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Changing patterns of precipitation at the Sooke Reservoir in British Columbia.

Introduction

General Overview

While the subject of climate change is vast, there is at least one topic within climate change that deserves urgent and systematic attention, and that is the changing pattern of precipitation around the world. If climate change is already underway, how is it affecting observed precipitation? The objective of this paper is to consider the changing pattern of observed precipitation at the Sooke water reservoir, which serves the city of greater Victoria in Canada. The city has a population of 331,000, which is expected to grow to half a million by 2026. The city is home to a large and growing number of retired persons and is a major tourist destination in Canada. Changing precipitation patterns have major implications for the drinking water supply, and the city is indeed very pro-active in planning ahead for the expected changes in precipitation patterns, some of which are already clear from the existing precipitation data at the Sooke Reservoir. Our analysis of past precipitation data already shows some emerging patterns that may be comparable to changes in other regions. For example, Dore (2005)carried out a survey of changes in global precipitation patterns. He found one systematic result over most of the world, and that is increased variance of precipitation everywhere. Consistent with this finding, we observe that at least for most of the Northern Hemisphere, wet areas become wetter, and dry and arid areas become more so. In addition, the following general global pattern is emerging: (a) increased precipitation in high latitudes in the Northern Hemisphere; (b) reductions in precipitation in China, Australia, and the small island states in the Pacific; and (c) increasingly variable equatorial regions, with increased variance.

From the beginning of the 20th century, global land precipitation has increased by 2% (Hulme et al. 1998; Jones and Hulme 1996). Though being statistically significant, this increase is neither spatially nor temporally uniform (Doherty et al. 1999; Karl and Knight 1998). Furthermore, Dai et al. (1997) found a global long-term increase in precipitation. Thus there appears to be strong evidence of increasing global precipitation accompanied by shifts in patterns.

The increase in precipitation in the mid and high latitudes contrasts with the decreases in the northern sub-tropics that were largely responsible for the decade-long reduction in global land precipitation from the mid-1980s through the mid-1990s. Since 1995, record low precipitation has been observed in equatorial regions, while the sub-tropics have recovered from their anomalously low values of the 1980s. This is the global picture in summary. Next, we consider North America.

Precipitation Patterns in North America

Over the 20th century, annual averaged precipitation increased by between 7% and 12% for the zones 30[degrees] North to 85[degrees] North (Dore 2005). The increase in the Northern Hemisphere is likely to be slightly biased because adjustments have not been made for the increasing fraction of precipitation falling in liquid as opposed to frozen form. The exact rate of precipitation increase depends on the method of calculating the changes, but the bias is expected to be small, because the amount of annual precipitation affected by this trend is generally very small. Nevertheless, this unsteady but highly statistically significant trend towards more precipitation in many of these regions is continuing. For example, in 1998, the Northern Hemisphere's high latitudes (55-N and higher) had their wettest year on record, and the mid-latitudes have had precipitation totals exceeding the 1961 to 1990 mean every year since 1995.

In the Northern Hemisphere's mid and high latitudes, precipitation has mostly been increasing, especially during the autumn and winter, but these increases vary both spatially and temporally. For example, precipitation over the United States has increased by between 5% and 10% since 1900, but this increase has been interrupted by multiyear anomalies such as the drought years of the 1930s and early 1950s (Karl and Knight 1998; Groisman et al. 1999). The increase is most pronounced during the warm seasons.

Using data selected to be relatively free of anthropogenic influences, such as ground water pumpage or land use changes, several recent analyses (Groisman et al. 2001; Lettenmaier et al. 1999; Lins and Slack 1999) have detected increases in streamflow across much of the contiguous United States, confirming the general tendency of increasing precipitation. However, Lins and Michaels (1994)found that in some regions, increased streamflow did not correlate well with an increase in rainfall. This has been further evaluated by Groisman et al. (2001), who show that changes in snowcover extent can also influence the timing and volume of streamflow. Regionally, Mekis and Hogg (1999) showed that precipitation in Canada has increased by an average of more than 10% over the 20th century. Zhang et al. (2000) report an increase in Canadian heavy snowfall amounts north of 55-N and Akinremi et al. (1999) found rainfall significantly increasing in the Canadian prairies from 1956 to 1995, with pockets of droughts. Regarding precipitation in the Canadian prairies, there are conflicting reports. For example, Akinremi et al. (1999) reported a significant increase from 1956 to 1995, but Gan (1998) reported a marginal increase in precipitation between 1949 and about the 1990s.

Description of the Data

Data used in this study were collected at the Sooke reservoir by the Regional Municipality of Greater Victoria (also known as the Capital Region District, or CRD) and consists of the set of daily precipitation in millimetres over the period 1914-2004, inclusive. These data were used to analyze past patterns in the precipitation of the region and to detect and date signals of climate change over the past 90 years. Figure 1 shows the annual daily average precipitation for the period 1914-2004.

It is clear that it is difficult to determine any climate change signals from the mean alone. Also, at least one feature of the data that becomes quite apparent when we calculate a five period moving average is that the data demonstrates the presence of a non-periodic but recurring cycle.

Figure 2 shows the annual average of precipitation in millimetres over the past 90 years and illustrates the annual precipitation pattern from observed data. The observed average annual precipitation from Fig. 2 shows the emergence of two very different precipitation seasons over the past 90 years. During the summer months, the appearance of a dry season can be observed. The dry season is characterized by a period of several months of very low levels of precipitation with little variance. However, during the winter and autumn months, the emergence of a wet season is noticeable. The wet season is characterized by a period of several months with high levels of precipitation with a significant increase in the variance of the precipitation.

Methods and Results of the Analysis

This section presents a complete analysis of observed data from the period 1914 2004 for the Sooke reservoir in British Columbia. We also determined whether or not there was a change in the Data Generation Process (DGP) by analyzing the four moments of the distribution: the mean, variance, skewness, and kurtosis.

Determining the Structural Break in the DGP

Consider as a start the variance of the data, shown in Fig. 3.

In order to determine exactly where the structural break had occurred, it was necessary to conduct testing procedures developed by Andrews (1993), which is based on Quandt's (1960) test for structural breaks. However, before we performed this test, it was necessary to estimate an autoregressive (AR) model using the estimation procedure from Box and Jenkins (1970). After estimating and analyzing a number of models, we concluded based on the [[bar.R].sup.2] statistic that the model that best fit our data was the AR {1}. The estimated model is given in Eq. (1), where [x.sub.t] is the fitted variance in period t and [x.sub.t-1] is variance in period t-1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Other statistics from the above AR {1} model can be found in Appendix A. From the t-statistic we can conclude that at the 6% level of significance, we can reject the null hypothesis that the coefficient on the AR {1} variable is zero. An examination of the error terms shows white noise residuals, indicating that we have an acceptable model.

Next, we used the Andrews-Ploberger test for structural break to determine whether there was a change in the Data Generating Process (DGP). The results of the test are shown in Table 1.

The results show that at approximately 10% level of significance, we can reject the null hypothesis of no structural break. Also, we are 90% confident that there is a change in the structure of the DGP between the years of 1964-1968.

Estimation of Probability Density Functions

Kernel Method

Probability density functions provide us with a natural description of the distribution of a data set. One form of such estimation is the kernel method which offers numerous advantages over the simple histogram. For a detailed comparison of kernel and histogram methods, see Silverman (1986). The definition of the kernel method of density estimation as stated by Silverman (1986) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where h is the smoothing parameter, n is the number of observations within the data set, [X.sub.i] is the ith observation within the data, and x is the mean. The kernel estimator can be normalized so that its area is equal to i by satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The kernel with an optimal h is given by the Epanechnikov kernel defined by Silverman (1986) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

For [absolute value of t] < [square root of 5], Otherwise K=0.

It is well known that the Epanechnikov kernel has certain efficiency properties as stated by Silverman (1986).

Probability Density Estimation: The Observed Distributions

From the Sooke reservoir data, we estimated the probability density functions of the 18 five year periods which spans nine decades beginning from 1914 and ending in 2004 in order to determine what was happening to the distribution of precipitation over the 90 year period. An Epanechnikov kernel was used to fit the probability density functions (PDFs) for each five-year period. Four of these PDFs of the five year periods are shown in Fig. 4. The horizontal axis shows daily precipitation in mm and the vertical axis shows the probability of occurrence of this precipitation event. For a more detailed view, please see the individual graphs in Appendix D.

Figure 4 indicates that the estimated unconditional distribution of precipitation has changed over the last 90 years in the Sooke region; it is clear that the distributions have become more peaked or leptokurtic over time. To investigate this change further, we employ a non-parametric test on the data called the Kolmogorov-Smirnov (KS) test. The KS test is used to test whether or not there are statistically significant changes between the distributions. The results of this test can be used to detect significant climate change signals within the data set.

The Kolmogorov-Smimov Test Theory

The K-S model

The Kolmogorov-Smirnov (KS) test (Chakravarti et al. 1967) is based on the empirical cumulative distribution function (ECDF). Given N ordered data points [Y.sub.1], [Y.sub.2], .... [Y.sub.n], the ECDF is defined as

[E.sub.n] = n(i)/N (5)

Where n(i) is the number of points less than [Y.sub.i] and the [Y.sub.i] are ordered from smallest to largest in value. This is a step function that increases by 1/N at the value of each ordered data point. The KS test detects significant changes in the distribution of data sets by calculating the maximum distance, D, between the two estimated ECDFs. The maximum D value calculated by the KS test method is defined as

D = [max.sub.l[less than or equal to]i[less than or equal to]N [absolute value of F([Y.sub.i]) - i/N]], (6)

where F is the theoretical cumulative distribution of the distribution. The calculated D statistic is then compared to a critical value in order to determine whether or not there is a significant change in the distribution of the data. The formula for calculating critical values is given by:

Z = [square root of [n.sub.1] + [n.sub.2]/[n.sub.1] x [n.sub.2] x [alpha], (7)

where [n.sub.1] is the number of observations within the first data set, [n.sub.2] is the number of observations within the second data set, and [alpha] is the critical level value. The null hypothesis of the KS test is that the two distributions are from the same data generating processes.

The Two Sample KS Test

The two-sample KS test is used to determine whether or not two separate distributions come from the same DGE The two sample KS test computes the maximum distance D between the two empirical distribution functions of the observed samples. Figure 5 illustrates how the D-statistic between any two ECDFs is calculated within the cumulative fraction plot. The cumulative fraction plot graphs the cumulative value of precipitation observations that lie below a certain critical value x.

Estimated KS Test Results

In order to carry out the two-sample KS test, we grouped the data into five-year periods beginning with the 1914-1918 period. The period 1914-1918 is also our base period, which was used to compare with each successive five-year period in order to determine if the distribution is changing significantly over time. The results of the five-year period D values are reported in Table 2 and illustrated in Fig. 6.

The results from the two-sample KS test show that at the 5% level, there are six periods where there are significant changes in the DGP of precipitation levels. (1) At the 10% level, there are ten periods in which there are significant changes in the DGP of precipitation levels compared to the base period of 1914-1918. These results indicate to us that we can state with a fair degree of confidence that there have been significant changes in the precipitation over the past 90 years since at the 10% level of significance, approximately 60% of the 5 year periods have a statistically different distribution than the base year. However, we cannot interpret the magnitude or the direction of climate change from the KS test D values alone. Further analysis of the data can help determine the nature of the change in climate in the Victoria area. For this reason, it is essential to analyze the moments of the distributions of five-year periods from 1914 to 2004.

Unpacking the Distribution of Precipitation

In this section, we examine the changing nature of the distribution of precipitation by analyzing the four moments of the data for the five-year periods. The four moments of the data are the mean, variance, skewness, and kurtosis (formulas are given in Appendix B). The plots of the four moments of the five-year periods over the 1919-2004 time-frames are shown in Figs. 7, 8, 9, and 10 and the associated values are given in Table 3.

Mean

Mean precipitation as shown in Fig. 7 is the average daily precipitation for each 5 year time period from 1914 to 2004. We can deduce from both the graph of the mean (Fig. 7) and the statistics presented in Table 3 that there is no distinguishable trend within the mean over time. However, of all the five-year base periods which were significant at the 10% level, only four periods had mean precipitation below the base period mean of 3.999 mm/day. This indicates that there have been significant changes in the mean level of precipitation over the past 90 years, although it is difficult to determine exactly by how much that precipitation has increased over the past 90 years.

Variance

At the beginning of this section we have shown that the variance of precipitation has been the most important and identifiable indicator of climate change within the Victoria region. The variance of precipitation describes the spread of the distribution over time. It can be observed from Fig. 8 that the most pronounced increase in the variance occurred in the 1960s. Thus we might imagine that the series can be broken up into two separate regimes. The variance appears to fluctuate in the band of 75-150 until the 1964-1968 period and then jumps into the 125-225 range from the 1964-68 period onward to 2004. This observation is consistent with results from the beginning of this section. In the section entitled Estimation of Probability Density Functions, it was estimated using the AR {1} model and the Andrews-Ploberger test that a structural break in the variance occurred in 1964 to 1968 period.

The variance also appears to be becoming larger compared to the base period. The variance in the last four five-year periods has been consistently larger than the variance observed in the base period. Although the increase in variance does not necessarily indicate whether precipitation is increasing or decreasing, it is strong evidence to support the proposition that the distribution is changing and is a strong signal of climate change.

Skewness

The skewness of the distributions was used as a measure of asymmetry about the distribution. The long tails observed in Fig. 4 show that there was an increase in skewness in the distributions over time. The statistics provided in Table 3 confirm the presence of large tails shown in Fig. 4 that are heavily skewed to the right. We also observe that for the last five five-year periods, the skewness levels were larger than the base year (see Table 3 and Fig. 9). Indeed, after the structural change in the 1964-1968 period, most of the skewness levels were greater than the skewness level observed in the base year.

Kurtosis

We examine the level of kurtosis in order to determine the extent to which the peakedness or flatness of the distribution has changed over time. Kernel density estimations from above showed that the estimated kernels were becoming increasingly peaked (leptokurtic). In order to determine whether or not the kurtosis levels are significant, tests were carried out using the Standard Error of Kurtosis (SEK) and the values were then compared. All values of kurtosis calculated were found to be significant, showing that distributions were tall or lepto-kurtic; a table of statistics can be found in Appendix C, More than 50% of the kurtosis levels after the 1954-1958 period was also larger than prior to that period while the most recent five-year period (1999-2003) in this study has an abnormally high kurtosis level. In Fig. 10, we observe a slight upward trend over the five-year periods after the 1954-1958 period, giving further evidence of the changing shape of distribution of precipitation over time. The changing shape of the estimated distributions becoming more leptokurtic is further evidence supporting the proposition that precipitation patterns are changing at the Sooke reservoir.

Conclusion

We have shown that the distribution of precipitation at the Sooke Reservoir has changed over time. For our structural analysis on the variance of precipitation, we selected an AR{1} model as our best model from a group of models and determined the structural break using the Andrews-Ploberger method. We found that the structural break in the variance lies within the 1964-1968 period. Next, we examined the four moments of the distribution and found evidence of change in at least three of the moments. In particular, we found that the estimated probability density functions are becoming more leptokurtic over time.

As might be expected, the water authorities in Greater Victoria has taken account of the key feature of climate change, namely an increase in variance which meant that wet months are getting wetter and the drier summer months getting even more dry. Accordingly, they have expanded the capacity of the Sooke Reservoir in 2002 by 78% at a cost of an investment of $23 million. (2)

Appendix A: Statistics for AR{1} Model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A.1)

Table 4 T-stats and significance
levels

                             Constant     [beta]1

Standard Error of Estimate   (10.219)     (0.111)
t-Statistic                   13.03838     1.91660
Significance Level             0.000000    0.058

Table 5 Model fit statistics

Durbin-Watson statistic    1.9997
Regression F(1,77)         3.6733
Significance Level of F    0.0600
Q(19-1)                   16.0733
Significance level of Q    0.5874


Appendix B: Formulas for the Four Moments of a Distribution

Mean: Average observation within the data set

u = [SIGMA] X/N (B.1)

Variance: The measure of how spread out a distribution is

[[sigma].sup.2] = [SIGMA] [(x - u).sup.2]/N (B.2)

Skewness: The measurement of degree of asymmetry about the distribution

Sk = [[SIGMA].sup.N.sub.i=1] [([Y.sub.i] - [bar.Y]).sup.3]/(N - l) [s.sup.3] (B.3)

Kurtosis: The measurement of the degree of peakedness about a distribution

Ku = [[SIGMA].sup.N.sub.i=1] [([Y.sub.i] - [bar.Y]).sup.4]/(N - l) [s.sup.4] (B.4)

Appendix C: Significance Values for Kurtosis

Table 6 Testing for significance of kurtosis

Year        Kurtosis   SEK       Critical value        Conclusion
                                 (two times the SEK)

1914-1918   29.7880    0.25642   0.51285               Significant
1919-1923   30.6019    0.25642   0.51285               Significant
1924-1928   32.0310    0.25642   0.51285               Significant
1929-1933   28.0123    0.25642   0.51285               Significant
1934-1938   52.2417    0.25642   0.51285               Significant
1939-1943   18.1640    0.25642   0.51285               Significant
1944-1948   18.2160    0.25642   0.51285               Significant
1949-1953   14.5790    0.25642   0.51285               Significant
1954-1958   42.3910    0.25642   0.51285               Significant
1959-1963   27.9510    0.25642   0.51285               Significant
1964-1968   32.7453    0.25642   0.51285               Significant
1969-1973   36.7750    0.25642   0.51285               Significant
1974-1978   22.3170    0.25642   0.51285               Significant
1979-1983   39.2761    0.25642   0.51285               Significant
1984-1988   56.5920    0.25642   0.51285               Significant
1989-1993   30.2391    0.25642   0.51285               Significant
1994-1998   27.1335    0.25642   0.51285               Significant
1999-2003   113.655    0.25642   0.51285               Significant


Appendix D: Estimated Unconditional Distributions

Acknowledgments This paper was financed by a research grant from the Social Sciences and Humanities Research Council of Canada. This research was also partly funded by the Municipality of Greater Victoria, known as the Capital Region District (CRD), and we express our thanks to Jack Hull, General Manager of Integrated Water Services at CRD. However, the authors alone are responsible for its contents.

DOI 10.1007/s11293-012-9346-y

Published online: 16 November 2012

References

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Doherty, R. M., Hulme, M., & Jones, C. G. (1999). A gridded reconstruction of land and ocean precipitation for the extended tropics from 1974-1994. International Journal of Climatology, 19(2),119-142.

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Groisman, P. Y., Karl, T. R., Easterling, D. R., Knight, R. W., Jamason, P. B., Hennessy, K. J., et al. (1999). Changes in the probability of heavy precipitation: important indicators of climatic change. Climatic Change, 42(1), 243-283.

Groisman, P. Y., Knight, R. W., & Karl, T. R. (2001). Heavy precipitation and high streamflow in the United States: trends in the 20th century. Bulletin of the American Meteorological Society, 82(2), 219-246.

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Mohammed H. I. Dore * Mariano Matilla-Garcia * Manuel Ruiz Marin

M. H. I. Dore ([mail])

Climate Change Lab, Department of Economics, Brock University, St Catharines, ON, Canada L2S 3A1

e-mail: dore@brocku.ca

M. Matilla-Garcia

Universidad Nacional de Educacion a Distencia, Madrid, Espania

M. R. Marin

Universidad Politeenica de Cartagena, Cartagena, Espania

(1) Cumulative fraction plots for the two sample KS test for each time slice can be made available upon request to the author.

(2) For detailed information, see: http://www.crd.bc.ca/water/enginecring/sookereservoir/description.htm

For budget information see: http://www.crd.bc.ca/reports/water /2002_/srepaw644/SREP-AW644.pdf-p.2

Table 1 Andrews-Ploberger test result

                     Date     Test value   P-value

Constant            1964:01     2.230       0.039
Y {l}               1964:01     1.503       0.098
All coefficients    1968:01     2.616       0.093
Residual variance   1966:01    -0.421       1.000

Table 2 Estimated five-year period
D-values from 1919 to 2004
against a 1914-1918 base

                                 5%
                              Critical
Year                D-value     value

Base Vs. 1919-23     .0603     .1006
Base Vs. 1924-28     .0521     .1006
Base Vs. 1929-33     .1041     .1006
Base Vs. 1934-38     .0849     .1006
Base Vs. 1939-43     .0685     .1006
Base Vs. 1944-48     .1342     .1006
Base Vs. 1949-53     .1123     .1006
Base Vs. 1954-58     .0904     .1006
Base Vs. 1959-63     .1096     .1006
Base Vs. 1964-68     .0959     .1006
Base Vs. 1969-73     .1260     .1006
Base Vs. 1974-78     .0962     .1006
Base Vs. 1979-83     .0822     .1006
Base Vs. 1984-88     .0899     .1006
Base Vs. 1989-93     .0466     .1006
Base Vs. 1994-98     .1452     .1006
Base Vs. 1999-03     .0986     .1006

Table 3 Statistics of four moments of precipitation data

Year        Mean    Variance   Skewness   Kurtosis   D-Value

1914-1918   3.999   106.062     4.647      29.788      --
1919-1923   4.687   137.219     4.736      30.601    0.0603
1924-1928   3.876    80.601     4.485      32.031    0.0521
1929-1933   4.296   105.895     4.526      28.012    0.1041
1934-1938   4.652   144.217     5.727      52.241    0.0849
1939-1943   3.796    76.359     3.806      18.164    0.0685
1944-1948   4.349   103.973     3.856      18.216    0.1342
1949-1953   5.038   121.639     3.440      14.579    0.1123
1954-1958   4.845   122.359     5.036      42.391    0.0904
1959-1963   4.913   115.405     4.208      27.951    0.1096
1964-1968   5.487   232.062     4.857      32.745    0.0959
1969-1973   3.908   137.173     5.234      36.775    0.1260
1974-1978   4.703   166.755     4.262      22.317    0.0712
1979-1983   4.958   200.722     5.326      39.276    0.0822
1984-1988   3.735   120.426     6.186      56.591    0.0438
1989-1993   4.180   124.495     4.796      30.239    0.0466
1994-1998   5.195   139.714     4.377      27.133    0.1452
1999-2003   4.250   141.985     8.022     113.650    0.0986
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Publication:Atlantic Economic Journal
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