s ingle gauge located
at the or igin of the coordinate system is shown in Fig. 5 and, again, is
identical in size, shape, 'and orientation to the rain cell for any given
rain rate. Locating an additional gauge at a separation distance of I
along both the positive x- and y- axes yields the geometry shown in
Fig. 6. The areas of influence associated with the gauges are again
equal for any given rain rate,
(9) '..;..' A p t (a ,b ,< j>) = "Aj.(a,M) = A2(a,M.j = A 3 ( a , b , 4 > )
1.0
0.8
0.6
Q.
0.4
0.2
20/
I I I I I I I
1.5 5 6 7 8 9 10 15 20
RATIO OF RAIN RATE PROBABILITY D ISTRIBUTION
FUNCTIONS vs. GEOMETRICAL PARAMETER
Fig, 4. Ratio of rain rate probability distribution
functions versus geometrical parameter y
RAIN CELL
RAIN CELL
CENTER
AREA OF INFLUENCE
Apt
ELLIPTICAL AREA OF INFLUENCE
Fig. 5. Elliptical rain cell area of influence.
And, the point rain rate probability density function is given by
(10)
 Ppt(R) = Ap PceV|(R)
The joint areas of influence are denoted by A,-,-(a,b,<j>,^) and are given
by J
/ i o \ « / L. \ ab • -i I L(12) A (a, b, <{,,£) = — sin x jl
' '
 1 2
 ' 2 I
sin2<|)
£ ; / , o o . o . o
-
 T ,;(b2cos2<() +• a2sin2<)2
 V - - --- --
/cos sin
8
3 GAUGE GEOMETRY
Fi g. 6. 3 gauge gepmetry.
,2
2,3
(13) A 1 3 ( a , b , ( j ) 5 £ > = j- sin'1
1 , .
 2(j)
P-"A
.J Va2
, COS2<|)\b2 )
' P /<;in2<)> rn<;z<JAI
/h2ci'n2A 4. a2rnc2A.^' 1 o2|S l n ^ 4- 9i
\ a 2 L Ob2
(14) A 2 3 (a ,b ,<M) = f5: sin "
1 (sin<j)-cos4))2
/(sintf)-cos(j))2
y ]| b2(sin<j)-cos(j))2+a2(sin<))+cos(|))2| 11-£2 I -———!
(sin<))+cos()>)2
The ij subscript notation here and in the following obviously implies
only the pairs: 1,2; 2,3; or 3,1. The three joint rain rate probability
density functions may now be expressed as
(15) /p^CRVO s Ajj(a,b,<M') pcell.(R) . .
Eliminating pce l l(R) from Eqs. (11) and (15) yields
P i j ^ o f : • • ^(16) — •*— = l\ sin-Vr^T-p- _
 Y../TI
Pp t(R) : *L • : - . ^ ^
where the geometrical parameters are given by
l(sin<j>-cos<f>)2 (sin(j>+cos<|>)2
r——— + —7
Isin2(() cos2<(>
Comparing Eqs. (8) and (16), it is seen that the ratios of the proba-
bility density functions have the same functional dependence on the
geometrical parameters for both models. Thus, given a set of joint and
point rain rate probability density functions, the geometrical parameters,
Y.JJ, may be determined from Fig. 4. Eqs. (17), (18), and (19) may now be
solved simultaneously to find a, b, and $ in terms of this set of geo-
metrical parameters, Y-|J:
(20) tan(2<(,) =
c o s c ( > - s i n < j )
[Y.122 cos2* - Y312 sin2o,.
10
sin2<i>-cos2<j> ,
(22) b = £ -
12
2
 sin2<j>-Y312 cos2<f>
Note that the order of solution here is important since Eqs. (17)-(19)
were not linear. The orientation angle,<j>, must be determined first from
Eq (20) and then used in Eqs. (21) and (22) to find the ellipse axes.
The orientation angle, <fr, ranges from•-n/2 to Ti/2; nevertheless, Eq. (20)
still admits two possible solutions. This does not lead to an ambiguity
since the value of a determined from these equations gives the length of
the axis along the particular orientation angle chosen. Thus, a is hot
necessarily the major axis of the ellipse, it is the axis oriented in
the direction of the orientation angle,*, chosen from Eq. (20).
Thus, given a set of joint and point rain rate probability density
functions and the gauge spacing, it is possible to deduce unique
effective ellipse axes and the ellipse orientation angle for each rain
rate. The conclusions regarding practical gauge spacings discussed in
the context of the circular model apply in this case as well.
V. SUMMARY i
A simple rain gauge method for the determination of effective rain
cell dimensions and orientation has been described. Both circular and
elliptical rain cell models have been treated. Either two or three rain
gauges are required depending upon the model chosen. Further, the gauges
should be spaced relatively closely, on the order of one-half kilometer.
And, finally, the data processing required to extract the desired infor-
mation from the rain rate statistics is quite simple and straightforward.
Thus, a relatively simple, inexpensive technique is available for the-
statistical determination of effective rain cell dimensions and orientation,
11
ACKNOWLEDGMENTS
The author would particularly like to thank Professor R. T. Gompton
for his generous and helpful discussions concerning some aspects of this
development.
12
REFERENCES
•1. "Earth-Satellite Propagation Above 10 GHz," L. J. Ippolito, NASA
GSFC Rept. X-751-72-208, May, 1972. .
2. "The Prediction of Millimeter Wavelength Precipitation
Attenuation Using a Cylindrical Storm Cell Model."
L. R. Zintsmaster and D. B. Hodge, Report 2374-10, June 1972,
The Ohio State University ElectroScience Laboratory, Department
of Electrical Engineering; prepared under Grant NGR-008-080 for
National Aeronautics and Space Administration.
3. "Rain on Earth-Space Paths," D. C. Hogg, 1971 G-AP International
Symposium Digest, P. 322, Los Angeles, California, September 1971.
4. "Point and Area Precipitation Probabilities," E. S. Epstein, Monthly
Weather Review, V. 94, P. 595, 1966.
13


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