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Keywords:

  • rainfall regime;
  • spatial distribution;
  • regression analysis;
  • kriging;
  • Fourier series;
  • Italy

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

We propose the use of Fourier series for representing the precipitation regime in a certain location and predicting it in ungauged locations, allowing for map production. We analyse monthly average precipitation data of 2043 gauging stations covering the Italian territory. The Fourier series allows to represent a curve as a sum of different sinusoidal components characterized by their period, amplitude and phase. Being the different harmonics not correlated, it is possible to fit them with stepwise multiple linear regressions. The Fourier series allows for a parsimonious representation of the regime, being usually the 12- and 6-month harmonics able to reproduce the observed values with little residuals [in this exercise the fitting gave an average monthly root mean square error (RMSE) of 9.21 mm and a correlation coefficient of 0.979]. Once the at-station harmonics parameters are obtained, it is possible to map them for predicting the regime in ungauged locations. Here we use ordinary kriging and the leave-one-out validation scheme for evaluating the amplitudes and phases of the harmonics of the 12- and 6-month periods and reconstructing the precipitation regime. We use the same scheme for the interpolation of the station data on a month-by-month basis, whose results are used as a benchmark. The analyses provide similar results, with overall RMSEs of 17.53 and 15.97 mm and correlation coefficients of 0.909 and 0.921, respectively. The spatial patterns of the reconstruction error are similar for the two cases. The stations having higher RMSE are clustered in the areas presenting high precipitation gradients, such as in the Appennines, or where major precipitation regime changes occur. For demonstrating that the Fourier series approach is more suitable for regionalization purposes, a k-means cluster analysis on the Fourier parameters was performed and the effect of such stratification on the mapping of the precipitation regime by applying regression kriging was assessed. Copyright © 2010 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

The problem of assessing the spatial behaviour of variables measured in a limited number of locations, and mostly at the point scale, is of major interest for several science branches. The advances in computer sciences and geographical information systems' tools, as well as the increased availability of spatial data, have facilitated and stimulated the research efforts in this field.

Despite the most common techniques being developed several decades ago, the discussion on definitions, ranges of applicability and standards for assessing the quality of the estimates and for representing the results is still open and productive. For instance, only recently Hengl et al. (2007) demonstrated that universal kriging and regression kriging (RK) are mathematically equivalent; moreover, it is common that authors using RK do not report the relative weight of the two components in the determination of the final result, or make use of predictors that, despite being statistically significant, are not meaningful for the process under investigation.

Large-scale spatial variability of climatic variables, such as temperature or rainfall, has recently received considerable attention (e.g. Zheng and Basher, 1996; Nalder and Wein, 1998; Prudhomme and Reed, 1999).

In several studies, the main efforts are spent in maximizing the efficiency of statistical spatial interpolation techniques (e.g. Nalder and Wein, 1998; Lapen and Hayhoe, 2003). However, the approaches that investigate the role of a given predictor on the spatial variability of a climatic variable have the advantage to provide a major insight in the processes controlling the phenomenon; moreover, the physical consistency of such estimates allows the recognition of potential spurious samples in the base data, which could not be detected by means of purely statistically driven estimates. Considering, for instance, the studies on air temperature (see e.g. Zheng and Basher, 1996; Agnew and Palutikof, 2000; Ninyerola et al., 2000; Gyalistras, 2003; Claps et al., 2008), the authors start from the assumption that elevation and latitude already explain much of its spatial variability and then proceed to evaluate other factors that can have a significant influence on it, including the position of the site with respect to seas and continents and, at small scales, terrain attributes (aspect and morphology), atmospheric factors (humidity, precipitation and wind) and maritime factors (configuration and aspect of coasts and effects of sea currents).

When dealing with climatic regimes, here meant as the sequence of the average monthly values of a climatic variable, it is usual to process the data separately. As a result, a separate model is defined for each month (see e.g. Zheng and Basher, 1996; Attorre et al., 2008; Fiorenzo et al., 2008). The separate monthly models, despite being able to represent thoroughly the data, have the major drawback of not being consistent within the regime, e.g. using different explanatory variables in regression models or different semi-variogram models.

In order to overcome this problem, Claps et al. (2008) proposed to represent the temperature regime at gauged locations by means of Fourier series and to obtain its spatial representation by interpolating the Fourier series parameters by means of regression with geographic and morphometric covariates. The amplitudes and phases of the harmonics of 12- and 6-month periods used in that study were able to represent the regime and demonstrated clear relations with the proposed covariates. The mean root mean square error (RMSE) was 0.53 °C, which is comparable to the results obtained by Attorre et al. (2008) and Fiorenzo et al. (2008) for Italy and Basilicata, respectively.

The scope of this article is to highlight the effectiveness of the Fourier series approach for representing the regime of climatic and environmental variables in general, of precipitation in particular, and as a support for map production.

After presenting the Fourier series framework and providing some hints on its usage, we evaluate its capabilities based on the rainfall regime of 2043 gauging stations in Italy.

The Fourier series potential in fitting the at-station observed regime is first assessed. We present the classification of the Italian precipitation regime as obtained by means of unsupervised clustering of the Fourier parameters.

We evaluate the effectiveness of the Fourier series approach in the framework of the estimation of the precipitation regime in ungauged locations. The monthly estimates obtained by reconstructing the series after interpolating the Fourier parameters by means of ordinary kriging and the leave-one-out scheme are compared with those obtained when processing the monthly data directly within the same interpolation scheme.

Finally, we investigate the possibility to make use of geographic and morphologic variables extracted from a digital elevation model within regression and RK frameworks, and we assess the skill of the stratification based on the regions previously obtained in improving the performance of such schemes.

2. Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

The curves representing the regime of climatic variables can be reproduced by means of Fourier series as the sum of sinusoidal curves having different periods:

  • equation image(1)

where j = month of the year (1/12); A0 = mean of V(j); τ( = 12) period of the cycle; Ti = period of the ith harmonic; ATi = amplitude of the ith harmonic; and ϕTi = phase of the ith harmonic.

For the estimation of the Fourier series parameters, it is convenient to decompose the cosine argument in Equation (1), obtaining the polynomial form:

  • equation image(2)

where equation image; equation image; and A0 are parameters that can be estimated by means of the least squares method. The amplitude and phase of the ith harmonic can then be obtained as:

  • equation image(3)

The phases are now represented in radians. It is possible to obtain their values in months as:

  • equation image(4)

As an example, we report in Figure 1 the A0 component and the harmonics of periods 12 and 6 months as obtained from the adaptation to the precipitation regime of the gauging station located in Potenza. The regime reconstructed with the two mentioned harmonics and the observed data are reported in Figure 2.

thumbnail image

Figure 1. Average monthly value (A0) and T = 12 and T = 6 months' harmonics as estimated for the data from the gauging station of Potenza. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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thumbnail image

Figure 2. Precipitation regime for the gauging station of Potenza, as observed (OBS) and predicted (PRED) by means of Fourier series with two harmonics. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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Once the Fourier parameters are evaluated, it is worth to perform some controls on the values obtained. First of all, the amplitudes having negative values can be set to positive by imposing a shift of half a period to the phase of that harmonic. It is then necessary to verify that the phase is comprised between zero and the period of the harmonic, eventually correcting it by adding or subtracting one period.

When working on datasets including several sampling points to be used for regional studies or interpolation, it is necessary to perform a further control on the phases in order to ensure that their distribution is well represented within the mentioned lower and upper bounds. Once their histogram is plotted, it can happen that the data present a cut in the distribution. In such cases it is possible to set a threshold value falling into an interval where no data are located and to transform the values lower than the threshold by adding one period, allowing for a better representation of the distribution. As an example, in Figure 3 the distribution of the phase of the 12 months' harmonics is transformed from the initial one, bounded between zero and 2π, to a new one more suitable for further processing.

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Figure 3. Histogram of the phase of the harmonics T = 12 months. (a) The data are bounded between zero and 2π; (b) the data have been adjusted for rebuilding the first mode Gaussian bell. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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Once the at-site estimates of the Fourier parameters are obtained, it is possible to use them for the production of the monthly precipitation maps. In particular, we decided to apply ordinary kriging within a leave-one-out validation scheme. The estimates obtained for the Fourier series parameters are used for reconstructing the monthly precipitation values through Equation (1). The same scheme has been applied to the observed data for the construction of 12 separate models of the monthly normals to be used as a benchmark in the analysis. Both the estimates are compared with the observed values through the RMSE and the correlation coefficient (R). The indices are calculated in both a station-by-station and a month-by-month sampling, which will be reported as maps and tables, respectively.

In order to demonstrate the suitability of the Fourier series approach for regionalization studies, we present the results of a k-means unsupervised classification based on the Fourier's parameters. The clusters are compared with prior climatic and geographic knowledge.

In a previous work (Claps et al., 2008), the parameters of the thermal regime resulted well correlated within a linear regression scheme with geographical and morphological indices extracted from a DEM, namely elevation (Z), the geometric average of the distance from the sea in the eight cardinal directions (M), an exposure index based on the direction and distance from the closest sea coast (E) and a measure of terrain concavity (IC), and the two location variables, latitude (Lat) and longitude (Lon), of the sampling points. Such approach would also be helpful for precipitation. However, the definition of morphological indices being meaningful over a large spatial domain is a critical issue that goes beyond the aim of this article. The independent variables proposed by Claps et al. (2008) have been used within a RK interpolation scheme in combination with the stratification provided by the regime classification for assessing the effectiveness of such classification in improving the reconstruction of the regime in ungauged locations. We expect that the stratification will play a major role in the selection of the most suitable covariates at the regional scale, i.e. the combination of the Lat and Lon variables by means of a proper coefficient set will allow to mimic the effect of the distance of the stations from the most relevant moisture source in a certain region.

2.1. Available data

The analyses were carried out on a dataset of 2043 gauging stations. Large part of the data have been made available by the Central Office for Agricultural Ecology (UCEA) within a coordinated collection of own and former National Hydrographic Survey (SIMN) stations, for a total amount of 1437 locations. Additional 983 stations, partly overlapping the UCEA ones, were made available within national research projects. The data have been harmonized by discarding the overlapping stations for which the normals have been calculated over shorter periods.

Unfortunately, the data do not cover the Italian territory evenly, being part of northern Italy less densely covered and the Sardinia Region almost ungauged (two stations covering 24 000 km2).

With respect to elevation, the distribution is quite even, apart from the mountainous part of the Italian territory, where the station density is lower (Table I). Unfortunately, the higher spatial variability of precipitation in such areas will produce higher estimation errors.

Table I. Distribution of the gauging stations of the database with elevation and related percentages of the area of the Italian territory
Elevation (m.a.s.l.)% Stations% Area
E < 1002023
100 < E < 8006454
800 < E < 12001111
E > 1200512

3. Results and discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

3.1. At-station Fourier series fitting

The Fourier parameters of the precipitation regimes of the 2043 gauging stations have been reconstructed by means of multiple linear regression. In Table II, the RMSEs of reconstruction of the monthly data with one to four harmonics, as well as the average for each combination, are reported. We have tested the harmonics of periods 12, 6, 4 and 3 months. RMSE decreases while increasing the number of harmonics. The major improvement is obtained when the 6-month harmonic is added to the 12-month harmonic.

Table II. Fourier series fitting of the precipitation regime
MonthRMSE (mm) after Fourier series fitting
 Ti = 12Ti = 12, 6Ti = 12, 6, 4Ti = 12, 6, 3Ti = 12, 6, 4, 3
  1. Stepwise analysis based on the harmonics of 12-, 6-, 4- and 3-month periods.

January16.269.127.978.688.20
February17.288.477.747.587.42
March9.538.186.657.866.24
April12.987.545.667.605.29
May16.827.085.537.545.07
June8.696.535.757.235.33
July20.418.368.396.485.56
August13.598.759.155.855.65
September10.648.717.487.425.93
October19.1911.419.778.636.68
November25.7414.0612.329.617.63
December15.0212.2610.609.598.26
Average15.519.218.087.846.44

We decided to consider only those two harmonics for further processing. This combination allows to obtain an overall R of 0.979 and RMSE of 9.21 mm, and to take advantage of the Fourier approach in terms of data compression: it is possible to reduce the number of variables to be investigated by means of interpolation techniques from 12 (the monthly values) to 5 (A0 and the two Ai − Φi couples).

The at-station RMSE after the Fourier series fitting is reported in Figure 4. It is possible to observe that the stations having higher RMSE are clustered in certain areas, mostly characterized by high topographic and precipitation gradients.

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Figure 4. RMSE after the Fourier series fitting. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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3.2. Classification of the precipitation regime based on Fourier series parameters

Once the check on the phase distribution is performed (the results for Φ12 are reported in Figure 3), the Fourier's parameters are processed by means of the k-means unsupervised classification algorithm. In Figure 5, the stations have been labelled with their class number; the precipitation regime of the centroids of each class is reported in Figure 6.

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Figure 5. Stations' clusters as obtained from Fourier series parameters unsupervised classification. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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Figure 6. Precipitation regime for the centroids of the clusters. Same legend as Figure 5. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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Despite the geographic position not being involved in the classification, the clusters are well separated and spatially consistent. Their distribution reproduces known features of the Italian topography, such as the Appennines chain in southern Italy (class 3) or the Padane and the coastal Adriatic plains in the north (class 5).

From Figure 6, it is possible to have a simplified insight into the main characteristics of the Italian precipitation regimes. In classes 2 and 3, the effect of the 12-month harmonic is dominant, with the rainy season occurring in November to January. The 6-month harmonic prevails in classes 4, 5 and 6, for which the rainfall regime is characterized by two relative maxima occurring in May and October. In class 1, the 12- and 6-month harmonics have a similar weight, with an absolute maximum occurring in November and a relative one occurring in April.

As a more general observation, it is worth to notice that the combination of the relative weights of the average and the amplitudes, and the phase shifting of the curves allows to represent thoroughly nearly any regime shape. Particular caution should be put in regions where no rainfall occurs in one or several months.

3.3. Reconstruction of the precipitation regime in ungauged locations by means of ordinary kriging

In order to assess the effectiveness of the Fourier series approach in representing the precipitation regime in ungauged sites, the Fourier series parameters by means of ordinary kriging within a leave-one-out scheme have been estimated. Such parameters have been used for reconstructing the monthly precipitation values through Equation(1). With the same interpolation scheme, we have produced 12 separate models for the reconstruction of the monthly normals from the at-station precipitation values, directly. The results obtained are used as a benchmark for evaluating the performance of the Fourier scheme. The monthly RMSE and correlation coefficient (R), and their average, as obtained within the leave-one-out validation framework, are reported in Table III. The Fourier-based and the benchmark estimation schemes produce similar results, with overall RMSEs of 17.53 and 15.97 mm and correlation coefficients of 0.909 and 0.921, respectively.

Table III. RMSE and R after the leave-one-out validation
MonthRMSE (mm) after krigingCorrelation after kriging
 Fourier (Ti = 12, 6)StationFourier (Ti = 12, 6)Station
January22.2821.360.8870.896
February19.5618.590.8660.880
March18.7017.770.8660.873
April16.7915.860.8950.903
May14.1613.790.9470.948
June11.1210.270.9660.969
July10.697.930.9610.970
August11.829.960.9630.966
September14.2912.130.9180.933
October19.9917.620.8820.909
November25.6723.120.8760.898
December25.2923.220.8870.900
Average17.5315.970.9090.921

In order to assess the spatial behaviour of the two models, the at-station RMSE maps are reported in Figures 7 and 8. The patterns of the reconstruction error are similar for the two cases. The stations having higher RMSE are clustered in the areas presenting high precipitation gradients, or where major precipitation regime changes occur. It is possible to observe the appearance of linear patterns, i.e. the Appennines' crest between Tuscany and Emilia Romagna, as well as the Ligurian ones, and the transition between lowland and mountainous regions in Friuli Venezia Giulia. Isolated stations being characterized by different regimes and located at the border of the study area (see e.g. the stations at the Italy–France border) suffer of high RMSE, as well. Such features are deemed to be a typical characteristic of the ordinary kriging approach, which is sensitive to border effects on samples located at the edge of the study areas, and where major gradients occur. We believe that the prior delineation of homogeneous regions could be beneficial for reducing such effect.

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Figure 7. Reconstruction of the precipitation regime by interpolating the Fourier parameters: RMSE after the leave-one-out validation. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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thumbnail image

Figure 8. Reconstruction of the precipitation regime by interpolating the station data: RMSE after the leave-one-out validation. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

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3.4. Reconstruction of the precipitation regime in ungauged locations based on RK and regional stratification

As mentioned in the ‘Introduction’, the use of suitable covariates within a RK approach could be beneficial. Precipitation patterns are triggered by the interaction of medium to large-scale circulation phenomena and local morphological features. A prior delineation of homogeneous regions could be helpful in this sense for highlighting the morphological features that control the patterns at the local scale.

The regions delineated by means of the unsupervised classification of the Fourier's parameters reported in Figure 5 have been used for stratifying the dataset within a RK interpolation scheme based on the geographic and morphologic variables derived from a DEM that have been described in the ‘Methods’ section. The results obtained after the stepwise model selection are reported in Table IV. For each Fourier parameter, we have chosen the model having the highest coefficient of determination (R2) among the ones that passed the Student's t-test for significance of the variables and the variance inflation factor test for multicollinearity. The R2 for the model including all the variables has also been reported, as well as the RMSE for the regression model (R), for the RK model obtained after applying kriging on the residuals of the regression and for the ordinary kriging (K) performed on the Fourier variables, which has been reported as a benchmark value.

Table IV. Results of the stepwise regression analysis
Fourier parameterClassRegression modelR2R2maxRMSE
  ZLonLatMEIC  RRKK
  1. The first column represents the Fourier variable under investigation, and second the set of stations considered. The final regression model variables, as defined in the methods section, marked with a X, and the related coefficient of determination (R2), as well as the one for the full model (R2max), are reported. The RMSEs for R, the RK and the ordinary kriging (K) are reported.

A0 (mm)AllXXXXXX0.2610.26128.2012.4012.82
 1X XXXX0.4880.48920.9611.3712.29
 2X X  X0.5720.58118.2912.8615.47
 3X  XXX0.1950.28425.9114.9316.59
 4  X   0.1780.23426.5218.1119.58
 5 XXX X0.6140.62114.2213.3110.63
 6 XXX X0.4080.40832.0414.5113.97
A12 (mm)AllX XXXX0.4120.41219.399.859.53
 1XX XXX0.2390.25312.476.266.26
 2X XXXX0.3110.31714.5210.2111.40
 3X  XXX0.4580.46121.5513.6214.47
 4   X  0.1330.23611.668.048.38
 5X  X  0.1960.2425.094.414.35
 6  XX  0.5300.5419.606.508.16
F12 (rad)AllX XX X0.5020.5030.6920.3340.317
 1X X XX0.1580.1600.1840.1340.134
 2X X XX0.3040.3040.1040.0840.087
 3 XX   0.1480.1640.1750.1180.136
 4  X   0.3850.4230.2730.2630.260
 5 XX   0.5450.5700.2160.2300.233
 6  XX  0.3840.4000.2140.1770.169
A6 (mm)AllXXXXXX0.2400.2408.555.525.37
 1XX  X 0.4630.4706.234.334.46
 2XXX XX0.3580.3586.455.915.98
 3X X XX0.1910.2376.325.855.95
 4   X  0.1730.22913.078.178.22
 5 X X X0.3390.3526.876.484.55
 6 XXX  0.2770.28211.716.676.19
F6 (rad)AllXX XXX0.2160.2170.2220.1920.192
 1XX XXX0.4540.4830.1010.0850.084
 2 XXX X0.1830.1950.1100.0990.098
 3   X  0.0610.0700.2900.2860.287
 4XXXXXX0.5310.5600.0940.1010.110
 5      0.0490.3200.3190.323
 6   X  0.0650.0980.1940.1230.123

From Table IV it is possible to observe that the regression variables occur rather evenly in the selected models, with Z, Lat, M and IC being present in more than half of the models. Lon and E have the lowest rates of appearance, as they are highly correlated, and hence filtered out for multicollinearity, due to the shape of the study area, with the peninsular part of Italy expanding in the north–south direction, with the distance of a certain point from the nearest coast being similar to the difference among the longitudes. A similar effect occurs also when considering Lat and E in the continental part of Italy. Elevation (Z) seems to be more effective in the peninsular classes (1, 2 and 3) than in the Alpine regions, where the distance from the sea (M) and the latitude (Lat) seem to play a major role. As hypothesized in the previous section, latitude and longitude have had a role in emulating the distance from the moisture source, and this effect is emphasized by the stratification.

The R2 values seem to indicate that the classes 3 and 4 are the ones presenting major problems. This is only partly confirmed from the RMSE. The values achieved through R are worse than those related to RK and K, apart from a limited number of Fourier variable-class combinations. RMSEs for RK and K are rather similar to each other. This result would indicate that the use of the selected covariates has provided a negligible improvement.

The Fourier parameters obtained within the leave-one-out validation framework have been processed for reconstructing the monthly precipitation estimates through Equation(1), which have been compared with the observed values. The monthly RMSE and its average for the ordinary kriging, the multiple linear regression and the RK interpolation schemes are reported in Table V.

Table V. RMSE of reconstruction of the monthly precipitation based on the Fourier parameters estimates obtained with kriging, regression and RK applied to the entire dataset and to the subdivision based on the classification reported in Figure 5
MonthRMSE (mm) entire datasetRMSE (mm) after stratification
 KRRKKRRK
January22.2842.5521.8822.6533.0521.11
February19.5637.2019.2120.3127.3519.07
March18.7036.7818.1619.7727.7718.45
April16.7933.0316.3718.0925.4416.99
May14.1630.4314.4015.5423.1514.73
June11.1224.6711.9412.5218.0011.83
July10.6920.8511.3411.9014.4611.06
August11.8224.5412.5013.1518.0412.58
September14.2933.1614.9515.5823.1715.01
October19.9945.4720.6621.6133.0021.02
November25.6748.9025.8027.7540.0426.33
December25.2947.1725.0725.7635.7224.09
Average17.5335.4017.6918.7226.6017.69

It is possible to notice that K and RK have similar RMSE, as reported also by Lloyd (2005) and Fiorenzo et al. (2008), based on studies in Great Britain and Basilicata, southern Italy, respectively, and by Prudhomme and Reed (1999), based on the mapping of extreme rainfall in Scotland. Both K and RK and are not sensitive to data stratification.

On the contrary, the statistics for regression depict an interesting situation: the stratification allows to obtain largely improved results, with average RMSE passing from 35.4 to 26.6 mm and the gap to K being reduced of one half of the one related to the entire dataset.

The independent variables here used are the ones proposed by Claps et al. (2008). Those variables were intended to be suitable for representing the spatial variability of average air temperature. The results of this study would benefit from the implementation of variables more suitable for the interpolation of precipitation. However, this goes beyond the scope of this work.

4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

The precipitation regime in ungauged locations reconstructed by means of the interpolation of the two harmonics Fourier series parameters derived from observed data has been demonstrated to be as accurate as the one obtained through the application of the month-by-month estimation based on observed data, with average RMSEs of 17.53 and 15.97 mm and correlation coefficients of 0.909 and 0.921, respectively.

The Fourier series approach allows to reproduce the regime in a more compact and consistent way and to reduce the complexity of the geostatistical problem by reducing the number of kriging operations from 12 (the monthly precipitation values) to 5 (the average and the amplitudes and phases of the Fourier series harmonics).

Moreover, it allows for a fast and meaningful delineation of homogeneous regions by means of simple unsupervised clustering of its parameters and to characterize in a compact way the precipitation regime, as highlighted by the representation of the mean curves of the regions.

The regions obtained have been used for the stratification of the dataset within a multiple linear regression approach based on geographic and morphologic variables. Despite the regression models being outperformed by ordinary kriging and RK, it has been possible to appreciate the effectiveness of the stratification in improving their performance and to assess the efficiency of the independent variables in the different regions.

The approach here proposed can be applied to a broad range of problems. We have successfully tested it on the representation of the regime of other climatic variables, as well as of monthly runoff and vegetation phenology as captured by remote sensing. It allows to simplify the study of complex processes, emphasize their main features and identify the relationships of complex systems.

Some caution should be used when dealing with datasets with not such a marked seasonal behaviour, such as in regions with almost flat precipitation regime. We would discourage the use of this approach when the signal has impulsive characteristics, such as in regions with a significant number of months with zero precipitation.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References

I would like to thank Prof. Pierluigi Claps for the attention and patience he has had in the recent and past collaborations.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. References
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