A model for interpolation of hydrometeorological data in Mali

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1 Scientific Prcedures Applied t the Planning, Design and Management f Water Resurces Systems (Prceedings f the Hamburg Sympsium, August 1983). IAHSPubl. n A mdel fr interplatin f hydrmeterlgical data in Mali ABDOULAYE DIAWARA Ryal Institute f Technlgy, Schl f Civil Engineering, Department f Hydraulics, S Stckhlm 70, Sweden ABSTRACT The netwrk f hydrmeterlgical statins is generally underdevelped in develping cuntries. As a cnsequence water resurces develpment prjects are ften lcated in basins with insufficient bservatin data r in many cases in ttally unbserved sub-basins. It appears that under these cnditins data transfer methds available tday must be made mre effective. This paper presents a mathematical interplatin methd based n the develpment f the riginal bservatin series int empirical rthgnal functins. Weights and amplitude functins are calculated frm the cvariance matrix f the riginal series and then analysed tgether with physigraphical parameters fr reference elements. Six climatical parameters were cnsidered in the study. The regressin equatins included in the mdel were established between weights f amplitude functins and latitude, lngitude and altitude. This prcedure allws interplatin n a cntinuus basis ver the entire studied area. Sme results and pssible applicatins f the methd are als discussed. Mdèle pur 1'interplatin de dnnées hydrmétêrlgiques au Mali RESUME La réseau de statins hydrmétérlgiques est suvent peu dévelppé dans les pays en vie de dévelppement. En cnséquence beaucup de prjets de dévelppement des ressurces hydrlgiques dans ces pays divent être réalisés pur des bassins versant avec peu u pint de dnnées de base. Cela indique un besin de rendre les prcédures d'interplatin aussi efficace que pssible. Cet article présente un mdèle mathématique d'interplatin basé sur le dévelppement de séries d'bservatin en "empirical rthgnal functins". Les cefficients et amplitudes snt calculés en partant de la matrice de variance-cvariance des dnnées initiales et analysés cnjintement avec des paramètres physigraphiques. L'étude a été cnduite avec six paramètres climatlgiques. Des équatins de régressin nt été établies entre les cefficients des fnctinamplitudes et, d'autre part, la latitude, la lngitude, et l'altitude. Cela a dnné la pssibilité d'interpler de manière cntinue dans l'espace dans tute la zne d'étude. Quelque résultats et pssibilités d'applicatin ffertes par la méthde snt aussi examinés. 35

2 36 Abdulaye Diawara INTRODUCTION The mdel presented in this paper has been develped as part f a study titled "methdlgy fr water resurces evaluatin in arid and semiarid areas, especially Mali". This study yielded the map f main hydrmeterlgical znes shwn in Fig.l (Diawara, 1982), The climatical parameters included in the study were: precipitatin, temperature, sunshine hurs, evapratin, humidity and wind speed. The bservatin series were btained frm 80 statins fr precipitatin and 17 synptic statins fr the remaining parameters. The develpment int empirical rthgnal functins was used in the wrk as a tl fr bth classificatin and interplatin. It is primarily the practical meaning f the methd under cnditins prevailing in develping cuntries like Mali that will be emphasized in this paper. Hwever, it is necessary t present briefly the mathematical basis f the methd in rder t effectively assess its applicatins. In the fllwing the develpment f the theretical basis f the methd is summarized, sme applicatins and results are presented, and in cnclusin its ptential fr applicatin, especially in cnditins similar t thse prevailing in the area studies, are discussed. THEORETICAL BASIS Given x-^ and -x.2% as bservatin series f tw variables where the mean values have been subtracted ut we are here interested in analysing variatins f x-^ and ^ i- n relatin t external factrs. Figure 2(a) shws an assumed representatin f the series with the FIG.1 Hydrmeterlgical reginalizatin f Mali (frm Diawara, 1982).

3 Interplatin f hydrmeterlgical data in Mali 37 (a) (b) FIG.2 Representatin f the variables. axes x-^ and x 2. In Fig.2(b) a new crdinate system has been intrduced by rtatin where 9 represents the angle f rtatin. The new axes (r variables) are y-, and y 2. The new variables can be expressed as yi = x-j^css X2SinG. y2 = XCOSt x^sinb (1) It appears frm the diagram that y2 has the smallest variance while yi has the largest. Using the sample data the angle 6 crrespnding t minimizing the variance f y 2 can be derived frm the expressin tan2 6 2x 1 x 2 /(x 2 1 x 2 ) (2) The hrizntal bar indicates an ensemble average. It appears frm (2) that the angle f rtatin is related t the cvariance f x^ and x 2. Mrever, as we are using the same riginal data, the ttal variance remains the same after transfrmatin, that is: var(y^) + var(y 2 ) = varfx^) + var(x 2 ) Thus minimizing var(y 2 ) is equivalent t maximizing var(y^) and cmpressing the maximum f infrmatin int the variable y-^. It can be shwn using frm example (1) and (2) that cv(y 1, y2) = 0, that is, the new variables yj and y2 are uncrrelated (Hlmstrm, 1977). S if the influence f external factrs is assumed linear the factrs underlying the variatins alng y-^ are uncrrelated t thse causing variatin alng y 2. This indicates a theretical pssibility f islating different external influences. That is the basic prperty which is fundamental t "factr analysis" r "principal cmpnent analysis" yet nt fr interplatin purpses as we will see later. The transfrmatin can be generalized t the multidimensinal case with M variables and N bservatins. The riginal values may then be represented as: Hj' 1,...,M and j = 1,,N As in the tw-dimensinal case we are here interested in

4 38 Abdulaye Diawara cmpressing the maximum f variatin nt a few uncrrelated new variables. The new variables may then be written as: Y = AX where A is a quadratic matrix (M x M) f cefficients chsen t be rthgnal. The variance-cvariance matrix f the riginal series and the new series are respectively: _C = XX*A N - 1) V = (AX)(AX)*/(N - 1) = ACA* As the new variables are uncrrelated ^V must be a diagnal matrix with elements V]_,..., VJJ. The equality V = ACA* leads t a determinant equatin and eigenvalue prblem where it appears that v^ are eigenvalues f the cvariance matrix; A is a matrix f eigenvectrs and the new variatin series; Y, are dented principal cmpnents, see fr example Gttschalk (1985). S in the multidimensinal case it appears that if we measure the variatin in ur data set alng the principal axes f the variancecvariance matrix we can cncentrate the infrmatin in a few variables. Here again, as in the tw-dimensinal case, it is the cvariance structure in the data that determines the angles f rtatin giving the new bservatin axes. In factr analysis the new variables are related t the main factrs underlying the variatins in the data and the mdel generally has the frm Xi = Z k=l W ik y k + e i ; i = 1,2,...,M K < M where w ik are called lading factrs, and e^ are the errr terms. When the riginal series are time series the principal cmpnents are als dented empirical rthgnal functins (Hlmstrm, 1969). The transfrmatin f the riginal time series by prjectin n principal axes gives eigenfunctins - als called amplitude functins. The mdel may then be given the frm X i (t) = \=1 W ik A k^) i = 1,2,...,M (3) where Ajj(t) are amplitude functins, and w., are weights. Such a develpment f riginal time series int empirical rthgnal functins (EOF) has imprtant applicatins with varius bjectives in many scientific fields. An example f applicatin fr data classificatin is given in Diawara (1982). Here we will cnsider the prcedure fr the purpse f interplatin.

5 Interplatin f hydrmeterlgical data in Mali 39 DEVELOPMENT INTO EMPIRICAL ORTHOGONAL FUNCTIONS (EOF) FOR INTERPOLATION PURPOSE Hydrmeterlgical time series derived frm a netwrk f synptic statins were used fr the analysis. The bjective here is t relate the variatins in the data t physigraphical variables in rder t btain a pssibility f interplatin. Fr that purpse the mdel given in (3) may be rewritten as: xj.(t) = 2 w ik A k (t) + R ± i = 1,2,...M m < M (4) K 1 where M is the number f bservatin statins; w ik are the weights; Rj_ is an errr term; and A k (t) are the amplitude functins. If the series are rapidly cnvergent then a few amplitude functins may give mst f the variatins at any statin with a negligible errr term R. If mrever the weight-cefficients fr a given statin can be identified and analysed in relatin t specific amplitude functins and physigraphical parameters then the mdel given in (4) may frm a cnvenient basis fr data interplatin. Nw the statins are cnsidered t be the variables, the bjects being the 12 mnthly mean values f the cnsidered parameter. The series frm statin i crrespnds then t clumn i in the riginal matrix X and the cefficient matrix will be dented W_. The clumns in \V are thus the eigenvectrs derived frm the variance-cvariance matrix f X. We then have A = XW AW -1 = XWW" 1 X. = AW -1 It is thus pssible t restre the riginal variatins f the statin frm the matrix f amplitude functins and weightcefficients. The matrix W l gives us the weights w ik which depend n the statin and can be related t the physigraphical parameters characterizing the lcatin f the statin. As was mentined earlier the variance-cvariance matrix f the new variables, here A, is a diagnal matrix with the eigenvalues A^, A2,...,Ajyj. S Zk=l ^i is equal t the ttal variance f the data (see Davis, 1977). Each amplitude functin then accunts fr a percentage f the ttal variance prprtinal t the crrespnding eigenvalue. The amplitude functins are cmmnly arranged in decreasing degree f explanatin f variance. Generally the first amplitude functin, bearing mst f the ttal variance, describes the main r dminant variatins in the data while the fllwing express particularities in the variatin pattern. If the series are rapidly cnvergent tw r three amplitude functins tgether will restre almst all the riginal variatins at any statin. In this study nt mre than three first amplitude functins needed t be cnsidered as the crrespnding trace-percentages (degree f explanatin) were generally greater than 95%, see Table 1. S in the mdel equatin (4) we have m < 3 and R < 5% fr all the six variables cnsidered in the study. The weight-cefficients f the included amplitude functins were then crrelated t

6 40 Abdulaye Diawara TABLE 1 Eigenvalues and ttal variance Ampli funct tude ins Percentage f trace Cumulative per cent f trace Precipitatin Temperature Sunshine hurs Evapratin Humidity Wind speed physigraphical parameters at recrding statins namely latitude, lngitude and altitude, see Table 2. The particular climatic variatin pattern f the Sahel with a well marked nrth-suth variatin is apparent fr precipitatin, temperature and evapratin. The influence f altitude generally negligible is nticable nly fr evapratin in agreement with the tpgraphy f the (mstly flat) investigatin area. Regressin equatins were then derived between weights f amplitude functins and the latitude, lngitude and altitude, see Table 3. Here again we see in the clumn giving the multiple crrelatin cefficients, R 2, that precipitatin, temperature and TABLE 2 Crrelatin cefficients between weights f amplitude functins (PC) and latitude (LA), lngitude (LO) and altitude (AL) Precipitatin: LA AL LO Temperature: LA AL LO Evapratin: LA AL L0 PC 1 PC Sunshine hurs: LA AL LO Humidity: Wind speed: LA AL LO LA AL LO PC PC PC

7 Interplatin f hydrmeterlgical data, in Mali 41 TABLE 3 Regressin equatins between weights f amplitude functins (PC) and latitude (LA), lngitude (LO) and altitude (AL) Regressin equatins PRECIPITATION PC 1 wl = LA LO AL PC 2 w2 = LA LO AL TEMPERATURE PC 1 wl = LA LO AL PC 2 w2 = LA LO AL EVAPORATION PC 1 wl = LA LO AL PC 2 w2 = LA LO AL SUNSHINE HOURS PC 1 wl = LA LO AL PC 2 w2 = LA LO AL PC 3 w3 = LA LO AL HUMIDITY PC 1 wl = LA LO AL PC 2 w2 = LA LO AL WIND SPEED PC 1 wl = LA LO AL PC 2 w2 = LA LO AL PC 3 w3 = LA LO AL evapratin are the parameters shwing a definable spatial variatin, It shuld be nted, fr precipitatin, that R 2 =0.30 is related t nly 1% f the ttal variatin (98% lay in the first cmpnent) which can be included in the errr term. Fr evapratin a regressin equatin was established between weights f the first amplitude functin n the ne hand and latitude, lngitude, altitude, precipitatin and temperature n the ther hand. The crrespnding R 2 was then abut 0.90 cmpared with 0.74 given in Table 3. Precipitatin and temperature values in the equatin were the bserved histrical values. Validatin calculatins shwed in additin that the increase in R 2 is significant and nt nly related t the intrductin f new variables in the equatin. Thus it becmes pssible t intrduce in the interplatin mdel a prcedure fr calculating weight-cefficients fr precipitatin, temperature and evapratin with R 2 > The prcedure, with niy precipitatin and temperature as additinal independent variables t latitude, lngitude and altitude, yielded n significant increase f R 2 -values fr the ther parameters. By means f the btained regressin equatins it became pssible fr any given lcatin with knwn lngitude, latitude and altitude, t derive the weight-cefficients fr the main amplitude functins and cnsequently the variatin series fr the lcatin. A mdel was calibrated fr that purpse fr the investigatin area. Fr a particular statin, weight-cefficients f the main amplitude functins prvide us with infrmatin abut the regime f

8 42 Abdulaye Diawara the statin, that is, its variatin pattern cmpared with the main r dminant variatins f the statin grup. Cnsequently the scatter diagram f weights f the first tw amplitude functins can be efficient in terms f separating statin grups which shw the same affinity. That gruping will als crrespnd t a reginal separatin if the weighting factrs are significantly crrelated t the physigraphy. An investigatin f scatter diagrams fr classificatin purpse have been carried ut fr all the parameters in Diawara (1982). The znal divisin is clear r at least sufficiently apparent fr precipitatin, temperature and evapratin in agreement with Table 2. SOME RESULTS Sme results frm calibratin calculatins fr precipitatin, temperature and evapratin will be shwn belw. Calibratinvalidatin prcedure is mre cnveniently discussed against the backgrund f the znal divisin given in Fig.l. The derived main hydrmeterlgical znes and the parameters have been analysed in rder t bserve particularities in their nature. Precipitatin, temperature and t a lwer extent evapratin shwed significant stability in their variatin pattern with minr differences due t lcal factrs. Evapratin shwed, in additin t a mre prnunced effect f lcal factrs, tw main variatin patterns fr lwer and heigher latitudes respectively. Zne I and zne IV shwed mre hmgeneity in ppsitin t zne II and III. Zne I influences bth amplitude functins while zne IV shws mre affinity fr the secnd amplitude functin. The effects f lcal factrs are als mre apparent n statins in zne I and zne IV. Validatin calculatins have been carried ut fr precipitatin, temperature and evapratin. All the statins were included in the develpment f riginal series in empirical rthgnal functins. Ten independent statins, scattered ver the investigatin area as shwn n Fig.l, were selected as validatin statins and excluded (ne at a time) frm the calculatins fr mdel calibratin. Their histrical series were then cmpared with the estimated values frm the mdel. Figures 3-5 shw the mdel utput in relatin t bserved values at the 10 statins. Fr precipitatin the mdel shws a tendency t underestimate r verestimate fr validatin statins in zne I r zne IV respectively. There is an apparent agreement between mdel estimates and bserved values fr statins in zne II and zne III as we can see n the figures where the E-values represent the relative errr f estimatin summed ver the 12 mnths. The values are gd measures f estimatin errrs fr mnths with large mean values while errr n mnths with relatively very lw mean values may be much greater. This is particularly valid fr precipitatin and als evapratin. Fr temperature we find largely the same mdel behaviur. Hwever, in this case, the differences in mean temperature are relatively small. Mdel estimates in zne II and zne III nce again shw significant agreement with bserved values at validatin statins.

9 Interplatin f hydrmeterlgical data in Mali 43 E H CD c! < ; a (ï m *> O " * E t 0) a; i ÏH- H

10 44 Abdulaye Diawara S s. y K, \ <^ MOPTI (Ml) c c. 03 ' J NIORO >J3 V\ TOMBOUCTOU "^ O O 0 H I g -u g O a, S. <JJ -C in OO *: < CD M3 3 t 0) ft; 0 O g ~" H

11 Interplatin f hydrmeterlgical data in Mali 45 mat ' w -*- c c 0 M 4J V M 0 OH 0} ï> 01 in 0 ' H 3 t 0) ft; - l CD H

12 46 Abdulaye Diawara Validatin calculatins fr evapratin gave, ne again, better estimates fr znes II and III. The estimatin errrs fr zne I are large while thse fr zne IV may be cnsiderable. In general the mdel gave reasnable estimates f the bserved values. This is particularly true fr precipitatin and temperature. The systematic underestimatin r verestimatin f bservatin values fr statins lcated at lwer r higher latitude respectively may be related t the basic assumptin f linearity in the relatinship between the variables and latitude. This assumptin may be less valid at the large scale under investigatin. Thus this suggests the develpment f sub-mdels fr every zne keeping the same mdel structure. Hwever, the need fr sub-mdels wuld depend largely n the purpse f interplatin. It is then mre cnvenient t examine the relative errrs in respective znes, with mre extensive basic data than was used in this study, and subsequently determine the necessity f develping zne-mdels fr varius purpses. CONCLUSION The mdel estimatin is best fr statins with bservatin values clse t the mean values fr the investigatin area. It is als clear that ther physigraphical parameters in additin t latitude, lngitude and altitude may be develped t be included in the equatins. An advantage f the EOF-interplatin methd is that the amplitude functins which are independent f the statins can be calculated frm series frm all the statins and used in validatin calculatin fr statins excluded frm regressin equatins. Mrever, traditinal interplatin methds based n calculatins f crrelatins between neighburing statins may face great difficulties in the prevailing cnditins when single bservatin series may be f very pr quality fr accurate interplatin. The EOF prcedure implies a smthing f all infrmatin btained frm the studied area as they are derived frm the cvariance structure f all the existing statins in the area. An additinal advantage f the EOF methd is that it is a practical prcedure f deriving series fr any lcatin with little calculatin effrt. It can be a practical tl fr a number f peratins such as autmatic mapping f islines, filling r extending series with missing data r the planning f measurement statin netwrk. The present study used 80 precipitatin statins but nly 17 synptic statins fr the remaining parameters. The results, hwever, appear satisfactry and prmising. With mre extensive and better quality data the EOF prcedure may be expected t be an efficient tl fr analyses fr bth classificatin and interplatin purpses in the studied area. ACKNOWLEDGEMENT The authr is grateful t Prfessr Lars Gttschalk fr his suggestins and critical cmments and Prfessr Klas Cederwall fr his valuable help. The financial supprt given by SAREC-SIDA which made the wrk pssible is als gratefully

13 Interplatin f hydrmeterlgical data in Mali 47 acknwledged. (SAREC: Swedish agency fr research cperatin with develping cuntries.) REFERENCES Davis, J.C. (1977) Statistics and Data Analysis in Gelgy. New Yrk. Diawara, A. (1982) Hydrlgical reginalizatin in Mali. T be published. Gttschalk, L. (1985) Hydrlgical reginalizatin f Sweden. Hydrl. Sci. J. 30 (1) (in press). Hlmstrm, I. (1969) Extraplatin f meterlgical data, SMH1, Ntiser ch preliminara rapprter, Série Meterlgi nr 28. Hlmstrm, I. (1977) On empirical rthgnal functins and variatinal methds. In: Prc. Wrkshp n the Use f Empirical Orthgnal Functins in Meterlgy (Reading, UK).