Skip to main content
SpringerLink
Log in

A stochastic model for the analysis of the temporal change of dry spells

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

In the present paper a stochastic approach which considers the arrival of rainfall events as a Poisson process is proposed to analyse the sequences of no rainy days. Particularly, among the different Poisson models, a non-homogeneous Poisson model was selected and then applied to the daily rainfall series registered at the Cosenza rain gauge (Calabria, southern Italy), as test series. The aim was to evaluate the different behaviour of the dry spells observed in two different 30-year periods, i.e. 1951–1980 and 1981–2010. The analyses performed through Monte Carlo simulations assessed the statistical significance of the variation of the mean expected values of dry spells observed at annual scale in the second period with respect to those observed in the first. The model has then been verified by comparing the results of the test series with the ones obtained from other three rain gauges of the same region. Moreover, greater occurrence probabilities for long dry spells in 1981–2010 than in 1951–1980 were detected for the test series. Analogously, the return periods evaluated for fixed long dry spells through the synthetic data of the period 1981–2010 resulted less than half of the corresponding ones evaluated with the data generated for the previous 30-year period.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

a mod(n):

Remainder of the Euclidean division of a by n

a 0, a j , b j :

Coefficients of the truncated Fourier series, with j = 1, 2,…,n h (days−1)

α 0, α j , β j :

Coefficients of the truncated Fourier series for the Monte Carlo simulations, with j = 1, 2,…,n h (days−1)

c :

Integration constant

D :

Period (in this case, average length of the year, days)

\( H_{0}^{{(n_{h} )}} \) :

Null hypothesis about the truncated Fourier series with n h harmonics

\( H_{0}^{{(\bar{m})}} \) :

Null hypothesis about the mean expected annual value \( \bar{m} \)

\( i_{n}^{'} \) :

First day of the nth sequence of no rainy days

int[u]:

Greatest integer number lower than u

inv f(x):

Inverse function of f(x)

K′ :

Number of consecutive time intervals with no rainfall (k′ = 0, 1, 2,…,)

\( k_{\hbox{max} }^{'} \) :

Maximum value of k′ over a time interval of length 1 year

\( k_{n}^{'} \) :

Number of no rainy days of the nth sequence of K′

\( k_{*}^{'} \) :

Threshold value of K′

λ(t):

Arrival intensity function (parameter of the Poisson process)

Λ(t):

Integral of the arrival intensity function λ(t)

L :

Number of rainy consecutive time intervals Δt belonging to a rainfall event started before the instant iΔt (l = 0, 1, 2,…,)

Lf :

Likelihood function

M :

Number of consecutive time intervals of equal length Δt (=1 day) with no rainfall, followed by at least one rainy day (m = 0, 1, 2,…,)

\( m_{*} \) :

Threshold value of M

M′:

Number of consecutive time intervals of length Δt (=1 day) without any arrival of rainfall events, followed by at least one rainy day (m′ = 0, 1, 2,…,)

\( \bar{m}_{X}^{(j)} \) :

Mean of the sample values of the variable X for the jth class

\( \bar{m}_{X}^{{({\text{year}}1 {-} {\text{year}}2)}} \) :

Mean of the sample annual values of the variable X for the period (year1–year2)

µ X :

Theoretical mean of the generic variable X

\( \bar{\mu }_{X,p}^{(j)} \) :

p-percentile of the mean value of the simulated series for the jth class of the variable X

\( \bar{\mu }_{X}^{{({\text{year}}1 {-} {\text{year}}2)}} \) :

Mean of the simulated annual values of the variable X for the period (year1–year2)

\( \bar{\mu }_{X,p}^{{({\text{year}}1 {-} {\text{year}}2)}} \) :

p-percentile of the mean value of the simulated series of the variable X in the period (year1–year2)

N c :

Number of sample data classes

N d :

Sample size of dry spells

N G :

Number of series generated through an embedded Monte Carlo procedure

n h :

Number of harmonics

\( \sigma_{X}^{2} \) :

Theoretical variance of the generic variable X

\( t_{R}^{{(n_{R} )}} \) :

Series of temporal arrival of sequences of consecutive no rainy days longer than 5 days with n R  = 1, 2,…,

T :

Return period

\( U^{{(n_{R} )}} \) :

Synthetic generation of uniformly distributed random values in the interval (0, 1) with n R  = 1, 2,…,

ω(t):

Arrival intensity function of the Monte Carlo simulation

Ω(t):

Integral of the arrival intensity function ω(t)

References

  • Abaurrea J, Asin J, Cebrian AC, Centelles A (2007) Modeling and forecasting extreme hot events in the central Ebro Valley, a continental-Mediterranean area. Glob Planet Change 57:43–58

    Article  Google Scholar 

  • Alpert P, Ben-Gai T, Baharad A, Benjamini Y, Yekutieli D, Colacino M, Diodato L, Ramis C, Homar V, Romero R, Michaelides S, Manes A (2002) The paradoxical increase of Mediterranean extreme daily rainfall in spite of decrease in total values. Geophys Res Lett 29:31-1–31-4

    Google Scholar 

  • AMS Statement (2004) AMS statement on meteorological drought. Bull Am Meteorol Soc 85:771–773

    Google Scholar 

  • Anagnostopoulou CHR, Maheras P, Karacostas T, Vafiadis M (2003) Spatial and temporal analysis of dry spells in Greece. Theor Appl Climatol 74:77–91

    Article  Google Scholar 

  • Bates BC, Kundzewicz ZW, Wu S, Palutikof JP (2008) Climate change and water. Technical Paper of the Intergovernmental Panel on Climate Change, Geneva

    Google Scholar 

  • Box GEP, Jenkins GM (1970) Time series analysis forecasting and control. Holden-Day, San Francisco

    Google Scholar 

  • Brunetti M, Caloiero T, Coscarelli R, Gullà G, Nanni T, Simolo C (2012) Precipitation variability and change in the Calabria region (Italy) from a high resolution daily dataset. Int J Climatol 32:55–73

    Google Scholar 

  • Buishand TA (1978) Some remarks on the use of daily rainfall models. J Hydrol 36:295–308

    Article  Google Scholar 

  • Buttafuoco G, Caloiero T, Coscarelli R (2011) Spatial and temporal patterns of the mean annual precipitation at decadal time scale in southern Italy (Calabria region). Theor Appl Climatol 105:431–444

    Article  Google Scholar 

  • Caloiero T, Coscarelli R, Ferrari E, Mancini M (2011a) Precipitation change in southern Italy linked to global scale oscillation indexes. Nat Hazards Earth Syst Sci 11:1–12

    Article  Google Scholar 

  • Caloiero T, Coscarelli R, Ferrari E, Mancini M (2011b) Trend detection of annual and seasonal rainfall in Calabria (southern Italy). Int J Climatol 31:44–56

    Article  Google Scholar 

  • Cancelliere A, Salas JD (2004) Drought length properties for periodic–stochastic hydrologic data. Water Resour Res 40:W02503

    Article  Google Scholar 

  • Capra A, Consoli S, Scicolone B (2013) Long-term climatic variability in Calabria and effects on drought and agrometeorological parameters. Water Resour Manag 27:601–617

    Article  Google Scholar 

  • Chang TJ, Kavvas ML, Delleur JW (1984) Daily precipitation modeling by discrete autoregressive moving average processes. Water Resour Res 20:565–580

    Article  Google Scholar 

  • Chapman T (1997) Stochastic models for daily rainfall in the Western Pacific. Math Comput Simul 43:351–358

    Article  Google Scholar 

  • Chapman T (1998) Stochastic models of daily rainfall: the impact of adjoining wet days on the distribution of rainfall amounts. Environ Model Softw 13:317–324

    Article  Google Scholar 

  • Cinlair E (1975) Introduction to stochastic processes. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Craddock JM (1979) Methods of comparing annual rainfall records for climatic purposes. Weather 34:332–346

    Article  Google Scholar 

  • De Luis M, Raventos J, Gonzalez-Hidalgo JC, Sanchez JR, Cortina J (2000) Spatial analysis of rainfall trends in the region of Valencia (East of Spain). Int J Climatol 20:1451–1469

    Article  Google Scholar 

  • Del Rio S, Herrero L, Fraile R, Penas AP (2011) Spatial distribution of recent rainfall trends in Spain (1961–2006). Int J Climatol 31:656–667

    Article  Google Scholar 

  • Deni SM, Jemain AA, Ibrahim K (2010) The best probability models for dry and wet spells in Peninsular Malaysia during monsoon seasons. Int J Climatol 30:1194–1205

    Article  Google Scholar 

  • Di Giuseppe E, Vento D, Epifani C, Esposito S (2005) Analysis of dry and wet spells from 1870 to 2000 in four Italian sites. Geophys Res Abstr 7:07712

    Google Scholar 

  • Dobi-Wantuch I, Mika J, Szeidl L (2000) Modeling wet and dry spells with mixture distributions. Meteorol Atmos Phys 73:245–256

    Article  Google Scholar 

  • Dracup JA, Lee KS, Paulson EG (1980) On the statistical characteristics of drought events. Water Resour Res 16:289–296

    Article  Google Scholar 

  • Feidas H, Noulopoulou Ch, Makrogiannis T, Bora-Senta E (2007) Trend analysis of precipitation time series in Greece and their relationship with circulation using surface and satellite data: 1955–2001. Theor Appl Climatol 87:155–177

    Article  Google Scholar 

  • Ferrari E, Caloiero T, Coscarelli R (2013) Influence of the North Atlantic Oscillation on winter rainfall in Calabria (southern Italy). Theor Appl Climatol 114:479–494

    Article  Google Scholar 

  • Grayson RB, Argent RM, Nathan RJ, McMahon TA, Mein RG (1996) Hydrological recipes: estimation techniques in Australian hydrology. Cooperative Research Centre for Catchment Hydrology, Melbourne

    Google Scholar 

  • Gregory JM, Wigley TML, Jones PD (1993) Application of Markov models to area-average daily precipitation series and interannual variability in seasonal totals. Clim Dyn 8:299–310

    Article  Google Scholar 

  • Gupta V, Duckstein L (1975) Droughts as extreme distributions from a point rainfall process. Water Resour Res 11:221–228

    Article  Google Scholar 

  • Guttman NB, Wallis JR, Hosking JRM (1992) Spatial comparability of the Palmer drought severity index. Water Resour Bull 28:1111–1119

    Article  Google Scholar 

  • Güven O (1983) A simplified semiempirical approach to probabilities of extreme hydrologic droughts. Water Resour Res 19:441–453

    Article  Google Scholar 

  • Hoerling M, Eischeid J, Perlwitz J, Quan X, Zhang T, Pegion P (2012) On the increased frequency of Mediterranean drought. J Clim 25:2146–2161

    Article  Google Scholar 

  • Hutchinson M (1990) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun Stat Simul Comput 19:433–450

    Article  Google Scholar 

  • Kiktev D, Sexton DMH, Alexander L, Folland CK (2003) Comparison of modelled and observed trends in indices of daily climate extremes. J Clim 16:3560–3571

    Article  Google Scholar 

  • Kutiel H (1985) The multimodality of the rainfall course in Israel, as reflected by the distribution of dry spells. Meteorol Atmos Phys 36:15–27

    Google Scholar 

  • Lana X, Burgueño A (1998) Daily dry–wet behaviour in Catalonia (NE Spain) from the viewpoint of Markov chains. Int J Climatol 18:793–815

    Article  Google Scholar 

  • Lana X, Fernandez Mills G, Burgueño A (1995) Daily precipitation maxima in Catalonia (NE Spain): expected values and their spatial distribution. Int J Climatol 15:341–354

    Article  Google Scholar 

  • Lana X, Martinez MD, Burgueño A, Serra C, Martin-Vide J, Gomez L (2006) Distributions of long dry spells in the Iberian Peninsula, years 1951–1990. Int J Climatol 14:1999–2021

    Article  Google Scholar 

  • Malamud BD, Turcotte DL (1999) Self affine time series: measures of weak and strong persistence. J Stat Plan Inference 80:173–196

    Article  Google Scholar 

  • Mathier L, Perreault L, Bobée B, Ashkar F (1992) The use of geometric and gamma-related distributions for frequency analysis of water deficit. Stoch Hydrol Hydraul 6:239–254

    Article  Google Scholar 

  • Millan J, Yevjevich V (1971) Probability of observed droughts. Hydrology Paper number 50. Colorado State University Publication, Fort Collins

    Google Scholar 

  • Mishra AK, Singh VP (2011) Drought modeling—a review. J Hydrol 403:157–175

    Article  Google Scholar 

  • Nastos PT, Zerefos CS (2009) Spatial and temporal variability of consecutive dry and wet days in Greece. Atmos Res 94:616–628

    Article  Google Scholar 

  • Piccarreta M, Pasini A, Capolongo D, Lazzari M (2013) Changes in daily precipitation extremes in the Mediterranean from 1951 to 2010: the Basilicata region, southern Italy. Int J Climatol 33:3229–3248

    Article  Google Scholar 

  • Salas JD (1993) Analysis and modeling of hydrologic time series. In: Maidment D (ed) Handbook of hydrology. McGraw-Hill, New York

    Google Scholar 

  • Salas JD, Smith RA (1981) Physical basis of stochastic models of annual flows. Water Resour Res 17:428–430

    Article  Google Scholar 

  • Santos MA (1983) Regional droughts: a stochastic characterization. J Hydrol 66:183–211

    Article  Google Scholar 

  • Sen Z (1977) Run-sums of annual streamflow series. J Hydrol 35:311–324

    Article  Google Scholar 

  • Sen Z (1980) Regional drought and flood frequency analysis, theoretical consideration. J Hydrol 46:265–279

    Article  Google Scholar 

  • Serra C, Martínez MD, Lana X, Burgueño A (2013) European dry spell length distributions, years 1951–2000. Theor Appl Climatol 114:531–551

    Article  Google Scholar 

  • Sharma T (1995) Estimation of drought severity on independent and dependent hydrologic series. Water Resour Manag 11:35–49

    Article  Google Scholar 

  • Shiau JT, Shen HW (2001) Recurrence analysis of hydrologic droughts of differing severity. J Water Resour Plan Manag 127:30–40

    Article  Google Scholar 

  • Sirangelo B, Ferrari E, De Luca DL (2011) Occurrence analysis of daily rainfalls through non-homogeneous Poissonian processes. Nat Hazards Earth Syst Sci 11:1657–1668

    Article  Google Scholar 

  • Smith JA, Karr AF (1983) A point process model of summer season rainfall occurrences. Water Resour Res 19:95–103

    Article  Google Scholar 

  • Snyder DL (1975) Random point processes. Wiley, New York

    Google Scholar 

  • Soule PT (1992) Spatial patterns of drought frequency and duration in the contiguous USA based on multiple droughts event definitions. Int J Climatol 12:11–24

    Article  Google Scholar 

  • Spirkl W, Ries H (1986) Non-stationary Markov chains for modelling daily radiation data. Q J R Meteorol Soc 112:1219–1229

    Article  Google Scholar 

  • Srikanthan R, McMahon TA (2000) Stochastic generation of climate data: a review. CRC for Catchment Hydrology, Monash University, Clayton

    Google Scholar 

  • Srikanthan R, Harrold TI, Sharma A, McMahon TA (2005) Comparison of two approaches for generation of daily rainfall data. Stoch Environ Res Risk Assess 19:215–226

    Article  Google Scholar 

  • Tase N (1976) Area–deficit–intensity characteristics of drought. Hydrology Paper number 87. Colorado State University Publication, Fort Collins

    Google Scholar 

  • Tolika K, Maheras P (2005) Spatial and temporal characteristics of wet spells in Greece. Theor Appl Climatol 81:71–85

    Article  Google Scholar 

  • Vicente-Serrano SM, González-Hidalgo JC, de Luis M, Raventós J (2004) Drought patterns in the Mediterranean area: the Valencia region (eastern Spain). Clim Res 26:5–15

    Article  Google Scholar 

  • Vicini L, Hotta LK, Achcar JA (2012) Non-homogeneous Poisson processes applied to count data: a bayesian approach considering different prior distributions. J Environ Prot 3:1336–1345

    Article  Google Scholar 

  • Waymire E, Gupta VK (1981) The mathematical structure of rainfall representations. 1. A review of the stochastic rainfall models. Water Resour Res 17:1261–1272

    Article  Google Scholar 

  • Wilks DS (1999) Interannual variability and extreme-value characteristics of several stochastic daily precipitation models. Agric For Meteorol 93:153–169

    Article  Google Scholar 

  • Woolhiser DA (1992) Modeling daily precipitation—progress and problems. In: Walden AT, Guttorp P (eds) Statistics in the environmental and earth sciences. Edward Arnold, London, pp 71–89

    Google Scholar 

  • Woolhiser DA, Pegram GGS (1979) Maximum likelihood estimation of Fourier coefficients to describe seasonal variations of parameters in stochastic daily precipitation models. J Appl Meteorol 18:34–44

    Article  Google Scholar 

  • Xoplaki E, Luterbacher J, González-Rouco JF (2006) Mediterranean summer temperature and winter precipitation, large-scale dynamics, trends. Nuovo Cim C 29:45–54

    Google Scholar 

  • Xu Z, Schumann A, Brass C, Li J, Ho K (2001) Chain dependent Markov correlation pulse model for daily stream flow generation. Adv Water Resour 24:551–564

    Article  Google Scholar 

  • Yevjevich V (1967) An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Paper number 23. Colorado State University Publication, Fort Collins

    Google Scholar 

  • Zelenhasìc E, Salvai A (1987) A method of streamflow drought analysis. Water Resour Res 23:156–168

    Article  Google Scholar 

  • Zolina O, Simmer C, Belyaev K, Sergey K, Gulev S, Koltermann P (2013) Changes in the duration of European wet and dry spells during the last 60 years. J Clim 26:2022–2047

    Article  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the anonymous reviewers for the precious remarks and comments which led to improve the initial version of this paper.

Author information

Authors and Affiliations

  1. Department of Environmental and Chemical Engineering, University of Calabria, Rende, CS, Italy

    B. Sirangelo & E. Ferrari

  2. Institute for Agriculture and Forest Systems in the Mediterranean (CNR-ISAFOM), National Research Council of Italy, Rende, CS, Italy

    T. Caloiero

  3. Research Institute for Geo-hydrological Protection (CNR-IRPI), National Research Council of Italy, Rende, CS, Italy

    R. Coscarelli

Authors
  1. B. Sirangelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Caloiero
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Coscarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Ferrari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Ferrari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sirangelo, B., Caloiero, T., Coscarelli, R. et al. A stochastic model for the analysis of the temporal change of dry spells. Stoch Environ Res Risk Assess 29, 143–155 (2015). https://doi.org/10.1007/s00477-014-0904-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-014-0904-5

Keywords

Search

Navigation