Elevation as a precipitation predictor

The spatial distribution of precipitation is heavily influenced by temperature, especially by its vertical lapse rate, which dictates the local level (height) and rate of condensation. In the absence of detailed (small-scale) temperature information, we use elevation as its surrogate, at least as a first approximation; this would be true for the case of a spatially constant lapse rate. Because we are interested in the spatial patterns of temperature rather than in its absolute magnitude (see below), this first approximation is adequate for all practical purposes. One alternatively could establish a regression function between coarse-resolution (large scale) temperature and fine-resolution (small scale) elevation information. The fine-resolution terrain then could be transformed into a regression-based temperature map, which would exhibit small-scale variations due to corresponding local terrain variations.

A 1-km-resolution digital elevation model (DEM) is available for this area and is shown in Fig. 1b; elevation values range from −17 to 1668 m, with a median of 126 m and an interquartile range of 398 m. The DEM grid size is 300 × 360 km² and coincides with the grid at which estimation–interpolation of rainfall will be performed. All subsequently derived rainfall predictors are available at this spatial resolution. The first step, then, is to determine precipitation values at the 77 grid nodes that are closest to the 77 rain gauges using the nearest-neighbor method. The rank order (Spearman) correlation coefficient between collocated precipitation and elevation values is 0.22 (here the term collocated connotes that the pairs of rainfall and elevation data used to compute that correlation are located on the same grid node, possibly after nearest-neighbor interpolation of the rain gauge value). This valve implies that, for the particular area and season, elevation is weakly correlated with precipitation, which is expected for relatively low mountains such as those found in the study region. We use the Spearman correlation coefficient instead of the ordinary Pearson correlation coefficient because the former is more robust to outliers. In addition, we interpret the relevance of elevation with precipitation in terms of its correlation with the relative rank of precipitation and less in terms of its correlation with precipitation magnitude. The nonlinear rank-order transformation involved in calculating the Spearman correlation coefficient allows capturing nonlinearities in the elevation–precipitation relationship.

To determine the scale of interaction between precipitation and elevation, the DEM-reported value at each 1-km cell was replaced by the elevation average over gradually enlarged square windows ranging from 3 × 3 km² to 21 × 21 km², with an increment of 2 km in each direction. Ten different averaging windows were applied to obtain 10 sets of averaged elevation values derived at the 77 rain gauges. Rank correlation coefficients were then calculated between the 10 sets of collocated precipitation and average elevation values corresponding to each window size. As indicated by the scatterplot of Fig. 1c, correlation increases gradually as the size of the averaging window increases, and a maximum of 0.36 is reached for a 13 × 13-km² window; for larger window sizes, correlation drops gradually. Window averaging is similar to low-pass filtering with a boxcar window, which smooths elevation spatial variability below a given window size. For the particular region shown in Fig. 1a, elevation exhibits the maximum relevance to precipitation, that is, can better inform its mapping, when small-scale (<13 km) elevation features are suppressed. Henceforth, the term elevation is used for this 13 × 13 km² window averaged elevation, whose map is shown in Fig. 1d.

Atmospheric variables as precipitation predictors

Lower-atmosphere state variables at the study region for the winter of 1981/82 include specific humidity, integrated from 850- to 1000-hPa levels, and the horizontal wind components at the 700-hPa level. More specific, the time averages of specific humidity and horizontal wind components over the period of interest were retained. Lower-atmosphere state variables are available at a coarse resolution of 2.5° × 2.5° from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset (Kalnay et al. 1996). The reanalysis products provide snapshots of the global atmospheric and surface fields at a uniform resolution every 6 h. NCEP–NCAR reanalysis variables are generated via data assimilation, whereby observational data around the globe are used to drive a global atmospheric model. Details of the data quality control and assimilation procedure are presented in Kalnay et al. (1996). Together with the reanalysis from the European Centre for Medium-range Weather Forecasts, NCEP–NCAR reanalysis variables are regarded as the most reliable (and widely available) representation of the instantaneous state of the global atmosphere.

We retained reported values at nine reanalysis nodes within and nearby the study region, which are shown in Fig. 2. Season-averaged specific humidity ranges from 14.81 to 16.68 g kg⁻¹ and is relatively smaller in the southeastern part of the study domain (Fig. 2a). Wind speed ranges from 7.02 to 10.90 m s⁻¹ and is relatively higher at the northwestern part of the region (Fig. 2b). Last, wind direction ranges from 86.58° to 110.61° from the north (Fig. 2c), indicating a dominant wind direction from the west. Lower-atmosphere state variables, although known at a very small number of locations, provide a picture of the large-scale state of the lower atmosphere, which is expected to be related to observed precipitation at the local (1 km) scale.

We use interpolated horizontal wind components (not shown) to calculate the vertical component of wind w_s due to orographic lifting at the local scale. This terrain-induced vertical motion is defined as the inner product of directional elevation gradients with the corresponding horizontal wind components, (e.g., Alpert 1986):

where v = (u, υ) denotes the wind vector with local horizontal components u and υ, and dh/dx, dh/dy denote the local gradients of elevation along the same directions.

Although the horizontal wind vector v is available at a very coarse resolution from the nine NCEP nodes and interpolation at a 1-km resolution provides very smooth maps of the horizontal wind components over the study region, the vertical wind component due to orographic lifting exhibits small-scale variations due to local terrain. In this paper, such terrain-induced vertical motion, which can also be regarded as the exposure of local terrain to local wind, is accounted for in the regional-scale mapping of rainfall.

Rainfall predictors are subsequently derived from the NCEP–NCAR reanalysis lower-atmosphere state variables and their interaction with local terrain in the following three steps.

1. Interpolation (via the inverse-distance-squared method) of specific humidity and horizontal wind components from the nine NCEP nodes to a 300 × 360 grid with 1-km² cell size is done. Inverse-distance interpolation is retained in lieu of other methods of objective analysis because of the very small number of NCEP nodes, which does not allow for reliable inference of structural characteristics, for example, a spatial correlation model for specific humidity. The vertical wind component at any grid cell is calculated using Eq. (1). Let x₁(u), x₃(u) denote the interpolated values of specific humidity and derived vertical wind component at any grid cell with (2D) coordinate vector u = (u₁, u₂), expressed, for example, in degrees of longitude and degrees of latitude, respectively. Let x₂(u) denote the elevation at u obtained from the low-pass filtering procedure described in section 2a.
2. The resulting values are transformed to follow a uniform distribution. For the case of specific humidity, for example, the transformed value y₁(u) at any location u is computed as
   y₁uF_X1x₁u(2)
   where F_X1( ) denotes the cumulative histogram of the interpolated specific humidity values. For the case of vertical wind component X₃, a zero value corresponds to the median of the distribution; that is, F_X3(0.0) = 0.5, which implies that a rank-ordered value y₃(u) of greater than 0.5 should be interpreted as uplift. In the more general case, one would first compute the probability corresponding to a zero value of the vertical wind component p⁽⁰⁾₃ = F_X3(0.0) and then interpret as uplift all rank-ordered values greater than p⁽⁰⁾₃. Rank-ordered values lower than p⁽⁰⁾₃ conversely would correspond to downslope vertical wind movement.
3. The interaction terms among the three predictors y₁(u), y₂(u), and y₃(u) are calculated. For example, the interaction between humidity and the vertical wind component at a grid cell u is calculated as y₅(u) = y₁(u)y₃(u). This product term is the rank-ordered amount of moisture uplift due to wind encountering the local terrain slope. The interaction term y₇(u) = y₁(u)y₂(u)y₃(u) is the (rank ordered) moisture uplift due to wind encountering the local terrain slope, modulated by the decrease of moisture availability with topographic height. The remaining interaction terms were calculated as y₄(u) = y₁(u)y₂(u) and y₆(u) = y₂(u)y₃(u). All the resulting interaction fields are also transformed to follow a uniform distribution as in Eq. (2).

The transformation procedure of Eq. (2) is applied for eliminating the effect of different histograms of predictor fields to precipitation mapping, given that all transformed fields lie in the interval [0, 1]. One should interpret the relevance of a predictor value at any grid cell in terms of its relation with the relative rank of precipitation rather than in terms of its correlation with precipitation magnitude. The magnitude of precipitation is dictated by the rain gauge observations themselves.

Maps of the final rank-transformed values of specific humidity y₁(u), elevation y₂(u), and vertical wind component y₃(u) at any 1-km² grid cell u are shown in Fig. 3. Note the extremely smooth spatial variation of humidity (Fig. 3a), as opposed to that of elevation (Fig. 3b) and vertical wind component (Fig. 3c). The patterns of spatial variability of the latter are mainly due to the elevation gradient fields, because the interpolated horizontal wind components (not shown) exhibit very smooth spatial variation attributable to the limited spatial resolution of NCEP–NCAR reanalysis. The vertical wind component field (Fig. 3c) exhibits the most complex spatial variability. Relatively large positive values of vertical wind, dark pixels corresponding to stronger uplift, are found on the windward side of the terrain (recall that the prevailing wind direction is from the west).

Maps of the final rank-transformed values of the interactions between humidity and elevation y₄(u), humidity and vertical wind y₅(u), elevation and vertical wind y₆(u), and humidity with elevation and vertical wind y₇(u) are shown in Fig. 4. One can appreciate the smooth spatial patterns inherited from humidity and to a lesser extent from elevation (Fig. 4a) and the complex spatial patterns inherited from the vertical wind component (Figs. 4b–d).

Last, we investigate whether the predictor values at the rain gauge stations are a representative sample of their respective populations. This is done by constructing quantile–quantile plots between the distributions of all predictors (over the entire domain) and their respective sample distributions (Fig. 5). A representative sample for, say, elevation would lead to a corresponding quantile–quantile plot aligned with the 45° line. Rain gauges tend to be located in relatively low elevations, although very low and very high elevations are adequately covered (Fig. 5a). Specific humidity and vertical wind, on the other hand, are adequately sampled from the spatial configuration of the available rain gauges (Figs. 5b–c). Similar graphs were constructed for the remaining predictors (not shown), and indicated representative samples for these predictors.

Correlation between precipitation and its predictors

In this paper, the relevance of the three predictors and their interactions to observed precipitation is quantified in terms of their correlation with the rain gauge data (Table 2, top row). Maximum correlation between a predictor and precipitation is reached for the interaction of humidity with elevation (ρ_ZY4 = 0.69), followed by humidity alone (ρ_ZY1 = 0.54) and by the interaction of humidity with elevation and vertical wind component (ρ_ZY7 = 0.45). Elevation alone is not strongly correlated with precipitation (ρ_ZY2 = 0.35), and the same holds true for the interaction of humidity with vertical wind (ρ_ZY5 = 0.30). The vertical wind component and its interaction with elevation are almost uncorrelated with precipitation (ρ_ZY3 = 0.05 and ρ_ZY6 = 0.19, respectively). These low correlation-coefficient values might be the result of the small time averaging period (NDJ), which cannot reveal a definite correlation between vertical wind component and observed precipitation. Such a relationship could be more evident if observed precipitation within different storm types (e.g., frontal vs synoptic) was compared with the vertical wind component occurring during the same storm type.

From Table 2, one can appreciate the importance of humidity in mapping precipitation, given that its (Pearson) correlation with the rain gauge data is ρ_ZY1 = 0.54. However, the spatial variation of humidity is very smooth and cannot distinguish local precipitation features. It is here where the role of the interaction between lower-atmosphere state variables and terrain characteristics becomes important. Indeed, the interaction of humidity with elevation is strongly correlated with precipitation (ρ_ZY4 = 0.69), and the interaction of humidity with elevation and vertical wind is moderately correlated with precipitation (ρ_ZY1 = 0.45). Both of these predictor fields exhibit nonsmooth patterns of spatial variability and potentially can resolve local precipitation features.

Next, we investigate whether a significant correlation exists between the predictor variables themselves, a situation termed mutlicollinearity in a regression context (Draper and Smith 1998). The original predictors (i.e., specific humidity Y₁, elevation Y₂, and the vertical wind component Y₃) constitute nearly independent variables, because the correlation between them is very low: ρ_Y1Y₂ = −0.17, ρ_Y1Y₂ = −0.02, and ρ_Y2Y₃ = 0.03 (see Table 2). The interaction variables (i.e., that of specific humidity with elevation Y₄, specific humidity with vertical wind Y₅, elevation with vertical wind Y₆, and specific humidity with elevation and vertical wind Y₇) are not independent, as indicated by the nonnegligible correlation coefficients between them (Table 2). In a multiple regression model, interpretation of the resulting regression coefficients is straightforward only in the case of independent predictors. When predictors are correlated, as are the derived precipitation predictors (interactions) in this dataset, the resulting regression coefficients should be interpreted in the light of multicollinearity.

Spatial interpolation using only rain gauge data

Consider the task of estimating the unknown precipitation value z(u) at a location u from n(u) < n surrounding values {z(u_α), α = 1, … , n(u)} within a neighborhood W(u) centered at u. In the stationary case, n(u) + 1 pieces of information are available: the known regional mean m, and the n(u) residual values {r(u_α) = [z(u_α) − m], α = 1, … , n(u)}.

The simple kriging (SK) estimate z^*_SK(u) for the unknown value z(u) is expressed as

where r^*_SK(u) denotes the SK estimate (with known zero mean) of the unknown residual r(u) and λ_α(u) denotes the weight assigned to the αth residual value r(u_α) when estimation is performed at location u.

The n(u) weights {λ_α(u), α = 1, … , n(u)} are obtained per solution of the system of normal equations or SK system:

where C_R(u_α − u_β) is the covariance between any two data locations u_α and u_β, and C_R(u_α − u) is the covariance between any datum location u_α and the location u where estimation is performed. Note that the weights do not depend on the residual values, only on their configuration through the covariance model C_R(h). In meteorology, simple kriging is termed objective analysis (Daley 1991), and in the statistical literature it is termed best linear unbiased predictor (Cressie 1993).

In reality, the regional mean m of precipitation is unknown; it could be estimated by a weighted average of the sample rainfall data using various weighting schemes. In practice, however, the average precipitation spatial variability is nonstationary; that is, it varies systematically from one location to another. Local variations of the trend component m can be accounted for by considering a mean m(u), which is constant within each neighborhood W(u) but varies from one neighborhood to another:

Estimation of the unknown value z(u) now proceeds by first estimating the local mean m(u) within the neighborhood W(u), using data only from that neighborhood, and then performing SK using the resulting residuals. The algorithm is known as ordinary kriging (OK) with moving neighborhoods (or moving windows).

Within any neighborhood W(u), the OK estimate z^*_OK(u) is written as

where r′(u_α) denotes a new residual value defined as: r′(u_α) = z(u_α) − m^*_OK(u_α).

The n(u) weights {ν_α(u), α = 1, … , n(u)}, which are different from the SK weights {λ_α(u), α = 1, … , n(u)}, hence the different notation, are obtained per solution of the OK system:

where C_R′(u_α − u_β) denotes the covariance between any two residual values r′(u_α) and r′(u_β), and μ_OK(u) is a Lagrange parameter due to the constraint on the weights imposed for estimating the local mean m^*_OK(u). In this local stationary case, the typical practice consists of assimilating the covariance C_R′(h) of the residuals to the covariance C_R(h) (Goovaerts 1997). This is the reason for using the same notation for the SK weights λ_α(u) in both Eqs. (5) and (7).

Kriging (either SK or OK) is an exact interpolator; that is, kriging estimates reproduce observed sample data values at their locations: z^*_OK(u_α) = z(u_α) for any sampling location u_α. This data-exactitude property of kriging is not shared by the traditional ordinary least squares (OLS) regression models. In the case of rain gauge data contaminated by measurement errors with known statistics (mean, variance, and covariance), interpolation should not reproduce the sample data values at their locations; this situation is not treated in this work. Measurement errors, however, could be filtered out in the interpolation procedure via factorial kriging (Wackernagel 1995; Goovaerts 1997).

We now present two geostatistical alternatives for incorporating several relevant predictors into the mapping of rainfall. In what follows, we do not adopt the geostatistical approach of cokriging (Hevesi et al. 1992), because of space limitations and for the following two reasons: (a) the inference effort required by cokriging increases with the number of predictors and (b) various investigators have reported that cokriging does not improve significantly the accuracy of precipitation estimates when compared (in terms of cross-validation statistics) with the two approaches presented in the next section (Goovaerts 2000).

Spatial interpolation accounting for rainfall predictors

The coefficient vector b is constant over the study region, because the regression function is determined using all n data values; this implies that the function f( ) does not account for local variations in the relationship between precipitation and its predictors. Any difference (spatial variation) between two estimates z^*_f(u) and z^*_f(u′) at two locations u and u′ stems solely from the difference of the corresponding predictor vectors y(u) and y(u′) at these locations. The estimate z^*_f(u) accounts for the relevance of the collocated predictor vector y(u) to the unknown precipitation value z(u); it does not explicitly account, however, for any nearby sample precipation data z(u_α) or for any predictor vector y(u′) at a nearby location u′.

A regression-derived precipitation estimate z^*_f(u_α) at a rain gauge location u_α typically differs from the sample precipitation value z(u_α) at that location by a residual amount that varies from one rain gauge to another. If the method of OLS is used to establish the coefficient vector b, the n residual values {r(u_α) = [z(u_α) − z^*_f(u_α)], α = 1, … , n} are assumed to have a Gaussian distribution and a purely random (white noise) spatial variation. This latter assumption of spatial independence reduces the applicability of hypothesis-testing procedures in a spatial setting, given that the residuals are usually correlated in space (see section 4).

Recall that, through the subjective decomposition of the RF model in Eq. (4), the observed value z(u_α) at any rain gauge location u_α can be viewed as the sum of a local trend component m_alg(u_α) and a local residual component r_alg(u_α). One could choose to identify the local trend component m_alg(u_α) to the regression-based precipitation estimate z^*_f(u_α) at that location; that is,

where r_f(u_α) denotes the residual from the particular trend model z^*_f(u_α).

Note that the decision to adopt a linear versus a nonlinear regression model for establishing the local trend component m_alg(u_α) is one possible (subjective) interpretation of the decomposition in Eq. (4). The use of forward stepwise regression instead of backward elimination for selecting the pool of most relevant predictors (Draper and Smith 1998) is similarly another possible (subjective) interpretation of the general decomposition in Eq. (4). Different decisions evidently result into different sets of predictor variables, which in turn result into a different trend component m_alg(u) and consequently a different residual component r_alg(u) at each location u. In any case, the residual component r_alg(u) is considered to be stationary with zero mean, and SK consequently can be used for mapping the residual spatial variation.

In the case of a spatially varying trend component z^*_f(u), SK provides an estimate r^*f_SK(u) for the unknown residual r_f(u) based on the n(u) residual values within a neighborhood W(u) centered at u. The information utilized by SK comprises the n(u) residual values {r_f(u_α) = [z(u_α) − z^*_f(u_α)], α = 1, … , n(u)} within W(u), as well as their spatial covariance C_Rf(h). This estimated residual r^*f_SK(u) is then added back to the regression-derived estimates z^*_f(u), and the combined procedure is termed detrended kriging (Switzer 1979; Chua and Bras 1982) or SK with varying local mean (SKLM; Goovaerts 1997). The corresponding SKLM estimate z^*_SKLM(u) is written as

where the weights ξ_α(u) are determined per solution of a SK system similar to that of Eq. (6):

the only difference being that instead of the covariance C_R(h) used in Eq. (6), the covariance C_Rf(h) of the new residual component R_f(u) is used here. This is also the reason for the different notation ξ_α(u) adopted to denote the corresponding weights, instead of λ_α(u) or ν_α(u).

Local variations in the relation between precipitation and its predictors can be accounted for a posteriori, that is, by locally deforming the regression-based trend component m^*_f(u) = z^*_f(u). A new trend component m*(u) can be defined as a local linear rescaling of the regression-based z^*_f(u) estimate, so that the latter conforms locally [within each neighborhood W(u)] to the n(u) nearby sample precipitation data {z(u_α), α = 1, … , n(u)}. The locally modified trend component m^*_KED(u) is provided via kriging with an external drift (KED) [for example, Wackernagel (1995)]:

where d₀(u) and d₁(u) denote local regression coefficients, which are constant within each neighborhood W(u) and are different from one neighborhood to another.

The n(u) weights {θ_α(u), α = 1, … , n(u)} are obtained per solution of the KED system:

where C_Rf′(u_α − u_β) denotes the covariance between any two residual values r_f′(u_α) and r_f′(u_β), with r_f′(u_α) = z(u_α) − m^*_KED(u_α); μ^KED₁(u) and μ^KED₂(u) are two Lagrange parameters due to the two constraints on the weights. As a first approximation, the residual covariance C_Rf′(h) could be identified to the covariance C_Rf(h) of the residual values obtained from the (global) regression. The corresponding KED estimate z^*_KED(u) for the unknown value z(u) is:

which, as in the previous section, amounts to adding to the KED-derived trend component m^*_KED(u) an SK estimate r^*f′_SK(u) of the corresponding unknown residual r_f′(u). Because C_Rf′(h) ≃ C_Rf(h), the same notation ξ_α(u) is used for the SK weights in SKLM [Eq. (12)] and KED [Eq. (16)].

An alternative (and more elaborate) procedure could be envisaged, whereby the local trend of precipitation is evaluated with respect to each predictor individually instead of collectively through their linear combination obtained by regression. For the case of KED, this approach would call for the determination of two coefficients for each variable, leading to a total of 2(K + 1) trend coefficients for the case of (K + 1) variables. In this paper, we do not pursue this alternative; we essentially regard the regression-based precipitation predictions as a secondary variable carrying all the available information content of the individual predictors.

The different geostatistical algorithms presented in this section for mapping precipitation are a consequence of the different decompositions of the RF model {Z(u), u ∈ D} given in Eq. (3). To be more specific, their difference lies in the complexity of the spatial trend component m(u) within each neighborhood W(u) centered at the location u where estimation is performed and the complexity of the resulting residual component. For SK, the spatial trend component m(u) is assumed to be constant; that is, m(u) = m, ∀u ∈ D. This assumption is oversimplified, and SK is not considered in what follows. For OK, m(u) is estimated using only nearby precipitation data based on a model of their spatial correlation; that is, m(u) = m^*_OK(u). For SKLM, m(u) is identified to the precipitation estimates based on the regression between lower-atmosphere state variables (including their interactions with local terrain) and observed precipitation; that is, m(u) = z^*_f(u). Last, for KED, m(u) is identified to local deformations of z^*_f(u); that is, m(u) = m^*_KED(u).

Recall that the resulting residual component R(u) is considered to have a constant (zero) mean and a stationary covariance model C_R(h) in all forms of kriging presented in this section. As a consequence, it is expected that the introduction of more realism in the spatial trend component, via consideration of relevant predictor variables, will render the assumption of a stationary residual less unrealistic. This implies that SKLM and KED, whose spatial trend components are the most complex, will lead to more realistic (physically less inconsistent) rainfall maps.

Both SKLM and KED can be viewed as corrections to the “first-guess” field derived via an OLS regression. Recall that OLS assumes that the regression residuals are spatially uncorrelated; hence SKLM and KED can be regarded as procedures for reintroducing such a residual spatial correlation to the mapping of precipitation. Both SKLM and KED corrections ensure data exactitude and could be viewed as (intermittent) data assimilation techniques (Rutherford 1972). The only difference is that in the meteorological definition of data assimilation, the first-guess field is a physically based prediction from a dynamic atmospheric model. In our case, the first-guess field is a statistical prediction based on relevant (in a physical sense) variables with observed precipitation. By construction, the regression-based estimates are independent of the resulting residuals, whereas this might not be true in the case of data assimilation involving the difference between atmospheric model predictions and observed precipitation.

The alternative geostatistical algorithms (apart from SK) presented in this section are subsequently applied for mapping rainfall over the study area shown in Fig. 1a.

Regression models between precipitation and its predictors

The map of precipitation estimates z^*₂(u) derived via Eq. (17) is shown in Fig. 6a. Note the smooth spatial characteristics of this regression-based field, which does not account for either orographic lifting or for humidity. The terrain pattern is imprinted on the map of regression estimates, but there is no rain shadow on the leeward slopes of the terrain nor any indication of more intense precipitation on the windward side (recall that the prevailing wind direction is from the west).

The map of precipitation estimates z^*₄(u) derived via Eq. (18) is shown in Fig. 6b. The regression-based field depicts higher precipitation values where elevation is higher and moisture is more available. The most pronounced difference between the patterns in Fig. 6a and Fig. 6b is found in the southeastern part of the study region, where the model in Eq. (18) yields low precipitation values. Although elevation is high in this area, specific humidity is low, resulting in less precipitation.

Last, we use three predictors that are determined as most significant using backward elimination. Backward elimination is a variable selection procedure whereby one starts with the entire set of predictors and subsequently drops the least significant variable (according to an F test) from this initial set. Variables are repeatedly eliminated from the initial set until no remaining variable can be dropped (again according to the F test). For details regarding backward elimination, and variable selection in general, the reader is referred to Draper and Smith (1998). In this way, we retained the vertical wind component, the interaction of specific humidity with elevation, and the interaction of specific humidity with elevation and vertical wind as the three most important predictors leading to Y₃ = [1 y₃ y₄ y₇]. The resulting regression equation is

where the subscript 347 implies that the predictors y₃(u), y₄(u), and y₇(u) were used in the regression model. For this regression model, R² = 0.52, a 333% increase from model (17), and rmse = 2.14 mm, a 26% reduction from Eq. (17).

Recall from Table 2 that, although the vertical wind component Y₃ is not correlated with the interaction between humidity and elevation Y₄(ρ_Y3Y₄ = 0.03), the all-interactions term Y₇ is significantly correlated with both Y₃ and Y₄ (ρ_Y7Y₃ = 0.61 and ρ_Y7Y₄ = 0.75, respectively). The regression coefficients of Eq. (19) therefore should be interpreted in the light of multicollinearity. In particular, the negative sign of the last regression coefficient (b₄ = −7.55) is a consequence of multicollinearity. In other words, the influence of the all-interactions predictor Y₇ is negative, once the effect of the other predictors Y₃ and Y₄ is removed. This effect could be also quantified in terms of the corresponding coefficient of partial determination. Note that principal components regression (Draper and Smith 1998), that is, multiple regression using orthogonal (uncorrelated) projections of the original correlated variables, is an alternative procedure for variable selection and regression building when multicollinearity is present. We did not pursue this alternative, because it calls for equally subjective decisions as backward elimination and because the interpretation of the resulting regression coefficients (not in terms of the orthogonal projections but in terms of the original predictors) is as difficult as that of Eq. (19).

The map of precipitation estimates z^*₃₄₇(u) derived via Eq. (19) is shown in Fig. 6c. The large-scale precipitation patterns of this regression-based field are similar to those of Fig. 6b, but the small-scale patterns differ locally and are more closely associated with terrain variations. In particular, there are obvious rain shadows on the leeward slopes of the terrain (recall that the prevailing wind direction is from the west).

The correlation coefficient between regression-based and observed precipitation values for the regression model in Eq. (19) is 0.72 (see Fig. 7a). Such correlation values between regression-based and rain gauge data are comparable with those obtained from predictions using deterministic regional models with limited inclusion of atmospheric physics and dynamics; for this latter case, such correlation values range between 0.61 and 0.84 (Alpert and Shafir 1989a; Sinclair 1994). The spatial organization of the resulting residual values {r₃₄₇(u_α) = [z(u_α) − z^*_f347(u_α)], α = 1, … , n} is shown in Fig. 7b. The normal probability plot of these r₃₄₇ values (Fig. 7c) indicates a quasi-Gaussian distribution. Note that among all precipitation residuals resulting from Eqs. (17)–(19), the r₃₄₇ values exhibit the smallest spatial correlation (see next section), and their histogram is the closest to a Gaussian distribution.

The three different maps of Fig. 6 represent three different first-guess fields, that is, three different interpretations of the relative importance of the seven predictor variables available over the study area. We now proceed by incorporating these three different maps, as well as the information brought by the precipitation residuals at the rain gauge stations (and their spatial correlation), into the geostatistical mapping of rainfall.

Geostatistical mapping of rainfall

All geostatistical interpolation procedures presented in the previous section call for a covariance (or, equivalent, a variogram) function that models the spatial continuity (or, equivalent, the spatial variability) of the original sample precipitation z data or that of the residual r values resulting from the particular trend function adopted. Figure 8 depicts the omnidirectional experimental and model variograms of the sample precipitation data (Fig. 8a), as well as those corresponding to the residuals from different trend functions (Figs. 8b–d).

Two nested variogram structures (with no nugget effect), corresponding to one small-scale and one large-scale spatial process, were used to model the sample variograms of the ith dataset, be it the original rain gauge z data or any set of residual r values:

where |h| denotes the modulus of a distance vector h, and C⁽ⁱ⁾(0) denotes the variance of the ith dataset. Parameters c⁽ⁱ⁾₁ and a⁽ⁱ⁾₁ denote the respective sill (contribution to the total variance of the ith dataset) and range of the small-scale variogram structure; c⁽ⁱ⁾₂ and a⁽ⁱ⁾₂ denote the respective sill and range of the large-scale variogram structure.

The parameters of the variogram models for the original precipitation z data and the residual r values from various trend functions were derived via cross-validation [see Isaaks and Srivastava (1989) and next section] and are tabulated in Table 3. Because a zero nugget effect was adopted for all cases, the sum of the sills of the two variogram structures should be approximately equal to the variance of the respective dataset. In the case of original rain gauge z samples, precipitation spatial variability is decomposed in a small-scale (40 km) process that explains 22% of the total variance (=9.18 mm²), and a large-scale (160 km) process that explains the remaining 78% of the total variance. Similar decompositions, although with different parameters, were adopted for the residual r datasets. In general, as the trend function becomes more complex, thus involving more predictor variables and explaining a larger proportion of the z sample variability, the variance of the corresponding residual r values [sum of c⁽ⁱ⁾₁ and c⁽ⁱ⁾₂] becomes smaller and the corresponding nested variogram structures exhibit smaller ranges [a⁽ⁱ⁾₁ and a⁽ⁱ⁾₂]; see Table 3.

The map of precipitation estimates z^*_OK(u) derived via OK using Eqs. 7 and 8 is shown in Fig. 9. Note the smooth pattern of spatial variability exhibited by the OK precipitation estimates and the reproduction of large-scale features found in the rain gauge map of Fig. 1a. A rain shadow on the leeward slopes of the terrain is hardly discernible in this case. The map of precipitation estimates z^*_SKLM2(u) derived via simple kriging with locally varying mean using Eqs. 12 and 13, with the regression-based trend component z^*₂(u) as local mean, is shown in Fig. 9b. The pattern of spatial variability in the resulting estimates of Fig. 9b is now less smooth than that of the OK estimates of Fig. 9a. Note that the underlying trend component of Fig. 6b can be distinguished in the map of Fig. 9b. The map of precipitation estimates z^*_SKLM4(u) derived via SKLM, with the regression-based trend component z^*₄(u) as local mean, is shown in Fig. 9c. Note the more complex spatial pattern imposed by the corresponding trend component of Fig. 6c. Last, the map of precipitation estimates z^*_SKLM347(u) derived via SKLM, with the regression-based trend component z^*₃₄₇(u) as local mean, is shown in Fig. 9d. The spatial distribution of the resulting estimates of Fig. 9c exhibits the most complex spatial patterns, especially at small scales. Note the strong rain shadow on the leeward slopes of the terrain, in accordance with the corresponding trend component of Fig. 6d.

The maps of precipitation estimates z^*_KED(u) derived by local deformation of the regression-based trend component z^*_f(u) via KED using Eqs. 16 and 15 are shown in Fig. 10. The trend components z^*_f(u) were those depicted in Fig. 6D and used for deriving the SKLM estimates of Fig. 9. The residual covariance model adopted was identified to that of the r′ residuals; that is, C_Rf′(h) ≃ C_Rf(h). The pattern of spatial variability is as complex as that found in the SKLM estimates of Fig. 9, in accordance with the corresponding trend components of Fig. 6, yet local differences between the corresponding SKLM and KED maps can be distinguished (compare, for example, the north-central parts of Figs. 9d and 10c); it is likely that KED leads to somewhat less variable precipitation estimates in areas away from the rain gauges.

The task now is to compare objectively these alternative rainfall maps and to evaluate the improvement (if any) brought by the predictors to the accuracy of the derived map product.

Comparison of alternative mapping procedures

Because all variants of kriging are exact interpolators, no estimation error occurs at rain gauge locations. The different geostatistical interpolation procedures are therefore compared via cross-validation [see, for example, Isaaks and Srivastava (1989)]. Cross-validation amounts to sequentially dropping a single precipitation value from the sample dataset and reestimating its value from the remaining samples using all other available information. At any sample location u_α, both the original precipitation value z(u_α) and its cross-validation-derived estimate z^*_cr(u_α) are available. Comparison of the various approaches therefore can be performed by examining the statistics and spatial patterns of the 77 cross-validation error values {e^*_cr(u_α) = [z^*_cr(u_α) − z(u_α)], α = 1, … , n} computed at the 77 rain gauge locations. A positive cross-validation error e^*_cr(u_α) indicates overestimation of the actual precipitation z(u_α) by the cross-validation estimate z^*_cr(u_α), whereas a negative cross-validation error indicates the reverse.

Table 4 gives selected cross-validation statistics for the interpolation algorithms considered in the previous section. Cross-validation statistics examined are the rmse, as well as the correlation ρ_Z^*crZ between the cross-validation estimates and the true (sample) values and the correlation ρ_E^*crZ between the cross-validation errors and the true (sample) values. For the rmse, the best case corresponds to near-zero values. For ρ_Z^*crZ, the best case is a value of 1, indicating an almost perfect correlation between cross-validation estimates and true values. For ρ_E^*crZ, the best case is a value of 0 indicating a lack of correlation between cross-validation errors and true data; this implies that there is no systematic underestimation or overestimation of high or low precipitation values. In Table 4, values in parentheses denote the relative difference in the corresponding cross-validation scores from the case of OK using only rain gauge data.

Ordinary Kriging is the least accurate, because it leads to the largest rmse and ρ_E^*crZ values and to the smallest ρ_Z^*crZ value, of all the algorithms considered (see Table 4). The best performance is achieved by SKLM₃₄₇ and KED₃₄₇, because they lead to the lowest rmse and ρ_E^*crZ values and to relatively high ρ_Z^*crZ values. Note also that SKLM₃₄₇ yields cross-validation errors with the smallest spatial correlation. In other words, out of all algorithms considered, SKLM₃₄₇ yields cross-validation residuals with the smallest variogram range (not shown). For OK, the high negative correlation value ρ_E^*crZ = −0.54 implies that negative (or positive) OK cross-validation errors are systematically associated with high (or low) precipitation values, which is a direct consequence of the smooth trend component m^*_OK(u). Low precipitation values tend to be systematically overestimated (leading to positive cross-validation errors), whereas high precipitation values tend to be systematically underestimated (leading to negative cross-validation errors). Such a smoothing effect is much less severe for SKLM₃₄₇ (ρ_E^*crZ = −0.37, a 31% reduction from OK) and for SKLM₄ (ρ_E^*crZ = −0.44, a 15% reduction from OK).

From Table 4, one can conclude that, overall, rainfall mapping via SKLM and KED, accounting for the maximum amount of relevant predictors and for spatial correlation of the resulting residuals, yields relatively better cross-validation scores for the particular dataset and study region. Although OK performs worst in terms of cross-validation scores, its performance is not dramatically different than the other algorithms considered in this work. Such relatively good cross-validation scores for the case of OK are largely due to the rain-gauge density and the large spatial correlation range (160 km) of precipitation.

To arrive at a more definite conclusion regarding the improvement brought by the lower-atmosphere state variables and terrain information, we compare their relationship with the various cross-validation error datasets that result from the different algorithms adopted. In particular, we investigate whether the entire set of available predictors (Table 2) could account for a portion of the spatial variability of the cross-validation errors. If, for example, a statistically significant regression function can be established between the predictors and the cross-validation errors, this would imply that such errors could be potentially reduced, had those predictors been included in the interpolation procedure. Relations between predictors and cross-validation errors were investigated for all the algorithms presented in the previous section.

The regression characteristics between cross-validation errors corresponding to different mapping algorithms and three precipitation predictors Y₃, Y₄, and Y₇ are shown in Table 5. A relatively high R² indicates that a higher proportion of the spatial variability of cross-validation errors can be accounted for by the three predictors. A relatively high F statistic associated with a small significance p value implies that the regression model between the corresponding cross-validation errors and the precipitation predictors is statistically significant.

From Table 5, one can see that statistically significant regression models between cross-validation errors and the three precipitation predictors Y₃, Y₄, and Y₇ can be established for the case of OK, SKLM₂, and, to a lesser extent, for the case of SKLM₄. For the case of OK, the percentage of variance of cross-validation errors accounted for by the regression on the predictors is 24%, whereas such a proportion drops to 19% in the case of SKLM₄. Note that KED and SKLM give similar results in all cases. One could argue that the R² values of Table 5 for the case of OK, SKLM₂, and SKLM₄ (or, equivalent, for the corresponding KED algorithms) are not very high. They are, however, important in that the three predictors Y₃, Y₄, and Y₇ can explain a nonnegligible proportion of the spatial variability of the cross-validation errors. Such results corroborate the importance of including lower-atmosphere state variables and their interaction with local terrain characteristics into the spatial interpolation of rainfall.

Last, we evaluate the performance of the various algorithms in terms of jackknife scores. To be more specific, we exclude sample precipitation values at the 15 stations marked with crosses in Fig. 1a and perform estimation at these 15 locations. We preferentially exclude the highest precipitation values for the jackknife, because accurate estimation of such values is critical in many hydrologic analyses. Most of the jackknife locations in the northwestern part of the study region are located outside the convex hull of the remaining rain gauges. This means that estimation at these jackknife locations is performed in extrapolation mode, in which case the local trend model m_alg(u) adopted is of paramount importance. The algorithm that will suffer most from this extrapolation setting is OK, because its local trend model is estimated from nearby data that are located (south)east from the jackknife stations (see Fig. 1a). This adverse setting for OK, however, is compensated by the fact that these highest sample precipitation values are not predicted as well as the other ones from the corresponding regression model (local trend; see Fig. 7a). It also should be noted that, for all the algorithms considered, we build the corresponding regression models and infer the resulting residual variograms using all n = 77 rain gauges. The resulting rmse of jackknife errors is shown in Table 6. Ordinary kriging yields the largest rmse value (4.57), whereas SKLM₃₇₄ and SKLM₄ yield the smallest rmse values (3.42 and 3.69, respectively), thus achieving a 25% and 19% reduction from the OK-based rmse. Again, the importance of including lower-atmosphere state variables and their interaction with local terrain characteristics into the spatial interpolation of rainfall is corroborated.