P. S. Eagleson‘ and R. F. Lariviere’
S U M M A R Y : The serial correlation coeKicient of monthly point precipitation at lag one is examined for a common fifty year period at thirty-seven stations spanning the North American continent.
The geographical distribution of these coefficients shows significant values along the Pacific (windward) coast which decay Eastward becoming insignificant near the Misskippi River.
This is interpreted as defining the spatial scale of maritime influence over precipitation on a windward land mass.
L‘kCHELLE D E L’INFLUENCE O C É A N I Q U E SUR LES PRECIPITATIONS CONTINENTALES.
RESUME : Le coefficient de corrélation d’ordre un d’une série de données de précipitation mensuelle en un point est examiné pour une période commune de cinquante ans à trente-sept stations s’étendant sur le continent nord-américain.
La distribution géographique d e ces coemcients montre des valeurs significatives le long de la côte du Pacifique (direction du vent); les valeurs diminuent vers l’Est et deviennent non signifi- catives pres du fleuve Mississippi. Ceci est interprété commc définissant la limite de l’influence maritime sur la précipitation d’une masse d’air située dans la direction du vent.
ESCALA D E LA INFLUENCIA OCEANICA SOBRE LA PREACIPITACI6N CONTINENTAL
R ~ S U M E N : En este documento se examina el coeficiente d e correlación serial de la precipitacih mensual en un punto en el primer retardo para un período común de 50 años en 37 estaciones distribuidas en el continente norteamericano.
La distribución geográfica de estos cocficieiites indica valores significativos a lo largo dc la costa del Pacífico (a barlovento) que decrecen hacia el este, llegando a ser insignificantes en las proximidades del rio Mississippi. Esto se interpreta como la definición de la escala espacial de la influencia marítima en la precipitacihn sobre masas terrestres a barlovento.
INTRODUCTION
It has been recognized for some time that certain observed long-period atmospheric persistence inay result from long-perjod ocean-atmosphere interactions. That this should be so seems eininently reasonable since thc mechanical aiid thcrmal inertias of the ocean I. Professor of Civil Engineering, Massachusetts Institute of Technology. U SA.
2. Engincer, Hydro-Québec, CANADA.
The Scule oj' Oceanic InJluence ori Coritirtentul Precipitation
are respectively lo3 and 10" times those of the atmosphere. Maritime air masses should therefore follow with considerable fidelity the slowly changing thermal state of their parent ocean. Where these air masses move over windward land surfaces, ground stations should experience a correspondingly slow change (i.e. " persistence") in certain meteoro- logical variables. This persistence should decay with an increase in the distance between the ground station and the upwind sea due to the gradual assumption, by the air mass, of continental in place of maritime characteristics. For continental ground stations near a lee coast the persistence should be much shorter due to the lower thermal incrtia of the up-wind land masses.
In this brief paper, the above erect is explored through the examination of serial correlation in point precipitation.
DEFINITION OF PERSISTENCE
Persistence will be defined here as the existence, at some lag k, of a serial correlation coef- ficient r, which is significantly greater than zero with some level of assurance. The correla- tion coefficient is defined by
N - k X i X i + k i = 1 (1)
N - k
rk
=
c
X?i = 1
where:
rk
At i,k N xi
serial correlation coefficient at time lag kdt;
time interval between values OF the discrete variable x(idt):
counting variables for time intervals;
number of observations of .u;
x(idt) = precipitation during ith period of time At as a deviation from the long- term mean.
For a truly random process, the serial correlation coefficient is of course zero exept fork
=
0. Correlation coeficients estimated from a sample function of length N taken from this process may be non-zero, however, and tests for the significance of these non-zero coeficients exist. In particular, the test of significance for the correlation between two random variables can be written (Brooks and Carruthers, 1953)where:
11
f p
r$
number of degrees of freedom = N- k
-
2 ;Student's variable at exceedance probability, I - P;
value of serial correlation coefficient which must be exceeded for significance at the chosen level P.
T o avoid the bias introduced into this test by series which are non-random, Bartlett (1935) recommends replacing v in equation (2) by the effective number of degrees of freedom v' calculated according to
P. S. Euglesori utid R. F. Lurivière
36
The Scnle of Ocrunic Itijlrretice on Continerircrl Precipitation
If w e assume the process to be a first-order Markov process then (Dawdy and Matalas, 1964) the modification of v depends only upon r , according to
v'
-
1-r:v ~ + r :
DATA SELECTCON
T w o criteria were used in the selection of the station records to be analyzed. Firstly, since the primary purpose is to study oceanic influence at continental scale, the stations should be numerous and should be distributed across the North American continent froin coast-to-coast. Secondly, the records should all be for a c o m m o n period of years and this period should be as long as possible.
The stations selected are shown on figure 1 and the c o m m o n period of record was 1910-1959 inclusive.
DATA A N A L Y S I S
The serial correlation coefficient at lag one (i.e. k = 1) was calculated for each station according to equation (I) for
A Z =
1 day, 1 month and 1 year. T h e respective time period and number of observation pairs are listed in table 1.TABLE 1. Tinte Series for Calculation of Serial Correlation Coefficient at Lag O n e
Time Interval Record Pcriod Number of Obser-
vation Pairs N-1 O n e D a y
O n e Month O n e Year
January-June, I958 I80 19 1 O- 1959 599
1919- 1959 49
The correlation coefficients r , and their 95%) significance levels r y are tabulated in table 2. The values of r y were determined from equation (2) with v replaced by v' through equation (4).
D I S C U S S I O N
Note from table 2 that at the 95% level, the serial correlation coefficient at lag one for annual precipitation has only two significant values and w e therefore have no evidence for persistence at this time scale. For daily precipitation it appears likely that persistence exists at several stations but no geographical pattern emerges. Monthly precipitation however provides overwhelming evidence of systematically varying persistence, a fact which could be anticipated from the observed seasonality of precipitation at many locations.
Roesner and Yevdjevich (1966) have shown that this persistence can be modeled very well by a Fourier series containing 12, 6, 4 and 3 month periodicities having amplitudes which decay exponentially with frequency.
TADLE 2. Serial Correlation Coeficientc at Lag One for Point Precipitation
Water balance
-
fndiuri O c e a nThe geographical distribution of the strength of the persistence is very revealing as is shown here in figure 1. Notice h o w the high values lie along the Pacific (windward) coast and decay inland becoming largely insignificant East of the Mississippi River. This pattern suggests the decay of maritime air mass characteristics as the air over-rides a continent, and the spatial rate oî this decay, as given by the distribution of significant monthly coefficients, m a y be used to measure the spatial scale of oceanic influence on continental precipitation.
REFERENCES
BROOKS, C.E.P. and C A R R U T H E R S , N. (1953): fIutidhook of Stutisticd Methods in Meteorology, London, Her Majesty's Stationery Office, p. 220.
BARLETT, M. S. (1935): Some aspects of the time-correlation problem in regard Lo tests of signifi- cance, f. Roy. Statist. Soc., vol. 98, pp. 536-543.
DAWDY, D. R. and MATALAS, N. C. (1964): Hundbook OfAppliedHydrology, editedhy V.T. Chow, N e w York. McGraw-Hill Book Co., Section 8-111, pp. 886-887 :
ROESNER, L.A. and YEVDJEVICH, V. M. (1966): Mathematical models for time series of monthly precipitation and monthly runoff, Hydrology Paper No. 15, Colorado State Ulriuersity, Fort Collins.