## Abstract

The authors use variograms of radar reflectivity as a summary statistic to describe the spatial continuity of Alpine precipitation on mesogamma scales. First, how to obtain such variograms is discussed. Second, a set of typical variograms of Alpine precipitation is found. Third, some examples are given on how these variograms can be used to tackle several questions such as, What spatial variation of precipitation rate can be found in Alpine catchments? What difference can be expected between the measurements at two points separated by a given distance? To what accuracy can areal precipitation be estimated from point observations? Are there preferred regions for convection in Alpine precipitation? Variograms are obtained using a method-of-moments estimator together with high-resolution polar reflectivity data of well-visible regions. Depending on the application, the variogram was determined in terms of linear precipitation rate, logarithmic reflectivity, or linear reflectivity. Spatial continuity was found to vary significantly both in time and space in the various types of Alpine precipitation analyzed so far. At a separation distance of 10 km, the expected difference of reflectivity ranges from 4 dB*Z* (factor of 2.5 in stratiform rain or snow) to about 13 dB*Z* (factor of 20 in a mesoscale convective system). In a 96-h period of heavy rain in the southern European Alps, maximum variation occurred in upslope regions (frequent convection), while close to the crest of the Alps the variation was relatively weak (persistent stratiform rain). The representativeness of a point observation, which can be quantified given the variogram, therefore depends on both the time and the location within the Alps and also on the integration time (integrated rainfall maps being less variable than instantaneous ones). For a 576-km^{2} basin and 40-min average rain, the fractional error of the basin precipitation estimated by a gauge measurement ranges from 11% (variogram of stratiform autumn rain) to 65% (variogram of a mesoscale convective system). Next steps will extend the variogram analyses to a larger space–time domain toward a climatic description of spatial continuity of Alpine precipitation.

## Introduction

Statistical studies of spatial continuity of radar reflectivity, in particular by means of the variogram (or, alternatively, the autocovariance function), play a key role in many practical questions related to the natural variability of precipitation. We use the word *continuity* rather than *variation* to place emphasis on the fact that close measurements are similar. Owing to the continuity, we can compare measurements of different locations [e.g., weighted multiple regression of Gabella et al. (2000)], extrapolate point observations to neighboring areas (e.g., when producing rainfall maps from gauge network data), or estimate vertical profiles of reflectivity in partly shielded regions (profile correction). An introduction to the analysis of spatial continuity can be found in Isaaks and Srivastava (1989). The first part of this paper presents the technique applied to obtain variograms of radar reflectivity in Alpine precipitation. In the second part, we give examples of how they can be used to tackle the following questions:

What spatial variation of precipitation rate do we find in the European Alps?

How does the variation decrease when averaging in time and space?

To what accuracy can we estimate the total water input in a catchment using a point measurement?

How can we compare precipitation measurements from two or more instruments in different locations? Does an observed difference lie within limits of meteorological variability, or is it significantly larger and must be interpreted as an instrumental difference?

Are there preferred regions for convection in Alpine precipitation? How does the frequency and the type of convection depend on the topography?

Can we use the nugget variance, that is, the discontinuity observed at the origin of reflectivity semivariograms, to estimate the uncertainty of single radar measurements?

These questions, covering a wide range of meteorological and hydrological topics, are directly related to each other. The representativity of point measurements depends on the degree of spatial variation, which varies in time and space and as a function of the integration time, as well as on the uncertainty of single measurements. With the uncertainty of single measurements, we mean the difference between the measurement and the desired quantity (e.g., remaining scatter caused by signal fluctuations). If we find the spatial variation of precipitation to be much weaker close to the crest of the Alps than in upslope regions, we must consider this when interpreting gauge data in the Alps. In a similar way, the comparison of measurements made in different locations (we ask whether two instruments agree) and the combination (radar–gauge adjustment) require the knowledge of the natural variability of precipitation at a specific location and time. These relations are the reason why we discuss the questions together.

Related work goes back to the beginning of quantitative radar meteorology. Austin and Houze Jr. (1970) combined reflectivity patterns with variance spectra of gauge data to investigate the relation between mesoscale precipitation areas and phenomena on larger and smaller scales. Zawadzki (1973) proposed an optical device to determine the space, as well as the Eulerian and Lagrangian time, autocorrelation function (ACF) of radar images. For the analyzed storm, he found that the Taylor hypothesis (e.g., Stull 1993) holds up to periods of 40 min. If so, time and space can be exchanged, and we can compare, for example, spatial variation in images of volume-scanning radars with time variation of data collected with vertical-pointing radars, the advantage of the latter being the high resolution in time and the vertical direction.

One problem of radar–gauge adjustment is the scatter introduced by the differences between the sample volumes. Different approaches have been proposed to quantify this effect. Zawadzki (1975) takes ACFs obtained from gauga data of a 10-yr period; Kitchen and Blackall (1992) determine the difference between point and areal values directly from rainfall maps drawn by eye on the basis of high-resolution gauga data. Ciach and Krajewski (1999) assume that radar–gauge differences can be partitioned into the error of the radar estimate and the lack of representativity of the gauge. To estimate the two components, they propose a procedure that requires the ACF at scales below the size of the radar pixel. The nugget parameter of the exponential function used to model the ACF is estimated from data of five gauges a few kilometers apart. A similar problem arises when verifying satellite-derived rainfall estimates with gauge data. Flitcroft et al. (1989) established a regression model that calculates the correction factor relating point measurements to areal averages. The underlying data is from a dense gauge network experiment.

The representativity of a point observation for area-mean values is a function of the spatial continuity. Ripley (1981) derives equations for the error variance for several standard schemes of spatial sampling given the ACF of the regionalized variable. Rodriguez-Iturbe and Mejia (1974a) discuss three techniques to relate point rainfall of a certain level of probability to areal rainfall with the same level of probability. They approximate the space ACF needed for one of the techniques using exponential and Bessel-type functions (Rodriguez-Iturbe and Mejia 1974b).

Another group of applications of precipitation variograms, not included in the six questions listed above, is rainfall modeling (e.g., Waymire et al. 1984) and kriging (e.g., cokriging radar-rainfall and rain gauge data) as proposed by Krajewski (1987) or detrended kriging considering the elevation dependence (Garen et al. 1994). Variograms and related functions also play an important role in space–time downscaling of rainfall and its evaluation. The downscaling technique proposed by Venugopal et al. (1999) uses quantities that describe the multiplicative variation of precipitation and preserves both the temporal and spatial correlation structure of rainfall.

In this paper, we use high-resolution reflectivity data of well-visible regions (not shielded, free from clutter) to determine the range of variograms in precipitation in the European Alps. We focus on scales ranging from 0.5 km to tens of kilometers and from 5 min to a few hours (mesogamma). This leaves the way open for several practical applications. Because of the influence of the orography on precipitation physics, we expect a complex picture of variograms in the Alps. Section 2 gives an introduction to the variogram analysis. Section 3 presents the technique used to calculate variograms of radar reflectivity, followed by a set of typical variograms of Alpine precipitation in section 4. In section 5, we discuss the representativeness of point measurements, quantify the effect of spatial averaging (change of support), and briefly explain how variograms have been used for an instrument intercomparison experiment. Section 6 shows how variograms can be used in the context of orographic precipitation; we look for the regions of maximum orographic triggering of convection, given 96 h of rain in the Lago Maggiore region (southern Alps). In section 7, we estimate the nugget variance by extrapolating the semivariogram to zero lag and give a meteorological interpretation for it.

Gauges are scarce in the Alps and suffer from site-specific attributes such as wind-induced errors (e.g., Sevruk 1989), losses caused by the wetting of the instrument or evaporation, and malfunction of the instrument. Weather radars, on the other hand, provide a high resolution in space and time. If the regions are carefully selected, contamination by measurement errors can be neglected. To what extent the integration of gauge data into the variogram analyses discussed in this paper will add new information has to be shown in future work.

Here, data are from the operational C-band radar network of MeteoSwiss (Joss et al. 1998) and from the Mesoscale Alpine Programme field experiment (MAP; Bougeault et al. 2001). Problems and solutions when applying weather radars in an Alpine context are discussed, for example, in Joss and Germann (2000).

## Spatial continuity

Spatial and temporal continuity are intrinsic in meteorological data. Measurements from neighboring stations tend to be more similar than those further apart. Even strongly variable properties such as rain rate in convective showers, show some spatial continuity. In general, correlation occurs whenever the phenomenon exhibits frequencies lower than the sampling frequency. We intuitively use this fact when planning a measurement campaign or designing a network of ground stations. For synoptic meteorology in a flat country, there is no use in measuring absolute air pressure on a grid with 1-km meshes, because there will be almost no variation on this scale. A 20-km mesh is likely to be sufficient.

Describing and interpreting the spatial information of each science data is the main goal of geostatistics, initiated by the theory of regionalized variables of Matheron (1962a, 1963) and his colleagues at the French Mining School and by Krige (1951), after whom one of the most powerful interpolation techniques was named (Cressie 1990). Matheron and Krige were both looking for statistical tools to improve ore-deposit evaluation using point measurements. Some terms, such as the nugget effect, still refer to the mining roots of geostatistics.

Matheron (1962a) introduced the term *regionalized variable* (variable régionalisée) to underline the spatial aspect of earth science data. In Matheron (1963), he writes, “A regionalised variable is, sensu stricto, an actual function taking a definite value in each point of space.” In contrast to a random variable, its variation is to some extent correlated. On average, the difference between samples increases with increasing separation distance. Extrapolation becomes straightforward if we can describe or approximate the variation by a mathematical law (e.g., that of the power distribution of the radar beam within the pulse volume or the solar radiation at the earth's surface given the terrain model). Usually there is no strict law (or we just do not know it). Then spatial continuity can be understood as a tendency or a probability. For neighboring samples, similar values are more probable than large differences, even if there is no strict relationship. Spatial variation is also a function of the *scale* (champ géométrique) and the *support* (support géométrique) of the data. The first term describes the extent of the whole region in which data has been collected. The second relates to the size, shape, and orientation of the volume of each measurement (pulse volume in radar meteorology). Increasing the scale and decreasing the measurement volume both usually result in increasing variance, since additional variation is taken into account.

### The variogram

The *variogram* is a tool to quantify spatial continuity of regionalized variables. It goes back to Matheron (1962a) and is defined as the variance of the difference between two values as a function of the separation lag vector **h**:

where 2*γ*(**h**) is the variogram of the regionalized variable *Z*(**x**) with actual values *z*(**x**) and var{ } is the variance operator. It estimates the average squared difference between two measurements separated by a lag or, more generally, tells us what spatial variation we have to expect for a given scale and support (see Fig. 1).

The definition of the variogram assumes that *Z* is *intrinsically stationary,* which means two things. First, the expectation of *Z* is constant throughout the whole area (stationarity in the mean):

where *μ* is the population mean and *E*{ } is the expectation operator. Second, 2*γ*(**h**) depends on the lag **h** only and not on the position **x**; that is, for any subsample *V*′ of the whole region *V,*

If the region *V* is selected such that Eq. (3) is far from being fulfilled, Eq. (1) estimates a nonexistent model parameter, and the result depends on the position of the measurements and is only in part representative for the rest of the region. The variogram determined from a set of gauges all located in valleys is unlikely to be a perfect estimate of that in the mountains in between. With almost evenly spaced radar data, it is less critical; if Eq. (3) is not valid, we just obtain the average variogram of the region.

An alternate expression for the variogram can be derived by expanding the right side of Eq. (1) as

From Eq. (2) it follows that the last term vanishes, and we obtain

What happens when the first assumption, the stationarity in the mean [Eq. (2)], is not satisfied? Then, the right side of Eq. (4) equals the sum of the variogram (stochastic difference between two points) and the squared difference of the means (systematic difference). Whether this is desired depends on the application.

We can determine the variogram for different directions, allowing us to find whether spatial continuity shows preferred axes (anisotropy). Often, anisotropy is weak or of secondary interest, and an *omnidirectional* variogram 2*γ*(|**h**|) is all we need. The variogram is called *isotropic* if it only depends on the distance and not on the direction of the lag, that is, 2*γ*(**h**) = 2*γ*(|**h**|). Obviously, 2*γ*(**h**) = 2*γ*(−**h**), and 2*γ*(**h**) is always nonnegative. Of particular interest are the values of the variogram as the lag distance approaches zero. By definition, 2*γ*(0) is zero, and one might expect that as |**h**| approaches 0 the variogram also tends to zero. However, in reality, we can often observe a discontinuity at the origin, and *γ*(**h**) approaches *c*_{0} (nonzero) as |**h**| approaches 0. This effect is called *nugget effect,* and *c*_{0} is the *nugget variance* (Matheron 1963; Cressie 1993).

Two factors may be responsible for the observed discontinuity: microscale variability and measurement error. To illustrate the first, let us assume you have found a gold nugget. However close to the nugget you go, the probability of finding another one will never rise to 1. In fact, it will usually be low. The measurement error includes instrumental problems, causing a difference between the true and the measured value. Examples from weather radar systems are hardware deficiencies, signal fluctuations, or the system recovering after transmitting, which contaminates the measurements at close ranges. The separation of the two factors contributing to the nugget effect is not straightforward, unless one of them can be neglected.

### Covariogram and correlogram

The definition of the autocovariance function *C*(**h**) (covariogram) and the autocorrelation function *ρ*(**h**) (correlogram) requires similar assumptions as those made for the variogram [see Eq. (2) and Eq. (3)]: first, stationarity in the mean and second, for any subsample *V*′ of the whole region *V,*

where cov{ } is the covariance operator. This assumption is stronger than Eq. (3), because here, in addition to Eq. (3), the population variance must be finite and constant (second-order stationarity). If Eq. (2) and Eq. (5) hold, the semivariogram *γ*(**h**), the covariogram *C*(**h**), and the correlogram *ρ*(**h**) are related as follows:

where *σ*^{2} is the population variance. According to Eq. (6), at large lags, *γ*(**h**) tends to the population variance *σ*^{2} = var{*Z*}.

In practice, the variance is often ill defined, because the analyzed dataset is just a sample of the population. Extending the dataset usually results in a larger variance. The lower end (small lags) of the covariogram is directly affected, but that of the variogram remains unchanged. An important advantage of the variogram is the fact that, to calculate it, no assumption about the population variance is required. This is why we use the variogram rather than the more common autocovariance function.

### Additivity

Let *Z*(*p*) be a regionalized variable measured in the scale *A* with support *p.* The variance of any new variable derived by averaging *Z*(*p*) over support *a* (with *p* < *a* < *A*) is given by the difference of the variances of *p* inside *A* and *p* inside *a*:

Equation (7) is known as Krige's formula (Matheron 1963). Provided that *Z*(*p*) is intrinsically stationary [Eqs. (2) and (3)], *σ*^{2}(*p, **a*) only depends on *γ*(**h**) of *Z*(*p*) inside *a,*

and not on the number of samples of support *p* within *a* as for a random variable. The vector **h** is the lag between the two area elements **x**_{1} and **x**_{2}. Equation (8) defines the within-block variance needed to estimate the kriging variance in block-kriging (Webster and Oliver 1990). Substituting Eq. (8) in Eq. (7) yields

which we will use in section 5 to determine the effect of spatial averaging. By combining Eqs. (9) and (6), we obtain

Thus, the expected autocorrelation between two randomly chosen points in a block of size *a* is exactly the ratio between the variances of *Z*(*p*) averaged over blocks of size *a* and *Z*(*p*). Note that the integral term never becomes zero, because *ρ*(0) is equal to 1. Rodriguez-Iturbe and Mejia (1974a) propose to use the square root of the integral term

as a correction factor to transform point rainfall rate with a given return period [and variance *σ*^{2}(*p, **A*)] to areal rainfall rate with the same return period [and variance *σ*^{2}(*a, **A*)]. A more general discussion on change of support can be found in Isaaks and Srivastava (1989).

### The semivariogram and *h* scatterplots

*h*

Why is *γ*(**h**), known as the *semivariogram,* defined as one-half the variance of *Z*(**x**) − *Z*(**x** + **h**)? Where does the factor 2 in Eq. (1) come from?

The covariogram (section 2b) is basic to the minimization of the kriging variance (Isaaks and Srivastava 1989; Matheron 1962b). This is because the variance of a weighted linear combination of random variables distributed in space is a function of the covariogram and the weights [Eq. (16)]. Provided the population variance is constant and finite, the covariogram equals the difference between the population variance and the semivariogram [see Eq. (6)], and one can be replaced by the other. Traditionally, the semivariogram rather than the covariogram is estimated and modeled for kriging.

There is another argument for the factor 2 in Eq. (1): *γ*(**h**), the *semivariance* at lag **h**, can be interpreted as the normalized moment of inertia about the 45° line of an **h** scatterplot showing all pairs of data separated by lag **h** (Fig. 2). It is given by

Thus, the semivariogram is a vector of normalized moments of inertia, each element being a summary statistic of an **h** scatterplot. Note the analogy between *I*(**h**) and the moment of inertia about an axis of rotation in mechanics.

An example of an **h** scatterplot is depicted in Fig. 2. Data are from Monte Lema radar at 0.5° elevation, 30 September 1998, 1000–1100 UTC. A well-visible region of 914 km^{2} to the south of the radar site with no clutter problems, that is, no remaining clutter, has been chosen: azimuth between 135° and 251° at ranges between 9.5 and 31.5 km (Maggiore region, see Fig. 14). The reflectivity discretization of 0.25 dB*Z* results from averaging 12 plan position indicators (1 h) originally having 3 dB*Z* class width. For the depicted data pairs, we obtain *I*(1148 m) = 0.27 dB^{2}*Z.* This value, multiplied by 2, appears in Fig. 1 as a single point of the 60-min-average reflectivity variogram (see diamond). As the lag increases, so does the fatness of the cloud (compare Fig. 2 with Fig. 3), as well as the variogram (see Fig. 1).

The cloud of points in Fig. 3 looks asymmetric; more points lie below the line than above. The reason for this appearance, however, is of no interest. Each data pair separated by the given lag is only considered and plotted once. Whether it appears to the upper-left side or the lower-right side of the 45° line is just a question of order of data processing. The asymmetry is irrelevant for our studies.

## Estimating the variogram of radar reflectivity

According to the definition given in Eq. (1), the variogram of radar reflectivity is the variance of the difference between the reflectivity at two locations separated by the lag **h**. To estimate the variogram from a given dataset, Matheron (1962a) proposes

which is a bias-free estimator of Eq. (4). Because the sample mean is not needed, we divide by *N*(**h**) and not by *N*(**h**) − 1.

Whether we use linear precipitation rate (mm h^{−1}), logarithmic reflectivity (dB*Z*), or linear reflectivity (mm^{6} m^{−3}) for variogram analysis depends on the application, the variogram estimator, and the characteristics of the underlying data.

The classical variogram estimator of Matheron (1962a) using second moments [Eq. (13)] shows little robustness: nonnormality (e.g., skewness and heavy tails) and outliers (e.g., remaining clutter) seriously affect the result. Cressie and Hawkins (1980) and Hawkins and Cressie (1984) discuss the influence of nonnormality and outliers and propose a robust estimator instead of Eq. (13). A summary can be found in Cressie (1993). As an alternative to robust estimators, we may transform the data to normality, carefully select the data window, and check for outliers (e.g., by means of **h** scatterplots). For reflectivity, the difference *z*(**x**_{i}) − *z*(**x**_{i} + **h**) is approximately lognormal; therefore, Eq. (13) is a robust variogram estimator if using logarithmic reflectivity. Variograms obtained this way describe the multiplicative character of variation. In Fig. 2, the fatness of the cloud, and thus the multiplicative variation of rain, is approximately independent of the intensity, which is an example of multiplicative behavior of precipitation.

In section 5a, however, we need variograms of linear precipitation rate, the probability distribution of which is highly skewed. Then the poor robustness of Eq. (13) can be overcome as follows: 1) select a representative dataset such that the tails of the distribution are well defined and no outliers are present and 2) average in time to reduce the skewness of the data (central limit theorem).

### Database and processing

Why do we use polar data (plan position indicators with 1 km × 1° × 1° resolution) for variogram analysis? First, polar data are clutter filtered and, at close ranges, provide high spatial resolution, which is required to estimate the variogram at short lags. Second, polar data have the same support as the original measurements. Resampling, interpolation, and averaging required for the deviation of Cartesian products, such as horizontal sections and vertical maximum projections, would affect the estimation of the variogram. Third, the knowledge of spatial variation of reflectivity helps to design profile correction schemes, most of which are also based on polar data. The methods presented in recent papers work with measured, retrieved, and idealized profiles determined and applied on *various* scales, ranging from climatological profiles (long-term averages) that depend on the season or the rain type to several kinds of mesoscale profiles [for an overview, see, e.g., Smyth and Illingworth (1998)]. Because the shape of the profile is correlated with reflectivity (Fabry and Zawadzki 1995) and its variation (Germann and Joss 1999), we suggest to use variograms to investigate the representativity of the profiles on these various scales and, further, to check the validity of the assumption of spatial homogeneity of the profile made in Andrieu and Creutin (1995) and Vignal et al. (1999). Studies of this kind, however, are beyond the scope of this paper.

We know that meteorological radars have both sources of radial (azimuth dependent) and concentric (range dependent) error patterns; beam-blocking is the most important source of the first type of error while ground clutter, the dependence of the pulse volume on the distance, and the vertical change of reflectivity, including brightband effects, belong to the second group. As a result, if variograms were calculated separately for different azimuths and distances, we would observe “polar” anisotropy. The sources of errors listed above also affect omnidirectional variograms; beam-blocking, mountain returns, and the vertical change of reflectivity result in overestimating the variogram, because artificial variation of reflectivity is added to the meteorological one. Changing the pulse volume by varying the distance to the radar means changing the support, with larger supports having smaller variance (see Krige's formula and section 5b).

We can avoid these problems 1) by selecting an appropriate space–time window in which the errors can be neglected and 2) by correcting for them. If, for example, the height of the measurements varies too much (high elevation angles and/or large range intervals), polar data must be corrected for the vertical reflectivity profile before variogram estimation. Close to the radar at low elevation angles, the vertical extent of the pulse volume and the variation of height within the space window are small. Furthermore, if the 0° isotherm is considerably higher than the selected data, the influence of the profile is negligible.

There are also meteorological reasons for anisotropy such as, for example, parallel rain bands or alignment of cells. In these cases, it might be interesting to look at variograms obtained for several azimuth intervals. Since we observed no dominant anisotropy, we used omnidirectional variograms 2*γ̂*(*h*) = 2*γ̂*(|**h**|) with the advantage of having more data pairs for variogram estimation.

In contrast to other statistical techniques, variogram analysis does not require data gap filling. The variogram is estimated on the basis of all valid, clutter-free, and well-visible data pairs; missing data are ignored. Daily clutter maps and long-term rain totals provide information on the visibility of a region of interest (Joss and Lee 1995).

We are interested in reflectivity variation *within* rain rather than at the borders. Therefore, we restrict our analysis to regions with reflectivity greater than 13 dB*Z* and select space–time windows that provide a minimum percentage of samples above this threshold (here 50%).

The distance between two measurements is determined from the centers of the pulse volumes. The term 2*γ̂*(*h*) is calculated from all data pairs with lag distance |**h**| ± 0.4 dB (±10%). The class width of 0.8 dB has been selected such that each class is based on a sufficient number of samples; however it is small enough to analyze small-scale variability of reflectivity, our major interest. Since the radial resolution of the underlying data is 1 km, at lags smaller than 1 km, 2*γ̂*(*h*) is based on data pairs at constant range. Such small lags, for instance, are required to estimate the decrease of the variance resulting from averaging a few neighboring pixels (see section 5b). As a reasonable maximum lag, we propose one-fourth of the extent of the space window.

### Time averaging

A common technique to reduce the stochastic part of any type of error is averaging in time. Its effect on reflectivity variation can be quantified by calculating variograms for different averaging intervals. Say we want to know the multiplicative variation of reflectivity at two points separated by a certain lag as a function of the averaging time. Figure 1 depicts the variograms of raw polar data and 10-, 20-, 40-, and 60-min-average reflectivity during stratiform rain in September of 1998. Time resolution of Monte Lema data is 5 min, thus 10, 20, 40 and 60 min correspond to averaging 2, 4, 8, and 12 images, respectively.

If the data pairs were independent, the variance would decrease with 1/*N,* where *N* is the number of integrated images. Variation at small lags and between successive volume scans is of the same order of magnitude. Therefore, the difference of reflectivity measured at two locations is almost independent from the one in the previous scan. In fact, at lags below 1 km, the variograms in Fig. 1 approximately decrease with 1/*N.* Note that this depends on the temporal and the spatial resolution of the underlying data. Rainfall is well known to have structure down to much smaller scales (Fabry 1996). Data pairs sampled with a higher resolution in time, for example, would be correlated, and the variance would decrease less rapidly. As a consequence of the decrease with 1/*N* at small lags, averaging a few pixels in time and/or space efficiently reduces the sample variance, and we obtain a better estimate for comparison with other point measurements. At larger lags, correlation in time becomes important, and time averaging is less efficient; 2*γ*(10 km), for example, decreases with 1/*N*^{*}, where *N*^{*} = *N*/(0.78 + 0.22*N*).

Again, **h** scatterplots provide deeper insight into the data and help to interpret the variogram. Using the same data and lag, we calculated **h** scatterplots with raw as well as 10- and 20-min-average reflectivity (Figs. 4, 5, and 6). The semivariance decreases from 2.61 dB^{2}*Z* with raw data to 1.32 dB^{2}*Z* (10-min average) and 0.69 dB^{2}*Z* (20-min average). (See also the triangles in Fig. 1.)

## Variograms of Alpine precipitation

Figures 7 and 8 depict a set of typical variograms of Alpine precipitation. We selected a stratiform event from winter 1997 and autumn 1998 and a mesoscale convective system (MCS) from the summer of 1998. Variograms of measurements in snow have been calculated for both the autumn and winter cases. For the autumn case, we used 4.5° elevation data of the Monte Lema radar [1625 m above mean sea level (ASL)] at 20-km range (3.2 km ASL, 500 m above the melting layer) and a region with a small radial extent (2 km × 116°) to avoid errors caused by the vertical reflectivity profile (see section 3a). The other five variograms base on 0.5° elevation data allowing a region with a much larger radial extent (Maggiore region in Fig. 14: 22 km × 116°, 1.8 km ASL). In the winter case, the 0° isotherm fell from 1.2 km down to 900 m and was well below the selected measurements at 0.5° elevation. The 1-h period of the MCS is at the time of maximum convection, and the 5-h period almost covers its whole life cycle in the selected region.

The variograms differ by more than a factor of 10. At a separation distance of 10 km, the expected difference of reflectivity during the hour with maximum convection of the MCS is on the order of 13 dB*Z*; on average within the MCS, 7.6 dB*Z*; in the stratiform rain, 4.3 dB*Z*; and, in the stratiform snow, 3.8 dB*Z.* Note the small difference between the two variograms in freezing conditions.

## Representativeness of point measurements

In this section we determine the accuracy of areal precipitation estimated from point measurements such as gauge data, quantify the effect of spatial averaging using Krige's formula (section 2c), and briefly communicate an idea about how variograms may help to interpret measurements from instruments in different locations.

### Mean basin precipitation

Let *G* be the precipitation rate measured with a gauge over period *t.* Say this single measurement is used to estimate the rate *R* averaged over a drainage basin of size *B* and period *t,* assuming a constant rate in *B.* We now divide the basin in *N* pixels of the same size as the support *p* for which we determined the variogram (Maggiore region: 0.2 km^{2}). Let *r*_{i}, with *i* = 1, …, *N,* be the true rate of the *N* pixels, the gauge lying in the pixel with *r*_{1}. Further, let *r̂*_{1} be *r*_{1} estimated from the gauge measurement using *r̂*_{1} = *G* with error variance *s*_{r̂1}^{2}. We assume that, for the selected space–time window, precipitation is intrinsically stationary (section 2a). The true areal precipitation rate is defined by

and is estimated from *R̂* = *r̂*_{1}. To find out how accurate *R̂* is, we use the expected squared difference *s*_{R̂}^{2} between *R* and *R̂,* which is equal to the variance of *R* − *R̂* because *R̂* is unbiased.

Using the equation for the variance of a weighted linear combination of random variables [e.g., Isaaks and Srivastava (1989), p. 217],

where *w*_{i} are the weights and *X*_{i} are the random variables, we can write

The argument of the first term is itself a linear combination of random variables; therefore,

The second term of Eq. (17) is

In case *G* is uncorrelated with *R,* the last term of Eq. (17) becomes zero and *s*_{R̂}^{2} is simply the sum of Eqs. (18) and (19). A 10-min measurement of a gauge during a thunderstorm, for example, often exhibits poor correlation with the average rain in a larger area. However, we usually select the period and the area such that the correlation is significant. Then *s*_{R̂}^{2} decreases by the variance explained by *G,* that is, by the third term of Eq. (17):

We can now rewrite Eq. (17) as follows:

If the support *p* is small and the integration time *t* is long, *s*_{r̂1}^{2} can be neglected, and we obtain the final expression for the expected squared error

or, alternatively, as a function of *γ*(**h**) of *r*(*p*)

where **h**_{ij} is the separation vector between the two area elements *i* and *j.* The first term is twice the expected value of the semivariogram between the gauge and an arbitrary point in the drainage basin and quantifies the representativity of the gauge for another point in the area. The second term is the expected value of the semivariogram between two randomly chosen points in the basin, correcting for the fact that the variance of the basin precipitation decreases with increasing area. The difference between the two gives the desired quantity.

Now we use Eq. (23) together with variograms of linear rain rate of the two events “stratiform autumn” and “MCS” discussed in section 4. Data are from the Maggiore region, Fig. 14. The rain rate has been averaged over 40 min, a reasonable period for hydrological applications. Values smaller than 0.16 mm h^{−1} have been omitted for variogram estimation because they are, for the most part, caused by clutter elimination and are not true zero-rain pixels. Thus, true zero rain is also ignored, and the resulting *s*_{R̂}^{2} estimates the lack of representativeness of a nonzero point measurement for a nonzero rain area. In the case of isolated thunderstorms, zero-rain areas become relevant and must be considered. Then, however, the important question may be different from the one discussed here, for example: How accurately can we estimate the *maximum local* 1-h rain amount from a set of point measurements?

Long-term accumulations show that, in the Maggiore region, variations in the mean can be neglected when compared with the variations in 40-min-average fields. Thus, Eq. (2) is fulfilled, and the calculated variograms describe the spatial variation of precipitation as defined in Eq. (1). Any nugget variance caused by measurement error is undesirable and must be subtracted from the observed variogram. Table 1 lists the values of the variograms at a lag of 5 km and the mean and variance of the underlying data.

We defined two rectangular basins with 576 and 144 km^{2}, respectively (Fig. 9). The size of the large one reflects the density of fast-response gauges in the southern Swiss Alps. The expected error of the gauge estimate of the average 40-min precipitation in the basin depends on the precipitation type (variogram), on the size and the shape of the basin, and on the location of the gauge. Table 2 shows the results. Together with Table 1, we find the following. First, the fractional error (*s*_{R̂}/mean) ranges from small (11%, variogram of stratiform autumn rain) to serious (65%, variogram of MCS rain, 1 h). On average, in the MCS (variogram of MCS 5-h period), the fractional error is 43%. Second, using the MCS variograms, we obtain, as one would expect, a smaller error for the smaller basin. In the stratiform autumn rain it is the opposite. This because the first term of Eq. (23) increases less than the second when switching from the small to the large basin. In other words, in the stratiform autumn rain, the uncertainty introduced by larger distances is less important than averaging over a larger area, which reduces the overall variance. Third, when discussing the representativeness of a gauge measurement, one usually bears in mind the first term of Eq. (23) but often misses the importance of the second.

The next steps will be to extend this analysis to larger integration times (e.g., daily precipitation), to consider *a set* of point measurement for area-mean estimates, and, by means of daily clutter maps, to distinguish between true zero rain and zero pixels that are caused by clutter elimination. The results will form an important basis of error analysis in climatological studies, such as the precipitation climatological description of the European Alps by Frei and Schär (1998).

### Spatial averaging

From Eq. (9), it follows that the second term of Eq. (23) is exactly the amount of variance reduced by averaging *r*(*p*) over the basin area *B.* Thus, the second term of Eq. (23) can be used to determine the effect of spatial averaging. Results for three square boxes and the same three variograms as above are listed in Table 3. Comparing these values with the variance of the underlying data given in Table 1 yields the relative reduction of variance. Averaging over the 16 km^{2} box, for example, results in a reduction of the variance by 8%, 28%, and 21%, respectively.

Another example of spatial averaging is the vectorial sum of the echoes of all scatterers in the pulse volume. We may ask about the influence of the increase of the pulse volume with range on the determination of the variogram. Within the Maggiore region, the size of a 1° pixel ranges from 0.17 km × 1 km (at 9.5 km) to 0.55 km × 1 km (at 31.5 km). Measurements at long ranges, being an average over larger pulse volumes, have smaller meteorological variance. Both a theoretical and an experimental analysis show that this influence can be neglected; for the theoretical estimate of the reduction of the variance, we again use Eq. (9). Here, variograms of *linear reflectivity* are needed, because reflectivity is linearly averaged in the pulse volume. The nugget variance attributed to signal fluctuations must be subtracted before using *γ*(**h**) in Eq. (9). Let *p* and *a* of Eq. (9) be the support at 9.5 and 31.5 km, respectively. In precipitation represented by the variogram of the MCS 5-h period, for example, *σ*^{2}(*p, **a*)/*σ*^{2}(*p, **A*) becomes 0.06; that is, the variances of measurements of linear reflectivity at 9.5 and 31.5 km would differ by roughly 6%. A second estimate of the influence of the size of the pulse volume is obtained with the following experiment: calculating variograms of two regions at ranges between 9.5 and 20.5 km and between 20.5 and 31.5 km, respectively, reveals small differences (<0.3 dB^{2}*Z*), much smaller than those found between the various precipitation types (tens of dB^{2}*Z*; see Fig. 7). To ensure that we observe the same precipitation in both regions, we select a 24-h period of stratiform rain and define two equally sized regions on a line along echo movement.

In section 6, we compare variograms of the same time period but of different regions. Take, for instance, the regions of Gridone and Vigezzo (Fig. 11) lying at 17 and 34 km, respectively. Using again Eq. (9) and the variogram of linear reflectivity of the MCS 5-h period, we obtain a decrease of the variance of approximately 7%, which is again negligible.

### Instrument intercomparison

The last topic of this section is the comparison of measurements of two or more instruments in different locations. Here, we just briefly describe the idea. We want to know whether an observed difference lies within the limits of meteorological variability or is significantly larger and hence must be interpreted as an instrumental difference. The uncertainty introduced by the fact that two observations are from different locations and precipitation varies in space can be estimated by means of reflectivity variograms. This approach has been followed to interpret the differences between vertical reflectivity profiles of two volume-scanning and two vertically pointing radars operated in the vicinity of Monte Lema during the intercomparison experiment preceding the special observing period of MAP (Bougeault et al. 2001).

## Upstream and upslope variograms

Stratiform precipitation is characterized by a homogeneous intensity distribution and weak horizontal gradients. Convection, on the other hand, typically produces regions with high reflectivity and strong horizontal gradients reflecting the alternation of updrafts and downdrafts (Houze Jr. 1997). Steiner et al. (1995) proposed a classification scheme based on horizontal reflectivity gradients to separate stratiform and convective rain. Here, we use reflectivity variograms of Alpine precipitation to find the regions with maximum orographic triggering of convection.

We compare variograms of regions in upstream and upslope conditions during heavy precipitation in the southern Alps. Six days in September of 1999 [17–21 and 25–26 September, i.e., MAP intensive observing periods IOP-02a, IOP-02b, and IOP-03; Bougeault et al. (2001)] with 96 h of rain have been selected. The gauge at Locarno–Monti registered a total amount of 394 mm. Operational wind profiles of Monte Lema radar (Germann 1999) provide information on the main wind flow (Fig. 10). During the selected period at 2 km ASL, the wind blew from SSE at 5 to 20 m s^{−1}, at 5.6 km from SSW with 10 to 40 m s^{−1}. Reflectivity profiles of Monte Lema and two vertically pointing radars [the S-band orographic precipitation radar (OPRA) from the University of Washington located at Locarno-Monti, and the X-band radar from the ETH-Zürich located at Macugnaga; Fig. 11] as well as radiosonde data of Milan are used to determine the range of height of the melting layer (Fig. 12).

Figures 11 and 13 depict the selected regions and the corresponding variograms using Monte Lema data. The inner three regions (Gridone, Valcolla, and Campo dei Fiori) are used together with 1.5° elevation data, the outer three with 0.5° data (Fig. 12: variograms in rain). They have been selected such that shielding and clutter contamination can be neglected and the highest measurement is considerably below the melting layer (top of 3-dB beam at 1.5° and 24.5 km is at 2.5 km ASL, whereas the bright band never fell below 2.7 km ASL; cf., section 3). The numbers of valid data pairs (both values > 13 dB*Z* and rain area > 50%) used to calculate the six variograms are listed in Table 4.

It might seem contradictory on the one hand to postulate intrinsic stationarity (section 2a) and on the other hand to look for different variograms in neighboring regions. It is not. First, it is sufficient if the regions for which we determine the variogram are small when compared with the overall extent of the analysis. Then, we found the variations of *μ* and *γ*(**h**) within the regions to be negligible. Second, in this section, the emphasis is on ranking variograms rather than on their absolute values.

The influence of the increase of the pulse volume with range on variogram estimation is considered to be small when compared with the discussed signal. In section 5b, we find the resulting difference of the variances in the regions of Gridone and Vigezzo to be approximately 7%. The estimate is based on data of the MCS 5-h period, which, to a first approximation, represents precipitation of the 96-h period discussed here. Range influences can be avoided by comparing variograms of regions in the same range interval (e.g., Gridone, Valcolla, and Campo dei Fiori).

In the Gridone region, reflectivity varied significantly more than elsewhere. The average squared difference between two measurements separated by a given lag is about twice as large as in upstream conditions (Sesto Calende region). This result indicates that it has been a preferred region for convective processes. High variation means also low representativeness of point measurements and difficulties in runoff modeling. In general, we observe an increase of variation downwind (see last column of Table 4).

To extend this analysis toward the crest of the Alps, we have to restrict the period to events with high 0° isotherms or, alternatively, to compare variograms calculated with data from above the melting layer (Fig. 12). Note that aloft the conditions are different; in the snow, variation is generally lower than in rain. In tall convective cells where large rain drops (and hail) are carried up, however, cores of high reflectivity are surrounded by weak echoes from snow, resulting in strong gradients. The convection triggered by the latent cooling of the air during melting (Atlas 1955; Atlas et al. 1969; Houze Jr. 1997) leading to fallstreaks from the bright band down to the ground, however, is not found in the snow region above.

During the six days, the height of the melting layer was too low to have measurements in *rain* across the Alps; consequently, we have to use data from aloft in the snow (Fig. 12). The selected regions are shown in Fig. 14 and the results are shown in Fig. 15 and Table 5. Maximum variation is again found in the same upslope area, here the region of Càmedo. We conclude that, in this area, upslope triggering of convection plays an important role in precipitation generation. The upstream regions at the edge of the plain of Padana take places 2 to 5 (upstream triggering of convection). The second column of Table 5, which shows the number of pairs in percent as a fraction of the total possible number, indicates that upstream rainfall was of short duration. It is interesting to note that the regions close to the main divide (Sempione and Piora) are characterized by weak reflectivity variation.

For comparison, Fig. 16 shows the square root of the values at 5-km lag of all variograms (below and above the melting layer). These values can be interpreted as the average difference (dB*Z*) between two points separated by 5 km. At a distance of 5 km, we expect large gradients in convection. Further, 5 km is small enough to obtain an accurate estimate of 2*γ*(*h*) for all regions. The same lag has also been selected for the Tables 4 and 5.

Figure 17 summarizes the results of this section, comparing variation of reflectivity (second moment) with gauge totals (first moment) along a section from SSE to NNW. Simplified for the 96 h of heavy rain, we can say that maximum variation approximately coincides with maximum precipitation amount (or with steepest gradient in precipitation?). To the south, variation is moderate and gauge totals are small (isolated upstream convection); to the north, variation is weak and gauge totals are moderate to high (persistent stratiform rainfall). Similar results are presented by Harris et al. (1996), who studied multifractal characteristics of orographic precipitation at different locations (gauges) along a transect from the coast to the main divide of the southern Alps of New Zealand.

A comparison with the partition of precipitation in stratiform and convective as output of mesoscale NWP models is planned. We want to know how correctly the model predicts the importance and the location of convection. Such a comparison, however, will be limited in space because of visibility problems, the vertical change of reflectivity, and range influences (see sections 3a and 5b). The limited visibility in topographically complex regions is addressed, for example, in Joss and Waldvogel (1990) or Westrick et al. (1999).

## Nugget variance

It is important to realize that the nugget variance of a regionalized variable *Z*(*p*) has nothing to do with var {*Z*(*p*)}. The variance of *Z*(*p*) is the value to which *γ*(**h**) tends as |**h**| approaches ∞ (see triangle in Fig. 8; cf. section 2). The nugget variance, on the other hand, is the discontinuity at the origin of the semivariogram [i.e., *γ*(**h**) as |**h**| → 0 (Fig. 18)]. It is one-half the average squared difference of two measurements separated by an infinitesimally small distance and thus represents the stochastic uncertainty of a single measurement. Only the stochastic part of the uncertainty contributes to the nugget; any systematic difference between the measured and the true reflectivity has no influence.

If we linearly extrapolate the variogram at small lags to zero lag, we get an estimate of the sum of the measurement error and small-scale variability. By small-scale variability, we mean the variation at lags smaller than the smallest separation distance in the selected region of samples of the size of the pulse volume. For more details, see the discussion of the scale of variation in Cressie (1993). Examining 12 variograms of logarithmic reflectivity of nine precipitation events to the north and south of the Alps, we found nugget variances ranging from 0.3 to 1.9 dB^{2}*Z.* The average value is 1.0 dB^{2}*Z,* providing a *rough* estimate of the average stochastic uncertainty of single measurements. Here, single measurements are polar pixels of 1 km × 1° × 1° at ranges between 9 and 32 km; the 3-dB beamwidth is 1°. Further averaging typically done for the generation of Cartesian products reduces this value.

The inphase and the quadrature-phase components of the vectorial sum of the complex signals of many drops moving in a random way in the pulse volume are normally distributed (central limit theorem). The standard deviation of logarithmic power of signals with normally distributed inphase and quadrature-phase components is 10*π*(6)^{−1/2} log*e* = 5.6 dB (Marshall and Hitschfeld 1953). If these fluctuations are the dominant factor contributing to the nugget variance, we can use it to estimate the number of independent samples in one pixel. The value of a polar pixel is the average of 396 samples, 33 pulses in azimuth (the center of the beam scans one beamwidth) times twelve 83-m gates in range (postdetection averaging). However, these samples are not independent. The square of 5.6 dB divided by 1.0 dB^{2} from above gives an average number of independent samples in one polar pixel of about 30. For a nugget variance varying between 0.3 and 1.9 dB^{2}*Z,* we get 103 (convective) to 16 (stratiform) independent samples, a reasonable result.

The variograms of Fig. 18 let us speculate on a correlation between the degree of variation and the nugget variance, convective precipitation having smaller nuggets. A reasonable explanation could be wind shear and turbulence, which are typically stronger in regions with convection. As a result the decorrelation time is shorter, and the average of successive samples in azimuth and range fluctuates less than in situations with a long decorrelation time. A more detailed analysis is needed to confirm or reject this speculation.

## Conclusions

Spatial variation of Alpine precipitation has been quantified by means of variogram analysis using high-resolution reflectivity data. In sections 2 and 3, we presented the background of this geostatistical technique and discussed the specific problems arising when dealing with radar data. A set of typical variograms of Alpine precipitation has been given in section 4. To summarize the answers to the questions listed in the introduction and discussed in the sections 4–7 are as follows:

The range of variation observed in the various types of Alpine precipitation is large. At a separation distance of 10 km, the expected difference of reflectivity ranges from 4 dB

*Z*in stratiform to about 13 dB*Z*in convective rain (Fig. 8).During the heavy rain in September of 1999, variation in terms of logarithmic reflectivity was weaker above the melting layer than below (Fig. 16 and Tables 4 and 5). Above the melting layer, 2

*γ*(5 km) varied from 14 to 46 dB^{2}*Z,*below in the rain it varied from 46 to 91 dB^{2}*Z.*Examining the variance in terms of logarithmic reflectivity is equivalent to looking at the coefficient of variation of linear reflectivity (ratio of the standard deviation to the mean). Thus, even by correcting measurements from aloft by the factor mean(below)/mean(aloft), we cannot fully explain the variation observed at the ground. That is to say, measurements from aloft in the snow only in part reflect what happens at ground level. This fact is an intrinsic limitation of linear profile correction schemes.Averaging is a common method to reduce the variance. We used variograms to quantify the reduction of variance when averaging measurements in space, integrating radar images in time, or increasing the pulse volume.

Averaging 40-min–average stratiform rain over 16, 64, and 256 km

^{2}reduces the variance by 8%, 13%, and 26%, respectively (Tables 1 and 3). In precipitation represented by the variogram of the MCS 5-h period, the relative reduction is larger and amounts to 28%, 44%, and 69%, respectively.During a 5-h period of a mesoscale convective system, the square root of 2

*γ*(10 km) decreases from 7.6 dB*Z*using raw data to 2.9 dB*Z*for 60-min-average reflectivity (Fig. 7). This change corresponds to a factor decreasing from 5.8 to 1.9.Because of the increasing pulse volume, the variance of radar measurements decreases with increasing range. To estimate the order of magnitude, we used Krige's formula together with a variogram of linear reflectivity. In precipitation represented by the variogram of the MCS 5-h period, the variances of measurements at 9.5 and 31.5 km, for example, would roughly differ by 6%. Since the variograms of Fig. 7 are based on data between the ranges of 9.5 and 31.5 km, we conclude that the influence of the pulse volume on the derived variograms can be neglected when compared with the observed differences between the various precipitation types.

As a consequence of findings 1 and 2, the representativeness of a point observation strongly depends on the type of precipitation (on the time and the location), on the integration time, and on the size of the basin. For the 576-km

^{2}basin and a 40-min period, the fractional error, that is, the expected difference between a point observation and the average basin rate normalized by the overall mean, varies from 11% (variogram of stratiform autumn rain) to 65% (variogram of MCS) (see Tables 1 and 2). In other words, the lack of representativeness of a gauge measurement for the average rainfall in a drainage basin ranges from negligible to serious. Analogously, we can calculate the fractional error for any variogram and basin.The variogram can be used to estimate the meteorological uncertainty for a given separation distance (section 5c). Say, for raw data and 60-min-average reflectivity, the square root of 2

*γ*(10 km) is 7.6 and 2.9 dB*Z,*respectively (see finding 2). Hence, when comparing hourly reflectivity profiles of this type of precipitation of two vertically pointing radars 10 km apart, one would still have to expect differences of about 3 dB*Z.*Given the 7.6 dB*Z*for raw data, comparing instantaneous profiles would be foolish.We used variograms to look for preferred regions for convection during 96 h of heavy rain in the southern Alps in September of 1999 (MAP special observing period). We found that close to the crest of the Alps variation was weak (persistent stratiform rain), while maximum variation (frequent convection) occurred in upslope regions (Figs. 13 and 15).

We determined the nugget variance of 12 variograms (for examples, see Fig. 18). We found values between 0.3 (convective) and 1.9 dB

^{2}*Z*(stratiform), giving a rough estimate of the stochastic uncertainty of single radar pixels.

The high-resolution information of radar images is unique for analyses of the spatial continuity of precipitation. The results show that, even under the difficult conditions in a mountainous region, radar data give quantitative answers to practical questions related to the spatial continuity. The next steps will be the integration of more events and the extension to a larger domain incorporating other radars and gauge data.

## Acknowledgments

This research was supported by MeteoSwiss and the Swiss NSF under Grant 21-52355.97. The authors thank Gianmario Galli for the fruitful discussions and Remo Cavalli for supporting our research. The authors also thank Bertrand Vignal and two anonymous reviewers for insightful and helpful comments that contributed to many improvements of this paper. For the two vertically pointing radars operated at Locarno-Monti in summer 1999, thanks go to the mesoscale group of the University of Washington and to the radar group of the ETH, Zürich.

## REFERENCES

## Footnotes

*Corresponding author address:* Urs Germann, Dept. of Atmospheric and Oceanic Sciences, McGill University, 805 Sherbrooke St. W., Rm. 945, Montréal, PQ H3A 2K6, Canada.urs@zephyr.meteo.mcgill.ca