Lorenzo Vergni1, Bruno Di Lena2, Enrico Maria Lodolini3
1 Dipartimento di Scienze Agrarie, Alimentari ed Ambientali, Università degli Studi di Perugia, Borgo XX Giugno 74 Perugia
2 Regione Abruzzo Centro Agrometeorologico Regionale - Scerni (Ch)
3 Centro di ricerca Olivicoltura, Frutticoltura e Agrumicoltura, Consiglio per la ricerca in agricoltura e l’analisi dell’economia agraria, via di Fioranello 52
Roma
*lorenzo.vergni@unipg.it
Abstract
In this work, the joint probabilities and return periods of two characteristics - Duration, D (days) and Severity, S (mm) - of the simulated water stress in olive orchards were modelled by a two-dimensional copula. First, the precipitation and temperature daily time series of some stations in central Italy were used in input to a “bucket” soil water balance model to simulate the corresponding dynamics of the soil water (SW) available for olive. Then, by applying the theory of runs to SW, with a threshold equal to the crop critical point, the water stress events were identified and characterized in terms of D (days) and S (mm). This last variable is given by the sums of the daily evapotranspiration deficit during the corresponding water stress event. A 2-parameter Gamma distribution was fitted to both D and S, whilst their dependence structure was modelled by a Student’s t copula.
Finally, the stations considered were compared in terms of joint probabilities and joint return periods for D and S, thus enabling the evaluation and the characterization of the risk of water stress related to olive in different areas.
Keywords: risk of water deficit, copula functions, joint probabilities, return periods
Parole chiave: rischio di deficit idrico, funzioni copula, probabilità congiunte, tempi di ritorno Introduction
As many other natural hazards, the crop water stress has a typical multivariate nature, i.e. it is characterized by the contemporary presence of multiple characteristics correlated with each other (e.g. duration, severity, peak, areal extension, etc.). In this situation, a risk analysis based on a traditional univariate approach may lead to misleading, inadequate or incomplete interpretations of the phenomenon.
The problem of the probabilistic joint analysis of two or more random correlated variables can be effectively solved by the copula functions (Nelsen, 1999), which have been widely applied in the last decade in the economical and hydrological contexts. Copulas are functions that join univariate probability distributions to form multivariate probability distributions, enabling to model the dependence structure among random variables independently of their marginal distributions (Joe, 1997). In this work, the joint probability and return periods of two characteristics of water stress related to olive orchards (Duration, D and Severity, S) have been modelled by a two-dimensional copula. The case study refers to some stations of the Umbria and Abruzzo regions. The climatic data were used to estimate the dynamics of the available soil water and to derive the characteristics of water stress D and S. The definition of a copula model for D and S allowed to derive the joint probabilities and the joint return periods and to perform a comparative analysis among the considered localities.
Materials and Methods
The climatic data (daily precipitation, and minimum and maximum daily temperature) for the stations considered were obtained from both the Italian National Hydrographic and Oceanographic Service and the Regional Hydrographic Services. The length of the available time series and some summary statistics for the stations selected are given in Tab. 1.
Daily reference evapotranspiration ET0 was estimated by the Hargreaves and Samani equation (Allen et al., 1998), using
only daily minimum and maximum temperature. The estimation reliability of this equation was considered adequate for the purposes of the study. The meterological data where used in input to a “bucket” soil-water balance model to simulate the daily dynamics of the available water for the olive orchards. The soil water balance model follows the scheme suggested in the FAO 56 paper (Allen et al., 1998). In particular, daily precipitation amounts lower than 5 mm were not considered effective and larger amounts were reduced by 20%. For simplicity’sake, a 1-m soil depth with 20 % of crop available water (Total Available Water, TAW= 200 mm) was assumed for all the stations. The olive crop coefficients for the calculation of the non-stressed crop evapotranspiration (ETm) were retrieved from the FAO 56 guidelines and a threshold level of 0.25
TAW was adopted as critical point (beginning of water stress, i.e. actual evapotranspiration ETa< ETm).
The hydrological balance model has been applied continuously throughout the time series starting from an initial condition of full recharge of the soil water (i.e. SWini=TAW), thus enabling the simulation of the soil water (SW, mm) dynamics. The
water deficit events were identified by applying the Theory of Runs (Yevevich, 1967) to the time series of SW, using the threshold level of 0.25 TAW (50 mm): a water stress event was indentified by a continuos period in which SW < 0.25 TAW. Each event has a duration D (numer of consecutive days in which ETa < ETm) and a severity S (sums of the deficits
ETm-ETaduring the event). The mean value of D, S and of the number of water stress events per year is given in Tab.1
Tab. 1: Mean values of some agrometeorological variables for the stations considered.
Tab. 1: Valori medi di alcune variabili agrometeorologiche per le stazioni considerate
The first step of the analysis consisted in the identification of adequate marginal distributions for the random variables considered. The variable D is a discrete variable, but this is an artefact dependent on the application of discrete (daily) soil water balance. Therefore it was assumed that both D and S are continuous random variables. Several 2-parameter continuous distributions have been considered for the description of D and S and their goodness of fit was compared in terms of RMSE (Vergni et al., 2016).
The most suitable copula model was selected by applying both subjective (graphical) and objective methods. The first was based on the analysis of the scatter plots of pseudo-observations plotted together with random pairs of cumulative probabilities generated from a given copula model. The objective evaluation was based on the Cramer-Von Mises statistics, Sn (Genest and Remillard, 2008) whose p-value was calculated by a bootstrap approach based on 1000 repetitions (Vergni
et al., 2015). The R package “Copula” (Hofert et al., 2017) was used for the calculations.
Therefore, the complete models (copula + marginal distributions) were used to estimate: the joint probabilities P (D ≤ d, S ≤ s); the conditional probabilities (P<=S|D>=d’), i.e. the cumulative probability of S given D has already reached a certain threshold d', the joint return periods (years) for the condition D³d and RS³rs.
Results and Discussion
The gamma distribution showed a better goodness of fit in comparison to other tested distributions (Exponential, Weibull) in terms of RMSE and it was therefore selected as marginal distribution for both D and S. The corresponding shape and scale parameters were estimated by the L-moments approach (Hosking, 1990).
Different copula functions were tested to describe the dependence structure between the correlated variables D and S. The best choice was represented by the Student’s t-copula whose parameter was estimated by the method of inversion of Kendall’s tau and the degrees of freedom were optimized by minimizing the statistic Sn (Vergni et al., 2015).
The complete copula model (dependence structure + marginal distributions) allowed to obtain the estimation of the joint probability P(D ≤ d, S ≤ s). A graphical example of this information is given in the perspective graphs of Fig. 1 for the Orvieto (Fig. 1a) and Gubbio (Fig. 1b) stations, which are characterized by very different climatic conditions (drier in Orvieto than in Gubbio, Tab.1). This determined the different shape of the perspective graphs.
Fig. 1: Joint probabilities P (D ≤ d, S ≤ s for two exemplificative stations (a: Orvieto; b: Gubbio).
Fig. 1: Probabilità congiunte P (D ≤ d, S ≤ s per due stazioni di esempio (a: Orvieto; b: Gubbio).
Another example of the results attainable from this type of analysis is shown in Fig. 2, which shows the conditional cumulative probability of S given the duration has reached at least 90 days. The shape of the curves indicates that, in this circumstance, the expected severity is higher for Umbria than Abruzzo stations. Very similar probabilities were observed for Spoleto and Gubbio stations. Fig. 3a and 3b show, respectively, the pairs (D, S) having 5-year and 20-year return periods (condition D ³ d and S ³ s) for olive in the stations considered. The stations characterized by less favourable meteorological conditions (Tab. 1) exhibit in Fig. 3 curves more distant from the axes origin (e.g. Orvieto and Todi). For example, the curve for the 5-year return period in Orvieto is similar to that of Gubbio for a 20-year return period. It is interesting to notice that the two stations Spoleto and Gubbio, characterized by very similar mean
a b
Station Time series precipitation (mm)Annual Number of rainfall events > 5mm evapotranspiration (mm)Reference
Olive maximum evapotranspiration (mm) D (days) S (mm) Water stress events per year
Chieti 1951-2015 777.1 40.0 893.7 475.3 40.3 35.1 1.7 Scerni 1951-2015 765.0 40.2 935.6 497.7 44.6 43.1 1.9 Penne 1951-2012 830.8 44.7 945.3 502.7 30.3 24.3 1.9 Spoleto 1951-2015 991.1 54.1 1074.0 571.1 32.9 33.5 2.2 Gubbio 1951-2015 1007.4 56.9 1016.1 540.2 27.5 27.4 1.9 Orvieto 1951-2015 818.1 46.2 1092.0 580.5 49.0 65.8 2.1 Todi 1951-2015 824.2 47.0 1060.3 563.8 38.3 43.5 2.4
0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 Co nd it io na l p ro ba bi lit y P(S <= s| D >= d’ ) Severity, S (mm)
Chieti Scerni Penne Spoleto
Gubbio Orvieto Todi
d'=90 days
values of D and S (Tab.1) and similar conditional probabilities (Fig. 2), are not so similar if compared in terms of return periods (Fig. 3). This is a consequence of the higher frequency of stress events in Spoleto (Tab. 1). Moreover, it can be also noticed that some curves intersect (e.g. Penne and Gubbio, Spoleto and Chieti): of course these are stations characterized by similar conditions, but the water stress events have a different characterization in terms of D and S. This type of information is only attainable from a joint probabilistic analysis of D and S.
Fig. 2: Conditional probabilities (P<=S|D>=90 days) in some stations of Abruzzo (dotted lines) and Umbria (dashed lines).
Fig. 2: Probabilità condizionate (P<=S|D>=d’) in alcune stazioni di Abruzzo (linee punteggiate) e Umbria (linee tratteggiate).
Conclusions
The use of multivariate probabilistic models is becoming common in hydrological studies. In this paper, a bivariate risk analysis of the Duration and Severity of water stress in olive has been illustrated in relation to some stations of Umbria and Abruzzo regions. This preliminary work was mainly addressed to illustrate the potentiality of this type of approach. The next step will be to improve the model's reliability through its calibration and validation with actual crop data. In this context it would be interesting to explore the possibility to introduce other variables in the joint analysis. For example, taking into account that the sensitivity to water stress varies with the phenological stage of olive, the date of occurrence of the water stress could be a useful variable in a 3D copula model.
Fig. 3: pairs (D, S) having a 5-year (a) and 20-year (b) return periods (condition D
³ d and S ³ s) for olive in
some stations of Abruzzo (dotted lines) and Umbria (dashed lines).
Fig. 3: coppie (D, S) aventi tempi di ritorno 5 (a) e 20 (b) anni (condizione D ≥ d e S ≥ s) in alcune stazioni di Abruzzo (linee punteggiate)
e Umbria (linee
tratteggiate).
References
Allen RG, Pereira LS, Raes D, Smith M (1998). Crop evapotranspiration. Guidelines for computing crop water requirements. Irrigation and drainage paper 56, FAO, Rome.
Genest C., Remillard B., (2008). Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models, Ann. I. H. Poincare-Pr., 44, 1096–1127.
Hofert M., Kojadinovic I., Maechler M. Yan J. (2017). Copula: Multivariate Dependence with Copulas. R package version 0.999-16 URL https://CRAN.R-project.org/package=copula
Hosking JRM (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52:105–124.
Joe H. (1997). Multivariate model and dependence concepts. Chapman and Hall, London. Nelsen R. B. (1999). An introduction to copulas, Springer, New York.
Todisco F., Mannocchi F., Vergni L., (2013). Severity–duration–frequency curves in the mitigation of drought impact: an agricultural case study Nat Hazards 65:1863–1881.
Vergni L., Todisco F., Mannocchi F. (2015). Analysis of agricultural drought characteristics through a two-dimensional copula Water Resources Management 29 (8) 2819–2835.
Vergni L. Todisco F., Di Lena. B., Mannocchi F. (2016). Effect of the North Atlantic Oscillation on winter daily rainfall and runoff in the Abruzzo region (Central Italy). Stoch Environ Res Risk Assess, 30 (7) 1901–1915.
Yevjevich V. (1967). An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology paper No. 23. Colorado State University, Fort Collins.
a 0 30 60 90 120 150 180 210 240 270 300 330 40 70 100 130 160 190 Se ve ri ty , S (m m ) Duration, D (days) Chieti Scerni Penne Spoleto Gubbio Orvieto Todi b 0 50 100 150 200 250 300 350 40 90 140 190 Se ve ri ty , S (m m ) Duration, D (days) Chieti Scerni Penne Spoleto Gubbio Orvieto Todi