Assessing the uncertainty associated with intermittent rainfall fields

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2004wr003740, 2006 Assessing the uncertainty associated with intermittent rainfall fields Eulogio Pardo-Igúzquiza Department of Geodynamics, University of Granada, Granada, Spain David I. F. Grimes and Chee-Kiat Teo Department of Meteorology, University of Reading, Reading, UK Received 15 October 2004; revised 19 September 2005; accepted 17 October 2005; published 24 January [1] In many practical situations where spatial rainfall estimates are needed, rainfall occurs as a spatially intermittent phenomenon. An efficient geostatistical method for rainfall estimation in the case of intermittency has previously been published and comprises the estimation of two independent components: a binary random function for modeling the intermittency and a continuous random function that models the rainfall inside the rainy areas. The final rainfall estimates are obtained as the product of the estimates of these two random functions. However the published approach does not contain a method for estimation of uncertainties. The contribution of this paper is the presentation of the indicator maximum likelihood estimator from which the local conditional distribution of the rainfall value at any location may be derived using an ensemble approach. From the conditional distribution, representations of uncertainty such as the estimation variance and confidence intervals can be obtained. An approximation to the variance can be calculated more simply by assuming rainfall intensity is independent of location within the rainy area. The methodology has been validated using simulated and real rainfall data sets. The results of these case studies show good agreement between predicted uncertainties and measured errors obtained from the validation data. Citation: Pardo-Igúzquiza, E., D. I. F. Grimes, and C.-K. Teo (2006), Assessing the uncertainty associated with intermittent rainfall fields, Water Resour. Res., 42,, doi: /2004wr Introduction [2] Accurate monitoring of rainfall is important in the fields of meteorology, hydrology, agriculture and climatology. Depending on the application, point or areal average values may be required. Methods of monitoring rainfall include rain gauge observations, radar signals, satellite images and weather model output. Even when rain gauges are not used for direct measurement, they are crucial for calibration and validation of instruments and models. In many cases, interpolation to ungauged locations or areas is desirable. The geostatistical technique of kriging has great advantages as an optimal interpolation procedure not least because it takes account of the spatial correlation structure of the rainfall and also allows an uncertainty estimate to be calculated for the interpolated values [Delhomme, 1978]. A number of studies [e.g., Lebel et al., 1987] have shown it to be superior to other interpolation methods provided certain criteria can be met. These include the specification of a variogram or correlation function which accurately represents the spatial correlation of the rainfall, and the assumption of a Gaussian distribution of rainfall amounts. The second criterion is clearly not met in the case of spatially intermittent or patchy rain. This problem has been addressed by the use of disjunctive kriging [Matheron, Copyright 2006 by the American Geophysical Union /06/2004WR ; Yates et al., 1986]. However a simpler solution was put forward by Barancourt et al. [1992] (hereafter referred to as BCR92) which makes use of indicator kriging to define a binary random function identifying the most likely rainy and nonrainy areas. The final interpolated rainfall field including intermittency can then be expressed as the product of the binary function and the ordinary kriged rainfall field. A problem identified but not solved in BCR92 is the assessment of the uncertainty or error associated with the binary random function. Without this information an uncertainty level cannot be assigned to the final rainfall amounts. An alternative approach was suggested by Seo [1998] but without addressing the issue of uncertainty. The contribution of this paper is to complete the BCR92 methodology by providing a means of estimating the uncertainty on the intermittent rainfall with a sound statistical basis. First, for completeness a brief review of the BCR92 estimation method is given. [3] Adopting the notation of BCR92 for consistency, we consider the two dimensional space < 2 within which positions are identified by the vector u =(x, y). We define N measurement points (in our case rain gauge observations) at locations u i (i =1,..., N) and we aim to calculate an estimate of rainfall within a target area c. [4] The BCR92 procedure then comprises these basic steps. [5] 1. Define an indicator variable I(u i ) which has the value 1 at rainy gauge locations and zero at nonrainy 1of13

2 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS gauges. By kriging these data we obtain the optimal indicator values I k (u) foru 2 c < 2. [6] 2. Apply a threshold value I c to the indicator kriging estimates to define a binary indicator value: I* ðuþ ¼ 8 < 1 if I k ðuþ > I c : 0 otherwise Rainy locations correspond to I* (u) = 1, nonrainy locations correspond to I* (u) = 0. To ensure an unbiased estimator, the threshold I c is obtained using the condition: ð1þ E I k ðuþ ¼ E½I* ðuþš u 2 c ð2þ where E [] is the mathematical expectation operator. [7] 3. Calculate an initial rainfall field by kriging using only the rainy gauges. This is F k ðuþ ¼ XN l i Fðu i Þ u 2 c ð3þ i¼1 where l i are the weights obtained from the ordinary kriging equations [Goovaerts, 1997] [8] 4. Calculate the final rainfall estimate Z*(u) as the product of the binary indicator I* (u) and the initial rainfall field F k (u): Z* ðuþ ¼ I* ðuþf k ðuþ u 2 c ð4þ In this paper, the aim is to attach an uncertainty measure to the estimator Z*(u) given in equation (4). One possibility is to calculate its estimation variance: s 2 Z* (u) as is the usual practice in geostatistics. In the following sections we show how this can be done in a relatively straightforward manner provided I* and F k can be assumed independent. However, Z*(u) will typically be non-gaussian which makes the variance difficult to interpret physically. We therefore also describe a more robust method for assessing the uncertainty on Z*(u) by calculating the local probability distribution function (pdf) for rainfall amount at each estimation point. This allows the uncertainty to be represented in terms of appropriately selected confidence intervals. An estimate of variance can also be computed from the local pdf if desired. 2. Maximum Likelihood Indicator Kriging [9] An intermediate result that feeds into both methods of uncertainty evaluation mentioned above is the calculation of the probability of the binary variable I*(u) having the value 1. We call this probability p* (u) and note that the probability of I*(u) taking the value zero is 1 p* (u). A first approximation for p* (u) is simply the indicator estimate I k (u) because it can be interpreted as I k ðuþ ¼ P½ZðuÞ > 0Š ð5þ where P[] represents probability [Journel, 1983]. [10] However this is not the best estimate because I k (u)is calculated from the experimental indicator data and as such has an associated uncertainty which needs to be considered. A more complete treatment requires knowledge of the probability density function (pdf) of I k (u). If we know this pdf, then we can determine p*(u) as the proportion of realizations for which I k (u) >I c. [11] The pdf of I k (u) can be easily calculated by a method similar to empirical maximum likelihood kriging for continuous variables [Pardo-Igúzquiza and Dowd, 2005a, 2005b]. The method adapted for indicator variables is developed in the following paragraphs. [12] Each indicator random variable (RV) I(u) may be seen as a Bernoulli RV which takes a value of 1 with probability E[I(u)] = m I (u) and 0 with probability 1 m I (u). For simplicity, we assume second-order stationarity, in which case m I is the same for all u. [13] Now N independent Bernoulli RVs follow a binomial distribution which is well approximated under very mild restrictions by a Gaussian distribution. The conditions for the approximation are Nm I (1 m I ) > 5 and 0.1 m I 0.9 or for any value of m I if Nm I (1 m I )>25[Evans et al., 1993]. However, in spatial statistics, the N Bernoulli RVs may be correlated and it is assumed that they may be approximated by a multivariate Gaussian distribution (MGD) such that if we consider a spatial pattern represented by the vector I(u) =[I(u 1 ), I(u 2 ),..., I(u N )] T, its probability is given by p½iu ð Þ; M I ; 2 I Š ¼ ð2pþ N=2 j2 I j 1 2 exp 1 ð6þ ½ 2 Iu ð Þ M IŠ T 2 1 ½Iu ð Þ M I Š where N is number of experimental data, I(u) isthen 1 vector of experimental indicator data. I(u) =[I(u 1 ), I(u 2 ),..., I(u N )] T, M I is the N 1 vector of indicator means (m Ii = m I,fori =1toN), and 2 I is the N N matrix of indicator variances and covariances. The MGD given by equation (6) is closely related to indicator kriging because at any given location u 0 the estimator obtained by maximizing the likelihood given by p[i(u); M I, 2 I ] in equation (6) is identical to the indicator kriging estimator. Given the indicator mean m I, the indicator variance-covariance matrix 2 I and the vector of experimental data I(u), equation (6) can be adapted to give the likelihood function L of the indicator I(u 0 ) for the new location u 0. Thus Nþ1 LI ½ ðu 0 Þ; m I ; 2 I ; Iu ð ÞŠ ¼ ð2pþ ð Þ=2 ~2 I 1 2 exp 1 2 ~ T Iu ð Þ ~M I ~2 1 ð7þ I ~IðuÞ ~M I Equation (7) is just the MGD from equation (6) with the N experimental data I(u) seen now as parameters (together with m I and S I ) and with the only unknown being the indicator RV I(u 0 ), and where ~I(u) isthe(n +1) 1 vector ~I(u) =[I(u 0 ), I(u 1 ), I(u 2 ),..., I(u N )] T, ~M I is the (N +1) 1 vector of indicator means m I.(~m Ii = m I,fori =1toN + 1), and ~2 I is the (N +1) (N + 1) matrix of indicator variances and covariances. (2 I is incremented by one row and one column to account for the variance of I(u 0 ) and the covariance between it and the experimental indicators I(u i )). In practical applications m I and 2 I are usually unknown and must be I 2of13

3 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS estimated from the experimental information I(u i ). Furthermore, 2 I (and ~2 I ) are not estimated directly but derived from a parametric model of spatial covariance C I (h), or variogram g I (h) where h is the separation vector between two locations. This procedure guarantees that the indicator covariance matrix will be positive definite and invertible. [14] The desired pdf for p* (u) can then be written as fðiðu 0 ÞÞ ¼ LIu ð ð 0Þ; m I ; C I ðhþ; Iu ð ÞÞ Z 1 0 LIu ð ð 0 Þ; m I ; C I ðhþ; Iu ð ÞÞdI where f(i(u 0 )) is the pdf of I(u 0 ) and L(I(u 0 ); m I, C I (h), I(u)) is the likelihood function. Note that in equation (8), dependence on the indicator covariance matrix has been changed to dependence on the indicator covariance function C I (h). The denominator in equation (8) is just a normalization factor to ensure that the pdf integrates to one. Equation (8) implies that, although the MGD was adopted in equation (6), the final pdf is, in fact, never Gaussian. [15] The method can be generalized further to take account of any other available information which may influence the uncertainty in I(u 0 ) by adopting a Bayesian approach. In the case of rainfall data, other available information might include topography, meteorological data or satellite imagery. The additional information may be used to form a prior probability density function (pdf) for I(u) so that equation (8) becomes fðiðu 0 ÞÞ ¼ Z 1 0 ð8þ f p ðiðu 0 ÞÞLIu ð ð 0 Þ; m I ; C I ðhþ; Iu ð ÞÞ ð9þ f p ðiðu 0 ÞÞLIu ð ð 0 Þ; m I ; C I ðhþ; Iu ð ÞÞ di where f p (I(u 0 )) is the prior pdf of I(u 0 ). [16] Finally, the desired probability value, p* (u) may be calculated as Z 1 p* ðuþ ¼ fðiðu 0 ÞÞdI ð10þ I c Having shown a method for estimating p*(u), its use in the assessment of the uncertainty of Z*(u) is described in sections 3 and Variance Calculation for Intermittent Rainfall [17] Equation (10) allows the calculation of the probability of rain at a given location. The calculation of the variance of the intermittent rainfall amount Z*(u) is then relatively straightforward provided that I* and F k can be taken to be independent variables. This assumption was also made in BCR92 and is justified here in order to complete their methodology. In physical terms, it is equivalent to assuming that the intensity of rainfall is unrelated to spatial location within the rainy area. While this may not be true for all rainfall spatial patterns, it is a reasonable approximation when the spatial scale of rainfall variability is much smaller than the spatial extent of the rainy area. With this assumption, we can write an expression for the variance of Z* from equation (4) using the normal formulation for the variance of the product of two independent variables [e.g., Kendall and Stuart, 1977] s 2 Z* ðuþ ¼ E2 ½I* ðuþšs 2 F ðu K ÞþE2 F k ðuþ s 2 I* ðuþ ð11þ E[I* (u)], E[F k (u)] and s 2 F K (u) are all derived directly from the ordinary kriging equations and it remains to determine s 2 I* (u). I*(u) is a binary variable and therefore its variance is given by s 2 I* ðuþ ¼ p* ð u Þ ½ 1 p* ð u ÞŠ ð12þ As we already know p*(u), we can determine s 2 I* (u). Then 2 (u) is calculated from equation (11) as required. s Z* 4. Uncertainty Evaluation for Intermittent Rainfall [18] Although the variance of Z*(u) can be calculated from equation (11), it is not necessarily easy to interpret in terms of a quantitative estimation of uncertainty on the rainfall amount. This is because Z*(u) typically has a mixed distribution with a finite probability mass at zero and a continuous part on the positive real axis. In this case the best way of describing the uncertainty of Z*(u) is by estimating its complete local distribution conditional on the experimental data. From the estimated local distribution measures of uncertainty such as variance or confidence intervals can be obtained directly. The easiest and most efficient way of estimating the local distribution is by using an ensemble approach to generate a large number of realizations from which the pdf can be built. The sequence of steps is described below. [19] 1. First, the rainfall observations {F(u i ); i =1,..., N} are transformed to a Gaussian distribution {G(u i ); i =1,..., N} by means of a standard one-to-one invertible mapping Gðu i Fðu i Þ ¼ jðfðu i ÞÞ ð13þ Þ ¼ j 1 ðgðu i ÞÞ ð14þ where j is the normal score transform and j 1 is its inverse. A description of the normal score transform can be found elsewhere [e.g., Goovaerts, 1997; Deutsch and Journel, 1992]. [20] 2. Ordinary kriging is then used to obtain values for G k (u) and s 2 G k(u) from G(u i ) using the covariance (or semivariogram) estimated from the normal score data. [21] 3. Then p* (u) is estimated as described in the previous section. [22] 4. For every target location u 0 we now have a triplet of values {p*(u 0 ), G k (u 0 ), s 2 G k(u 0 )}. A Monte Carlo approach is used to choose a rainy or nonrainy realization at u 0 by generating a random number R j (0 R j 1) and then assigning Z* j (u 0 )=0forR j > p*(u 0 ) and Z* j (u 0 )>0for R j p*(u 0 ). [23] 5. A realization of the normal transformed rainfall G k j (u 0 ) is computed for all Z* j (u 0 ) > 0 by generating a second random number from a normal distribution with mean G k (u 0 ) and variance s 2 G k(u 0 ). 3of13

4 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 1. Simulated intermittent rainfall field. Zero rainfall is shown white and accounts for 52% of the whole area. The spatial measurement unit is arbitrary, but the rainfall distribution is consistent with daily rainfall and a spatial scale in km. The legend shows rainfall amounts in mm. [24] 6. For rainy realizations only, Z* j (u 0 ) is calculated by applying the inverse transform of equation (14) to G j k (u 0 ). [25] 7. Steps 4 6 are repeated for j =1toM where M is a large number (say, 1000). [26] It may be noted that the use of kriging in combination with a normal score transform as described above is a standard technique used by many workers [e.g., Deutsch and Journel, 1992; Goovaerts, 1997] to obtain a representation of a non-gaussian distribution within the kriging paradigm. [27] The set of M realizations for values for Z*(u 0 )isa Monte Carlo evaluation of the local conditional distribution of Z*(u 0 ). In principal we could represent the uncertainty at every estimation point by giving its complete pdf. This would usually be impractical, therefore we seek summary statistics that best characterize the distribution. For example, we can calculate the estimation variance as while uncertainty implies a measure of the statistical uncertainty determined as outlined in the preceding sections. 5. Case Study Using a Simulated Random Rainfall Field [29] The technique of rainfall estimation within intermittent random fields can, in principle, be applied to any s 2 Z* ð u 0Þ ¼ 1 X M M i¼1 ðz i * ðu 0 Þ Z* ðu 0 ÞÞ 2 ð15þ where Z*(u 0 ) is the mean value of Z* i (u 0 ). Alternatively, confidence intervals including a specified proportion of realizations can be calculated directly from the local pdf. The exact specification of the confidence intervals is a matter of user choice. In the remainder of the paper we present two case studies illustrating one way in which such intervals may be defined. [28] To avoid confusion we here adopt the terminology that error refers to the difference between estimates calculated as in BCR92 and observed or simulated values, Figure 2. Location of 100 rain gauges randomly selected from the simulation grid showing rainy (circles) and nonrainy (squares) locations. 4of13

5 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 3. Map of final estimated rainfall Z*(u). The legend shows rainfall amounts in mm. rainfall data set. Maps of rainfall estimates Z*(u), confidence intervals, estimation variance s 2 Z* (u) etc. can be easily obtained by the methods described above. As a first example, we apply the procedure to a simulated rainfall field. The advantage of using simulated data is that the target is perfectly known so that the kriged estimates may be compared with true values. Rainfall fields with appropriate spatial structures and point statistics may be conveniently represented using a geostatistical random function [Bras and Rodríguez-Iturbe, 1985; Lebel et al., 1987]. Rainfall simulation is carried out by generating a single realization of such a function. For the example here, a spectral method was implemented [Mantoglou and Wilson, 1982; Dietrich and Newsam, 1993] using an approach described by Pardo-Igúzquiza and Chica-Olmo, [1994]. Rain gauge observations were obtained by selecting values at random from the simulated field. [30] There are different factors that can be varied in the simulation experiment such as percentage of nonrainy area, number of rain gauges, rain gauge location, spatial structure Figure 4. Plot of the posterior distribution f(i(u 0 )) for a spatial indicator pattern at a location within the simulated rainfall field. The indicator threshold is I c = 0.51, giving a probability value p*(u) = of13

6 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 5. Map of the probability p*(u) for simulation case study. of variability inside the rainy areas, rainfall accumulation time, size of the study area, etc. but to analyze the influence of those factors in detail is outside the scope of this paper. Therefore we have used, for illustrative purposes, a rainfall field (shown in Figure 1) that could be typical of daily convective rainfall totals. The field was constructed so that approximately 50% of the area was rainy. A spherical variogram was used with zero nugget and range equal to one sixth of the side of the area of observation. The simulated field comprised grid points which for convenience we assume to be spaced at 1 km intervals. The variogram range is then taken as 20 km. In Figure 1 intermittency is clearly visible with the nonrainy areas (in white) accounting for 52% of the total area. [31] From this field, measurements from a sparse network of 100 rain gauges were simulated by selecting locations at random from the grid. The positions of the selected rain gauges are shown in Figure 2. Figure 2 Figure 6. Local conditional distribution as estimated by Monte Carlo simulation with N = 3000 and for the same location as in Figure 4. Estimation variance and confidence intervals are calculated from this histogram. 6of13

7 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 7. Map of standard uncertainty s Z* (u) calculated via Monte Carlo simulation. may also be interpreted as the experimental indicator data with 1s in the rainy locations and 0s in the nonrainy locations. [32] The selected network was then used to derive variograms for both indicator and ordinary kriging. Again, using only data from the 100 selected gauges, the indicator kriging map I k (u) was computed. From equation (3) the hard threshold for delineating the rain/no rain boundary was found to be I c = This was then applied to the indicator field to give the map of rainy and nonrainy areas. Following BCR92 the rainfall field F k (u) was estimated from the values at the rainy gauges. The final rainfall estimate Z*(u) = I*(u) F k (u) is presented in Figure 3. Probabilities p*(u) were then calculated via the indicator maximum likelihood method giving the pdf of I k (u) for each location. Figure 4 shows this for a particular location, u 0. The probability of location u 0 being rainy (I k (u 0 )>I c ) is represented by the shaded area under the curve to the right of the threshold indicator value I c. The map of p*(u) is shown in Figure 5. Also as described in the previous section the local conditional distribution was calculated by Monte Carlo simulation for each experimental location (a particular case for the same location as Figure 4 is shown in Figure 6). From these local distributions the estimation variance s 2 Z* (u) associated with the final estimated rainfall map of Figure 3 was obtained via equation (15) and its square root (the standard uncertainty (SU)) is depicted in Figure 7. The SU was also calculated via equation (11). [33] The proportion of validation points lying within one and two SU of the true value is given in Table 1. It can be seen that the results are very similar. This is also shown by Figure 8 which compares both estimates of standard uncertainty for all data points. [34] The expected proportions for a Gaussian distribution lying within one and two SU are 68% and 95% respectively. Comparison with Table 1 demonstrates that the calculated variance overestimates the uncertainty because of the asymmetric, non-gaussian pdf of the rainfall. Comparison with Figures 1 and 2 shows that the highest variances are in areas of high rainfall and there are patches of low variance around the observation points of the sparse network. Thus the variance map is useful in that it gives a realistic assessment of relative error within the rainfall field. [35] Confidence limits were calculated from ensemble of values generated by the Monte Carlo simulation. In general we define a confidence interval x between bounds Z 1, Z 2 such that PfZ 1 Zðu 0 Þ Z 2 g ¼ x ð16þ The definition of Z 1 and Z 2 is not unique and because of the asymmetry of the distribution functions in this example we choose Z 1 and Z 2 such that the intervals (Z*(u) <Z j (u) <Z 2 ) and (Z 1 < Z j (u) <Z*(u)) contain equal numbers of ensemble members. Upper and lower limits for 68% and 95% confidence levels were calculated in this way. The 68% Table 1. Proportion of Validation Data Lying Within One or Two Standard Uncertainties of the Measured Values as Calculated From Equations (11) and (15) Equation (11), % Equation (15), % Simulation 1SU SU Kenyan network 1SU SU of13

8 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 8. Comparison of variances calculated from equations (11) and (15). limits are shown in Figures 9a and 9b. It can be seen that the greatest uncertainty (as represented by the confidence interval) corresponds closely with the highest variances shown in Figure 7, justifying the use of variance as a relative measure of estimate reliability. Inspection of Figures 1 and 2 shows that the widest confidence intervals also correspond to a region of high rainfall with no gauges. Taking the validation locations all together, the proportion of estimates within the 68 and 95% limits for rainy locations were respectively 71% and 96% suggesting that they are indeed realistic. 6. Case Study Using Rain Gauge Data [36] The methodology has also been tested using actual rainfall observations from a rain gauge network in Kenya. The problem with real data is that the true rainfall field remains unknown making validation more difficult. In our case study, the network is located in the Kenyan highlands close to Lake Victoria and comprises about 190 gauges lying between 34 E and 36 E and between 1.8 S and 1.2 N (an area of 70,000 km 2 ). The topography and proximity of the lake make this a challenging region for rainfall interpolation. The density of the network ensures sufficient rain gauges for splitting into two groups; one for modeling and a second for validation. As an example, we have chosen rainfall observations for the 15th July, The 164 rain gauges operational on that day were randomly divided into groups of 104 rain gauges for modeling and 60 rain gauges for validation. Figures 10a and 10b show modeling and validation groups, respectively. The modeling group data were used to calculate the necessary kriging parameters. Kriged rainfall values with associated uncertainties were calculated for the locations of the target gauges. [37] The experimental variogram data and modeled variograms for indicator values and rainfall amounts (respectively g I (h) and g F (h)) were estimated and are shown in Figures 11a and 11b, respectively. The models were fitted to the experimental data by weighted least squares [Rice, 1995] (program VARFIT, see Pardo-Igúzquiza [1999]). The indicator variogram was adequately modeled with a nugget variance representing 66% of the total variability and a spherical structure function representing the remaining 34% with a range of 0.5. The high nugget variance is to be expected from the small-scale intermingling of rainy and nonrainy rain gauges seen in Figures 10a and 10b, suggesting a correlation structure that cannot be resolved by this network density. This is consistent with the likely scale of the convection cells giving rise to the rainfall. However the behavior of a cluster of such convective cells is captured by the correlated part of the indicator variogram. [38] The range of the variogram for modeling rainfall inside the rainy areas (Figure 11b) is much shorter and 0.2 has been used as most likely value. The relatively small range of g F compared to g I provides some justification for the assumption of independence of F K and I* as argued earlier. Following the BCR92 procedure, the indicator kriging map I k (u), was derived, then the estimated map I*(u) of rainy/nonrainy areas, followed by the kriging estimates F k (u), and finally the desired rainfall estimate Z*(u) =I*(u)F k (u). The local conditional distribution was then evaluated by the Monte Carlo procedure described earlier. Variances were calculated both from the Monte Carlo simulation (equation (15) and via equation (11)). The proportion of validation points within the one and two SU bounds are shown in Table 1. Again, the two methods give very similar results. In this case the results are close to those expected from a Gaussian distribution. [39] Upper and lower limits for the 68 and 95% confidence limits have also been calculated. The proportion of validation points lying within these ranges are 64 and 96%, indicating that the pdfs calculated via the Monte Carlo simulation again give a realistic representation of the actual uncertainty. The 68% limits have been plotted for all validation gauges in Figure 12 together with the BCR92 kriged estimates and the actual validation measurements. First, it can be seen that the distinction between nonrainy and rainy locations is well made by the BCR92 algorithm, 8of13

9 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 9. (a) Map of the lower limit of the 68% confidence interval. (b) Map of the upper limit of the 68% confidence interval for the simulated data set. 9of13

10 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 10. Kenyan rain gauge locations. (a) Group of 104 rain gauges for modeling. (b) Group of 60 rain gauges for validation. In both cases, circles are rainy gauges, and squares are nonrainy gauges. 10 of 13

11 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 11. (a) Experimental indicator variogram and fitted model g I (h) for Kenyan data set. (b) Experimental variogram using rainy gauges only and fitted model g F (h) for Kenyan data set. 11 of 13

12 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Figure 12. Rainfall observations and estimates together with 68% upper and lower confidence limits ranked by observed rainfall amount for the Kenyan validation gauges. especially given the small-scale variability of rainfall illustrated in Figure 10. Second, although this small-scale variability inevitably leads to inaccuracy in the estimate of rainfall amount at some locations, the estimated uncertainties adequately encompass these large errors, showing that the method works as expected. 7. Conclusions [40] Barancourt et al. [1992] presented a simple and effective approach based on kriging for delineating rainfall intermittency and estimating rainfall in the presence of intermittency. This was an important result as intermittency is frequently encountered in rainfall fields, particularly for aggregations over time intervals of one day or less. However, the BCR92 approach was limited by the lack of a method of estimating the uncertainty of the estimates. This is usually done in geostatistics by calculating the estimation variance (or the standard uncertainty). In this paper, we have shown how the uncertainty can be assessed provided the probability of rain at each location is correctly specified. The variance may then be calculated by assuming that rainfall intensity and spatial location within the rainfall field are independent quantities. A more rigorous approach using an ensemble method has also been used to calculate the local pdf for each estimate from which both variance and confidence intervals covering a specified range may be easily obtained. Case studies using both simulated and real data sets show that the variances calculated by both methods give similar results. The non-gaussian characteristics of the rainfall distribution mean the variances must be interpreted with caution, nevertheless they provide a reasonable relative assessment of uncertainty which may well be an overestimate depending on the precise nature of the rainfall pdf. For strongly non-gaussian distributions, the uncertainty is much better expressed in terms of confidence intervals which are independent of the structure of the local pdf. The precise definition of the confidence interval bounds is a matter of user choice. For both the case studies presented here, the proportion of nonzero rainfall amounts from validation locations lying within the upper and lower limits of a specified confidence interval is close to the predicted amount, suggesting that the calculated confidence intervals are realistic. More case studies are needed covering different rainfall climates. [41] For simplicity, this study has focused on point rainfall estimation. A useful application is in assessing the uncertainty of areal rainfall means in the case of intermittency. Practical examples include the uncertainty on daily mean rainfall quantities for river catchments or agricultural land, allowing rigorous evaluation of the propagation of rainfall measurement uncertainty in rainfall-runoff and crop yield models. This is the subject of ongoing research. [42] Acknowledgments. The first author is a Ramon y Cajal Grant Holder from the Ministry of Science and Education of Spain. The work was carried out while the first author was working as a Visiting Fellow with the TAMSAT research group at the Department of Meteorology, University of Reading, UK. We are grateful to the Kenyan Meteorological Service for providing the rain gauge data for the second case study. We would like to thank the reviewers for their constructive criticism. References Barancourt, C., J. D. Creutin, and J. Rivoirard (1992), A method for delineating and estimating rainfall fields, Water Resour. Res., 28(4), Bras, R. L., and I. Rodríguez-Iturbe (1985), Random Functions and Hydrology, 559 pp., Addison-Wesley, Reading, Mass. Delhomme, J. P. (1978), Kriging in the hydrosciences, Adv. Water Resour., 1, Deutsch, C. V., and A. G. Journel (1992), GSLIB: Geostatistical Software Library and User s Guide, 340 pp., Oxford Univ. Press, New York. Dietrich, C. R., and G. N. Newsam (1993), A fast and exact method for multidimensional Gaussian stochastic simulations, Water Resour. Res., 29, of 13

13 PARDO-IGÚZQUIZA ET AL.: UNCERTAINTY OF INTERMITTENT RAINFALL FIELDS Evans, M., N. Hastings, and B. Peacock (1993), Statistical Distributions, 170 pp., John Wiley, Hoboken, N. J. Goovaerts, P. (1997), Geostatistics for Natural Resources Evaluation, Oxford Univ. Press, New York. Journel, A. G. (1983), Nonparametric estimation of spatial distributions, Math. Geol., 15, Kendall, M., and A. Stuart (1977), The Advanced Theory of Statistics, vol. 1, 4th ed., 472 pp., Oxford Univ. Press, New York. Lebel, T., G. Bastin, C. Obled, and D. Creutin (1987), On the accuracy of areal rainfall estimation: A case study, Water Resour. Res., 23(11), Mantoglou, A., and J. L. Wilson (1982), The turning bands method for simulation of random fields using line generation by a spectral method, Water Resour. Res., 18(5), Matheron, G. (1976), A simple substitute for conditional expectation: the disjunctive kriging, in Advanced Geostatistics for the Mining Industry, NATO ASI Ser., Ser. C, vol. 24, edited by M. Guarascio, M. David, and C. Huijbregts, pp , Springer, New York. Pardo-Igúzquiza, E. (1999), VARFIT: A Fortran-77 program for fitting variogram models by weighted least squares, Comput. Geosci., 25, Pardo-Igúzquiza, E., and M. Chica-Olmo (1994), SPECSIM: A program for simulating random fields by an improved spectral approach, Comput. Geosci., 20, Pardo-Igúzquiza, E., and P. A. Dowd (2005a), Empirical maximum likelihood kriging: The general case, Math. Geol., 31, Pardo-Igúzquiza, E., and P. A. Dowd (2005b), EMLK2D: A computer program for empirical maximum likelihood kriging, Comput. Geosci., 31, Rice, J. A. (1995), Mathematical Statistics and Data Analysis, Duxbury, Belmont, Calif. Seo, D. J. (1998), Real time estimation of rainfall fields using rain gauge data under fractional coverage conditions, J. Hydrol., 208, Yates, S. R., A. W. Warrick, and D. E. Myers (1986), Disjunctive kriging: 2. Examples, Water Resour. Res., 22(5), D. I. F. Grimes and C.-K. Teo, Department of Meteorology, University of Reading, Reading RG6 6BB, UK. (d.i.f.grimes@reading.ac.uk) E. Pardo-Igúzquiza, Department of Geodynamics, University of Granada, E Granada, Spain. 13 of 13