Coupled Spatial Distribution of Rainfall and Temperature in USA

Procedia Environmental Sciences 19 ( 2013 ) 178 – 187

1878-0296 © 2013 The Authors. Published by Elsevier B.V Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of the Scientific Committee of the Conference doi: 10.1016/j.proenv.2013.06.020

Four Decades of Progress in Monitoring and Modeling of Processes in the

Soil-Plant-Atmosphere System: Applications and Challenges

Coupled spatial distribution of rainfall and temperature in

USA

Francesco De Paola

, Maurizio Giugni

a_{Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II, Naples 80125, Italy}

Abstract

In the last decades, the international scientific literature has focused on the assessment and quantification of trends in average temperature and rainfall, underlining changes in time and space sometimes relevant. The changes concern the yearly average values, the seasonal distribution, the daily values, with marked differences both at a continental and national scale.

In this paper, with reference to a database containing the records of about 8000 thermometric stations and 12000 rainfall stations for the period 1895-1997 for the USA, available on the website of the Institute for Mathematics Applied to Geosciences (http://www.image.ucar.edu/Data/US.monthly.met/), an analysis on the spatial correlation between the yearly average rainfall hc and the mean of the yearly average ground temperature T was performed. In particular, with the aid of the copula distribution function, a stochastic model to define the conditional probability distribut*_{ion P(h}

c(mm)|T(°C)) was developed, analyzing the dependence of the yearly average rainfall on the thermometric changes.

© 2013 The Authors. Published by Elsevier B.V.

Selection and/or peer-review under responsibility ofthe Scientific Committee of the conference.

Keywords: Rainfall; Temperature; Spatial distribution; Copula function; Climate change.

1. Introduction

Over the past decades, the risk related to climate change is one of the most discussed topics in scientific literature. In particular, the climate change could create impacts both on physical and economic and social development.

* _{* Corresponding author. Tel.: +39-081-768-3420}

E-mail address:[email protected]

© 2013 The Authors. Published by Elsevier B.V Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of the Scientific Committee of the conference

T Yearly average temperature C(u, v) Copula distribution function

Copula parameter Gamma function

z Incomplete gamma function

H Hurst coefficient

K Kendall’s rank correlation coefficient

Laplace integral

S Mann-Kendall trend statistic test , Parameters of the normal distribution , Parameters of the gamma distribution P Probability distribution function

S Spearman’s rank correlation coefficient

Particular attention was paid to the statistical correlation between temperature and precipitation, variable as a function of geographical areas, for which it's possible to find different situations moving from a latitude to another. For example, several studies of precipitation in Italy over the last 120 years [1, 2, 3] showed a temperature rise with a slight decrease of the annual accumulated rainfall height, although in summer months there is an increasing trend. At the same time, a sharp decline in the number of rainy days was highlighted, associated with a significant increase of intensity. Rajagopalan & Lall (1998) [4] analyzed daily rainfall measures for a period of about 60 years in seven rain gauge stations, located between Idaho and Arizona. They highlighted that rainfalls in recent years did not follow a trend but were characterized by the typical El Niño Southern Oscillation, which includes periods of flooding alternating with periods of drought with an average frequency of five years, and in any case between three and seven years.

This work aims to investigate in the USA the spatial correlation of the average time series of rainfall and temperature. Such analyses are frequently adopted in literature to investigate different hydrological processes, such as the behaviour of structural protection measures [5], or quality and quantity pollutant assessment [6, 7].

Data supplied by the Institute for Applied Mathematics to Geosciences (http://www.image.ucar.edu/Data/US.monthly.met/) were considered, with reference to a database containing records of about 8000 monthly thermometric stations and about 12000 monthly rain gauge stations, for the period 1895 to 1997 (103 years). In the present study, only the stations with available rainfall data for the entire observation period were considered. Thus the sample was reduced to 286 stations, shown in Figure 1, which provides a fairly homogeneous coverage of the USA territory, with a higher density in the central-eastern area.

Fig. 1. Location of the analyzed stations

For these stations, with reference to the entire sample (103 years), the mean total annual rainfall and the mean annual average temperature were evaluated. In particular in Figure 2 are shown the spatial distribution (obtained by means of a spline surface FIELDS package in R) of the mean annual rainfall hc

(Figure 2a) and of the corresponding mean annual average temperature T, for the period 1895 to 1997 (Figure 2b). In figure 2c is also presented the box-plot of the annual average rainfall related to the mean of the annual average temperatures. The diagram highlights the greatest scattering for temperatures ranging 15 and 17 °C, while for T<15 °C the box-plot points out a statistical dependence between these variables. The Spearman and Kendall tests were applied, obtaining the following values of the rank correlation: S = 0.518 and K = 0.374, thus highlighting a statistical correlation of ordinal type, at least in

spatial terms.

2. Conditional distribution function evaluation by means of Copula approach

In order to seek the joint probability distribution P (hc, T), the approach of Copula function [8, 9, 10,

11, 12, 13] was used. Applying the test of Genest, Remillard and Beaudoin (2009) [10], the Gumbel copula was estimated as the best ,whose parameter was = 1.607. The Gumbel copula has the following expression: / 1

))

ln(

(

))

ln(

(

exp

)

,

(

u

v

u

v

C

distribution with parameters = 334.82 mm and = 920.69 mm, and a gamma distribution with parameters = 7.571 and = 1.469 °C.

Therefore, the resulting probability distribution function is the following:

607 . 1 / 1 607 . 1 / 607 . 1 ) ( ) ( ln ln exp ) , ( c T c h T h P (2)

in which is the integral of Laplace, is the Gamma function and z is the incomplete Gamma function.

The distribution function of the conditional probability P(hc | T) is obtained by means of the partial

derivative of the Copula function through the following expression:

v v v u v u v v u C v V u U P ) 1 ( 1 1 / 1 ( ln()) )) ln( ( )) ln( ( )) ln( ( )) ln( ( exp ) , ( ) | ( (3)

In order to obtain the value of the mean annual rainfall conditioned to the average of mean annual temperatures it is necessary to equal the (3) to the value of the sample frequency, expressed through the plotting position of Weibull, i.e. to write an equation of congruence:

) ( ) | ( * c c t T Fh h P (4)

In this way, it's possible to evaluate the values hc* to be compared with the values of annual

precipitation hc shown in fig.1. The analysis highlighted that the function of conditional probability

expressed by the (3) provides results quite satisfactory (deviation ± 15%, including more than half of USA territory).

On the basis of these results, an increase of the mean annual average temperatures T =+2 °C and T =+ 4 ° was considered and the corresponding spatial distribution of total annual rainfall was plotted (Fig. 3).

The suggested increase of temperature is consistent with the agreement signed in Copenhagen during the World Climate Conference in December 2009, between the United States and some developing countries (India, China, Brazil and South Africa): the agreement providesthat all Member States will put in place the necessary measures to maintain the temperature rise of the planet below two Celsius degrees [14].

Figure 3 shows that the rainfall variations related to the temperature increase do not indicate a clear trend. At least from a statistical point of view, it is not possible to forecast clearly for the whole USA territory the rainfall changes (increase or decrease) connected to a temperature increase of 2 °C. In fact, some computations pointed out that the increase of the total annual rainfall is equal to about 22% for the US territory. This percentage is increased to 38% considering an increase in the average temperature of +4 °C.

(a)

(b)

(c)

Fig. 2. (a) Yearly average rainfall hc(mm) spatial distribution; (b) Spatial distribution of mean of yearly average temperatures T(°C); (c) box-plot for conditional distribution for hc by binning on T.

(a) T=+2°C

(b) T=+4°C

Fig. 3. Spatial distribution of the rainfall obtained through (3) and (4), for an increase of temperature T: (a) 2 °C and (b) 4 °C.

Another investigated aspect is the possible presence of a trend in the rainfall series. The Mann-Kendall non parametric test [15, 16] is the most widely used statistical test to detect the presence of temporal trends in rainfall data. It evaluates the association between two variables X and T as the proportion of concordant pairs minus the proportion of discordant pairs in the samples. Two bivariate observations, (Xi, Ti) and (Xj, Tj), are called concordant whenever the product (Xi-Xj)(Ti-Tj) is positive, and discordant when the same product is negative. The Kendall’s statistic S is the difference between the numbers of concordant and discordant pairs. If X and T are independent and randomly ordered, S has a mean equal to zero and a variance that is a function of the sample size. One of the problems associated with the Mann-Kendall test is that the results can be affected by the presence of serial and cross correlations [17, 18, 19, 20, 21]. Even though the use of average annual values should avoid the issue of serial correlation, it is suitable to compute the autocorrelation function and check whether the lag 1 value is significantly different from 0. If the lag 1 autocorrelation is not significant, the Mann-Kendall test can be applied to the series. If a change point is detected, the autocorrelation function is computed separately on the series before and after it.

Figure 4 shows the spatial distribution of the Mann Kendall test for the territory of the United States, highlighting a singularity in the values of S in the areas close to New Orleans. In fact there are two portions of the territory in which the value of S assume opposite values (-0.4 and +0.4) denoting a great variability in the trend.

Fig. 4 – Spatial distribution of the Mann-Kendall trend test statistic S for the yearly average rainfall hc (average spatial value S=+0.051 with 2-sided p-value equal to +0.274).

A two sided p-value was evaluated to analyze the reliability of the S value over the USA territory, obtaining a value equal to +0.274, that it’s quite significant.

3. Long-memory analysis

The existence of long-term persistence in precipitations is a well-known aspect that could have direct consequences in water resources management. When climatic series are analyzed, distinguishing between long-term fluctuations and non-stationarity (trend) is not a simple task, mainly because of the limited extension of the available data sets. Nevertheless, such a distinction would be extremely interesting. In fact, the presence of long-term climatic fluctuations, rather than non-stationarity, would imply that the climate patterns could likely be attributed to a cyclical behaviour rather than to irreversible tendencies. An approach to analyze this issue is the detection of the presence of long memory in the data.

A way to perform such analysis is the evaluation of the Hurst exponent, which is used as a measure of long term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases.

The name "Hurst exponent" or "Hurst coefficient," derives from Harold Edwin Hurst (1880–1978) [22], who was the leader researcher in these studies; the use of the standard notation H for the coefficient relates to his name.

A value in the range 0 <H< 0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the following value will tend to be high, with a tendency to switch between high and low values lasting a long time into the future.

cases.In the calculations we have analyzed the total annual rainfall for the entire period of 103 years (1895-1997) for all of the 286 station considered.

4. Conclusive remarks

Climate change is one of the most important issues of the scientific community with a special focus on the combined effects of anthropogenic modifications of the atmosphere and the natural climatic cycles. Various scenarios and many models have been formulated in order to forecast the global atmospheric circulation and consequently the local variability of the global distribution of air temperature and rainfall. The effects of climate change have been analysed with respect to several natural risks such as desertification, droughts and floods, remaining mainly limited to the atmospheric and surface components of the hydrologic cycleChanges in warm extremes generally follow changes in the mean summertime temperature. Cold extremes occur faster than warm extremes by about 30%–40%, globally averaged. With the exception of northern polar latitudes, relative changes in the intensity of precipitation extremes generally exceed relative changes in annual mean precipitation, particularly in tropical and subtropical regions. Consistent with the increased intensity of precipitation extremes, waiting times for late-twentieth-century extreme precipitation events are reduced almost everywhere, with the exception of a few subtropical regions. The multimodel multiscenario consensus on the projected change in the globally averaged 20-yr return values of annual extremes of 24-h precipitation amounts is that there will be an increase of about 6 % with each kelvin of global warming, with the bulk of models simulating values in the range of 4%–10% K 1 _[25].

In the paper the attention is focused on the coupled spatial distribution of rainfall and temperature in USA, analyzing the spatial statistic dependence of the the mean annual total rainfall versus mean of average annual temperatures. The dependence structure is studied through the Copula approach defining a function to estimate the conditional probability of the mean annual total rainfall over the mean of average annual temperatures. The study showed that for an overall increment of the mean of annual average temperatures of + 4 °C the global increase of precipitation will be about 38%, while for an increment of +2°C the increase will be about 22 %.

Therefore the impact of climate change both on extreme events and on surface and groundwaters hydrological regime is a challenging topic. In addition, the analysis carried out with the Mann-Kendall test pointed out that the analyzed area is also characterized by a strong instability in climatic conditions. It is interesting observe from this point of view that the area of the United States most affected by tornado is located at E - SE (Fig. 6). Finally, the long - memory analysis carried out highlighted that in the area around New Orleans, the Hurst exponent is approximately equal to 0.50, indicating a cyclic and chaotic behaviour (instability) of the rainfall series.

(a)

(b) Fig. 5 – Spatial distribution of the Hurst exponent evaluated with : (a) R/S method and (b) Periodogram method.

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