Spatial Interpolation of Rainfall in the Dry Zone of Sri Lanka

One of the problems which often arises in climatology is either data at a given site is missing or the site is ungauged. In this study, a spatial interpolation model was aeveloped to estimate the weekly rainfall of the Dry zone of Sri Lanka at ungauged sites assuming that the spatial continuity of rainfall at two neighbouring locations are exponentially correlated. Twenty years of weekly rainfall data from six stations located in the Dry zone was used in the study. To support the methodology, the results of the exponential model were compared with the other two methods of spatial interpolation techniques, namely, the local mean and the inverse distance methods. The results of the study indicate that the exponential correlation model is a promising candidate for estimating mean weekly rainfall of the Dry zone. However, the local mean and the inverse distance methods compare quite well along with the exponential model, indicating that more complex models have no particular advantage over simple models for estimating rainfall in the Dry zone of ,)ri Lanka. Nevertheless, the results point towards the relative importance 'of the exponential model as opposed to the other two models when the neighbouring locations do not have long series of historical records.


INTRODUCTION
The complex, interacting atmosphcric processes which give rise to rainfall make it a variable phenomenon across the landscape.Therefore, recorded rainfall from a rain gauge usually represents only an extremely small area of the catchment.Rain gauges in the Dry zone are usually separated by several kilometres.Therefore, the existing networlr: of gauges may not be sufficient to estimate the parameters that are needed for hydrological and climatological applications.This problem is further aggravated by the frequent missing data in the observed rainfall sequences.Thus, there is a need for a methodology of spatial interpolation of rainfall which uses only minimum available data.
Spatial interpolations of data available at other sites are being used in the field of hydrology and climatology to generate the data for ungauged locations.In most cases, simple methods of point estimation are applied.The availability

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of computing facilities has encouraged the development of advanced methods of interpolation.As a result, a number of spatial interpolation tcchniques are available today with varying degrees of complexity such as local mean, Thiessen polygon, inverse distance, inverse square distance, isol~yetal and krigging.'" Some of them are very simple with limited applicability while others iilvolve complex mathematical frameworks and need a large number of data points to obtain a reasonable level of accuracy.

METHODS AND MATERIALS
Spatial continuity exists in most earth science data sets and two data sets close to each other are expected to have closcr values than those that are far apart." function can be developed to describe the continuity of the relationship bctween the valse of one variable at a point ancl the value of the same variable a t another point, a given distance away.]Correlation, covariance and variogram functions have been used to express the spatial continuity of a random variable.Similar assumptions have been made about rainfall phenomena over an area, and estimation methuds used in earth science have been applied to rainfall data to estimate the values at ungauged sites.
The spatial correlativn models for rainfall have been presented in inversc power and exponential forms4 : Assuming homogeneity and isotropicity," study was undertalten to determine the appropriateness of point cstimation of weekly ramfall by an exponential spatial correlation model for the Dry zone's climatic environment.In this study, two distinctive regions of the Dry zone were considcred, the northcentral part and the southern part of the Dry zonc (Figure 1).Both rcgions exhibit fairly similar physiography of gently-undulating to rolling, with 3 to 4% slopes.However, some geographical features are not alike.The north-central part of the Dry zone, (abbreviated NCDZ), is generally an inland rcgxon.The soutllern part of the Dry zone, (abbreviated SDZ), resembles an area that is closer to the ocean.Therefore, the amount of water vapour in the atmosphere, available to become cloud with the chance of subsequently becoming rain, may not be comparable in the two I-egions.Thus, the corrclatlon structure of the rainfall process could be different in the two regions.This necessitates the evaluation of the spatial correlation model for the two regions separately to meet the assumptio~is m a d c on tbc isotropicity and homogeneity.The selected rainfall recording stations from the NCDZ region are located a t Maha-llluppallama, Pelwehera and Maradankadawala.Out of' these three stations, Maradankadawala which lies in between the other two stations was considered as the location for estimation of rainfall values.The areal distances from Maradankadawala to Maha-Illuppallama and from Maradankadawala to Pelwehcra are 17 km and 25 km respectively, while thc areal distance between Maha-Illuppallama and Pelwehera is 38 km.From the SDZ region which represents a coastal area, Angunakolapellessa, Ambalantota and Weerawila were selected for the study.In this region, Ambalantota which lies in between the other two stations was considered as the location for estimation of rainfall values.The areal distances from Arnbalantota to Angunakolapellessa and from Ambalantota to Weerawila are 15 lrm and 27 km respectively, while the areal distance between Angunakolapellessa and Weerawila is 38 km.In the selection of the rainfall recording statioi~s, care was given to select the locations with reliable data with a maximum number of rccord lengths to be on par with the guidelllles stipulated by the Hydrology and Watcr Resources Program, Department of Civil Engineering, Colorado State University.Vhe said guidelines prescribe that data records wit11 Inore t l ~a n 30 years should be used.But the available length of the records from tllc sclectcd locations were 20 years.Although there arc some other locatlous 111 the D1.y zone wh~ch have the minimum of 30 years of records, a large nurnl~er of lnisslng data and ~~nreliability of the measurements forced us not to selcct lhcm Li)r the study.

SCALE
In addition, the models were also evaluated for situations with short series of l~istorical data and when the stations are located relatively Car away.Wcckly rainfall values for Angunalrolapellessa were interpolated using 10 years of historical rainfall records fi-om two neighbourillg locations, J3rnbilipitjya and Tangallc.As the reliable rainfall data from the immediate v~cinity of a rainfall station in the Dry zone was unavailable, these two locations we1 c sclcctcd from the ile~ghbouring Intermediate zone (Figure 1).The aerial cl I stances fi.0131 Embilipitiya to Angunalrolapellessa and from Angu~~alrolapellessa to 'I'angallc are 18 km and 20 km respectively, while thc aerial distance bctwecrl Embilipitlya and Tangalle is 35 km.An evaluation of the interpolation modcls lo1 the stat~ons located farther apart was carried out by interpolation of wcekly rainfall at Weerawila using Tangalle and Lahugala (eastern part of'the D1.y zone).Thc areal distance from Lahugala to Tangalle is 143 km whereas distances fiaom Lahugala to Weerawila and from Weerawila to Tangalle are 93 and 58 km respectively.
The spatial correlation coefficient Yab for weekly rainfhll v;llues ciul bc clctcrmlned using contemporaneous observation pairs Iron? siatiot-1s A and B. Using the calculated Yab and tbc distai~cc between s t a t ~o n s A and l3, the coefficient ( a ) of equation ( 2) can be round.The least squares regression for equation ( 4) can be wriitcn in matrix notatio~~: The matrix C consists of'the covariancc value ofrainfall betwcc~i t l ~c t w o san? plc locations.The vcctor ' )' collsists ol' the covariancc values of rainfall bctwccn two sample locatlolls and the location where wc need the estimation.T l ~c vcctor w consists ofthe weight given to each location and the Lagrmgc_parainctc~.11. 1 To solve for the weights, multiply both sides of' equation ( 5) by C -l.
Based on the homogeneity and isoti.opicityassumptions, thc cstlmated mean rainfall and the standard deviation of the rainfall at station C can be calculated using the following linear estimation :

RESULTS
The validity and applicability of the foregoing lnterpolatlon model was examined by comparing the model o~ltput with the observed data from Maradankadawala and Arnbalantota.In addition, a f ~~r t h e r comparison of the model output was made with the other two interpolation techniques, namely, d ~e local mcan method and thc inverse distancc method.Use of local mean 01. the aritl~metic mean In spatial interpolation is the most simplistic approacl~ It assumes equal weight from all nearby sample locations, using the sample mean as the estimate.Invcrse distance method is a technique which gives more weight to the closest samples and less to those that are farthest away.Thus, wcight for each sample is inversely proportional to its distance from the point being estimated: where A R = estimate of rainfall fbr ungauged location V; = observed value a t the it" location cZ; = distance from each location to the point being estimated

Comparison of estimated and observed rainfall
Flgures 2 and 3 show the mean estimated and observed rainfall in each wcelr for Maradankadawala and Arnbalantota respectively.Typically, we wail t a set of estimates that comes as close as possible to the true values.Thus, we would prefer the results shown in Figures 2 and 3 .There was no significant difference between the observed values and the estimated values a t both Ambalantota and Maradankadawala.The standard deviations of the observed sequences of rainfall were comparable with the estimated sequences of rainfall from the exponential model (Tables 1 and 2).However, most of the timc the variability of the estimated values from the exponential model was less than that of the observed variability.This trend was more apparent a t Ambalantota in the SDZ region.Reduced variability of estimated values is often referred to as "smoothing" and is a coilsequence of combining two or more samplc valucs to form an estimate."As morc sample values arc incorporated in a weigl~tecl l i ~~e a r combination, thc resulting estimates geilcrally become less variable.Figure 4 shows the performance of the exponential model in interpolating weekly rainfall a t Angunakolapellessa with a short series of h~storical data from neighbouring two locations.It is clear that the estimatcd data do not represent t h e observed d a t a well compared to the e s t ~m a t i o n s a t Maradankadawala and Arnbalantota.Ilowever, tlie differences between the estimated mean values and the observed mean values were not significant at the 5% probability level.The estimated mean weekly rainfall at Weerawlla from the historical data of Tangalle and Lahugala are shown j 11 Figure 5.It is clear that there is a distinct deviation of the estimated mean values from thc observed mean values.In general, these meail deviatioi~s were significant at the 5% probability level, especially during the second half of tbc year.The results of the otl3er two interpolation metliods described in, a precctling section, of this paper were compared with the o~ztcome of exponerl.ti.al, corr~lation moclil..As the first criterion for comparing the d.if'f'erent methods, the nieans in each week were computed.Figures 6 and 7 show the mcans of weekly interpolated rainfall values from the three methods for Marada~7kadawala and Ambalantota, respectively.It may be seen that practically al.1 ofthe iilterpolatio~~ techiiiques reproduce the means well.None of these means welye significant1.ydi.f'ferent from each other and also from the observed va:lues.The estj.mated values from all three mode1.sa t Maradanlradawala are almost identical ((r'b I rure 6).At Ambalantota, though it is not significant, a small discrepancy between estimated values from the three models is noticeable during the two dry periods and during the Yala season, mid-March to mid-May, (Figure 7).The differences between estimated values fi-om the three models a t both Anguna'kol.ape'l.lessaand Weerawila were also not significant.However, most of the time the closest value to the observed value was found wit11 the exponential moclel.Another way of checking the appropriateness of the model is to ca'lculate the correlation coefficient between the observed and the estimated values.It is a good index for summarisii~g how close the points on a scattkr plot come to falling on a straight line, and therefore can be made use of' to compa:re' different estimation models.The correlation coefficient between th.c observed and the estimated values fiom each model was calculated for every week of the year.These values were averaged over four different time periods oi'the year, namely, first dry season ( early February to mid March), second dry season (late March to late September), Yala season (mid.March to mid May) ancl Maha season (early October to ].ate January).At both Maradan.kadawalaant1 Am.balautota, the seasonal correlation coefficients were always above 0.65 escept &uring the Yala season at Ambalan.tota (Table 3).I t is interesting to note that when the correlation between the estimated and the observed values is low, it is consistent with all the three mode1.s.

Ambalantota
The correlation coefficients between the estimated ancl the observed values a t Angunakolapellessa were always above 0.5 with all the tlll-ee models where the estimation was based only on 10 years of historical data (Table 4).I-Iowever, a t Weerawila, where the estimation was based on two neighbouring locations separated by over 100 km distance, only the first dry season exceeded the 0.5 boundary.During the Maha season, correlation coefficient was closein to 0.5 with the exponential model.But, during both theYala and the second dry seasons, correlation coefficients were below 0.40 with all the three models (Table 4).

DISCUSSION
Overall., the results show that mean weekly ra:in.fall of both Ambalailtota and Maradankadawala are well preserved.However) the discrepancy between the observed and. the estimated values at Maradankadawala is less than the same at Ambalantota.The correlation of rainfall between any two locations is highest for places which are close to each other, in flat country an.d, away from the coast.Vhe areal distances between the two sample locations at both regions are almost equal.The topography of the two regions is also couiparable to each other.Thus, closeness to the ocean could be the main determining factor for the small discrepancy between the observed and. the estimated values a t Ambalantota in SDZ region.However, the performance of the expoioential.model.based on a short series of historical data (i.e. 10 years) is not colnparablk to its performance with long historical data.This confirms that the more historical data avail.able,the better the estimation will be.The situation becomes worse when the neighbouring locations are further apart (i.e. 100 krn or in excess).This could be attributed to the fact that the correlation structure is bein.gweakened when the stations are farthest away.
The exponential method and the inverse distance method should give a better estimation of weekly rainfall data compared to the local mean mcthod.Because, th&,:give a varying weight depending on the distance apart ratlier than a eq&1 weight as in the case of local mean method.Thereforc, it is reasonable I to expect an improvement to the correlation with thb exponential method and the inverse distance method over the local mean method.However, the correlation coefficient values of three models are almost similar a t both Maradankadawala and Arnbalantota (Table 3) suggesting that performances of all the three models are similar under the given environments.Thus, if one is .interested only in mean rainfall, as is often the case in climatological applications, then there is no particular advantage in computing a complex exponential relationship; rather a simple inverse distance or local mean will suffice.
Although the difference between the correlation coefficient values among the three models is not large, both at Weerawila and Angui~aliolapellessa, the resulting correlation coefficient values from the exponential model were always higher than that from the other two models exccpt during the Yala season at Weerawila (Table 4).Thus, when the neigl~bouring locations are farther away or especially with short series of historical data, thc individual estimations of the cxponential model would approximate the real values comparatively bettcr than the other two models.
In conclusion, the results of this study suggest that the exponential correlation model is a promising candidate for estimating weelrly rainfall of the Dry zone of Sri Lanka.However, the less sophisticated local mean and inverse distance methods rate quite well along with the exponential model.There is no particular basis to claim that the exponential model is significantly better than the other two methods tested under the given environments.Nevertheless correlation analysis shows an improvement to the estimates with the exponential model, especially when the neighbouring locations do not have long series of historical data.
Yab = spatial correlation coefficient between stations A and I3 Ct = a coefficient c = a power coefficient d = distance between the pair of stations

Figure 1 :
Figure 1: Location of the reference rainfall stations used in the study.
The observed spatial correlation coefficients between the t w o stations, A and B, and the value for a from equation (3) can be used in equation (2) t o estimate the correlatioil coefficients between stations A and B wit11 the station C (yac and Y b c ).Let the unbiased, linear estimator for the normalised rainfall at station C be: where R: = estimated normalised rainfall at station C = observed normalised rainfall at station A R = observed normalised rainfhll at station.B Wa = weight assigned to the station A I/Vb = weight assigned to the station I3 where kc = estimated mean rainfall at station C jia = observed mean rainfall at station A -R b = observed mean rainfall at station B A oc = estimated standard deviation of rainfall at station C oa = observed standard deviation of rainfall at station h ob = observed standard deviation of rainfall at station B AB = distance between stations A and B in km AC = distance between stations A and C in km Once the above parameters are determined from the observed data, equation (4) can be used to estimate the rainfall in eacl~ week of the year.

Figure 2 :Figure 3 :
Figure 2: Observed and estimated rainfall at Maradanltadawala in the Dry zone of Sri Lanlca.

Figure 4 :Figure 5 :
Figure 4: Observed and estimated rainfall at Angunalrolapellessa in the Dry zone of Sri Lanka.

Figure 6 :Figure 7 :
Figure 6: Estimated weekly rainfall from three models at Maradanlradawala in the Dry zone of Sri Lanka.