A preliminary investigation was made to determine estimates of the number of raingauges needed in order to measure the variability of rainfall in time and space over small areas (approximately 40 sq miles). The literature on rainfall variability was examined and the types of empirical relationships used to relate rainfall variations to meteorological and catchment-area characteristics were considered. Relations between the coefficient of variation and areal-mean rainfall and area have been used by several investigators. These parameters seemed reasonable ones to use in any future study of rainfall variations. From a knowledge of an appropriate coefficient of variation (determined by the above-mentioned relations) the number rain gauges needed for the precise determination of areal-mean rainfall may be calculated by statistical estimation theory. The number gauges needed to measure the coefficient of variation over a 40 sq miles area, with varying degrees of error, was found to range from 264 (10% error, mean precipitation = 0.1 in) to about 2 (100% error, mean precipitation = 0.1 in)

Runnels, R. C.

English

NASA Technical Reports Server

L• | : ,
i( VARIABILITY OF RAI:_ALL OVER SMALL AREAS
N86-14098
Robert C. Runnels
Assistant Professor
Department of Meteorology
:_ ; Texas A&MUnlversity
College Station, Texas
ABSTRACT
A preliminary investigation was made to determine estlmatls of the number of
:: ralngauges needed in order to measure the variability of zainfall in time and
apace over small areas (approximately 40 mi2). The literature on rainfall varia-
bility was examined and the types of empirical relatlonsh_ps used to relate rain-
fall variations to meteorologlca_ and catchment-area characteristics were considered.
Relations between the coefficient of variation and areal-mean rainfall and area
have been used by several investigators. These parameters seemed reasonable
ones to use in any future study of rainfall variations. From a knowledge of
an appropriate coefficient of variation (determined by the above-men:ioned
relations) the number of rain gauges needed for the precise determination of
areal-mean rainfall may be calcualted by statistical estimation theory. The
number of gauges needed to measure the coefficient of variation over a 40 mi 2
area, with varying degrees of error, was found to range from 264 (10% error,
mean precipitation - 0.i in) to about 2 (100% error, mean precipitation - 0.i in).
Center Research Advisor: Dr. David E. Pitts
n-z ,i
1986004609-321
https://ntrs.nasa.gov/search.jsp?R=19860004629 2020-03-20T16:34:11+00:00Z
+
i
5 IN_fRODUCTION
'i This paper discusses some preliminary considerations necessary for planning
a more-extenslve study on _he determination of the spatial and temgoral varia-
bility of rainfall over small areas. These variations may be important in
affecting the variability of soil moisture that, in turn, may affect crop yield.
_"q%eforecasting of crop yield by the remote sensing of the status of crops
can provide routine assessments over larger geographical areas than measurements
made at a few sample plots on the ground. Relations are sought between the
!
amount and frequency of energy reflected, scattered, and emitted (the signature)
back to the sensor from the crop _t a particular stage of crop growth. The
factors that can cause variations in signatures are numerous and the reduction
of these faators to measureable prcportlons i_ an essential aim of remote-
7
sensing research. Some of these factors are planting date, row direction, row
+:- width, leaf area index, spectral properties of individual leaves, the geometry
&d
- of the canopy, soil background, crop stage, and crop condition ( Pitts et al.,
19s3),
Weather-sensitive factors may produce significant correlations between yield
and particular meteorological elements but these factors may not act in a direct
. cause-effect relation (Munn, 1970). Insect infestations and crop diseases
are usually related to weather conditions but their impact may not be immediate.
Many investlgator_ feel that soll moisture is the best environmental predictor
of crop yield (Munn, Ibi____d,).However, once adequate rainfall has occurred,
variation in rainfall amount seems to have a small effect on crop yield. Pitts
e__tal, (1983) compared rainfall _mounts with the prediction of the date when _
spring wheat would be ripe. These dates were obtained from a sprlng-wheat phenology
model developed by Doraiswamy and Thompson (1982). Across small areal segments
(5X6 nautical miles) the influence of rainfall on the phenology of spring wheat
did not vary appreciably when rainfall amounts accumulated for periods of a day
or longer were used. Variations, of the elapsed days to the ripe stage were
o_ly 1 to 2 days out of approximately 225 days necessary to attain the ripe stage.
However, some slight variability, found within a few segments, did appear to be
related to variations in rainfall. Changnon and Neill (1968) compared the
association of corn yields from 60 farms with 10 weather and 4 agrlcu]tural
: variables, Nine years of data were used and the authors found that, among the
meteorological variables, monthly rainfall amounts did not seem to be too
+:+: 21-2
I,
9 +++.+,+. ++®
]986004609-322
h important. For a growing season extending from May until August., the best
rainfall-related correlation coefficient was between August rainfall and yield.
_ The value of this coefficient was 0.29. Rainfall amounts during July, June, and
the preseason months correlated more poorly.
_ The increasing use of sensors that can resolve spectral signatures of ever smaller
areas suggests the desirability of understanding variations of rainfall within
: areas smaller than 40 square miles. A preliminary understanding of these varia-
° _ ;ions is the object of this study.
Previous studies of rainfall variability were examined and the emplrica] relation-
ships, from these studies, were chosen that showed the association among rain-
_ ; fall variability and mean precipitation, size of sampling area, season, type of
_:_
_ precipitation, and synoptic weather type. When these relationships were combined
with statistical estimation theory, a simple expression was obtained that indi-?
cares the number of rain gauges needed in order to measure rainfall variability,
within certain error bounds, at a predetermined level of confidence.
THEORY
t
In order to find approximate empirical relationships between rainfall variability
aRd meteorological quantities, the extensive studies performened by the Illinois
State Water Survey (Huff and Neill, 1957, Huff and Shlpp, 1968, and _uff, 1970)
J
were chosen as being representative ones of rainfall during the growing season
over relatively flat terrain. Catchment areas ranged from I0 to 550 square miles
and they have been studied by this organization since 1955. Gauge densities
ranged from 0.12 to 0.50 gauges per square mile. Uniform spacing between gauges
, was used. Such spacing is preferable when rain is produced by convective
motions (Rodda, 1972). Convective rainfall is the type most likely to occur dur-
ing growing seasons.
The coefficient of variation (V) is a useful statistic for showing both variation
about the mean and variability relative to the mean.
V= p
!
• where s is the sample standard deviation and P is the sample mean rainfall amount.
_1-3 _.
1986004609- 2
l_ff and Neill (1957) have shown the following relations involving V_
V o&I/P,
V _ A,
where A is area,
V _rainfall gradient,
and
V _I/D_
where D is the duration of the rain event.
-
_ Rainfall gradient and duration are strongly proportional to mean rainfall.
Thus, to a first approximation, the natural variation of rsinfall, expressed
by V, is
log V = log k + a log P + b log A, (I)
where k, a, and b are constants obtained by regression analysis. Logarithms
of P and A were used sin a they range over several orders of magnitude. Typical
values, for shower-type precipitation, are k = 9, a = -0.44, and b = 0.15 when
V is in per cent, P is in inches, and A is in square miles. The correlation
coefficient of Eq. (i) is about 0.7. _
To relate the coefficient of variation, obtained by Eq. (I), with the minimum
number of gauges needed for adequate measurements, the following expression for
confidence intervals, for V, i3 used, v__iz.,
_ Z V
v 2, (2)
where Z is the standardized value of V and N is the number of observations (rain
gauges) (Spiegel, 1961). This expression gives the interval for (]-_)100%
confidence.
21-4
I
1986004609-:324
i__v _ _. _< •
Solution of Eq. (2) for N gives
N - + 2V2, (3)
where d is the percentage uncertainty in the measurement of V that is acceptable.
To be 95% confident, i -_ = 0.95, and Zo_ 1.96
RESULTS
?
An illustration of the results anticipated by the method explained above is shown
in Table I, An area of 40 square miles was chosen to correspond to the sizes
I
of areas recently observed by the Landsat Multi-Spectral Scanner (Pitts et al.,
1983).
: _ Table I. Required number of gauges (N) and gauge density (G_ tc measure the
variation of rainfall over an area of 40 square miles with a confidence
of 95%.
P V N G !
Miles 2inches - - --
Gauge
0.1 0.43 264 0.15
0.5 0.21 209 0.19 d- 0.1V1.0 0.15 202 0.20
2.0 0.11 198 0.20 _ _
0.I 0.43 66 0.61
0.5 0.21 53 0.75
d - 0.2V
, 1.0 0.15 51 0.78
2.0 0.ii 49 0.82
0.I 0.43 ii 3.6
0.5 0.21 9 4.4
1.0 0.15 8 5.0 d ,,0.5V
2.0 0.11 8 5.O
0.1 0.43 3 13.3
0.5 0.21 2 20.0,
1.0 0.15 2 _0.0 d = 1.0V
r 2.0 0.Ii 2 20.04"
21-5
1986004609-325
, .L
j
CONCLUSIONS
!
The relationship betwaen V and P and A, from the Illinois State Water Survey, is
consistent with results of other studies of the characteristics of shower-type
rainfall_ The effect of mean rainfall i_ greater _han the effect of area in the e.m-
plrical determination of V.However, area of rainfall is not independent of rain-
fall mean. i_.(1) with a correlation coefficient of 0.7 and a coefficient of
l
determination of 0.49 means that only 4% of the variation is explained by a
!
' linear relation of logarithms. It remains to be seen whether or no_ non-linear . [
relations and/or the incorporat'on of o.[.er variables will produce higher
correlations, f
d
Huff(1970) indicates that Eq. (I) may have an uncertainty of 2 orders of magnitude.
This range of uncertainty is xtotunusual for many geophysical measurements but it
:_ may be large enough to affect the estimate of the minimum number of gaugesd
determined by Eq. (i).
The regression coefficients for Eq. (I) vary, to some extent, with locality.
This varlation, in the coefficients, _uggests a survey of rainfall characteris-
tics in a variety of crop-growing regions during the growing season.
)
Continuation of this research will consist of an investigation of rainfall data
collected during concomitant surveys of crops by remote sensing from Landsat. '_
Sites from several climatic zones were surveyed with each site having dimensions
of 5 x 6 v mi. Terrain is relatively smooth and the number of rain _uges varies i
from 1 to 23 per site. The rainfall data can be stratified in several ways and
regressl )n relationships other than linear will be investigated. Since the
ralnfall-produclng processes are complex, an initial investigation, as suggested t
here, is needed to de_act some of Lhe predominant relations between these
processes and the res,lting rainfall characteristics.
I'
z1-6
]986004609-326
i REFERENCES
I. Changnon, Jr., S.A., and J.C. Neill, "Corn-Weather Response on Cash-Grain
b Farms," Journal of Applied Meteorology,, Vol. 7, pp 94-104, February, 1968. ,_
2. Doraiswamy, P.C., and D.R. Thompson, "A Crop Moisture Stress Index for Large
Areas and Its Application in the Prediction of Spring Nheat Phenology,"
_: A_ricult'_ral Meteorology, Vol. 27, pp 1-15, I_82. I
3. Huff, F.A., "Sampling Errors in Measurement of Mean Precipitation," Journal
I,
_- o fg Applied_ Meteorology, Vol. 9, pp 35-44, February, 1970.
__ 4. and J.C. Neill, Rainfall Relations on Small Areas in i,
_, Illinois, Illinois State Water Survey, Urbana, 1957, 61 p.
_J p
5. _, and W.L. Shipp, '_esoscale Spatial Variability in Mid- }
western Precipitation," Tournal of Applied Meteorology, Vol. 7, pp 886-891,
_ October, 1968.
6. }_nn, R.E., Biometeorolo_ical Methods, Academic Press, N.Y., Chapter 11, 1970.
!
•- )
7. Pitts, D.E._ D.R. Thompson, K.E. Henderson, and M. Wise, "The Effect of Rain-
e
fall Variability on Spring Wheat Growth Stages," Preprint Volume of Extended
.
_ Abstracts, 16th Conference on Agricultural and Forest Meteorology, American
Meteorological Society, Boston, pp 159-162, April, 1983. "_4
S
8. Rodda, J.C., "Precipitation," Casebook on Hydrological Network Design Practice, I
!
• Workd Meteorological Organization, Geneva, pp i-1.2-i - i-1.2-15, 1972.
9. Spiegel, M.R., _ and Problems of Statistics, Schaum Publishing Co., N.Y.,
"4
Chapter 8, 1961.
J , !
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]986004609-327


Variability of rainfall over small areas

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