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The areal extension of rainfall records: An alternative model
Journal of Hydrology 7 (1969) 404-414; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE AREAL EXTENSION OF RAINFALL RECORDS:
AN ALTERNATIVE M O D E L D. J. UNWIN Department of Geography University College of Wales, Aberystwyth, Wales Abstract: Attention is drawn to the trend surface model as a possible alternative to the
more conventional regression methods as a predictor of rainfall totals. A test of this model using data from Snowdonia, North Wales, shows that it is slightly the more efficient. Introduction
In hydrological and climatological research it is often necessary to form an objective assessment of the overall variation of rainfall over the study area. Several methods have been developed to enable such areal extension of precipitation records, ranging from subjective isoplething 1) through various graphical methods 2) to simple and multiple linear regression analysis which treats the observed gauge data as a statistical sample. Rodda 3) provides an illustration of these latter methods together with a review of available techniques, whilst Ruhe 4) has applied a regression technique in palaeoclimatic reconstructions, This paper presents an alternative regression model based upon the techniques of trend surface analysis. Trend surface analysis
The method has been used extensively in the geological sciences 5, 6) and is finding increasing application in geography 7), but as yet does not seem to have been applied in climatology. A full description of the mathematical development of the model and its terminology has been given in a number of reviews 8-10) and is outside the scope of this work. In its simplest form trend surface analysis uses the least squares criterion to fit polynomials of successively higher order to areally distributed data: X. = bo + b , U + b2V
Linear surface
X n = c 0 -~- ClU 2t- c2V 2i- c3 U2 -.~ c4UV -~ c5 V2
Quadratic surJace
X. = do + d l U + d2V + daU 2 + d4UV + d5V 2 + d6 U3 + dvU2V + daUV 2 + d9 V3
Cubic surface
404
THE AREAL EXTENSIONOF RAINFALLRECORDS
405
WHERE X, is the value of the areally distributed variable and U, V are the geographic co-ordinates. Each polynomial defines a trend surface which is an expression of the broad regional variability inherent in the data, whilst the deviations of observed values from this surface, or residuals, are assumed to be an expression of strictly local factors. The degree of fit given by the polynomial, and hence its effectiveness in describing the areal variability of the data, is given by the percentage reduction in sums of squares it achieves. In certain circumstances a very high percentage reduction indicating a very close fit to the observed data can be obtained suggesting that the method can be used as a predictive technique for the extension of rainfall records in place of the more conventional regression methods. This alternative approach has a number of distinct advantages. The use of a polynomial rather than the planes fitted by conventional regression allows areal variations
Fig. 1. Mean annual rainfall (after Bransby-Williams).
406
D.J. UNWIN
to be followed more readily and there are considerable theoretical advantages to be gained in the separation of regional from local components in rainfall totals.
Rainfall prediction by trend surface analysis The area chosen for investigation was that of Snowdonia, North Wales. Several general climatologies of the area have been produced 11-13) but the best available information on rainfall amount and character is that published by Bransby-Williams14), a part of whose map is reproduced here (Fig. 1). In addition there are the published averages of rainfall for Great Britain and Northern IrelandlS). Mean annual totals for 47 stations were collected from both of these sources (Table 4) but of necessity these records vary both in quality and quantity. The difficulties of accurate precipitation measurement in mountain areas have often been commented upon and need no further discussion here, but further difficulties arise because of the differing time periods used by these sources. Thirty-three station means were taken from the Averages of Rainfall (1916-1950). In order to extend the areal cover a further 14 means for the period 1881-1915 were added from Bransby-Williams' list. The validity of this combination of records was examined by comparison of the 20 station means which are common to both sources. Table 1 lists these values together with summary statistics and the percentage change between the two periods at each station. Neither the sample means nor the sample standard deviations of these two sets of records differ significantly at the 99 9/00confidence level and no consistent pattern could be distinguished within the percentage changes at each station. It was concluded that the records could be combined without seriously affecting the analysis. Several stations listed by Bransby-Williams but not listed in Averages of Rainfall (1916-1950) were rejected for reasons of a suspected site change, insufficient length of record or a change in gauge type. The completed data file of 47 station means shows a complex pattern of rainfall distribution (Fig. 1). The effect of altitude can be clearly discerned but a secondary effect is that of gradually decreasing rainfall totals from south-west to north-east across the area in the direction of the prevailing rain bearing south-westerly winds. A common approach to the problem of rationalising this pattern and assessing the relative contribution of each effect to the observed totals would be the multiple correlation of rainfall with elevation and other factors such as position expressed by the latitude and longitude. The alternative trend surface method outlined here treats the areal variation of rainfall which is independant of station altitude, the equivalent precipitation, as the regional component whilst residuals from this are ascribed to the effects of altitude.
407
THE AREAL EXTENSION OF RAINFALL RECORDS TABLE 1 Comparison of station means for the standard periods 1881-1915 and 1916-1950 for 20 selected stations Site
Colwyn Bay Blaenau Ffestiniog (1) Blaenau Ffestiniog (2) Llandudno Snowdon (Crib Goch) Snowdon (Llydaw) Snowdon (Cwm Dyli) Snowdon (Llewedd) Snowdon (Delta) Snowdon (Teryn) Snowdon (Copper Mine) Llyn Eigiau Melynllyn Llyn Dulyn Plas Dulyn Conway Bethesda Aber. (U.C. Farm) Beddgelert Trefarthen
Height in Ft
1881-1915 Average
1916-1950 Average
~o change
118 1100 700 13 2340 1480 310 1485 1435 1080 1480 1244 2065 1632 509 50 527 60 133 30
29.4 108.0 109.3 28.1 172.1 151.4 128.5 153.0 171.8 155.7 159.7 80.3 125.0 102.7 55.2 30.5 65.8 41.5 88.6 36.2
29.4 115.7 110.0 28.8 171.1 157.9 140.5 153.5 170.1 156.0 157.9 80.3 119.0 105.3 55.3 30.5 64.6 41.6 90.2 36.3
0.00 + 7.13 + 0.64 -t- 2.49 -- 0.20 + 4.29 + 9.34 + 0.37 - 0.99 + 0.19 - 1.10 0.00 -- 4.80 + 2.53 + 0.18 0.00 -- 1.22 + 0.20 + 4.81 + 0.28
Means = 99.6 Sample standard = 50.11 deviation
100.7 49.70
I n o r d e r to derive the regional c o m p o n e n t the o b s e r v e d gauge d a t a was first s t a n d a r d i s e d to an equivalent p r e c i p i t a t i o n at c o n s t a n t altitude following the m e t h o d outlined b y Sutcliffe a n d C a r p e n t e r t e ) . I n this s t u d y the c o n s t a n t altitude selected was m e a n sea level a n d equivalent values for p r e c i p i t a t i o n were derived b y s u b t r a c t i n g a c o r r e c t i o n f a c t o r b a s e d u p o n a p r e c i p i t a t i o n lapse rate o f 5.5~/100 ft o f ascent. This value was established f r o m the simple linear regression o f rainfall against height for all stations within the area. Similar regressions using the station d a t a for m o r e restricted subareas s h o w gradients in the r a n g e 3.3 to 7 . C / 1 0 0 ft, i n d i c a t i n g t h a t whilst this p a r a m e t e r is b y no m e a n s c o n s t a n t over the whole a r e a the value used can be t a k e n as a fair a p p r o x i m a t i o n for an overall analysis, tT) T h e new set o f r e c o r d s was then subjected to t r e n d surface analysis using a m o d i f i c a t i o n o f the c o m p u t e r p r o g r a m m a d e available b y WhittenlS).
408
D . J . UNWIN
The results for the linear, quadratic and cubic surfaces are presented in Table 2 which also shows the percentage reductions given by randomly chosen values within the range of the original data. The percentage reductions obtained are sumciently high to indicate a strong regional trend, whilst the low reductions given by the random values indicate that this trend is a TABLE 2
Results of trend surface analysis using data reduced to sea level Percentage reductions in sum of squares obtained Surface Variable
Corrected rainfall Data Random values
Linear
Quadratic
Cubic
31.40
43.19
55.98
4.62
6.27
21.15
"real" one which could not have arisen by chance. Figure 2 shows the mapped cubic surface. To eliminate boundary effects this map was constructed for a smaller area than was used in the computer analysis. Its form is of interest in that it confirms the impression of a declining regional component in rainfall totals towards the east and north, from more than 75" in the south-west to less than 25" in the north-east. The result of this analysis is a series of equivalent precipitation values at mean sea level, but for comparison purposes these trend values at each station were recorrected for station altitude to produce the predicted station means tabulated in Table4(a). The percentage reductions in sums of squares given by these values are 82.51 for the quadratic and 84.89 for the cubic (Table 4(b)), indicating that a large proportion of the variance associated with the original station means can be accounted for by this simple trend surface model.
Rainfall prediction using regression analysis In order to test the trend surface model it was thought desirable to examine the same data file using normal regression methods. Table 3 shows the results of such an analysis for the observed rainfall means against altitude, latitude and longitude. Both latitude and longitude were expressed in arbitrary units north or east of an origin located to the south-west of the area. The matrix of simple linear correlations (Table 3(a)) shows that
409
THE AREAL EXTENSION OF RAINFALL RECORDS
N
/ PEET CT NSUM0rS0 ESS98
t
• SampLeVaLue 0
Fig. 2.
1
2 Mi~e5
3
4
Cubic trend surface for rainfall standardised to sea-level.
whilst the correlations between rainfall and height, and rainfall and latitude are significant at the 95% confidence level there is no evidence of a simple linear correlation between precipitation and longitude. In no case does a logarithmic transformation significantly improve the regression. Table 3(b and c) gives the results of sequential multiple regression using the same variables. The analysis was completed using a standard computer library procedure19). Step (1) entered the height variable into the regression, step (2) added to this the latitude, and step (3) further added longitude to produce the prediction equations shown. As Table 3(b) shows the third step added little to the efficiency of the regression. The overall reduction in sums of squares given was 79.3% (Table 4) with a multiple correlation coefficient of 0.891 indicating that the prediction equation is a reasonably efficient one.
D. J. UNWIN
410
TABLE 3 Results of regression analysis a) Matrix of simple linear correlation coefficients
X1 Latitude X2 Longitude X3 Station Height )(4 Rainfall
X1
X2
Xa
X4
1.000
-- 0.141 1.000
0.470 -- 0.030 1.000
0.705 -- 0.128 0.810 1.000
Underlining indicates coefficients which are significant at the 9 5 ~ confidence level. b) Results o f multiple regression Multiple Step 1. 2. 3.
Variables entered Height )(3 Height, Latitude Xa, X1 Height, Latitude, Longitude, X3, X1, X2
Increase
R
Rsq
in R
F-Value
0.810 0.889
0.6565 0.7903
0.6565 0.1339
85.98 28.10
0.891
0.7930
0.0027
0.56
F-Value at the 95 % confidence level, step 3 is 2.800. c) Prediction equations Step 1. 2. 3.
Equation X4 : 41.57 + 0.055 3(3 X4 = 13.87 + 0.175 Xa + 0.042 )(1 X4= 19.14+0.171 X3 + 0.042 )(1--0.247 X~
The efficiency of the methods T h e relative merits o f b o t h the a b o v e m e t h o d s m a y be c o m p a r e d d~rectly by m e a n s o f the percentage reductions in sums o f squares they give. Table 4 presents a s u m m a r y o f the o b s e r v e d data, values p r e d i c t e d by surface analysis, the residuals associated with this m e t h o d together with the residuals over the multiple regression a n d s u m m a r y statistics. I n b o t h cases there is a c o n s i d e r a b l e s p r e a d o f residual values b u t t a k e n overall the t r e n d surface m e t h o d with its 82.51 a n d 84.89 reductions in sums o f squares c o m p a r e s f a v o u r a b l y with the 79.30% given b y direct regression. These values indicate t h a t in the a r e a studied a n d with the available d a t a the t r e n d surface m e t h o d is slightly the m o r e efficient as a p r e d i c t o r for the areal extension o f rainfall totals. F u n d a m e n t a l l y this arises because the p o l y n o m i a l s used in this
THE AREAL EXTENSION OF RAINFALL RECORDS
411
TABLE 4 A.
Rainfall totals, stations and residuals over regressions
Station
Colwyn Bay Tal-y-Cafn Ffestiniog Blaenau Ffestiniog (1) Biaenau Ffestiniog (2) Cowlyd (A) Llugwy Cowlyd (B) Cowlyd (C) Llyn Eigiau Melynllyn Llyn D u l y n Plas Dulyn Conway Llandudno Nantlle (Dorothea) Bethosda Bangor Bangor Aber. (U.C. F a r m ) Penmaenmawr Beddgelert S n o w d o n ( C w m Dyli) S n o w d o n (Lliwedd) S n o w d o n (Delta) S n o w d o n (Teryn) S n o w d o n (Llydaw) S n o w d o n (Copper Mine) S n o w d o n (Crib G o c h ) Trefarthen Plas C a d n a n t Treffos Gladdllys Cottage Bettws-y-Coed Pentre-Foelas Trefriew Dolcrum Waunfawr Caernarvon
Height ft. O.D.
Rainfall Inches
Rainfall Predicted by Cubic Surface
118 170 610 1100 700 1340 1760 1140 1200 1244 2065 1632 509 50 13 400 527 275 30 60 170 133 310 1485 1435 1080 1480 1480 2340 30 120 290 340 118 640 25 100 520 180
29.4 38.4 76.3 115.7 110.0 90.9 100.8 89.4 90.4 80.3 119.0 105.3 55.3 30.5 28.8 63.4 64.6 47.0 42.3 41.6 36.5 90.2 140.5 153.5 170.1 156.0 150.7 157.9 171.7 36.3 43.5 43.7 43.8 50.1 57.7 47.2 47.2 55.7 36.2
44.5 31.6 84.0 131.1 108.5 117.4 138.2 98.4 101.1 101.8 146.1 120.7 50.9 23.9 29.2 76.1 63.6 42.5 29.3 26.2 28.5 96.5 84.9 153.3 148.5 127.7 149.4 149.4 196.5 19.6 36.5 52.2 56.7 47.1 54.0 35.9 27.0 71.0 33.5
Residual
Residual f r o m Mult. Regression
15.1 -- 6.8 7.7 15.4 -- 1.5 26.5 37.4 9.0 10.7 21.5 27.1 15.4 -- 4.4 -- 6.6 0.4 12.7 -- 1.0 -- 4.5 -- 13.0 -- 15.4 -- 8.0 6.3 -- 55.6 -- 0.3 21.6 -- 28.3 -- 1.3 -- 8.5 24.8 -- 16.7 -- 7.0 8.5 12.9 -- 3.0 -- 3.7 -- 16.4 -- 20.2 15.3 -- 2.7
-- 38.4 3.7 -- 1.4 -- 9.7 1.4 - 20.2 --26.1 -- 8.3 - 9.3 -- 19.9 -- 15.0 -- 9.4 -- 4.4 3.4 10.7 -- 26.6 -- 6.9 -- 6.5 -- 2.3 3.3 2.7 5.1 59.9 22.1 41.8 43.1 21.3 28.5 6.3 -- 16.7 0.8 -- 2.6 - 3.7 -- 15.4 -- 36.4 2.1 8.8 --28.1 - - 29.0
412
D. J. UNWIN TABLE4 (Continued)
Station
Bangor Llanfairfechan Snowdon (Glaslyn) Hafod Tan-y-graig Penmon Pen-y-Parc Llandudno (2)
Height ft. O.D.
Rainfall Inches
Rainfall Predicted by Cubic Surface
Residual
Residual from Mult. Regression
150 120 2050 310 60 372 71
50.7 39.9 154.7 129.8 39.0 43.1 30.3
34.5 27.0 182.3 98.1 45.8 51.1 35.2
-- 6.2 -- 12.9 27.6 -- 31.7 6.8 8.0 4.9
-- 4.3 2.9 0.6 41.8 13.7 -- 6.1 11.8
B. Summary Statistics Procedure
~ reduction in sums of squares
Multiple linear regression Quadratic trend surface Cubic trend surface
79.30 82.51 84.89
m e t h o d are better descriptors of areal variation than are the best fit planes fitted by conventional regression. In the present analysis it was found necessary to resort to an awkward standardisation procedure to correct for the effects o f station altitude but surface analysis methods which include such a term within the polynomials are available and might profitably be used in future analyses s0). Moreover the m o u n t a i n area chosen for the analysis here presented, over which altitudinal variations in precipitation predominate, constituted a very severe test for the surface model. Its performance when applied to such a difficult area indicates that it would be of even greater value if applied to areas over which relief variations are less marked. Whilst it is true that the addition o f other factors such as aspect and distance from the watershed might substantially improve the multiple regression model it is t h o u g h t that the method o f trend surface analysis is one which is worthy o f further attention in all studies concerned with the areal variation o f climatological parameters.
Acknowledgements M u c h of the work presented here was completed whilst the author was a research student in the Department o f Geography, University College
THE AREAL EXTENSION OF RAINFALL RECORDS
413
L o n d o n . I a m p a r t i c u l a r l y grateful to Professor B r o w n a n d the D e p a r t m e n t o f G e o g r a p h y , University College L o n d o n , for p r o v i d i n g access to c o m p u t i n g facilities. Miss P . M . Page supervised the multiple regression analysis. A c k n o w l e d g e m e n t s are also due to Dr. H. O. S l a y m a k e r , University College o f Wales, A b e r y s t w y t h , a n d Dr. J. C. R o d d a o f the Institute o f H y d r o l o g y b o t h o f w h o m critically r e a d the initial draft.
References 1) J. Glasspoole, The Rainfall of Norfolk. British Rainfall (1928) 271-2 2) V. Conrad and L. W. Pollak, Methods in Climatology. Harvard (1962) p. 222 et seq. 3) J. C. Rodda, An objective method for the assessment of areal rainfall amounts. Weather, 17 (1962) 54--59 4) R. V. Ruhe, An estimation of palaeoclimate in Oahu Hawaii. Amer. J. Sci. 262, 9 (1964) 1098-1115 5) D. F. Merriam and N. C. Cocke (eds.), Computer applications in the Earth Sciences: colloquium on trend analysis. Computer Contribution, State Geol. Surv., University of Kansas, 12 (1967) 62 pp. 6) W. C. Krumbein, Trend surface analysis of contour-type maps with irregular control point spacing. J. Geophys. Res. 64 (1959) 823-34 7) R. J. Chorley and P. Haggett, Trend surface mapping in geographical research. Trans. Inst. Brit. Geog. 37 (1965)47-67 8) W. C. Krumbein and F. A. Graybiil, An introduction to statistical models in geology. London (1965) 320--357 9) C. A. M. King, An introduction to trend surface analysis. Nottingham University, Department of Geography, Bulletin of Quantitative Data for Geographers, 12 (1967) 10) E. H. T. Whitten, A surface fitting program suitable for testing geological models which involve areally distributed data. O.N.R. Geogr. Branch. Tech. Rep., 2 of Task No. 389-135 (1963) 11) A. Lockwood, Meteorological Notes, in: The Mountains of Snowdonia (Ed. Cart and Lister) London (1925) 211-215 12) D. A. Ratcliffe, The Vegetation of the Carneddau, North Wales. J. Ecol. 47 (1954) 371--413 13) D. F. Ball, The soils and land use of the district around Bangor and Beaumaris. H.M.S.O. (1963) 6-9 14) G. Bransby-Williams, Some characteristics of mountain rainfall illustrated in northwest Wales. Water and Water Engineering, 58 (1954) 289-304 15) H.M.S.O. Averages of rainfall for Great Britain and Northern Ireland 1916-1950. London (1958) 16) J. V. Sutcliffe and T. G. Carpenter, The assessment of runoff from a mountainous and semi-add area in Western Iran, in: Hydrological Aspects of the Utilisation of Water. General Assembly of Bern (1967) 383-394 17) The method outlined here could be adapted to take account of more marked areal variations in precipitation gradient. 18) See reference 10. The analyses were completed using either Elliot 4130 or IBM 360 computers 19) W. J. Dixon, BMD: Biomedical Computer Programs. University of California. (1967)
414
D.J. U N W I N
Pub. in Automatic Computing 2. The program used was BMD 02 R and the analysis was completed using IBM 360 computer at University College London. 20) J. W. Harbaugh, 1964. A computer method for four-vairable trend analysis illustrated by a study of oil-gravity variations in south-eastern Kansas, Bull. State Geol. Surv. Kansas, 171 (1964) 58 pp.
THE AREAL EXTENSION OF RAINFALL RECORDS:
AN ALTERNATIVE M O D E L D. J. UNWIN Department of Geography University College of Wales, Aberystwyth, Wales Abstract: Attention is drawn to the trend surface model as a possible alternative to the
more conventional regression methods as a predictor of rainfall totals. A test of this model using data from Snowdonia, North Wales, shows that it is slightly the more efficient. Introduction
In hydrological and climatological research it is often necessary to form an objective assessment of the overall variation of rainfall over the study area. Several methods have been developed to enable such areal extension of precipitation records, ranging from subjective isoplething 1) through various graphical methods 2) to simple and multiple linear regression analysis which treats the observed gauge data as a statistical sample. Rodda 3) provides an illustration of these latter methods together with a review of available techniques, whilst Ruhe 4) has applied a regression technique in palaeoclimatic reconstructions, This paper presents an alternative regression model based upon the techniques of trend surface analysis. Trend surface analysis
The method has been used extensively in the geological sciences 5, 6) and is finding increasing application in geography 7), but as yet does not seem to have been applied in climatology. A full description of the mathematical development of the model and its terminology has been given in a number of reviews 8-10) and is outside the scope of this work. In its simplest form trend surface analysis uses the least squares criterion to fit polynomials of successively higher order to areally distributed data: X. = bo + b , U + b2V
Linear surface
X n = c 0 -~- ClU 2t- c2V 2i- c3 U2 -.~ c4UV -~ c5 V2
Quadratic surJace
X. = do + d l U + d2V + daU 2 + d4UV + d5V 2 + d6 U3 + dvU2V + daUV 2 + d9 V3
Cubic surface
404
THE AREAL EXTENSIONOF RAINFALLRECORDS
405
WHERE X, is the value of the areally distributed variable and U, V are the geographic co-ordinates. Each polynomial defines a trend surface which is an expression of the broad regional variability inherent in the data, whilst the deviations of observed values from this surface, or residuals, are assumed to be an expression of strictly local factors. The degree of fit given by the polynomial, and hence its effectiveness in describing the areal variability of the data, is given by the percentage reduction in sums of squares it achieves. In certain circumstances a very high percentage reduction indicating a very close fit to the observed data can be obtained suggesting that the method can be used as a predictive technique for the extension of rainfall records in place of the more conventional regression methods. This alternative approach has a number of distinct advantages. The use of a polynomial rather than the planes fitted by conventional regression allows areal variations
Fig. 1. Mean annual rainfall (after Bransby-Williams).
406
D.J. UNWIN
to be followed more readily and there are considerable theoretical advantages to be gained in the separation of regional from local components in rainfall totals.
Rainfall prediction by trend surface analysis The area chosen for investigation was that of Snowdonia, North Wales. Several general climatologies of the area have been produced 11-13) but the best available information on rainfall amount and character is that published by Bransby-Williams14), a part of whose map is reproduced here (Fig. 1). In addition there are the published averages of rainfall for Great Britain and Northern IrelandlS). Mean annual totals for 47 stations were collected from both of these sources (Table 4) but of necessity these records vary both in quality and quantity. The difficulties of accurate precipitation measurement in mountain areas have often been commented upon and need no further discussion here, but further difficulties arise because of the differing time periods used by these sources. Thirty-three station means were taken from the Averages of Rainfall (1916-1950). In order to extend the areal cover a further 14 means for the period 1881-1915 were added from Bransby-Williams' list. The validity of this combination of records was examined by comparison of the 20 station means which are common to both sources. Table 1 lists these values together with summary statistics and the percentage change between the two periods at each station. Neither the sample means nor the sample standard deviations of these two sets of records differ significantly at the 99 9/00confidence level and no consistent pattern could be distinguished within the percentage changes at each station. It was concluded that the records could be combined without seriously affecting the analysis. Several stations listed by Bransby-Williams but not listed in Averages of Rainfall (1916-1950) were rejected for reasons of a suspected site change, insufficient length of record or a change in gauge type. The completed data file of 47 station means shows a complex pattern of rainfall distribution (Fig. 1). The effect of altitude can be clearly discerned but a secondary effect is that of gradually decreasing rainfall totals from south-west to north-east across the area in the direction of the prevailing rain bearing south-westerly winds. A common approach to the problem of rationalising this pattern and assessing the relative contribution of each effect to the observed totals would be the multiple correlation of rainfall with elevation and other factors such as position expressed by the latitude and longitude. The alternative trend surface method outlined here treats the areal variation of rainfall which is independant of station altitude, the equivalent precipitation, as the regional component whilst residuals from this are ascribed to the effects of altitude.
407
THE AREAL EXTENSION OF RAINFALL RECORDS TABLE 1 Comparison of station means for the standard periods 1881-1915 and 1916-1950 for 20 selected stations Site
Colwyn Bay Blaenau Ffestiniog (1) Blaenau Ffestiniog (2) Llandudno Snowdon (Crib Goch) Snowdon (Llydaw) Snowdon (Cwm Dyli) Snowdon (Llewedd) Snowdon (Delta) Snowdon (Teryn) Snowdon (Copper Mine) Llyn Eigiau Melynllyn Llyn Dulyn Plas Dulyn Conway Bethesda Aber. (U.C. Farm) Beddgelert Trefarthen
Height in Ft
1881-1915 Average
1916-1950 Average
~o change
118 1100 700 13 2340 1480 310 1485 1435 1080 1480 1244 2065 1632 509 50 527 60 133 30
29.4 108.0 109.3 28.1 172.1 151.4 128.5 153.0 171.8 155.7 159.7 80.3 125.0 102.7 55.2 30.5 65.8 41.5 88.6 36.2
29.4 115.7 110.0 28.8 171.1 157.9 140.5 153.5 170.1 156.0 157.9 80.3 119.0 105.3 55.3 30.5 64.6 41.6 90.2 36.3
0.00 + 7.13 + 0.64 -t- 2.49 -- 0.20 + 4.29 + 9.34 + 0.37 - 0.99 + 0.19 - 1.10 0.00 -- 4.80 + 2.53 + 0.18 0.00 -- 1.22 + 0.20 + 4.81 + 0.28
Means = 99.6 Sample standard = 50.11 deviation
100.7 49.70
I n o r d e r to derive the regional c o m p o n e n t the o b s e r v e d gauge d a t a was first s t a n d a r d i s e d to an equivalent p r e c i p i t a t i o n at c o n s t a n t altitude following the m e t h o d outlined b y Sutcliffe a n d C a r p e n t e r t e ) . I n this s t u d y the c o n s t a n t altitude selected was m e a n sea level a n d equivalent values for p r e c i p i t a t i o n were derived b y s u b t r a c t i n g a c o r r e c t i o n f a c t o r b a s e d u p o n a p r e c i p i t a t i o n lapse rate o f 5.5~/100 ft o f ascent. This value was established f r o m the simple linear regression o f rainfall against height for all stations within the area. Similar regressions using the station d a t a for m o r e restricted subareas s h o w gradients in the r a n g e 3.3 to 7 . C / 1 0 0 ft, i n d i c a t i n g t h a t whilst this p a r a m e t e r is b y no m e a n s c o n s t a n t over the whole a r e a the value used can be t a k e n as a fair a p p r o x i m a t i o n for an overall analysis, tT) T h e new set o f r e c o r d s was then subjected to t r e n d surface analysis using a m o d i f i c a t i o n o f the c o m p u t e r p r o g r a m m a d e available b y WhittenlS).
408
D . J . UNWIN
The results for the linear, quadratic and cubic surfaces are presented in Table 2 which also shows the percentage reductions given by randomly chosen values within the range of the original data. The percentage reductions obtained are sumciently high to indicate a strong regional trend, whilst the low reductions given by the random values indicate that this trend is a TABLE 2
Results of trend surface analysis using data reduced to sea level Percentage reductions in sum of squares obtained Surface Variable
Corrected rainfall Data Random values
Linear
Quadratic
Cubic
31.40
43.19
55.98
4.62
6.27
21.15
"real" one which could not have arisen by chance. Figure 2 shows the mapped cubic surface. To eliminate boundary effects this map was constructed for a smaller area than was used in the computer analysis. Its form is of interest in that it confirms the impression of a declining regional component in rainfall totals towards the east and north, from more than 75" in the south-west to less than 25" in the north-east. The result of this analysis is a series of equivalent precipitation values at mean sea level, but for comparison purposes these trend values at each station were recorrected for station altitude to produce the predicted station means tabulated in Table4(a). The percentage reductions in sums of squares given by these values are 82.51 for the quadratic and 84.89 for the cubic (Table 4(b)), indicating that a large proportion of the variance associated with the original station means can be accounted for by this simple trend surface model.
Rainfall prediction using regression analysis In order to test the trend surface model it was thought desirable to examine the same data file using normal regression methods. Table 3 shows the results of such an analysis for the observed rainfall means against altitude, latitude and longitude. Both latitude and longitude were expressed in arbitrary units north or east of an origin located to the south-west of the area. The matrix of simple linear correlations (Table 3(a)) shows that
409
THE AREAL EXTENSION OF RAINFALL RECORDS
N
/ PEET CT NSUM0rS0 ESS98
t
• SampLeVaLue 0
Fig. 2.
1
2 Mi~e5
3
4
Cubic trend surface for rainfall standardised to sea-level.
whilst the correlations between rainfall and height, and rainfall and latitude are significant at the 95% confidence level there is no evidence of a simple linear correlation between precipitation and longitude. In no case does a logarithmic transformation significantly improve the regression. Table 3(b and c) gives the results of sequential multiple regression using the same variables. The analysis was completed using a standard computer library procedure19). Step (1) entered the height variable into the regression, step (2) added to this the latitude, and step (3) further added longitude to produce the prediction equations shown. As Table 3(b) shows the third step added little to the efficiency of the regression. The overall reduction in sums of squares given was 79.3% (Table 4) with a multiple correlation coefficient of 0.891 indicating that the prediction equation is a reasonably efficient one.
D. J. UNWIN
410
TABLE 3 Results of regression analysis a) Matrix of simple linear correlation coefficients
X1 Latitude X2 Longitude X3 Station Height )(4 Rainfall
X1
X2
Xa
X4
1.000
-- 0.141 1.000
0.470 -- 0.030 1.000
0.705 -- 0.128 0.810 1.000
Underlining indicates coefficients which are significant at the 9 5 ~ confidence level. b) Results o f multiple regression Multiple Step 1. 2. 3.
Variables entered Height )(3 Height, Latitude Xa, X1 Height, Latitude, Longitude, X3, X1, X2
Increase
R
Rsq
in R
F-Value
0.810 0.889
0.6565 0.7903
0.6565 0.1339
85.98 28.10
0.891
0.7930
0.0027
0.56
F-Value at the 95 % confidence level, step 3 is 2.800. c) Prediction equations Step 1. 2. 3.
Equation X4 : 41.57 + 0.055 3(3 X4 = 13.87 + 0.175 Xa + 0.042 )(1 X4= 19.14+0.171 X3 + 0.042 )(1--0.247 X~
The efficiency of the methods T h e relative merits o f b o t h the a b o v e m e t h o d s m a y be c o m p a r e d d~rectly by m e a n s o f the percentage reductions in sums o f squares they give. Table 4 presents a s u m m a r y o f the o b s e r v e d data, values p r e d i c t e d by surface analysis, the residuals associated with this m e t h o d together with the residuals over the multiple regression a n d s u m m a r y statistics. I n b o t h cases there is a c o n s i d e r a b l e s p r e a d o f residual values b u t t a k e n overall the t r e n d surface m e t h o d with its 82.51 a n d 84.89 reductions in sums o f squares c o m p a r e s f a v o u r a b l y with the 79.30% given b y direct regression. These values indicate t h a t in the a r e a studied a n d with the available d a t a the t r e n d surface m e t h o d is slightly the m o r e efficient as a p r e d i c t o r for the areal extension o f rainfall totals. F u n d a m e n t a l l y this arises because the p o l y n o m i a l s used in this
THE AREAL EXTENSION OF RAINFALL RECORDS
411
TABLE 4 A.
Rainfall totals, stations and residuals over regressions
Station
Colwyn Bay Tal-y-Cafn Ffestiniog Blaenau Ffestiniog (1) Biaenau Ffestiniog (2) Cowlyd (A) Llugwy Cowlyd (B) Cowlyd (C) Llyn Eigiau Melynllyn Llyn D u l y n Plas Dulyn Conway Llandudno Nantlle (Dorothea) Bethosda Bangor Bangor Aber. (U.C. F a r m ) Penmaenmawr Beddgelert S n o w d o n ( C w m Dyli) S n o w d o n (Lliwedd) S n o w d o n (Delta) S n o w d o n (Teryn) S n o w d o n (Llydaw) S n o w d o n (Copper Mine) S n o w d o n (Crib G o c h ) Trefarthen Plas C a d n a n t Treffos Gladdllys Cottage Bettws-y-Coed Pentre-Foelas Trefriew Dolcrum Waunfawr Caernarvon
Height ft. O.D.
Rainfall Inches
Rainfall Predicted by Cubic Surface
118 170 610 1100 700 1340 1760 1140 1200 1244 2065 1632 509 50 13 400 527 275 30 60 170 133 310 1485 1435 1080 1480 1480 2340 30 120 290 340 118 640 25 100 520 180
29.4 38.4 76.3 115.7 110.0 90.9 100.8 89.4 90.4 80.3 119.0 105.3 55.3 30.5 28.8 63.4 64.6 47.0 42.3 41.6 36.5 90.2 140.5 153.5 170.1 156.0 150.7 157.9 171.7 36.3 43.5 43.7 43.8 50.1 57.7 47.2 47.2 55.7 36.2
44.5 31.6 84.0 131.1 108.5 117.4 138.2 98.4 101.1 101.8 146.1 120.7 50.9 23.9 29.2 76.1 63.6 42.5 29.3 26.2 28.5 96.5 84.9 153.3 148.5 127.7 149.4 149.4 196.5 19.6 36.5 52.2 56.7 47.1 54.0 35.9 27.0 71.0 33.5
Residual
Residual f r o m Mult. Regression
15.1 -- 6.8 7.7 15.4 -- 1.5 26.5 37.4 9.0 10.7 21.5 27.1 15.4 -- 4.4 -- 6.6 0.4 12.7 -- 1.0 -- 4.5 -- 13.0 -- 15.4 -- 8.0 6.3 -- 55.6 -- 0.3 21.6 -- 28.3 -- 1.3 -- 8.5 24.8 -- 16.7 -- 7.0 8.5 12.9 -- 3.0 -- 3.7 -- 16.4 -- 20.2 15.3 -- 2.7
-- 38.4 3.7 -- 1.4 -- 9.7 1.4 - 20.2 --26.1 -- 8.3 - 9.3 -- 19.9 -- 15.0 -- 9.4 -- 4.4 3.4 10.7 -- 26.6 -- 6.9 -- 6.5 -- 2.3 3.3 2.7 5.1 59.9 22.1 41.8 43.1 21.3 28.5 6.3 -- 16.7 0.8 -- 2.6 - 3.7 -- 15.4 -- 36.4 2.1 8.8 --28.1 - - 29.0
412
D. J. UNWIN TABLE4 (Continued)
Station
Bangor Llanfairfechan Snowdon (Glaslyn) Hafod Tan-y-graig Penmon Pen-y-Parc Llandudno (2)
Height ft. O.D.
Rainfall Inches
Rainfall Predicted by Cubic Surface
Residual
Residual from Mult. Regression
150 120 2050 310 60 372 71
50.7 39.9 154.7 129.8 39.0 43.1 30.3
34.5 27.0 182.3 98.1 45.8 51.1 35.2
-- 6.2 -- 12.9 27.6 -- 31.7 6.8 8.0 4.9
-- 4.3 2.9 0.6 41.8 13.7 -- 6.1 11.8
B. Summary Statistics Procedure
~ reduction in sums of squares
Multiple linear regression Quadratic trend surface Cubic trend surface
79.30 82.51 84.89
m e t h o d are better descriptors of areal variation than are the best fit planes fitted by conventional regression. In the present analysis it was found necessary to resort to an awkward standardisation procedure to correct for the effects o f station altitude but surface analysis methods which include such a term within the polynomials are available and might profitably be used in future analyses s0). Moreover the m o u n t a i n area chosen for the analysis here presented, over which altitudinal variations in precipitation predominate, constituted a very severe test for the surface model. Its performance when applied to such a difficult area indicates that it would be of even greater value if applied to areas over which relief variations are less marked. Whilst it is true that the addition o f other factors such as aspect and distance from the watershed might substantially improve the multiple regression model it is t h o u g h t that the method o f trend surface analysis is one which is worthy o f further attention in all studies concerned with the areal variation o f climatological parameters.
Acknowledgements M u c h of the work presented here was completed whilst the author was a research student in the Department o f Geography, University College
THE AREAL EXTENSION OF RAINFALL RECORDS
413
L o n d o n . I a m p a r t i c u l a r l y grateful to Professor B r o w n a n d the D e p a r t m e n t o f G e o g r a p h y , University College L o n d o n , for p r o v i d i n g access to c o m p u t i n g facilities. Miss P . M . Page supervised the multiple regression analysis. A c k n o w l e d g e m e n t s are also due to Dr. H. O. S l a y m a k e r , University College o f Wales, A b e r y s t w y t h , a n d Dr. J. C. R o d d a o f the Institute o f H y d r o l o g y b o t h o f w h o m critically r e a d the initial draft.
References 1) J. Glasspoole, The Rainfall of Norfolk. British Rainfall (1928) 271-2 2) V. Conrad and L. W. Pollak, Methods in Climatology. Harvard (1962) p. 222 et seq. 3) J. C. Rodda, An objective method for the assessment of areal rainfall amounts. Weather, 17 (1962) 54--59 4) R. V. Ruhe, An estimation of palaeoclimate in Oahu Hawaii. Amer. J. Sci. 262, 9 (1964) 1098-1115 5) D. F. Merriam and N. C. Cocke (eds.), Computer applications in the Earth Sciences: colloquium on trend analysis. Computer Contribution, State Geol. Surv., University of Kansas, 12 (1967) 62 pp. 6) W. C. Krumbein, Trend surface analysis of contour-type maps with irregular control point spacing. J. Geophys. Res. 64 (1959) 823-34 7) R. J. Chorley and P. Haggett, Trend surface mapping in geographical research. Trans. Inst. Brit. Geog. 37 (1965)47-67 8) W. C. Krumbein and F. A. Graybiil, An introduction to statistical models in geology. London (1965) 320--357 9) C. A. M. King, An introduction to trend surface analysis. Nottingham University, Department of Geography, Bulletin of Quantitative Data for Geographers, 12 (1967) 10) E. H. T. Whitten, A surface fitting program suitable for testing geological models which involve areally distributed data. O.N.R. Geogr. Branch. Tech. Rep., 2 of Task No. 389-135 (1963) 11) A. Lockwood, Meteorological Notes, in: The Mountains of Snowdonia (Ed. Cart and Lister) London (1925) 211-215 12) D. A. Ratcliffe, The Vegetation of the Carneddau, North Wales. J. Ecol. 47 (1954) 371--413 13) D. F. Ball, The soils and land use of the district around Bangor and Beaumaris. H.M.S.O. (1963) 6-9 14) G. Bransby-Williams, Some characteristics of mountain rainfall illustrated in northwest Wales. Water and Water Engineering, 58 (1954) 289-304 15) H.M.S.O. Averages of rainfall for Great Britain and Northern Ireland 1916-1950. London (1958) 16) J. V. Sutcliffe and T. G. Carpenter, The assessment of runoff from a mountainous and semi-add area in Western Iran, in: Hydrological Aspects of the Utilisation of Water. General Assembly of Bern (1967) 383-394 17) The method outlined here could be adapted to take account of more marked areal variations in precipitation gradient. 18) See reference 10. The analyses were completed using either Elliot 4130 or IBM 360 computers 19) W. J. Dixon, BMD: Biomedical Computer Programs. University of California. (1967)
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D.J. U N W I N
Pub. in Automatic Computing 2. The program used was BMD 02 R and the analysis was completed using IBM 360 computer at University College London. 20) J. W. Harbaugh, 1964. A computer method for four-vairable trend analysis illustrated by a study of oil-gravity variations in south-eastern Kansas, Bull. State Geol. Surv. Kansas, 171 (1964) 58 pp.