An initial model of the relationship between rainfall events and daily rainfalls

Journal of Hydrology, 75 (1984/1985) 357--364 Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands

357

[4] AN INITIAL MODEL OF THE RELATIONSHIP BETWEEN RAINFALL EVENTS AND DAILY RAINFALLS

K.J.A. REVFEIM Meteorological Service, Wellington 1 (New Zealand) (Received February 16, 1984; accepted for publication May 21, 1984)

ABSTRACT Revfeim, K.J.A., 1984. An initial model of the relationship between rainfall events and daily rainfalls. J. Hydrol., 75: 357--364. A simplistic model of mutually independent rainfall occurrences, amounts and durations is used to relate the statistical properties of rainfall accumulations over long and short time intervals. The assumption of exponentially distributed amounts and durations leads to a partitioning of rainfall events into a three-component distribution of daily amounts. Properties of this distribution are compared with some characteristics of daily rainfall data used in more complex stochastic models.

INTRODUCTION A d v a n c i n g b e y o n d p u r e l y statistical d e s c r i p t i o n s o f a n n u a l or m o n t h l y rainfall t o t a l s t h e r e are t w o a p p r o a c h e s w h i c h recognise s o m e o f t h e s t o c h a s t i c p r o p e r t i e s o f t h e rainfall process. T h e first a p p r o a c h , p r o b a b l y initiated b y Le C a m (1961), considers t h e p o i n t - p r o c e s s a p p r o x i m a t i o n o f t h e o c c u r r e n c e a n d a m o u n t o f rainfall events. By events is m e a n t t h e p r e c i p i t a t i o n arising f r o m s o m e shower, or cluster o f showers, d u e t o a f r o n t a l passage, c o n v e c t i v e b u i l d u p or o t h e r fairly c o h e r e n t a t m o s p h e r i c s i t u a t i o n . T h e s e c o n d a p p r o a c h considers t h e s t o c h a s t i c s e q u e n c e o f daily or h o u r l y rainfalls w i t h less a t t e n t i o n t o t h e p h y s i c a l a t m o s p h e r i c processes u n d e r l y i n g t h e d a t a (see Stern a n d Coe, 1 9 8 4 , f o r references). T h e essential d i f f e r e n c e b e t w e e n these t w o a p p r o a c h e s is t h e t i m e scale o n w h i c h t h e s e o b s e r v a t i o n s are m a d e . In t h e first a p p r o a c h t h e m o n t h l y or a n n u a l rainfall t o t a l s can be a s s u m e d t o c o m p r i s e w h o l e rainfall e v e n t s a p a r t f r o m t h e relatively small c o n t r i b u t i o n f r o m t h e p r o b a b i l i t y o f p a r t events at t h e b e g i n n i n g or e n d o f m o n t h or year. In t h e s e c o n d a p p r o a c h w i t h daily or h o u r l y rainfalls t h e r e is a relatively high p r o b a b i l i t y o f rainfall events being p a r t i t i o n e d into t w o or m o r e c o n s e c u t i v e periods. T h e r e l a t i o n s h i p between models for these two approaches depends on the duration and timevarying i n t e n s i t y o f t h e rainfall events. 0(}22-1694/84-85/$03.00

© 1984 Elsevier Science Publishers B.V.

358 Eagleson ( 1 9 7 8 ) observed that the e x p o n e n t i a l distribution gave a reasonable fit to " s t o r m d u r a t i o n s " where storms were defined by a m i n i m u m d r y - p e r i o d separation o f at least 2 hr. This could be some f o r m o f sub-clustering within an overall cluster o f showers making up an event. Storm o c c u r r e n c e and d u r a t i o n were f o u n d to be u n c o r r e l a t e d ( " i n d e p e n d e n t " ) b y Grace and Eagleson (1966). The a m o u n t and d u r a t i o n o f rainfall events do n o t a p p e a r to be m o r e than weakly c o r r e l a t e d and Eagleson ( 1 9 7 8 ) m a d e a similar a s s u m p t i o n t h a t average intensity ( a m o u n t / d u r a t i o n ) and d u r a t i o n were " i n d e p e n d e n t " . T o d o r o v i c and Yevjevich ( 1 9 6 9 ) assumed an e x p o n e n t i a l (first-order gamma) distribution o f a m o u n t s for finite-period rainfalls. Buishand ( 1 9 7 7 ) f o u n d t h a t daily rainfall a m o u n t s shifted b y the observation t h r e s h o l d fitted a g a m m a distribution o f a b o u t second order. Daffy rainfalls e x a m i n e d b y Stern and Coe {1984) gave the best fit to g a m m a distributions of o r d e r less t h a n 1. Katz ( 1 9 7 7 ) also f o u n d g a m m a distributions o f order 0.8 fitted the a m o u n t s o f b o t h wet-following-dry days and wet-following-wet days. The latter t w o conclusions are s o m e w h a t i n c o n g r u o u s since the implication is t h a t the m o s t f r e q u e n t a m o u n t s o f rainfall on wet days are infinitesimally small. Because o f the classification o f wet days o n e would e x p e c t some nonzero m o d a l i t y o f a m o u n t s . It seems p r o b a b l e in the a b o v e - r e f e r e n c e d cases and similar studies t h a t the fitting of a single distribution is inappropriate. The rainfall event a m o u n t s themselves m a y be f r o m a m i x t u r e o f two or m o r e t y p e s of event and the partitioning b y observation times into whole events, c o n t i n u i n g events, and initial or terminal tails o f events, is likely to f u r t h e r c o m p l i c a t e the distribution o f daily a m o u n t s . T h e a p p r o x i m a t i o n o f e x p o n e n t i a l l y distributed event a m o u n t s with i n d e p e n d e n t o c c u r r e n c e as a Poisson process was described by Le Cam {1961) as "vastly oversimplified". However, this m o d e l has been applied with reasonable success to m o n t h l y or annual rainfall totals by Buishand (1977), O z t d r k {1981), A l e x a n d e r s o n (1982), Revfeim (1982) and T h o m p s o n {1984). In this paper the simple m o d e l is e x t e n d e d t o include an i n d e p e n d e n t e x p o n e n t i a l distribution o f durations which gives a relationship b e t w e e n a m o u n t s per event and per rainy day, and also a m i x e d distribution o f daily rainfall amounts. This s t u d y is m e a n t as an initial guide as to how little extra i n f o r m a t i o n might be r e c o v e r e d f r o m a r e c o r d o f daily rainfall totals c o m p a r e d with a r e c o r d o f m o n t h l y rainfall totals.

MODEL Assuming t h a t events o c c u r as a Poisson process, at rate p, with m u t u a l l y i n d e p e n d e n t e x p o n e n t i a l l y distributed a m o u n t s , o f m e a n p, a n d durations, of m e a n 5, t h e n it is easy to derive t h e d e n s i t y f u n c t i o n o f the t o t a l a m o u n t

359 in time t as:

h t ( x ; p , p , 5) = x -l e x p ( - - p t - - x / p )

~

(pt2x/~p) ~

r=l

× ~Fl{r, 2r + 1 ; ( p - - 1/5)t}/(r-- 1 ) ! ( 2 r ) !

(1)

w h e r e 1F1 (a, fl, z) is a c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n ( S n e d d o n , 1956). T h e density f u n c t i o n o f t h e point-process a p p r o x i m a t i o n , w h e n event d u r a t i o n s are assumed t o be relatively insignificant c o m p a r e d with t h e t i m e b e t w e e n events, has a similar f o r m (l~evfeim, 1 9 8 2 ) :

h t ( x ; p , g ) -= X -1 e x p ( - - p t - - x / y )

~, (ptx/g)r/(r - 1 ) ! r ! r=l

which can be w r i t t e n as

ht (y;.p) = exp ( - - p t - - y 2 / 4 p t ) I i (y)

(2)

w h e r e y = 2 (ptx/p) 1/2; and 11 is a m o d i f i e d Bessel f u n c t i o n . T h e e s t i m a t i o n o f p and p f r o m eq. 2 is fairly s t r a i g h t f o r w a r d and can easily be e x t e n d e d to h a r m o n i c m o d e l s o f seasonality in the p a r a m e t e r s f o r analysing m o n t h l y rainfall totals. J o i n t e s t i m a t i o n o f p, g and 6 f r o m eq. I has as y e t n o t b e e n successful b u t it is intuitively obvious t h a t t h e e f f e c t o f including d u r a t i o n will be t o decrease the e s t i m a t e d r e c u r r e n c e rate t h e r e being less t i m e for events t o occur. H o w e v e r , the e s t i m a t e d m e a n a m o u n t will increase since t h e t o t a l s are c o n s t r a i n e d t o be equal t o p g b y the assumption of independence. L e t us n o w consider w e t d a y s so t h a t o c c u r r e n c e is no longer a relevant p a r a m e t e r and let us f u r t h e r assume t h a t : (1) t h e t i m e o f o n s e t o f rainfall in an e v e n t is u n i f o r m l y d i s t r i b u t e d over t h e unit scale (0, 1) daily interval, i.e. n o diurnal p a t t e r n ; and (2) t h e i n t e n s i t y o f rainfall is u n i f o r m over t h e d u r a t i o n o f an event. If a m o u n t s o f an e v e n t have d e n s i t y f u n c t i o n : f(x;g)

= ( l / p ) exp ( - - x / y )

t h e n a p r o p o r t i o n p will be d i s t r i b u t e d as f(x;pg). T h e n given t h e start o f an event at t i m e (1 -- t), (0 < t < 1), the d u r a t i o n s m a y be divided into t h r e e classes: (a) d u r a t i o n u ~ t and a m o u n t d i s t r i b u t e d f(x;g); (b) d u r a t i o n u ~ t with a m o u n t s in t h e o n s e t interval d i s t r i b u t e d as f(x;gt/u), and, a f t e r enduring t h r o u g h k w h o l e intervals, the a m o u n t in t h e t e r m i n a t i o n interval d i s t r i b u t e d as f{x ;g(u -- t - - k ) / u } ; and (c) k a m o u n t s in the w h o l e intervals d i s t r i b u t e d as f(x;g/u). Considering t h a t t is u n i f o r m l y d i s t r i b u t e d o n (0, 1) the p r o p o r t i o n a l f r e q u e n c i e s in t h e classes are:

360 1

(a)

P~ = I- [1

exp(--t/8)]dt -

1--5[1--exp(

1/6}]

0 1

(b)

P2 = 2 ~ e x p (

t/6)dt = 2811--exp(

1/5)]

0 t+2) + 2(0 t+2

q- 3 ( 0 t + 3

0 1

P3 = f

(c)

[(0 t + '

0 t+3)

-- o r + 4 ) . .

I~d t

0

= 6exp(

1/6)

whereO =exp( 1/6) If we assume t h a t n o m o r e t h a n one event can o c c u r on the same day t h e n t o t a l f r e q u e n c y o f daily rainfalls is (Pl + P2 + P3), i.e. (1 + 8) times the f r e q u e n c y o f events. Since the t o t a l a m o u n t o f rain is the same for all events and all rainy days the m e a n rainfall on wet days is a p p r o x i m a t e l y p/(1 + 8). This gives a simple m e t h o d for an a p p r o x i m a t e estimate o f the d u r a t i o n o f rainfall events if we assume t h a t t h e estimated m e a n a m o u n t ignoring d u r a t i o n is little d i f f e r e n t f r o m the estimated m e a n a m o u n t including d u r a t i o n . Then we can estimate the m e a n d u r a t i o n f r o m :

8

=

~/~

1

where ~ is the m e a n n u m b e r o f raindays per year; and T the mean annual rainfall. F o r example, the estimated mean event d u r a t i o n s for f o u r New Zealand sites, using the /~ given in Revfeim and Hughes ( 1 9 8 3 ) and ~ and f r o m station records, are given in Table I. The estimated d u r a t i o n s are in a g r e e m e n t with the n a t u r e o f the local rainfall climates. Since m o r e t h a n one event m a y s o m e t i m e s o c c u r o n o n e d a y the estimated d u r a t i o n is an upper bound. TABLEI Meanduration, 6, ofrainfalleventsatfourNew Zealandstations Station location

Auckland Wellington Christchurch Dunedin

/4 (mm)

"F (ram/yr.)

ff (wet days/yr.

(days)

13.7 14.0 13.5 11.4

1,184 1,215 663 937

184 158 126 160

]. 13 0.82 1.57 0.95

Similarly, t h e c o n t r i b u t i o n s t o the f r e q u e n c y d i s t r i b u t i o n o f daily rainfall a m o u n t s f r o m the t h r e e classes are: [1 :

(Pl/P) exp ( - - x / p )

361 l

oo

= 2s (S
xu, , du}d,

t

x l # + 116

= (28/p)

[

0 -1 exp (--0)d0

x / la

- - ( 2 / p ) exp [--(x/# + 1/5)]/(x/# + 1/5) 1

t +2

t +3

0

/+1

t+2

t +4

f3 = S { f+2 ~ +3 ~ +...}(u/pS)exp[--u(xlp+liS)]dudt /+3

= ( l / p ) exp [--(x/p + 1/5)] [1 + 2/(x/p + 1/5)]/5(x/p + 1/5) 2 which combine to give a mixedKtensity function of all daily rainfalls. Now the first c o m p o n e n t of f2 above is a difference of two exponential integrals and has an infinite asymptote at the origin. The remaining contributions from (fl + f2 + f3) are a weighting of the exponential density (lip) exp (--x/#). Hence as the assumptions are not altogether unrealistic, it is not surprising that Katz (1977) and Stern and Coe (1984) should identify a gamma distribution of order less than 1 in fitting a single distribution to this mixture. The asymptote on the vertical axis from the mixture is sufficiently large to give the result that these authors obtained. The probability of rain continuing till day (i + j ) given that the event starts on day i is: 1

Pu = j exp [ - - ( t + j - - 1 ) / 5 ] d t

= 5 e x p [--(] - - 1 ) / 5 ] [ 1 - - exp (--115)]

0

--- exp (-- ll~)pi,j_ 1 The probability of rain commencing on any day is the daily recurrence rate; ~ p/30.5 if the model was fitted to m o n t h l y rainfall totals. If a seasonal model has been fitted then p will vary for different months.

DISCUSSION

The previous section has concentrated on the partitioning of events into different days but there is an equally important concept of daily observation times covering or linking different events. There is a possibility of two fairly independent events occurring within one daily observation period; for example, a frontal band of rain may be followed by a complete clearance then within 12 hr. a thermal trough behind the front can produce convective showers. Other synoptic situations may also give rise to separate frontal, local convective or orographic rainfall events being included with the tail

362 o f an e v e n t on t h e initial or t e r m i n a l d a y o f a 2-day or longer sequence. It is also necessary to r e c o g n i z e t h a t an " i n t e r i o r " d a y o f a s e q u e n c e m a y be t h e s u m of t w o tails, or the single tail o f succeeding events, or a s e p a r a t e event b e t w e e n t w o tails, or o n e e v e n t within a s e q u e n c e o f s e p a r a t e events. These possibilities will alter t h e simple p a r t i t i o n i n g into classes o f o b s e r v e d isolated, initial/terminal and interior days. A l t h o u g h in a m i n o r i t y , such possibilities are also likely to r e p r e s e n t t h o s e situations w h e r e e v e n t o c c u r r e n c e , d u r a t i o n and a m o u n t m a y n o t be u n c o r r e l a t e d . These situations m a y f u r t h e r require a m o d e l o f s e p a r a t e r e c u r r e n c e rates, size distributions, and d u r a t i o n s f o r a m i x t u r e o f f r o n t a l and c o n v e c t i v e e v e n t types. Buishand (1977, p.196b} f o u n d t h a t m e a n rainfall on isolated wet d a y s was smaller t h a n t h e m e a n o f first a n d last d a y s o f a w e t spell, which again was smaller t h a n t h e m e a n o f interior w e t d a y s o f a sequence. T h e possibility o f m o r e t h a n o n e e v e n t on a d a y and t h e m e a n o f f~ suggest t h a t the mean rainfall on isolated w e t d a y s should be at least p which is greater t h a n the overall m e a n p/(1 + 5). The weight given to large a m o u n t s o f rain within a limited d u r a t i o n such as 1 d a y is p r o b a b l y t o o unrealistic. It is equally possible t h a t isolated 1-day rainfalls m a y have a smaller m e a n size and d i f f e r e n t l y s h a p e d d i s t r i b u t i o n t h a n 2oday or longer events. While a m o u n t s c o n t r i b u t i n g to t o t a l rainfalls f r o m a m i x t u r e o f convective and f r o n t a l events m a y be satisfactorily a p p r o x i m a t e d by a single e x p o n e n t i a l d i s t r i b u t i o n , this m a y not be a d e q u a t e for w e t days. C o n v e c t i v e events have s h o r t e r average d u r a t i o n t h a n frontal events and also do n o t e x t e n d into the u p p e r range o f a m o u n t s o f f r o n t a l events. At certain t i m e s o f the y e a r at s o m e tropical l o c a t i o n s t h e r e is a higher p r o b a b i l i t y o f c o n v e c t i v e events within the latter p a r t o f t h e diurnal heating cycle. F o r e x a m p l e (Finkelstein, 1967t, 55% o f Nandi (Fiji) rainfall falls b e t w e e n 12500 ~ and 20h00 m and o n l y 20% in t h e p r e c e d i n g 8 hr. In this rainfall regime m a n y events w o u l d o c c u r entirely within the latter p a r t o f the diurnal heating cycle. This could lead to smaller mean rainfalls on isolated w e t d a y s d u e to t r a d i t i o n a l early m o r n i n g o b s e r v a t i o n times in support of Buishand's observation. T h e m o d e l w o u l d be m o r e realistic if t h e a m o u n t and d u r a t i o n o f an event were c o r r e l a t e d so t h a t very large a m o u n t s c o u l d n o t be realized in very small d u r a t i o n s . Also it w o u l d be natural to i n t r o d u c e s o m e t i m e variability into t h e i n t e n s i t y b u t with a t e n d e n c y to m o r e u n i f o r m i n t e n s i t y as t h e m e a n i n t e n s i t y o f an event increases. K a t z ( 1 9 7 7 ) f o u n d little d i f f e r e n c e b e t w e e n t h e d i s t r i b u t i o n o f a m o u n t s in w e t - f o l l o w i n g - d r y a n d w e t - f o l l o w i n g - w e t days. B o t h o f these d i s t r i b u t i o n s w o u l d have an equal y - a s y m p t o t e c o n t r i b u t i o n f r o m f2 d u e to t h e first a n d last d a y s o f a rainfall event. CONCLUSIONS A simple m o d e l has b e e n p r e s e n t e d which challenges t h e s u p p o s i t i o n t h a t daily rainfall m o d e l s m u s t be m o r e i n f o r m a t i v e s i m p l y b e c a u s e t h e r e are

363 m o r e data. T h e m o d e l is a c o n j e c t u r e o f some e l e m e n t s o f t h e relationship b e t w e e n m o n t h l y rainfall p a r a m e t e r s and t h e characteristics o f daffy rainfall. T h e r e w o u l d seem t o be s o m e advantages in p a r t i t i o n i n g events into c l o c k h o u r s since this w o u l d r e m o v e t h e p r o b l e m o f m o r e t h a n o n e event cont r i b u t i n g t o a m o u n t s w i t h i n 1 hr. However, a 2--3-hr. " d e a d " t i m e in t h e r e c u r r e n c e process s h o u l d f o l l o w each event. F r o m this f r a m e w o r k it is n o t difficult to see t h e d i r e c t i o n t o take incorp o r a t i n g m o r e realistic p r o p e r t i e s such as diurnal p a t t e r n s o f rainfall initiation a n d c o r r e l a t i o n b e t w e e n d u r a t i o n and a m o u n t ( b o t h o f w h i c h m a y vary seasonally). O n l y in this w a y can t h e e f f o r t o f pursuing the marginal benefits o f daily rainfall analysis be justified.

ACKNOWLEDGEMENT The a u t h o r t h a n k s Mr. C.G. Revell, New Zealand Meteorological Service, for c o m m e n t s o n the t y p e s o f rainfall events t h a t m i g h t o c c u r within the same o b s e r v a t i o n period.

REFERENCES Alexanderson, H., 1982. A stochastic model of precipitation with applications, extensions and examples. Meteorol. Inst. K. Univ., Uppsala, Rep. No. 69. Buishand, T.A., 1977. Stochastic modelling of daily rainfall sequences. Meded. Landbouwhogesch. Wageningen, No. 77-3. Eagleson, P.S., 1978. Climate, soil and vegetation, 2. The distribution of annual precipitation derived from observed storm sequences. Water Resour. Res., 14: 713--721. Finkelstein, J., 1967. Diurnal variation of rainfall amount on tropical Pacific islands. N.Z. Meteorol. Serv., Tech. Info. Circ. No. 119. Grace, R.A. and Eagleson, P.S., 1966. The synthesis of short-time increment rainfall sequences. Hydrodyn. Lab., Mass. Inst. Inst. Technol. (M.I.T.), Cambridge, Mass., Rep. No. 91. Katz, R.W., 1977. Precipitation as a chain dependent process. J. Appl. Meteorol., 16: 671--676. Le Cam, L., 1961. A stochastic description of precipitation. In: J. Neyman (Editor), Proc. 4th Berkeley Syrup. on Mathematical Statistics and Probability. University of California Press, Berkeley, Calif. (Sztiirk, A., 1981. On the study of a probability distribution for precipitation totals. J. Appl. Meteorol., 20: 1499--1505. Revfeim, K.J.A., 1982. Comments "On the study of a probability distribution for precipitation totals". J. Appl. Meteorol., 21:1942--1945 (corrigendum and addendum, 22: 502). Revfeim, K.J.A. and Hughes, H.S., 1983. Physically meaningful parameters that characterise rainfall totals and rainfall extremes. N.Z.J. Sci., 26: 443--445. Sneddon, I.N., 1956. Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh, 164 pp. Stern, R.D., and Coe, R., 1984. A model fitting analysis of daily rainfall data. J. R. Star. Soc., Ser. A, 147: 1--34.

364

Thompson, C.S., 1984. Homogeneity analysis of rainfall series: An application of the use of a realistic rainfall model. J. Climatol. (In press.) Todorovic, P. and Yevjevich, V., 1969. Stochastic process of precipitation. Colo. State Univ., Fort Collins, Colo., Hydrol. Pap. No. 35.