RELATIONSHIPS BETWEEN EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN A MOUNTAINOUS REGION: A CASE STUDY IN SCOTLAND

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 18: (1998) RELATIONSHIPS BETWEEN EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN A MOUNTAINOUS REGION: A CASE STUDY IN SCOTLAND CHRISTEL PRUDHOMME* and DUNCAN W. REED Institute of Hydrology, Wallingford, Oxfordshire, OX10 8BB, UK Recei ed 16 September 1997 Re ised 23 March 1998 Accepted 6 April 1998 ABSTRACT The spatial pattern of precipitation is known to be highly dependent on meteorological conditions and relief. But the relationships between precipitation and topography in mountainous areas are not very well known, partly because of the complex topography in these regions, and partly because of the sparsity of information available to study such relationships in high elevation areas. Moreover, studies are usually focused on mean annual precipitation, and so the patterns of extreme precipitation at short time steps, like daily, remain difficult to model. Daily annual maximum precipitation for 1003 gauges in Scotland, the most mountainous region of the United Kingdom, are studied to investigate the relationships between the median of the daily rainfall annual maximum, RMED, and the topography. A set of 14 topographical variables, some of them defined with respect to one of eight cardinal directions, are calculated from a 1 1 km digital terrain model (DTM). A particular effort has been made to improve the definition of some of the topographical variables suggested in the literature, either to provide a better physical definition or to better reflect the spatial variability of the topography. Single and multiple regression analyses have been made in some parts of the Highlands, leading to a 4-parameter model. This model is a mixture of geographical parameters (distance from the sea in opposing directions) and of topographical parameters (obstruction against the prevailing winds, and roughness between the main moisture source and the gauge). Special care has been taken to define a model whose physical sense is consistent with the meteorological conditions and whose parameters are not too interdependent Royal Meteorological Society. KEY WORDS: extreme rainfall; annual maximum daily rainfall; Scotland; mountain climatology; regression analysis; topography precipitation relationships 1. INTRODUCTION Rainfall patterns are usually not very well known in mountainous regions. One part of the problem is due to the strong topography of mountainous areas, which leads to complex precipitation patterns. A second part of the problem comes from the lack of information in these regions. The raingauge networks are typically sparse and of uneven density. Moreover, most of the gauges are situated in valleys, where the precipitation mechanisms and patterns are likely to be somewhat different to those operating at higher elevations. Very few gauges are situated in high elevation areas, and the information available is inevitably biased towards the low elevations. Radar measurements could provide useful information on topographic effects on precipitation (Collier, 1996), but they can be difficult to calibrate, especially in mountainous areas where the presence of significant relief causes errors such as ground echoes and screening effects due to the topography. Moreover, radar records may not be long enough to encourage their use in general mapping. * Correspondence to: Institute of Hydrology, MacLean Building, Crowmarsh Gifford, Wallingford, Oxfordshire, OX10 8BB, UK; tel: ; C.Prudhomme@ioh.ac.uk Contract grant sponsor: Scottish Office CCC /98/ $ Royal Meteorological Society

2 1440 C. PRUDHOMME AND D.W. REED Because of the strong relationships between precipitation and topography, it should be possible to use a topographical description to improve the mapping of precipitation extremes. In this paper, relationships between extreme rainfall and topography in Scotland are investigated. Here rainfall is taken to represent precipitation measured by daily raingauges. In conditions of snowfall, as in other circumstances, gauge measurements taken by daily observations may not always represent precipitation depths fully. However, the allowance for systematic deficiencies in daily precipitation measurements in upland areas is beyond the scope of this paper. The median of the annual maximum of daily rainfall, called RMED, is studied here. It is the standardisation variable of the FORGEX rainfall frequency method (Stewart et al., 1995; Reed et al., 1998) developed for the flood estimation handbook study, whose goal is to provide design engineers with rainfall and flood frequency estimates at any location in the UK (Reed, 1994). FORGEX method involves two stages: mapping an index of extreme rainfall, chosen to be RMED, and deriving rainfall growth curves. The purpose of this paper is to understand the behaviour of RMED with the topography in order to develop later a methodology for its mapping. The subsequent step will be reported in a future paper. After a brief description of the study area, various topographical variables are presented and calculated for a network of 1003 daily raingauges in Scotland. Relationships between topographic variables and extreme rainfall are investigated through regression analyses in Northwest Scotland, and validation tests are made for the whole of Scotland. 2. BACKGROUND Relationships between precipitation and topography, and especially elevation, have been studied for a long time. For example, Bleasdale and Chan (1972) review research in Great Britain and Ireland from the late 19th century. One of the best known relationships is the orographic effect: rainfall increases with elevation. This very simple relationship has to be moderated by a secondary phenomenon, the rainshadow effect (Flohn, 1969). In addition to the elevation and shadow effects on rainfall, other factors, such as the direction and distance to moisture sources, as well as the synoptic climatology of a given region, considerably complicate and, hence, weaken precipitation elevation relationships (Konrad, 1996). Numerous examples of elevation precipitation studies can be found in the literature, with various degrees of success. For example, Konrad (1996) found that elevation explains only about 3% of the variance of annual precipitation totals in the southern Blue Ridge mountains, whereas Chuan and Lockwood (1974) obtained better results in the Pennines, where about 50 60% of the variance in annual precipitation was explained by gauge altitude. In addition to the simple precipitation enhancement due to the elevation, the geographical environment plays an important part in determining rainfall amounts. The distance from the moisture source is an obvious factor studied for example by Griffiths and McSaveney (1983) and Konrad (1996). The direction of the prevailing winds bringing the frontal systems and the presence or absence of important relief in the trajectory of these frontal systems are other factors which may influence rainfall amounts. Considering more complex topographical variables may also improve the description and mapping of rainfall extremes (Basist et al., 1994; Leblois and Desurosne, 1994). 3. STUDY AREA The study area is Scotland, the most northern part of the United Kingdom, and its most mountainous region (Figure 1). Glasgow and Edinburgh lie in the central lowlands of Scotland. To the north are the upland areas of the North West Highlands and the Grampian Mountains which are sparsely inhabited: they are collectively referred to as the Highlands. The North West Highlands are characterised by numerous glens and lochs (i.e. valleys and lakes) oriented to the west or southwest, and corresponding to former glacial valleys. They are separated from the Grampian Mountains by the Caledonian Canal which

3 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND 1441 bisects the Highlands along an axis from Fort William to Inverness and connects the Atlantic Ocean to the North Sea. The highest peaks in Scotland (and the UK) are Ben Nevis (1343 m) and Ben Macdui (1309 m) in the Grampian Mountains. Although less massive, the North West Highlands also include many peaks over 1000 m. The Southern Uplands are relatively less mountainous, reaching 842 m at the Merrick. According to HMSO (1989) in over 6000 square kilometres of Scotland the annual rainfall is less than 800 mm but, because high hills cover a large area and thanks to the proximity to the Atlantic Ocean, the mountainous region in the northwest is amongst the wettest in the UK, and even Europe. Green (1972) suggests two main reasons: (i) the mountains are sufficiently high to induce a large orographic increase in precipitation; but (ii) at the same time they do not form a sufficient barrier to significantly divert travelling depressions, which therefore usually travel right across Scotland. A related factor is the closeness of the region to depression tracks in a variety of synoptic situations. Weston and Roy (1994) analysed the distribution of rainfall amount associated with different airflow types. West-south-westerly flows formed the most prevalent pattern, inducing rainfall totals on mountainous areas of western Scotland four to eight times greater than in eastern Scotland. The next most frequent pattern, west-north-westerly flow, occured four times less often and showed similar orographic characteristics with the western coast again experiencing higher rainfall amounts than eastern Scotland. Figure 1. Study area

4 1442 C. PRUDHOMME AND D.W. REED Figure 2. Gauge network available for annual maximum daily precipitation study in Scotland The highest annual rainfalls are in the mountainous areas close to the west coast. There is a marked seasonal variation in average monthly rainfall in the west of Scotland, with the total rainfall for the 5-month period February June being only ca % of that for the 5-month period September January (HMSO, 1989). The seasonal difference is less marked for the east of Scotland, where, on average, July and August are the wettest months. 4. DATA USED 4.1. Topographical information A digital terrain model (i.e. elevation data) from the UK national water archive/uk Meteorological (Met.) Office was available on a 1 1 km grid. The information is held in a standard relational database and the corresponding map (Figure 2) was drawn with a locally produced mapping package. The accessibility of the data allows easy use of the topographical information with FORTRAN programs and routines.

5 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND Rainfall information Unlike most earlier studies of precipitation patterns, this study concerns an index of extreme rainfall, i.e. the median of the annual maximum of daily rainfall, called RMED. One thousand and three raingauges with at least 10 years of daily record were available in Scotland, from which annual maxima were extracted and RMED calculated. The gauge network extends across most of Scotland but with an uneven gauge density (Figure 2). The gauge density is highest in southern Scotland, and lowest for the Grampian Mountains. In a study of rainfall distribution in two catchments in Scotland, Johnson (1995) quotes UK Met. Office sources that the number of operational gauges in Scotland during 1991 was 993, of which 368 were above 200 m and only 6 above 500 m. The highest gauge reported was at an altitude of 605 m although many mountain summits are above 1000 m Data representati eness Because of the uneven spatial distribution of the gauges, the study is initially focused on the North West Highlands where the significant variation in elevation makes it highly relevant to the investigation of rainfall topography relationships. Later, the Grampian Mountains, where topography is still important but the gauge density is poor, and the rest of Scotland, are used as validation regions. A total of 161 gauges with at least 10 years of rainfall records were available in the North West Highlands. Ballantyne (1983) notes that in the Scottish Highlands, the great majority of official raingauges are located near settlements and consequently often 500 m or more below the altitudes of adjacent mountains and plateaux. Therefore, it is inevitable that the sample available is not fully representative of the geography. 5. METHODOLOGY 5.1. Deri ed topographic ariables From the digital terrain model (i.e. DTM) data, it is possible to derive various topographical variables for use in a regression analysis of the precipitation topography relationships. Some of the variables have been mentioned in the literature, notably in Basist et al. (1994) and Konrad (1996), whereas others have been specifically developed for this study. (i) Elevation is the topographical variable most often used in rainfall studies. ELEV is defined as the elevation of the nearest grid point of the DTM to the gauge. Hereafter, this position is referred to as the gauge grid point, ggp. The distance between ggp and the real geographical position of the gauge is up to 0.71 km. The difference between the true elevation (altitude given by the rainfall database) and ELEV can sometimes be up to 100 m, due to the particularly steep relief encountered in Scotland and the relatively large-scale DTM used (1 km grid interval). However the error is not considered too significant and is ignored in the rest of the study. (ii) Average elevation is an alternative elevation suggested by Konrad (1996). Two average elevations are calculated: (i) ELEV4, arithmetic mean elevation of the typically 25 DTM grid points in a 4 4 km square centred on ggp; (ii) ELEV10, arithmetic mean of the typically 121 grid-point elevations ina10 10 km square centred on ggp. (iii) Geographical position is represented by the EASTING and NORTHING in km of ggp on the British grid reference system. (iv) Distance from the sea, SEA, is calculated in each of eight cardinal directions (N, E, S, W and NE, SE, SW, NW). MINISEA corresponds to the minimum of the eight SEA values; MAXISEA is the maximum. See the Appendix for more details. (v) SLOPEi is the mean slope between ggp and a point i km away in each of the eight cardinal directions. In this study, SLOPE2, SLOPE5 and SLOPE10 are calculated.

6 1444 C. PRUDHOMME AND D.W. REED (vi) Two average slopes are evaluated in each of the cardinal directions: (i) arithmetic slope ASLOPE= (SLOPE2+SLOPE5+SLOPE10)/3; (ii) weighted slope WSLOPE=(2 SLOPE2+5 SLOPE5+10 SLOPE10)/17. Unlike in most of the previous studies, the slope variable is considered at several distances and in several directions. Basist et al. (1994) define an exposure variable, denoted as the distance between a station and an upwind blocking barrier whose elevation is at least 500 m higher than the station, while Konrad (1996) uses a criterion of 150 m higher than the station. After trying the two different options, the intermediate value of 200 m was adopted. The exposure variable, EXPO, is calculated in each of the eight cardinal directions and is assigned a maximum value of 150 km if no such barrier exists. Note that Schermerhorn (1967) prefers to define a variable called barrier as the average of the highest elevations along the azimuths 210, 220, 230, and 240 between a point 4 miles (6.4 km) southwest of the station and the coast (the winter precipitation occurring mostly from the southwest). (vii) The bumpiness of the terrain is represented by the variable BUMP adapted from the up and down variable of Leblois and Desurosne (1994). The absolute difference in elevation between a grid point and the next grid point is calculated. The sum of these values from ggp to the sea in a given direction (for example, the north direction) is the BUMP variable in this direction. (viii) Two product terms, modified from Konrad (1996), are tested: the product of ELEV10 and EXPO in each of the eight directions, ELEVEXPO, and the product of the weighted slope WSLOPE and the exposure EXPO, SLOPEXPO. Although an important advance, the definition of EXPO has a number of weaknesses. It uses two arbitrary thresholds (200 m and 150 km) which can lead to discontinuities in the variable. For example, a point whose elevation is 200 m higher than ggp will be considered to form a barrier, but one 199 m higher will not. A second weakness is the unidirectional definition of EXPO. This problem is quite general and exists also for the SEA and BUMP variables. Only one direction is considered at a time (one of the eight cardinal directions) but it is easy to imagine that, even if there is no barrier in a straight line from ggp in a given direction, the existence of a barrier in a slightly different direction (a few degrees different) is likely to be no less influential. An additional weakness of the BUMP variable is that it gives equal weight to elevation changes irrespective of their distance from the gauge, whereas one expects the relief close to the gauge to have more influence on rainfall at the gauge than that 150 km away. Four new variables were defined to address these shortcomings: (i) AVSEA is the average distance from the sea; (ii) OBST is the angle subtended by the highest barrier; (iii) BARRIER is the distance to that barrier; (iv) SHIELD is a distance-weighted version of the BUMP variable. All the new variables are defined by reference to a 90 sector centred on the relevant cardinal direction. Full details of the sectoral variables are given in the Appendix. The new variables are a spatial average of values computed in several different directions and can perhaps be termed spatialised variables. Because the slope variables (ASLOPE, WSLOPE) are already subject to averaging, albeit along a particular linear transect, it is thought less important to develop sectoral versions of these variables Regression analysis A standard statistical package (SAS Institute, 1989) is used for the analysis. Univariate regressions between RMED and the various topographical variables are first carried out. Then multiple regression analysis is performed using the above topographic variables to explore variation in RMED. Criteria to be borne in mind are that the explanatory variables should not be highly correlated and, as much as possible, they should respect physical explanations for RMED variations. For example, considering that the prevailing winds in the study region comes from the west and the southwest, explanatory variables oriented W E or SW NE are more likely to be physically meaningful than variables oriented N S or NW SE.

7 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND 1445 A cautious approach to modelling the relationship between RMED and topography is adopted, the regression analysis being performed on 161 gauge records in the North West Highlands. Subsequently, the performance of the model is evaluated by reference to the full data set of 1003 sites throughout Scotland. 6. RESULTS AND DISCUSSION 6.1. Uni ariate regression As a first step, it is helpful to explore the relationship between RMED and the topographic variables on an individual basis. The percentages of variance in RMED explained by different topographical variables (R 2 ) are presented in Table I and Figure 3. None of the variables found in the literature review adequately explains the spatial variation in RMED (Table I). Generally, variables aligned to the prevailing wind direction (i.e. oriented W E or SW NE) are the most useful (for example EXPO NE or BUMP E ). Considering an average slope (ASLOPE or WSLOPE) leads to some improvement over use of a single slope (such as SLOPE5). One can make the same remark for the average elevation which gives better results than the grid elevation ELEV. Among the variables Table I. Percentage of RMED variance explained by given variable in simple linear regression Variable Direction R 2 (%) Variable Direction R 2 (%) Variable Direction R 2 (%) ELEV 1.5 ASLOPE North 21 EXPO North 10 ELEV4 17 Northeast 15 Northeast 29 ELEV10 22 East 18 East 23 EASTING 22 Southeast Southeast 14 NORTHING 18 South 19 South 14 MINISEA 4.6 Southwest Southwest 1.3 MAXISEA 1.1 West 15 West 4.3 Northwest 10 Northwest 8.2 SLOPE2 North 17 WSLOPE North 21 SEA North 6.2 Northeast Northeast 18 Northeast 19 East 14 East 26 East 21 Southeast 6.2 Southeast 13 Southeast 25 South 8.0 South 16 South 2.3 Southwest 13 Southwest 6 Southwest 18 West 14 West West 7 Northwest 8.7 Northwest 7.6 Northwest 1.4 SLOPE5 North 13 BUMP North 18 SLOPEXPO North 19 Northeast 12 Northeast 15 Northeast 21 East 15 East 27 East 15 Southeast 9.3 Southeast 29 Southeast 10 South 21 South 0.6 South 22 Southwest 8.6 Southwest 14 Southwest 14 West 12 West 0.7 West 24 Northwest 6.7 Northwest 2.1 Northwest 5.7 SLOPE10 North ELEVEXPO North 1.8 BARRIER North 0.1 Northeast 15 Northeast 8.2 Northeast 2.7 East 4.8 East 3.1 East 0.7 Southeast 6.9 Southeast 3.4 Southeast 0.6 South 23 South 2.2 South 3.0 Southwest 0.03 Southwest 0.2 Southwest 10.5 West 3 West 0.01 West 4.0 Northwest 2.2 Northwest 1.9 Northwest 0.7 Regressions significant at the 0.05 level appear in bold.

8 1446 C. PRUDHOMME AND D.W. REED Figure 3. Comparison of RMED representativity (in terms of R 2 ) between unidirectional variables and spatialised variables: (a) EXPO and OBST; (b) BUMP and SHIELD; (c) SEA and AVSEA suggested in Basist et al. (1994) or Konrad (1996), EXPO and ELEVEXPO give disappointing results (only two of the 16 variables explaining more than 20% of RMED variance), whereas SLOPEXPO W explains up to 24% of the variation. Generally speaking, these results are comparable to the ones published by Konrad (1996) where (for cool season precipitation events) the heavy event frequencies display the fewest significant relationships with the topographical parameters....no significant relationship is found between heavy rain frequencies and elevation alone. In general, the performance of the new spatialised variables is better than their unidirectional equivalents. Figure 3(a) confirms that the sectoral variables indexing obstruction (OBST) explain a much greater percentage of variation in RMED than their directional counterparts (EXPO variables). The better physical definition of OBST as compared to EXPO is also an important factor which is expected to improve the modelling of RMED. Similarly, all but one of the SHIELD variables are found to have greater explanatory power than the BUMP variables they replace (Figure 3(b)). The spatialisation of the variable distance from the sea does not lead to a general improvement, although there is an increase in the highest R 2 values (N, E, NE) (Figure 3(c)). Nevertheless, the better physical definition is considered to be a sufficient reason to favour AVSEA over SEA. The variables which individually explain the highest percentage variation in RMED are SHIELD SE (46%), SHIELD E (44%) and SHIELD NE (43%). The next highest ranking variables are OBST NE, OBST E and OBST SE. RMED is associated with a positive coefficient for each regression made on one of the eastern SHIELD variables (SHIELD SE, SHIELD E or SHIELD NE ). Thus, RMED is higher at the gauge if SHIELD, in an eastern direction is high compared to if SHIELD is low. In other words, the rainfall is expected to be high if the terrain near the gauge is very rough in the eastern sector, but not so high if the relief is smooth towards the east, the southeast or the northeast, at least in the gauge neighbourhood. Relationships between obstruction and RMED behave in the same way; RMED increases with the obstruction to the east, the southeast or the northeast. Both of these results are physically meaningful in a region in which most of the rainbearing weather system approach from the west. Most surprising is the absence of correlation between RMED and AVSEA in a western direction, where the proximity to the moisture source could be expected to be a good indicator of RMED at the gauge. At first sight it is surprising that BARRIER is not representative of RMED; the best variable (BARRIER SE ) explains just 10% of the variance of RMED. However, a small BARRIER value represents either a small distance to an important obstruction or, for points near the coast, the non-existence of a barrier between the gauge and the sea. Because the variable is a mix of different populations, one cannot expect a clear correlation with the rainfall.

9 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND Multiple regression analysis An initial multiple regression analysis on RMED revealed systematic structure in the residual error in estimating RMED. This undesired structure in the residuals diminished when RMED was replaced by 1000/RMED. Only the results for the transformed variable are presented here. The analysis is again focused on data for the 161 sites in the North West Highlands. The main technique used was stepwise multiple regression. However, attention was paid to avoiding the appearance of highly mutually correlated variables in the predictor terms. The preferred model found is: 1000 RMED = 32.4 (1.676) OBST NF (0.024) SHIELD SW (0.008) AVSEA SW AVSEA NE (0.102) (0.008) This explains 55.6% of the variance of 1000/RMED. S.E. values are listed beneath the corresponding predictor variables. All the variables are significant at a 5% level. Only 1% of the variation in the squared residuals is associated with variation in the predicted 1000/RMED values, which does not constitute enough statistical evidence that 1000/RMED variation is not the same for all observations. Interestingly, this almost unfettered stepwise procedure applied to the totality of the topographical variables yielded a model that makes reasonable physical sense. The model is interpreted as follows: RMED increases with OBST NE. OBST is a variable reflecting the importance of the obstruction presented by the relief. The increase of RMED with OBST NE suggests a barrier effect: the wind coming from the southwest transports the fronts to the northeast, but they are impeded by the obstruction to the northeast, increasing the rainfall at the gauge as the movement of the frontal system slows; RMED increases as SHIELD SW increases. SHIELD is a variable reflecting the roughness of the relief. A very variable relief near the gauge indicates a mountainous region, where the rainfall is expected to be higher than everywhere else. Therefore, the increase of SHIELD SW leads unsurprisingly to an increase of RMED. Moreover, frontal systems might be delayed in areas of complex surface topography (Smithson, 1969) and high precipitation might still occur after a frontal system has passed the crest of a mountain (Hanson, 1982), therefore where SHIELD SW is the highest; RMED decreases as AVSEA SW increases. Others factors being equal, the highest precipitation occurs near the west coast, close to the main moisture source of the Atlantic Ocean; RMED increases as AVSEA NE increases. This is a complementary factor to AVSEA SW, indicating that RMED is the smallest towards the northeastern coasts E aluation The regression model fitted to the North West Highlands has been tested in three regions: (i) the Highlands, as a whole, and its two sub-regions: the North West Highlands (fitting region) and the Grampian Mountains; (ii) southern Scotland; (iii) the whole of Scotland. The performance in the whole Highlands is acceptable for the fitted model (50% of the variance explained), but rather mediocre in southern Scotland, where it explains about 21% of the variance (first row of Table II). Away from the fitting region (North West Highlands), the errors of the regression model seem to be dependent on the estimated values; the error structure suggests an increase of the residuals with 1000/RMED, more noticeably in southern Scotland (Figure 4). The use of all the information available Table II. Percentage of variance explained by various models in application to five regions (1) Model NW Highlands Grampians S. Scotland Highlands Scotland Model 1 a Model 2 b a Model fitted on the North West Highlands b Model fitted on the Highlands region as a whole

10 1448 C. PRUDHOMME AND D.W. REED Figure 4. Residuals of the 4-parameter North West Highlands model for 1000/RMED: (a) North West Highlands; (b) Grampian Mountains; (c) southern Scotland in the Highlands might lead to a better estimation over the complete range of observed values. For example, low values are rare in the North West Highlands, but are observed in the east of the Grampian Mountains.

11 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND Recalibration The same four variables are used to fit a new 4-parameter model on the 581-observation sample available for the entire Highlands. Compared to the model fitted for the North West Highlands, R 2 increases for the southern Scotland region (lowland regions), and consequently for Scotland as a whole, as well as for the Highlands region (second row of Table II). Less than 1% of the variation in the squared residuals is associated with variation in the predicted 1000/RMED values. Therefore, there is no statistical evidence that 1000/RMED variation is not the same for all the observations. Figure 5 presents the error structure given by the model fitted on the entire Highlands for the three distinct regions: North West Highlands, Grampian Mountains and southern Scotland. No pecularity exists for the southern Scotland sample, situated outside the fitting area. The final model is then: 1000 RMED = (0.851) OBST NE (0.013) SHIELD SW ( AVSEA SW AVSEA NE (0.004) (0.004) (2) S.E. values are listed beneath the corresponding predictor variables. All the variables are significant at a 5% level. 7. CONCLUSION Detailed maps of statistics of extreme rainfalls are important to the assessment of flood risk. The spatial distribution of extreme rainfalls in the mountainous region of Scotland has been studied by way of RMED, the median of the annual maximum daily rainfalls. The spatial distribution of RMED was found not to depend on the elevation in a simple way, but to reflect more complex relationships between relief and position relative to moisture source. The literature gives few examples of spatial studies of rainfall extremes in mountainous areas, being more often focused on the study of mean annual rainfall. Several geographical or topographical variables found in the literature have been tested in this study. Some, such as elevation and slope, have been found to give unsatisfactory results. Others have been considered not precise enough and have therefore been modified. A considerable effort has been made to define spatialised topographical variables as a refinement of the unidirectional variables usually encountered in the literature. For example, the distance from the sea in one direction has been modified and is here defined as an average distance to the sea in a sector of 90 centred on the main direction. Finally, entirely new variables like obstruction have been introduced. It appears that, in Scotland, the modified or new variables are more representative of RMED than the ones found in the literature. The spatialisation of variables, such as distance from the sea and a roughness index (SHIELD), significantly improves explanation of the spatial distribution of RMED. A 4-parameter regression model has been chosen and fitted to the Highlands area. To reduce the bias in the estimation of the rainfall variable (dependence of the residuals on the variable), the final model estimates the inverse of RMED. The model is a mixture of geographical parameters (average distance from the sea in opposing directions) and of topographical parameters (obstruction against the prevailing wind, and roughness between the main moisture source and the gauge). Special care has been taken to define a model whose physical sense is consistent with the meteorological conditions existing in the region, and whose input variables are not too interdependent. Because of the physical definitions of the parameters used in the model, one can suppose that a similar model can be fitted to other mountainous regions of the UK, providing of course that the meteorological conditions are quite similar (i.e. prevailing weather systems approaching from the W or SW).

12 1450 C. PRUDHOMME AND D.W. REED Figure 5. Residuals of the 4-parameter Highlands model for 1000/RMED: (a) North West Highlands; (b) Grampian Mountains; (c) southern Scotland ACKNOWLEDGEMENTS The research was funded by the Scottish Office. The study forms part of a programme to provide new generalisations of rainfall and flood frequency for the UK Flood Estimation Handbook. The co-operation of the Met. Office in supplying data under memoranda of understanding is gratefully acknowledged. The authors thank Alice Robson, David Jones and an anonymous reviewer for helpful suggestions.

13 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND 1451 APPENDIX A. CALCULATION DETAILS FOR SELECTED TOPOGRAPHICAL VARIABLES SEA: SEA d is defined as the distance between the gauge grid point (ggp) and the sea in the direction d. Because sea lochs are coded in the raster database in the same way as open sea, the routine considers that the sea is effectively reached only if at least n successive grid points with a sea code are encountered (where n is chosen to correspond to a distance of about 15 km). OBSTRUCTION: let d be the studied direction. The variable is defined by reference to an angular sector, A, defined by a 90 angle centred on d (Figure A1). For any point of A, the excess h gx is calculated, defined by: h gx =h x h g, where h g =ggp elevation (m); h x =grid point elevation (m). If h gx is positive (i.e. the grid point is higher than ggp), the value is retained. If not, it is set equal to zero. The angle is defined as the angle that the barrier subtends at the gauge site, given by: tan = h g x d gx, where d gx denotes the distance in km between the gauge (ggp) and the grid point x. Note that h gx is taken in m and d gx in km. For practical purposes, the sector A is divided into sub-directions from 45 to +45 ; each sub-direction is characterised by the angle between d and this sub-direction. For each sub-direction, the maximum value of tan is retained, called obst. It corresponds to the maximum angle formed by an imaginary line between ggp and the highest grid point. Each of the -obst. values is then weighted by cos in order to give more weight to values corresponding to a direction close to d (i.e. is close to 0) and less weight to values whose direction is very different to d. Thus, for a given direction d, OBST d is defined as: Figure A1. Definition of a spatialised variable: OBST

14 1452 C. PRUDHOMME AND D.W. REED OBST d = (cos i obst i ) cos i BARRIER: for the ith sub-direction used in the OBST d calculation, barr i is defined as the distance from ggp to the point that defines obst i. The final value is then weighted by cos i, i.e. BARRIER d = (cos i barr i ) cos i AVSEA: distance from the sea, sea i, is calculated for each of the sub-directions in the sector A. The final AVSEA d is then taken as: AVSEA d = (cos i sea i ) cos i SHIELD: for each sub-direction i, updown i is calculated; this is sum of the ups and downs in elevation in this sub-direction, divided by the distance between the gauge and the furthest point. If i denotes the angle between the direction i and d, SHIELD d is taken as: SHIELD d = (cos i updown i ) cos i REFERENCES Ballantyne, C.K Precipitation gradients in Wester Ross, North-West Scotland, Weather, 38, Basist, A., Bell, G.D. and Meentemeyer, V Statistical relationships between topography and precipitation patterns, J. Climate, 7, Bleasdale, A. and Chan, Y.K Orographic influences on the distribution of precipitation, in Proceedings, Distribution of Precipitation in Mountainous Areas, World Meteorological Organisation, Geneva, 31 July 5 August, Geilo, Norway, 326(II), pp Chuan, G.K. and Lockwood, J.G An assessment of topographical controls on the distribution of rainfall in the Central Pennines, Meteorol. Mag., 103, Collier, C.G Application of Weather Radar Systems. A Guide to Uses of Radar Data in Meteorology and Hydrology, 2nd edition, John Wiley, p Flohn, H Climate & Weather, World University Library, p. 53. Green, F.H.W Characteristics of precipitation in the Scottish highlands, World Meteorological Organisation, Geneva, in Proceedings, Distribution of Precipitation in Mountainous Areas, 31 July 5 August, Geilo (Norway), 326(II), pp Griffiths, G.A. and McSaveney, M.J Distribution of mean annual precipitation across some steepland regions of New Zealand, N. Z. J. Sci., 26, Hanson C.L Distribution and stochastic generation of annual and monthly precipitation on a mountainous watershed in Southwest Idaho, Water Resour. Bull., 18(15), HMSO, The Climate of Scotland: Some Facts and Figures, HMSO Publications, London, p. 23. Johnson, R.C Rainfall distribution in two catchments in Scotland, in Proceedings, BHS 5th National Hydrology Symposium, 4 7 September, Edinburgh (UK), Konrad, C Relationships between precipitation event types and topography in the southern Blue Ridge mountains of the southeastern USA, Int. J. Climatol., 16, Leblois, E. and Desurosne, I Regionalisation of rain intensities on the French Alpine area: A methodological test for interpolation in mountainous areas, in Proceedings, De elopment in Hydrology of Mountainous Areas, Sept. 1994, Stará Lesná (Slovakia), UNESCO Publications, pp Reed, D.W Plans for the Flood Estimation Handbook, in Proceedings, MAFF Conference of Ri er and Coastal Engineers, Loughbourough (UK), July 1994, Reed, D.W., Stewart, E.J. and Faulkner, D.S The FORGEX method of rainfall growth estimation, Part II: Description, Submitted to HESS journal.

15 EXTREME DAILY PRECIPITATION AND TOPOGRAPHY IN SCOTLAND 1453 SAS Institute, SAS/STAT User s guide, Version 6, 4th edition, Cary, NC, SAS Institute, p. 846, p Schermerhorn, V.P Relations between topography and annual precipitation in Western Oregon and Washington, Water Resour. Res., 3(3), Smithson, P.A Regional variations in the synoptic origin of rainfall across Scotland, Scot. Geogr. Mag., 85, Stewart, E.J., Faulkner, D.S. and Reynard, N.S Rainfall Frequency Estimation in England and Wales, Phase 1b: Pilot Study, Report to National Rivers Authority, R&D Note 478, p Weston, K.J. and Roy, M.G The directional-dependence of the enhancement of rainfall over complex orography, Meteorol. Appls., 1,