INTERNATIONAL JOURNAL OF CLIMATOLOGY, VOL. 14, 403-414 (1994)

551.577.21:551.508.77:551.502.3

ESTIMATING CONTINENTAL AND TERRESTRIAL PRECIPITATION AVERAGES FROM RAIN-GAUGE NETWORKS CORT J. WILLMOTT Center for Climatic

Research, Department

of Geography,

University

of Delaware,

Newark,

Delaware

19716,

USA

SCOTT M. ROBESON Department

of Geography,

Indiana

University,

Bloomington,

Indiana 47405,

USA

AND JOHANNES J. FEDDEMA Department

of Geography,

University

of California,

Los Angeles,

California

90024,

USA

Received 13 November 1992 Accepted 14 July 1993

ABSTRACT Influences of varying rain-gauge networks on continental and terrestrial precipitation averages (derived from data observed on those networks) are evaluated. Unsystematically and systematically designed station networks are considered, the latter being represented by the NCAR World Monthly Surface Station Climatology, which contains hand-picked but time-varying networks that date back to the 1800s. Biases arising from spatially uneven and temporally variable precipitation-observing networks can be significant. For all the continents, except South America, sparse rain-gauge networks produce overestimates of continental mean precipitation. Mean precipitation for South America, in contrast, is underestimated substantially by low densities of observing stations. Sampling errors tend to be large in areas of high precipitation and in regions with strong spatial precipitation gradients (e.g. in the Sahel). These patterns occur whether the station network has been selected systematically (as in the NCAR network) or unsystematically. Systematic sampling of mean precipitation (at the NCAR station locations), however, suggests that many yearly NCAR station networks are adequate for estimating continental average precipitation. As early as 1890, NCAR networks for Australia resolve continental average precipitation accurately. Not until 1960, however, do NCAR networks for South America begin to resolve continental mean precipitation adequately. Regional and continental NCAR network errors also tend to cancel one another, often giving accurate yearly estimates of terrestrial mean precipitation. KEY WORDS

Precipitation averages

Rain-gauge networks

INTRODUCTION C o n t i n e n t a l a n d terrestrial averages of p r e c i p i t a t i o n a r e key c o m p o n e n t s of t h e global hydrological cycle, yet completely reliable estimates of these spatial m e a n s are n o t available ( W i l l m o t t a n d Legates, 1991). Even less is k n o w n a b o u t the seasonal a n d i n t e r a n n u a l variability in c o n t i n e n t a l a n d terrestrial precipitation averages. A l t h o u g h global climate m o d e l ( G C M ) s i m u l a t i o n s a n d remotely sensed estimates of large-scale precipitation are i m p r o v i n g (Legates a n d W i l l m o t t , 1992), historical r a i n - g a u g e r e c o r d s a n d n e t w o r k s continue t o comprise the bases for the m o s t credible e s t i m a t e s (Legates a n d W i l l m o t t , 1990; H u l m e , 1992). Large-scale spatial averages m a d e from historical d a t a , in o t h e r w o r d s , c o m p r i s e o u r best u n d e r s t a n d i n g of the spatial, seasonal a n d i n t e r a n n u a l variability in c o n t i n e n t a l a n d terrestrial averages of precipitation. T h e reliability a n d variability of c o n t i n e n t a l a n d terrestrial p r e c i p i t a t i o n a v e r a g e s t h a t have been derived from the historical rain-gauge record are the subject of this p a p e r . C C C 0899-8418/94/040403-12 © 1994 by the R o y a l Meteorological Society

40-

C. J. WILLMOTT, S. M. ROBESON AND J. J. FEDDEMA

:

Africa

-

Australia

; ;

r'ji

f i

Eurasia North America

South America

Terrestrial

>

Surface

:

\ i • 300

^

A

i i

:

200

\j :

1870

1890

1910

1930

1950

1970

E z

1990

Year Figure 1. N u m b e r of stations per year in the N C A R W o r l d M o n t h l y Surface Station Climatology (Spangler and Jenne, 1988) for the terrestrial surface (left axis) a n d for each continent (right axis) from 1881 to 1987.

Of all the climate variables, precipitation has been s a m p l e d m o s t extensively (Mintz, 1981; Legates and W i l l m o t t , 1990). Although m o r e station records are available for precipitation than for air t e m p e r a t u r e or pressure, it also is true that spatial variability of p r e c i p i t a t i o n is m u c h greater. Higher resolution station n e t w o r k s , therefore, are required to sample precipitation a d e q u a t e l y . Rain-gauge samples of precipitation, however, have been spatially uneven a n d historically variable. T h e N C A R World M o n t h l y Surface Station C l i m a t o l o g y (Spangler a n d Jenne, 1988), for instance, c o n t a i n s 636 stations for 1900 whereas 2327 stations are available for 1960 (Figures 1 a n d 2). These c h a n g i n g station networks have injected considerable u n c e r t a i n t y into estimates of terrestrial m e a n p r e c i p i t a t i o n (Willmott a n d Legates, 1991; Willmott et al, 1991). Averaging over the entire terrestrial surface has r e d u c e d the influence of regional- a n d continental scale u n d e r - a n d overestimates. Terrestrial averages, c o n s e q u e n t l y , are m o r e accurate than continental and regional averages. O u r p u r p o s e here is t o d e t e r m i n e the variability a n d reliability of spatially averaged estimates of precipitation. Station-network influences are e x a m i n e d at continental a n d terrestrial scales t h r o u g h computer-intensive sampling from a high-resolution climatology (Legates and Willmott, 1990). The a d e q u a c y of n u m e r o u s network densities a n d station d i s t r i b u t i o n s is described. In addition, the adequacy of the N C A R W o r l d M o n t h l y Surface Station C l i m a t o l o g y is investigated by resampling the Legates and W i l l m o t t high-resolution climatology at c o n t i n e n t a l scales.

SPATIAL S A M P L I N G , I N T E R P O L A T I O N , A N D A V E R A G I N G Spatial averages derived from any irregularly d i s t r i b u t e d s t a t i o n network contain biases. Bias m a y arise from t h e spatial interpolation used to estimate h o w t h e variable behaves between observations. E r r o r s additionally m a y arise from the influence of ill-conditioned station n e t w o r k s ; t h a t is, errors result when intrinsic spatial variability is n o t resolved adequately. It usually is assumed that the variable is spatially c o n t i n u o u s and that linear c o m b i n a t i o n s of the o b s e r v a t i o n s c a n be used to estimate values at unsampled locations (see L a m , 1983; T h i e b a u x a n d P e d d e r , 1987; Daley, 1991). Since time-averaged precipitation is highly variable in space and occasionally d i s c o n t i n u o u s , sparse station networks a n d i n a d e q u a t e interpolation algorithms can combine t o p r o d u c e non-trivial a v e r a g i n g errors. Spatial averaging typically involves t w o steps: the interpolation from an irregularly spaced n e t w o r k of station values to a regular grid a n d then

Figure 2. Spatial distribution of precipitation stations in the N C A R World Surface Station Climatology for three selected years: (a) 1900, (b) 1930, and (c) 1960.

406

C. J. WILLMOTT, S. M. ROBESON AND J. J. FEDDEMA

the weighted (by area) s u m m i n g of the gridded estimates. O u r analyses also m a k e use of this a p p r o a c h which is discussed in m o r e detail by Willmott et al (1985a) a n d b y W i l l m o t t a n d Legates (1991).* Although a variety of spatial interpolation m e t h o d s are available, w h e n interpolating over large sections of the Earth's surface, spherical (or even ellipsoidal) geometry s h o u l d b e used (Willmott et al, 1985a). Several spherically based interpolation procedures have been i m p l e m e n t e d (e.g. W a h b a , 1981; R e n k a , 1984; Willmott et al, 1985a); however, inverse distance-weighting algorithms h a v e b e e n used m o s t frequently in precipitation studies (Bradley et al, 1987; D i a z et al, 1989) a n d several a r e q u i t e reliable (Legates, 1987; Bussieres and Hogg, 1989; Weber and Englund, 1992). O n e such p r o c e d u r e , a spherical version of Shepard's (1968) inverse-distance weighting m e t h o d (Willmott et al, 1985a), is used h e r e t o interpolate precipitation d a t a to continental and terrestrial grids. The relative reliability of this i n t e r p o l a t i o n m e t h o d allows us to separate out and evaluate errors arising from ill-conditioned station n e t w o r k s — t h e theme of this paper. Interpolations are made to a spherical grid of 1° of latitude by 1° of longitude. T h i s lattice was chosen as a compromise between c o m p u t a t i o n a l efficiency and accurate spatial depiction. O n c e precipitation d a t a a r e interpolated, cosine-of-latitude weighting is used to integrate precipitation o v e r t h e c o n t i n e n t a l and terrestrial d o m a i n s .

CONTINENTAL AND TERRESTRIAL P R E C I P I T A T I O N AVERAGES Influences of precipitation measurement networks on spatial a v e r a g i n g may be analysed at a variety of scales. O u r interest here, however, is on large-scale a v e r a g i n g a n d therefore we restrict o u r analyses to precipitation networks at continental a n d terrestrial scales. O c e a n i c precipitation is not considered because b o t h the in situ measurements and observational n e t w o r k s a r e m u c h less reliable t h a n their terrestrial counterparts (Legates and Willmott, 1990). T w o sampling strategies are used t o assess precipitation n e t w o r k biases. In the first instance, s u b n e t w o r k s for six characteristic station densities are unsystematically ( r a n d o m l y ) a n d repeatedly sampled from the Legates and Willmott (1990) high-resolution climatology (hereafter referred to as the L W climatology). This procedure essentially follows a similar experiment by Willmott a n d Legates (1991). Illustrating for the lowest density network (of the six), 50 subnetworks (each at a s t a t i o n density of 2-5 stations per 1 0 k m ) were selected randomly (with replacement) from the L W climatology. Legates and Willmott's a n n u a l m e a n precipitation estimates at the station locations within each s a m p l e were then interpolated to the nodes of a 1° by 1° grid. The gridded values, in turn, were weighted b y cosine-of-latitude and s u m m e d to obtain each of the 50 spatial averages. Variability a m o n g the 50 s p a t i a l m e a n s is an indication of the natural variability induced by rain-gauge networks at this station density. Differences between each of the 50 sample means a n d the mean c o m p u t e d from all station values in the L W climatology are estimates of the accuracy of the 2-5 network density. This same experiment was r e p e a t e d for s u b n e t w o r k s of 5, 10, 15, 20, a n d 25 stations per 10 k m . These densities are typical of t w e n t i e t h - c e n t u r y precipitation station networks. The second strategy involved systematically sampling the L W climatology at all the station locations in the N C A R climatology for each year since 1881. Values of a n n u a l mean precipitation from the L W climatology at these nodes were then interpolated t o the 1° by 1° grid a n d averaged (as above) to obtain an estimate of annual m e a n precipitation associated w i t h each year's N C A R n e t w o r k . T h e difference between a spatial mean derived from a yearly N C A R n e t w o r k a n d a c o r r e s p o n d i n g m e a n o b t a i n e d from all the stations in the L W climatology is one measure of t h e efficacy of that year's N C A R network. 6

6

2

2

Several promising interpolation methodologies (e.g., that make use of a n o m a l y or frequency-distribution parameter transformations of the raw precipitation data) are not used or evaluated in this paper because they require one or more additional realizations of the held of interest (often at a high spatial resolution). Applications—such as ours—also require estimates of precipitation magnitudes rather than of derived statistics, e.g., anomalies. At the heart of our analyses is interpolation from a single realization of a field on a single station network. More precisely, we focus on evaluating (i) how well individual station networks adequately resolve a mean precipitation field over an extensive spatial domain and (ii) how accurately large-scale spatial averages can be constructed from observations on a single station network. This involves the spatial interpolation of precipitation averages from an irregularly spaced station network to a regular grid—the most common application of interpolation in climatology—and, in turn, the numerical integration of the gridded precipitation field.

407

ESTIMATING PRECIPITATION AVERAGES

Precipitation station networks from five continents—Africa, Australia, E u r a s i a , N o r t h America, a n d South A m e r i c a — a n d from the entire terrestrial surface (excluding G r e e n l a n d a n d Antarctica, w h e r e few stations were available) are evaluated separately. B o t h the u n s y s t e m a t i c a n d systematic e x p e r i m e n t s w e r e performed separately for each continent as well as for the terrestrial surface. E r r o r s in e s t i m a t i n g c o n t i n e n t a l and terrestrial m e a n precipitation are a n a l y s e d statistically a n d graphically t h r o u g h the use of box p l o t s a n d m a p s of s a m p l i n g errors. Africa Precipitation station distributions within Africa a r e sparse, relative to well-represented areas such as E u r o p e a n d the U S A (see Figure 3(a)). H i g h e s t s t a t i o n densities are found in relatively m o i s t a r e a s a l o n g the M e d i t e r r a n e a n coasts of M o r o c c o a n d Algeria, in s o u t h e r n Africa, coastal western Africa, a n d t h e east

o

300 '

'

eoT^ ™ 1

1 - 1 1 1

™ ?'''''''' 1 1

Figure 3. Spatial distribution of (a) the over 24000 precipitation stations and (b) a n n u a l mean precipitation (mm) from the climatology of Legates a n d Willmott (1990).

408

C. J. WILLMOTT, S. M. ROBESON AND J. J. FEDDEMA

5

10

Sample

O

5

Size

10

Sample

800

15

20

(Stations

15

Size

20

25

(Stations

25

per

1 0

30

35

per

10

k

s

40

45

km )

s

2

4

0

5

Sample

10

Size

15

(Stations

20

per

25

1 0

6

30

km ) 2

Figure 4. Annual mean precipitation derived from resampling the L W high-resolution climatology using (i) unsystematic ( r a n d o m ) sampling (box plots) and (ii) systematic sampling at the yearly N C A R station networks (open circles) for (a) Africa, (b) Australia, (c) Eurasia, (d) North America, (e) South America, and (f) the terrestrial surface, excluding Greenland and Antarctica. E a c h b o x plot—showing minimum, maximum, lower and upper quartiles, and median—is derived from 50 r a n d o m samples of the climatology. The dashed line is the value estimated from all the stations in the L W climatology

ESTIMATING PRECIPITATION AVERAGES

409

410

C. J. WILLMOTT, S. M. ROBESON AND J. J. FEDDEMA

Table I. Number of stations in each continent corresponding to sampling densities (number of stations per 10 km ) 6

2

Sampling density

Africa

Australia

Eurasia

North America

South America

Terrestrial surface

2.5 5.0 10.0 15.0 20.0 25.0

73 146 292 438 584 729

19 38 76 115 153 191

125 250 500 749 999 1249

50 99 198 297 396 496

44 88 175 263 351 439

328 656 1311 1967 2622 3278

African h i g h l a n d s ( K e n y a a n d U g a n d a ) . S a m p l i n g p r o b l e m s a r e m o s t prevalent in Africa's desert regions (e.g. interior S a h a r a , K a l a h a r i , a n d in Somalia). T o a lesser extent, the upper reaches of the Zaire basin also a r e u n d e r - r e p r e s e n t e d . Overall, Africa's d r y regions are u n d e r s a m p l e d . R a n d o m s a m p l i n g (with s u b s e q u e n t i n t e r p o l a t i o n a n d averaging) from the African station network (Figure 3(a)), therefore, results in a wet-region bias at all s a m p l i n g densities (Figures 3(b) a n d 4(a)—box plots). B o u n d a r i e s b e t w e e n m o i s t a n d arid areas (e.g. in the Sahel) also a r e misplaced (discussed below). Moist a r e a s typically are i n t e r p o l a t e d t o b e t o o far i n t o dry regions. F o r t h e lowest station density considered (2-5 stations per 10 k m o r 73 stations for all of c o n t i n e n t a l Africa; T a b l e I), the m e d i a n of the 50 r a n d o m s a m p l e s is 140 m m higher t h a n the c o n t i n e n t a l m e a n p r e c i p i t a t i o n estimated from all stations in the L W c l i m a t o l o g y (the d a s h e d line in F i g u r e 4(a)). As the station density increases, sample-derived m e a n s a s y m p t o t i c a l l y decrease a n d a p p r o a c h (but never reach) the L W climatology mean. Consider also that the L W climatology itself m a y o v e r e s t i m a t e c o n t i n e n t a l m e a n p r e c i p i t a t i o n because the large expanse of the S a h a r a D e s e r t is s a m p l e d sparsely. M o s t yearly N C A R n e t w o r k s , by c o n t r a s t , a p p e a r t o estimate African mean precipitation well (Figure 4(a)—circles), for all b u t t h e sparsest s t a t i o n n e t w o r k s (i.e. prior to 1920). Integrations using N C A R n e t w o r k s p r i o r to 1900 a r e n o t s h o w n o w i n g t o small n u m b e r s of s t a t i o n s ( < 3 0 ) in the 1800s. T h e N C A R station n e t w o r k s after 1920, however, perform better t h a n r a n d o m s a m p l i n g . This is n o t surprising because the N C A R n e t w o r k s were chosen to have spatially uniform station distributions. 6

2

Australia Australia's relatively small size, l o w relief, a n d low a n n u a l m e a n precipitation result in less spatial variability in precipitation t h a n occurs over any o t h e r c o n t i n e n t (Figure 3(b)). As in Africa, areas with high s t a t i o n densities (Figure 3(a)) a r e found in relatively moist r e g i o n s — a l o n g the south-east a n d south coasts of Australia. T h e wet n o r t h e r n coast a n d arid interior are sampled sparsely, with the dry interior representing a m u c h larger l a n d area. Yearly N C A R station n e t w o r k s for A u s t r a l i a also are n o t distributed evenly. The N C A R station densities in Australia, nevertheless, are m u c h higher t h a n for the o t h e r continents, especially in s o u t h - e a s t e r n Australia (Figure 2). R a n d o m s a m p l i n g at low s t a t i o n densities p r o d u c e s a large r a n g e of estimates for Australian m e a n precipitation (Figure 4(b)). M u c h of this variability is c o n t r o l l e d by whether or n o t stations from the very wet regions in the n o r t h are s a m p l e d . W h e n wet tropical regions a r e sampled, t h e estimated moist region extends t o o far i n t o the dry interior w h e r e few s t a t i o n s exist. R a n d o m sampling of stations, n o n e the less, c a n lead to either over- o r u n d e r e s t i m a t e s of Australia's m e a n precipitation. Overestimates, however, are much more common. T h e N C A R station n e t w o r k s represent A u s t r a l i a n m e a n p r e c i p i t a t i o n adequately after 1890, although estimates are always a little high (Figure 4(b)). T h e interior of t h e continent, however, is undersampled in b o t h t h e L W c l i m a t o l o g y a n d in t h e N C A R n e t w o r k s . N C A R station densities along the Australian

411

ESTIMATING PRECIPITATION AVERAGES

c o a s t — p a r t i c u l a r l y in the s o u t h - e a s t — a r e h i g h e r t h a n in a n y o t h e r region in t h e w o r l d (well over 25 s t a t i o n s per 10 k m for m o s t years). T h e relative w e t n e s s of this region c o n t r i b u t e s to the slight b u t consistent o v e r p r e d i c t i o n of the continental average. 6

2

Eurasia S t a t i o n n e t w o r k s in p a r t s of E u r a s i a a r e extremely dense, especially in western E u r o p e , w i t h a g r a d u a l decline t o w a r d s the east (Figure 3(a)). T h e c o n t i n e n t a l interior (particularly Siberia, M o n g o l i a , a n d the G o b i Desert), the A r a b i a n peninsula, a n d t h e n o r t h e r n c o a s t a l areas a r e n o t as well represented in t h e L W climatology. As in Africa a n d Australia, sampling p r o b l e m s in E u r a s i a o c c u r a l o n g high p r e c i p i t a t i o n g r a d i e n t s , with m o i s t areas being better represented t h a n arid regions. O n c e again, s p a r s e s t a t i o n n e t w o r k s give rise t o overestimates of the continental m e a n ( F i g u r e 4(c)). A variety of factors c o n t r i b u t e t o t h e s a m p l i n g p r o b l e m s a l o n g s t r o n g spatial gradients (e.g. t h e influence of o r o g r a p h y , c o n t i n e n t a l i t y , a n d t h e spatial e x t e n t of t h e south-east Asian m o n s o o n ) . Nevertheless, s t a t i o n densities a r e sufficient t o o b t a i n a d e q u a t e (albeit slightly high) estimates of E u r a s i a n m e a n p r e c i p i t a t i o n (Figure 4(c)). Even at m o d e r a t e s t a t i o n densities (e.g. 10 s t a t i o n s per 10 k m or ca. 500 s t a t i o n s ; F i g u r e 4(c)), a c c u r a t e m e a n s c a n be c o m p u t e d . S p a r s e N C A R n e t w o r k s (prior to 1950) similarly p r o d u c e high estimates w h e r e a s m o d e r a t e - r e s o l u t i o n N C A R n e t w o r k s yield quite a c c u r a t e estimates (Figure 4(c)). 6

North

2

America

A r e a s receiving relatively high a m o u n t s of p r e c i p i t a t i o n within N o r t h A m e r i c a (e.g. central M e x i c o , t h e eastern U S A , a n d s o u t h e r n C a n a d a ) c o n t a i n dense s t a t i o n n e t w o r k s . N o r t h e r n C a n a d a , interior Alaska, a n d the n o r t h e r n deserts of Mexico (areas of relatively low p r e c i p i t a t i o n ) conversely c o n t a i n r a t h e r s p a r s e s t a t i o n distributions. U n s y s t e m a t i c s a m p l i n g a t s t a t i o n s from the L W climatology, therefore, leads to o v e r e s t i m a t i o n of N o r t h American m e a n p r e c i p i t a t i o n ( F i g u r e 4(d)). L a r g e variability in the e s t i m a t e d m e a n s occurs at low station densities b e c a u s e of t h e t r e m e n d o u s r e g i o n a l variability in N o r t h A m e r i c a n precipitation (Figure 3(b)). Estimates of c o n t i n e n t a l a v e r a g e p r e c i p i t a t i o n v a r y by nearly 600 m m at the lowest station density (Figure 4(d)). Even t h e largest s a m p l e size (25 s t a t i o n s per 10 k m ) p r o d u c e s a r a n g e greater t h a n 150 m m . Interquartile r a n g e s a r e m u c h smaller. Yearly N C A R s t a t i o n n e t w o r k s p r o d u c e r e a s o n a b l e estimates for m o s t station densities, p a r t i c u l a r l y after 1920. 6

South

2

America

E s t i m a t i n g average precipitation for S o u t h A m e r i c a is quite different from a n y of t h e o t h e r c o n t i n e n t s . Historical rain-gauge n e t w o r k s are u n u s u a l l y s p a r s e in S o u t h A m e r i c a a n d t h e m o s t p o o r l y s a m p l e d a r e a s a r e wet (Figure 3). N o r t h - w e s t e r n a n d s o u t h e r n sections of S o u t h A m e r i c a a r e b e t t e r s a m p l e d t h a n t h e interior, a l t h o u g h the sampling is dense a l m o s t n o w h e r e . U n l i k e virtually all o t h e r c o n t i n e n t a l i n t e r i o r s , t h e interior of S o u t h A m e r i c a — p a r t i c u l a r l y the A m a z o n b a s i n — h a s very few r a i n - g a u g e s a n d receives large a m o u n t s of rainfall. T r o p i c a l Africa is q u i t e wet b u t relatively well-sampled. Sparse s a m p l i n g of large areas with high p r e c i p i t a t i o n p r o d u c e s c o n t i n e n t a l m e a n p r e c i p i t a t i o n e s t i m a t e s t h a t are b o t h highly variable a n d grossly u n d e r e s t i m a t e d for all s a m p l i n g densities (Figure 4(e)). At low sampling densities, underestimates as large as 200 m m are c o m m o n . E v e n at densities of 20 s t a t i o n s p e r 10 km , estimates of S o u t h American m e a n p r e c i p i t a t i o n can b e in e r r o r by m o r e t h a n 100 m m . Sampling of S o u t h American p r e c i p i t a t i o n is especially p o o r within t h e N C A R c l i m a t o l o g y ( F i g u r e s 1 a n d 2). U n d e r e s t i m a t e s range from n e a r l y 700 m m t o 100 m m ( F i g u r e 4(e)). S t a t i o n densities w i t h i n t h e N C A R archive never exceed 15 s t a t i o n s per 1 0 k m . 6

2

6

2

412 Terrestrial

C. J. WILLMOTT, S. M. ROBESON AND J. J. FEDDEMA

surface

S t a t i o n distributions for m o s t p a r t s of the terrestrial surface are described above. O c e a n i a a n d several island nations, however, also are included in our analysis of the terrestrial surface. A l t h o u g h O c e a n i a a n d island n a t i o n s represent a relatively small p r o p o r t i o n of the total land area, a few islands d o c o n t a i n a large n u m b e r of stations. Such densely sampled areas include J a p a n , the Philippines, Indonesia, a n d G r e a t Britain a n d t h e y can exert a greater influence on terrestrially averaged precipitation estimates t h a n is w a r r a n t e d by their size. R a n d o m sampling of stations from the LW climatology (Figure 3(a)) gives rise to large variability in estimates of terrestrial m e a n precipitation until sampling densities of 15 stations per 1 0 k m are reached (Figure 4(f)). R a n d o m sampling also does considerably better t h a n the yearly N C A R n e t w o r k s at low s a m p l i n g densities. The N C A R n e t w o r k s a p p e a r to give reliable estimates when station densities exceed ca. 7 stations per 10 k m . Previously described continental-scale findings suggest that large u n d e r e s t i m a t e s in S o u t h American precipitation tend to cancel overestimates from other continents. Additional understanding of how uneven and variable station distributions influence regional precipitation estimates can be obtained by examining error maps. E r r o r statistics can be generated by s u b t r a c t i n g grid-point values interpolated from the 50 r a n d o m samples of the L W climatology from c o r r e s p o n d i n g grid-point values obtained from all the stations in the L W climatology. M o r e specifically, the m e a n a b s o l u t e error (MAE) at a grid-point is 6

6

2

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MAE = A T

JV

£ \P -P\

1

(1)

t

where P is the grid-point value associated with the entire L W climatology, P is a g r i d - p o i n t value interpolated from one of the 50 r a n d o m samples of the L W climatology, a n d N = 50 (Willmott et al, 1985b). Error m a p s are shown for t h e two extreme cases of 2-5 and 25 stations per 10 k m (Figure 5). Estimates of M A E are large in areas of relatively high precipitation and in u n d e r s a m p l e d regions, as is evident in m a p s of both 2 5 a n d 25 stations per 10 k m (Figure 5). Estimates of a b s o l u t e p r e c i p i t a t i o n e r r o r are roughly correlated with m e a n precipitation because precipitation has a lower b o u n d (zero) but n o u p p e r b o u n d . Central tendency a n d variance, therefore, tend to covary. Large zones within the lower latitudes have sampling errors in excess of 400 m m , especially in South America. A n u m b e r of sparsely s a m p l e d m o u n t a i n o u s a n d arid areas also exhibit large precipitation errors. Desert regions sometimes have low M A E s , simply because they have low precipitation variability. L a r g e errors (i) occur where steep precipitation gradients are coincident with steep station-density gradients a n d (ii) result from the extrapolation of the relatively well-sampled region's precipitation characteristics into the adjacent, sparsely gauged region. T h e extent to which the higher-resolution samples (25 stations per 1 0 k m ) better resolve the spatial variability in these transition zones is evident in the i m p r o v e m e n t s in Figure 5(b) relative to Figure 5(a). t

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2

SUMMARY AND CONCLUSIONS Precipitation averages a n d spatial variability of precipitation have been investigated at c o n t i n e n t a l a n d terrestrial scales. Effects of uneven a n d variable precipitation networks were investigated by s a m p l i n g from a high-resolution climatology. Both randomly selected station n e t w o r k s a n d stations within t h e N C A R W o r l d M o n t h l y Surface Station Climatology were evaluated. Repeated r a n d o m sampling (at a variety of station densities) revealed h o w continental- and terrestrial-scale estimates of mean precipitation c a n be affected by irregular station networks. By sampling from the N C A R station network, the a d e q u a c y of systematically selected station n e t w o r k s also was investigated. F o r all the continents except S o u t h America, low rain-gauge densities give rise t o overestimates of a n n u a l m e a n precipitation. Annual m e a n precipitation in South America, in contrast, is vastly u n d e r e s t i m a t e d by low station densities. M a p s of the m e a n absolute error also show t h a t sampling errors t e n d t o be high in areas of high precipitation, a n d in areas with strong precipitation gradients a n d station-density gradients.

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Sampling the L W climatology at the N C A R station locations suggests that many yearly N C A R station densities (particularly the more recent ones) are a d e q u a t e for estimating continental mean precipitation. Regional- and continental-scale errors, however, often cancel one another to give what appears to be reliable estimates of terrestrial average precipitation as early as 1920. Our results also imply that future rain-gauge deployments—particularly in currently undersampled regions of the world—should carefully take into account the spatial and temporal variability of precipitation within the domain of interest. Higher resolution deployments should coincide with high-frequency (in the spatial domain) precipitation variability. Spatial shifts in the precipitation field with time (on seasonal and interannual time-scales, for example) also should be considered in the station-network design. Samples of a time-varying field obtained from a fixed n e t w o r k (when the network systematically underrepresents portions of the field for periods of time) can produce temporally varying biases in the estimates of the field. ACKNOWLEDGEMENTS

Portions of this research were supported by N A S A (grants NAG5-853 and NAGW-1884) and UCLA (Faculty Senate G r a n t 4-563867-1990-07). REFERENCES Bradley, R. S., Diaz, H. F , Eisheid, J. K., Jones, P . D., Kelly, P . M . and Goodess, C. M. 1987. 'Precipitation fluctuations over Northern Hemisphere land areas since t h e mid-19th century', Science, 237, 171-175. Bussieres, N. and Hogg, W. 1989. ' T h e objective analysis of daily rainfall by distance weighting schemes on a mesoscale grid', Atmos. Ocean, 27, 521-541. Daley, R. 1991. Atmospheric Data Analysis. C a m b r i d g e University Press, New York. Diaz, H. F., Bradley, R. S. and Eischeid, J. K. 1989. 'Precipitation fluctuations over global land areas since the late 1800s', J. Geophys. Res., 94, 1195-1210. Hulme, M. 1992 'A 1951-1980 global land precipitation climatology for the evaluation of general circulation models', Climate Dynamics, 7, 57-72. Lam, N. S-N. 1983 'Spatial interpolation m e t h o d s : a review', Am. Cartogr., 10, 129-149. Legates, D. R. (1987) 'A climatology of global precipitation', Pub. Climatoi, 40(1), 85 pp. Legates, D. R. and Willmott, C. J. 1990. ' M e a n seasonal a n d spatial variability in gauge-corrected, global precipitation', Int. J. Climatoi, 10, 111-127. Legates, D. R. and Willmott, C. J. 1992. 'A comparison of G C M - s i m u l a t e d and observed mean January and July precipitation', Global Planet. Change, 97, 345-363. Mintz, Y. 1981. 'A brief review of the present status of global precipitation estimates', in Atlas, D. and Theile, O. W. (eds), Report of the Workshop on Precipitation Measurements from Space, N A S A G o d d a r d Space Flight Center, Greenbelt, M D . Renka, R. J. 1984 'Interpolation of d a t a on t h e surface of a sphere', Association For Computing Machinery Trans. Math. Software, 10(4), 417-436. Shepard, D. 1968. ' A two-dimensional interpolation function for irregularly spaced data', Proceedings of the 23rd National Conference, Association For Computing Machinery, pp. 517-523. Spangler, W. M. L. a n d Jenne, R. L. 1988. World Monthly Surface Station Climatology, National Center for Atmospheric Research, Scientific Computing Division, Boulder, C o l o r a d o . Thiebaux, H. J. and Pedder, M. A. 1987. Spatial Objective Analysis, Academic Press, New York. Wahba, G. 1981 'Spline interpolation a n d smoothing on the sphere', SI AM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comp., 2(1), 5-16. Weber, D. and Englund, E. 1992. 'Evaluation and c o m p a r i s o n of spatial interpolators', Math. Geol, 24, 381-391. Willmott, C. J., Rowe, C. M. and Philpot, W. D. 1985a. 'Small-scale climate maps: a sensitivity analysis of some common assumptions associated with grid-point interpolation a n d c o n t o u r i n g ' , Am. Cartogr., 12, 5-16. Willmott, C. J., Ackleson, S. G., Davis, R. E., F e d d e m a , J. J., Klink, K. M., Legates, D. R., O'Donnell, J. and Rowe, C. M. 1985b. 'Statistics for the evaluation and c o m p a r i s o n of models', / . Geophys. Res., 90(C5), 8995-9005. Willmott, C. J., Robeson, S. M. and Feddema, J. J. 1991. 'Influence of spatially variable instrument networks on climatic averages', Geophys. Res. Lett., 18(12), 2249-2251. Willmott, C. J. and Legates, D. R. 1991. 'Rising estimates of terrestrial and global precipitation', Climate Res., 1, 179-186.