Hydrological Sciences-Journal-des Sciences Hydrologiques, 45(6) December 2000
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Mapping average annual runoff: a hierarchical approach applying a stochastic interpolation scheme ERIC SAUQUET Hydrology-Hydraulics Research Unit, Cemagref, 3 bis Quai Chauveau CP220, F'-693'36 Lyon Cedex 09, France e-mail:
[email protected]
LARS GOTTSCHALK Department of Geophysics, University of Oslo, PO Box 1032 Blindern, N-0315 Oslo, Norway e-mail:
[email protected]
ETIENNE LEBLOIS* Hydrology-Hydraulics Research Unit, Cemagref, 3 bis Quai Chauveau CP220, F-69336 Lyon Cedex 09, France e-mail:
[email protected] Abstract A novel approach for mapping river runoff is presented. It is based on a disaggregation of the mean annual streamflow measured at the outlet of a basin to estimate water depths on elements of an exact partition of this basin. The developed technique is based on geostatistical interpolation procedures to which a global constraint of water balance has been added. The methodology is illustrated by a case study from a tributary to the Rhône River, France. The results were compared to an established method—the nested approach, and a cross-validation was performed for each mapping technique. The disaggregation approach appears to give the most consistent results. Finally, two gridded maps were derived by applying the disaggregation twice to assess water depth on an increasingly finer grid mesh. The global constraint of water balance was applied to each element of the coarser mesh to give estimates for the finer one.
Cartographie des écoulements interannuels: une approche désagrégative stochastique Résumé Cet article présente une nouvelle méthode de cartographie des écoulements annuels mesurés en rivière. Il s'agit de redistribuer le volume annuel moyen écoulé à l'exutoire d'un bassin sur les éléments d'une partition arbitraire du bassin considéré. La méthode reprend les techniques classiques de géostatistique et ajoute une contrainte supplémentaire de respect du bilan à l'exutoire du bassin principal. Elle a donné lieu à une application sur un affluent du Rhône. Nous avons pu comparer les résultats obtenus à ceux produits par une méthode déjà éprouvée. Au terme d'une validation croisée effectuée sur chaque technique, la méthode de désagrégation présentée ici semble fournir de meilleurs résultats. Deux cartes sont établies pour illustrer la possibilité d'une application itérative de la méthode. Ceci permet d'estimer des lames d'eau sur des découpages du bassin de plus en plus fins, chaque application fournissant des lames d'eau désagrégées pour en obtenir la carte suivante.
INTRODUCTION There are three main issues to be considered when choosing methods for the construction of runoff maps (Gottschalk & Krasovskaia, 1998): the method to be used " Corresponding author. Open for discussion until 1 June 2001
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for interpolation, the scale of fundamental units on the map, and the available observations that can be used to resolve the variability at different spatial scales. The method for interpolation can be either manual contouring (subjective methods in the meteorological terminology), or automatic interpolation (objective methods). The automatic interpolation can, in its turn, be divided into deterministic and stochastic approaches. In both cases a formula representing a weighted average is applied. Weighted averages include a wide class of methods from a simple averaging of point observations to stochastic interpolation with local support considering the extension of drainage basins. In this latter case it is assumed that the total area to be mapped is divided into fundamental units by means of subdividing a larger drainage basin into sub-basins or into a regular grid network. The second topic that needs attention is the scale. On a macro scale, drainage basins used for interpolation can become small in comparison with the total area to be mapped. They can therefore be approximated by points in space. This usually also implies that the simplified "vertical perspective" on runoff (or rather rainfall excess) is accepted and runoff is mapped in the same way as, for instance, precipitation. The simplest method is to use an average of the runoff from all the small basins which fall within a grid cell. A fundamental condition is, of course, that all cells contain observation points. Arnell (1995), for example, has applied this method across the FRIEND region. To overcome the problem of having empty cells and to allow a more sophisticated consideration of the difference in geographical location, a deterministic interpolation method can be utilized. Bishop & Church (1992) and later Arnell (1995) have applied the TIN method with this purpose (i.e. linear interpolation within the facets of the Triangulated Irregular Network defined by the gauging station considered as nodes). Conventional stochastic interpolation is also appropriate at the macro scale. Such methods are standard methods for interpolation of stochastic fields in meteorology and climatology (objective methods, Gandin interpolation) (Gandin, 1963; Daley, 1961). They are parallels to kriging, widely applied to interpolation problems in hydrogeology (Matheron, 1965; Delhomme, 1978). Kriging and Gandin interpolation are also of wide use for interpolation and integration of precipitation fields (Lenton & RodriguezIturbe, 1977; Creutin & Obled, 1982; Tabios & Salas, 1985; Dingman et al, 1988; Barancourt et al, 1992). There are also examples of the application of such methods to simplified assumptions for interpolation of runoff as a point process (Villeneuve et al, 1979, Hisdal & Tveito, 1993). If this approach is used properly, only data from small drainage basins can be applied so that a "point" covariance model can be constructed. On meso and micro scales (say, grid cell sizes in the range 10 km x 10 km to 1 km x 1 km and less than 1 km x 1 km, respectively), the area of drainage basins needs to be taken into account in the interpolation procedure, which has several advantages compared to the "point" interpolation. When basins are considered as "points" in a continuous space, the lateral aspects of the runoff process are neglected. Therefore, one cannot expect that runoff in this case, when integrated over a river basin, coincides with measured streamflow in the main rivers. These observations in the main rivers are, as a rule, avoided and not included in the data set. The information that they can add to the variation pattern of runoff is thus lost. A further practical aspect is that small basins are often situated in the headwaters to a river system leading to an overestimation of the total runoff (Arnell, 1995).
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In a collaborative paper on grid estimation of runoff (HASA, 1990), the catchmentbased area-weighted average method (the "nested approach", NA) is proposed for the calculation of runoff for fundamental units (grid cells) as weighted averages with a consideration of the drainage basin areas. Two cases are distinguished. In the first case, at least one basin with observations is within the grid cell. The runoff estimate is calculated from all measured runoff values in the grid cell as a weighted average (aggregation). In the second case, one drainage basin with observed runoff covers more than one grid cell. A disaggregation of the observed runoff for the basin has to be made. The runoff for each grid cell fully within the basin is found by interpolation of runoff (disaggregation) utilizing the relationship between the area of the whole basin and that of a grid cell and eventually other factors. Predeek & Isele (1992) calculated runoff for the Aller River (tributary of the Weser River) by applying this method. The drainage basin area was subdivided into grid cells of 0.5° x 0.5°. The method has also been applied to the FRIEND database (Arnell, 1995). It is a general opinion that, in most cases, the catchment-based area-weighted average method gives the closest estimate to observed runoff in comparison with the methods referred to above. Gottschalk (1993a,b) has developed an alternative stochastic approach for the interpolation of runoff. It takes full account of the fact that runoff is to be integrated to streamflow, thus considering the hierarchical structure of the basin drainage system. To achieve this, distance is measured along the river network and the covariogram for points must be replaced by a covariogram for the drainage basins, i.e. a covariogram model for the whole river system needs to be developed. The third topic to be considered is the type of observations at hand to resolve the variability across space at different scales. The estimated spatial variability from a regional set of observations can be expected to depend on the size of the basins involved—the higher the variability, the smaller the basins. This fact has two implications: the first is that the size of fundamental units of a map and the size of basins used for interpolation must be of a comparable scale and the second is related to the number of fundamental units on the map with respect to the number of basins available as a background for the interpolation procedure. If exactly the same set of runoff observations is used for interpolation to different fundamental units, the basic difference is in the estimation error. It will be larger, the smaller the fundamental units are. The eventual larger detail that a map, based on small fundamental units, reveals is counterbalanced by a larger estimation error. A hierarchical approach for interpolation is elaborated herein, with a consideration of the specific topics discussed above. The point of departure is the stochastic interpolation procedure developed by Gottschalk (1993a,b). The territory (major drainage basin) to be mapped is divided into sub-basins in a hierarchy of scales. The number of levels in this hierarchy is determined mainly by the amount of available observations, which also indicates the level of detail that can be achieved (size and number of fundamental units of the map). The first level in a larger drainage basin is usually already well defined by existing observation stations in the main rivers constituting the first level of sub-basins. These basins are in their turn divided into a second level of sub-basins (or grid cells), and observation stations with appropriate basin scales are chosen as the background for the interpolation. The interpolation procedure guarantees that the water balance equation is satisfied so that the sum of runoff from this second level of basins is equal to that of the first order basin
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accommodating them. The procedure can be repeated to a third level and so on. At each step new information must be added. Auxiliary runoff values to supplement or replace observed runoff values can be calculated for points in space in a regular or irregular pattern by means of empirical relationships and water balance models. In this way information from precipitation stations and on topography and other physiographic conditions can be included to resolve small-scale variability not covered by regular runoff observation networks. The stress in the presentation is put on the methodological aspects developed in the following section of the paper. The suggested methodology is illustrated by a case study from a basin in the upper part of the Saône River (a tributary to the Rhône River, France). This sub-basin constitutes the second order level in the Rhône basin. Two examples are elaborated. The first one considers sub-basins as the fundamental units, which allows a cross-validation against observed data and also a comparison with the "nested approach" referred to above. In the second example the fundamental units are grid cells. Here the interpolation is brought one level further down in the hierarchy by subdividing the cells into smaller ones keeping the water balance constraint and adding information from smaller drainage basins. THE STOCHASTIC INTERPOLATION SYSTEM Runoff with the dimension of flow per unit area is, as a rule, considered as a point process q(u) continuous in space. Intuitively the point runoff process is interpreted as a contribution from an arbitrary point in the basin with area A to the observed streamflow Q(A) from the basin. In its simplest way it is determined by dividing the observed streamflow by the corresponding drainage area. This creates a spatial step function constant over drainage basins. Another possible way of defining runoff would be to subtract the estimated actual point évapotranspiration from the estimated point precipitation, thus creating a continuous spatial process of net precipitation q(u). The integrated value of this spatial function over a basin area A should, in principle, coincide with the observed streamflow Q(A) for a given time period. A formula (weighted average) for interpolation of g(uo) as a point process at location uo from regional observations of this variable g(u,) (i = 1,..., N) is: c1(u0)=fjXiq{ul)
= ^Q
(la)
where À, (i = 1, ..., N) are weights. When the drainage basin area is taken into consideration the formula is changed to:
4)=iMA)=A 7 e
(ib)
where a0 is the area of a fundamental unit of the map and A,, and i = I, ..., N are the areas of drainage basins with observations. For both formulas Q is the column vector of observations and A7 is the transposed column vector of weights À, (/= 1,..., AO associated with the TV observations. In the case of stochastic interpolation, optimal weights in equation (la) are found by minimizing the estimation variance. Adopting an assumption of local second-order
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stationarity of the process and under the condition of unbiasedness this leads to the following linear equation system for the calculation of weights: CA = C0 where
(
Cov(u„u 0 )
Var(u,)
Cov(u 2 ,u 0 )
Cov(u 2 ,u,)
c0 =
CovCupU,) • • Var(u 2 )
Cov(u,,u„)
• • Cov(u 2 ,u w )
r
~K K
i
C=
A=
Cov(u N ,u 0 )
Cov(u, v ,u,)
1
1
Cov(u w ,u 2 ) • 1
I
Var(u w )
i
K
1
0
.^
and ji is a Lagrange multiplier. The unbiasedness is warranted with unit sum weights, i.e.:
J>>>=1
(;=1,...,M)
(3)
The optimal weights, i.e. the formal solution to equation (2) are calculated from: A = C'Cn
(4)
The elements in the matrix C represent the values of the fitted covariance function between each pair of data values located inside the network, while the elements of the column vector Co are the values of the fitted covariance function between the location of interest and each of the stations. Referring now to the case when the drainage basins are taken into account and equation (lb) is applicable, the point of departure is a drainage basin AT where the mean annual discharge Qj at the outlet point is known from measurements or estimation. In the following this value will be treated as a known constant. The area AT is approximated by a regular grid of m fundamental square cells of area a, so that AT = nTa. As only QT is available, the way in which the mean flow is distributed within the drainage basin is unknown and some assumptions are needed. Following the Laplace principle of "insufficient reason" (i.e. "if you have no justified reason, make it simple"), it is assumed that the distribution across each fundamental unit is uniform. 1
7
1
Therefore, discharge data are converted into 1 s~ km" or mm year" using the decomposition of each basin in grid cells: qT =
Q^=QT
AT
nTc
(5)
The total area AT can be subdivided into M non-overlapping areas AA, (i = 1, ..., M). The aim is to estimate the specific discharge 10000 km 2
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Fig. 2 Drainage network deduced from DEM and locations of the discharge stations.
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— ùr
Empincai covariogram
—m— Cov(d) — • - P o i n t process covanance function
•_.=-T=*^^
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Fig. 3 Empirical covariogram from catchment data and aggregated fitted covariogram.
Fitting of a covariance model The first step is the calculation of distances between each pair of independent drainage basins following the hierarchy of the drainage network. With these distances as a background, an experimental covariogram can be drawn. This step is followed by a selection of possible theoretical models for the point process covariance function Covp. Aggregated values were assigned to each couple of independent drainage basins applying equation (13) and compiled for distance classes to be able to draw a theoretical aggregated covariogram. The choice of the best point process model is based on a graphical comparison between Cov and the experimental covariogram. For the present data, a theoretical spherical covariance function with the following expression: Cov(A.,A,) = K
•
ff
>'
A
A
Cov
\duAduA
(14)
11A,,A,
with Cov p (rf) = 0
d>T
C
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Fig. 4 Average annual runoff estimated by the disaggregation procedure.
possibility for cross-validation and a comparison with another confirmed method (the "nested approach")- The target partition includes the headwater watersheds of the observation network. The runoff balance constraint is set so that the total sum of runoff for all of the fundamental units is equal to the discharge measured at the Lechâtelet gauging station. Estimated runoff ranges from 235 to 1424 mm and is consistent with the expected runoff pattern. The largest runoff values are found in the northeastern part of the basin, i.e. on the west side of the Vosges Mountains, whereas the lowest ones are located in the southwestern part of the basin, i.e. in a chalky sector with temporary flows or in the alluvial plain of the Saône River (Fig. 4). Validation The validation of the interpolated map is done in two steps: a comparison with the "nested approach" and a cross validation. Comparison with the "nested approach" The selected technique for comparison is the "nested approach" (IIASA, 1990) described in the introduction. This target partition into sub-basins is imposed by the drainage network. The elements of the partition are portions of basin between two or more gauging stations or uppermost headwaters of a basin where no hydrological observations are available upstream. Runoff generated by each area is derived by subtracting the mean annual discharge(s) measured upstream from the mean annual value observed downstream and expressed as depth over the fundamental unit, making reference estimations available for comparison with the procedure described herein.
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The interpolation system gives exactly the same values when the observation network is composed of N stations within the main catchment and when the target partition is composed of nested sub-basins defined by these N stations. Consider the sub-basin j , the downstream part of the basin Aj defined by the station j and the nearest upstream station k (AA, = Aj- Ak). The weights if,• (z = 1, ..., AO for this sub-basin y are equal to: T kI / = — ^ - w i t h A A , = A , - A AA ; ' '
k = l...,N
^=Tr
(15)
AAj
r\j = 0.0
i = l,...,M
4* • Thus, the condition > tif =
i*k,
i± j
A, A: — +—— = 1, / = 1,...,/V is fulfilled.
The nested approach respects the water balance at the outlet of each catchment of the observation network; therefore the global constraint with the weights r^t (i= 1, ..., N) expressed in equation (10) is fulfilled too. Furthermore, the covariance between elements has the following mathematical property: 1 Cov(4,^.)=——- jjCov p (|[« 4 Mj
1
A,{M,+Ak) A
-u^jdujdu
A„A,
\jCovp^uAj-uMiljduAduMi
+ JjCov p\\u 4< - it At\\)du
Adu
A-M
- [Mj • C o v ( 4 , AA. )+Ak- Cov(A,, Ak )]
and finally: A.. COV(A,,AAJ)=^-COV(A,,AJ)-A^COV(AI,AJ=JT1/COV(A,,A,)
(16)
The kriging system (equation (11)) admits a single solution and for this particular case; the weights given by the resolution of the matrix system are exactly those given by the nested approach (with the Lagrange multiplier ixT equal to 0), whatever the spatial structure is supposed to be. Cross-validation The analysis consists of excluding one gauging station in turn from the network and then estimating runoff at this site by applying the interpolation procedure to the remaining stations with the covariance model Covp fitted to the full data set. The cross-validation analysis gives valuable insight in the real influence of the global constraint on runoff assessment. For each removed station, the observed runoff is compared to the estimated runoff produced by the watershed upstream this point.
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This last calculation implies withdrawing the sub-basins upstream from the excluded station in the target partition. The performance of the method is studied using regression analysis between the withheld, observed and the calculated runoff (Fig. 5). The correlation coefficient is equal to 0.95. The results are quite sound, but the regression line indicates an underestimation for high values (i.e. the headwater catchments). The algorithm gives
200 200
400 600 800 1000 1200 1400 Observed mean annual runoff (mm)
1600
Fig. 5 Cross-validation of interpolated estimates of mean annual runoff with observations (the regression line (dashed) and the one-to-one line (black) are both represented).
1600
200 200
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1000
1200
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Observed mean annual runoff (mm) Fig. 6 Comparison of estimates of mean annual runoff according to the nested approach with observations in cross-validation (the regression line (dashed) and the one-to-one line (black) are both represented).
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poor results when the observation removed from the data set is one of the extreme values (highest or lowest). The highest absolute deviation is for the Ognon basin (-21%) followed by that for the Semouse basin (-20%). The mean error of prediction, which measures the average difference between the true runoff value and its estimates, is equal to -23 mm. A similar cross-validation analysis was performed simultaneously to the nested approach (Fig. 6). The correlation coefficient is lower (0.80), and the regression line deviates from the one-to-one line. The mean error of prediction (98 mm) is far higher. A comparison between the two analyses shows that the kriging system gives better estimates, especially for a headwater catchment. Indeed, no spatial information is introduced in the nested approach and a water balance constraint is not sufficient to estimate correctly water depths. A runoff map for grid cells The developed approach allows assessing runoff yielded by arbitrary areas (either administrative or geometric). In this second example, the interpolation procedure is applied to derive grid estimates of mean annual runoff in two steps in accordance with the following scheme: 1. Define the large-scale area where runoff is known and for which the budget constraint should be satisfied. 2. Divide this area into fundamental units (here grid cells). 3. Select gauging stations to define an observation network consistent with the scale of the fundamental units of the target partition. 4. Construct a covariance function consistent with the spatial structure observed on the data set. 5. Interpolate runoff to each element so that the runoff balance constraint is satisfied for the large-scale level identified in step 1. 6. The estimated values for fundamental units are now the new starting points at step 1. Steps 1-5 are repeated as long as new information on a smaller scale is available to define data sets for the finer resolution. To illustrate the principle of the disaggregation procedure, the interpolation scheme is applied to a target partition defined by the superimposition of a regular 32 x 32 km grid (1024 km2) over the main catchment boundaries. The runoff budget constraint on the main basin is achieved (Fig. 7(a)). The complete data set of 20 gauging stations and the spherical covariance function previously fitted for the point process above are kept. The interpolation procedure to assess runoff on 16 x 16 km cells (256 km2) (Fig. 7(b)) is applied to the full data set and the interpolation constraint is kept within each 32 x 32 km cell so that the sum of runoff from the smaller cells equals the runoff from this larger one. The sum of runoff from the larger cells in its turn equals the runoff from the total drainage basin at Lechâtelet. The expected pattern of runoff structure is reproduced on the two maps. The highest values are located in the northeastern part of the basin for the two gridded maps whereas the lowest ones concern the southeastern sector. The disaggregation procedure could in principle be continued in a third step to still smaller grid cells using data resolving small scale spatial variability (e.g. discharge records measured on experimental small watersheds, net rainfall or rainfall excess).
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(a) 2350
2300
2250
\
? 200
_ 2200
450
I
(Lambert coordinates)
750
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700
, ,
850
i
>1200mm :
. , , i , , .
900
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(b) 2350
&*, ^ 2300
:
i - .
I I
, n ^
i-
J
r
r
X ,/~ -~ f" 1 •-,
2250
n „ j •'"
I
i
']
j '
,^y
! / --
/s
J
^
^j
J
200
2200
ir
,nr
f^-
450
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>1200mm
; (Lambert coordinates)
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Fig. 7 Gridded maps of average annual runoff with (a) 32 x 32 km resolution and (b) 16 x 16 km resolution, with respect to water balance on 32 x 32 km grid cell estimation based on disaggregation procedure.
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CONCLUSIONS Streamflow (annual mean values) is an integrated process that follows the hierarchical structure of the river system. It must be so that the streamflow in a larger basin is the sum of streamflow from its sub-division into smaller parts. A method is suggested herein for creating maps of river runoff, based on annual discharge data and an objective interpolation scheme, that take into consideration these basic features of the streamflow process. The method is based on a hierarchical disaggregation principle and can assess runoff for elements of an arbitrary partition of a gauged drainage basin like sub-basins and grid cells. The method has been applied to the upper part of the Saône River, a French tributary to the Rhône River. The results of this application have been compared to another mapping technique, the nested approach. Both methods are able to fulfil the water balance constraint and their performances were tested through cross-validation. The disaggregation procedure developed here seems to give more reliable runoff estimates than the nested approach. The most noticeable difference between the two mapping techniques is that no information about the spatial structure is introduced in the nested approach whereas the covariance function in the disaggregation procedure fully respects the spatial runoff pattern and therefore allows better estimates in cases of extrapolation. The hierarchical principle allows the calculation of gridded maps for finer and finer resolution satisfying the water balance. A first map was derived from the disaggregation of the mean annual discharge generated by the main basin and a second one was the result of the disaggregation of the runoff estimates yielded by each element of the first map. The accuracy of the method observed on sub-basins partition is only expected on any partition that does not strictly respect the catchment delineation. Indeed, no objective validation can be proposed because of the lack of reliable measurement and the maps are analysed on visual agreement with observed runoff patterns. The presented methodology, as well as other methods based on water balance, is not applicable in watersheds where actual drainage area differs significantly, usually for geological reasons, from that which can be deduced from the topographic analysis. Modelling of the dynamic features of streamflow would be required theoretically to map runoff at a shorter time scale than mean annual values. The developed technique cannot be applied to interpolate extreme annual values either: due to the non-systematic concomitance of the flood events upstream of the main confluences, flood characteristics do not have the property of summing up along the river net. Acknowledgements This research is a part of the French project of hydrological modelling GEWEX-Rhône and is supported by grants from the Leonardo da Vinci programme of the European Community through the international association TECFIWARE. The authors wish to express their thanks to the Steering Committee of the French discharge database HYDRO for supplying the discharge data used in this study.
REFERENCES Arnell, N. W. (1995) Grid mapping of river discharge. J. Hydrol. 167, 39-56. Barancourt, C , Creutin, J. D. & Rivoirard, J. (1992) A method for delineating and estimating rainfall fields. Wat. Resoiir. Res. 28, 1133-1144.
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Bishop, G. D. & Church. M. R. (1992) Automated approaches for regional runoff mapping in the northeastern United States. J. Hydroi. 138, 361-383. Burrough. P. A. (1986) Principles of Geographical Information Systems for Land Resource Assessment. Clarendon Press. Oxford, UK. Creutin, J. D. & Obled, C. (1982) Objective analysis and mapping techniques for rainfall fields an objective comparison. Wat. Résout: Res. 18, 413-431. Daley, R. (1961) Atmospheric Data Analysis. Cambridge University Press. Delhomme, J. P. (1978) Kriging in the hydrosciences. Adv. Wat. Resour. 1, 251-266. Dingman, S. L., Seely-Reynolds, D. M. & Reynolds, R. C. (1988) Application of kriging to estimating mean annual precipitation in a region of orographic influence. Wat. Resour. Bull. 24, 329-339. Gandin, L. S. (1963) Objective Analysis of Meteorological Fields. Gidrometeoizdat, Leningrad (in Russian). Gottschalk, L. (1993a) Correlation and covariance of runoff. Stochasl. Hydroi. Hydraul. 7, 85-101. Gottschalk, L. (1993b) Interpolation of runoff applying objective methods. Stochasl. Hydroi. Hydraul. 7, 269-281. Gottschalk. L. & Krasovskaia, 1. (1998) Development of Grid-related Estimates of Hydrological Variables. Report of the WCP-Water Project B.3, WCP/WCA, Geneva, Switzerland. Hisdal, H. & Tveito, O. E. (1993) Generation of runoff series at ungauged locations using empirical orthogonal functions in combination with kriging. Stochasl Hydroi. Hydraul. 6, 255-269. 1IASA (1990) Planning meeting on grid estimation of runoff data. Annex G CP-90-09, Laxenburg, Austria. Lenton, R. L. & Rodriguez-Iturbe, I. (1977) Rainfall network system analysis: the optimal equation of total areal storm depth. Wat. Resour. Res. 13, 825-836. Matheron, G. (1965) Les variables régionalisées et leur estimation. Une application de la théorie des fonctions aléatoires aux sciences de la nature. Masson, Paris, France. Predeek, A. & Isele, K. (1992) A study on the transformation of point measured runoff data into grid based data. Document presented at the second planning meeting on "Grid estimation of runoff data", WCP-Water Project B.3., (Warsaw, Poland, 6-9 May). Tabios, G. Q. & Salas, J. D. (1985) A comparative analysis of techniques for spatial interpolation of precipitation. Wat. Resour. Bull. 21,365-380. Villeneuve, J. P., Morin, G., Bobée, B., Leblanc, D. & Delhomme, J. P. (1979) Kriging in the design of streamflow sampling networks. Wat. Resour. Res. 15, 1833-1840. Received 16 April 1999; accepted 26 June 2000
