Newcastle University ePrints - eprint.ncl.ac.uk

Lombardo F, Volpi E, Koutsoyiannis D, Serinaldi F.

A theoretically consistent stochastic cascade for temporal disaggregation of

intermittent rainfall.

Water Resources Research 2017

DOI: http://doi.org/10.1002/2017WR020529

Copyright:

This is the peer reviewed version of the following article: Lombardo F, Volpi E, Koutsoyiannis D, Serinaldi

F. A theoretically consistent stochastic cascade for temporal disaggregation of intermittent rainfall. Water

Resources Research 2017, which has been published in final form at

http://doi.org/10.1002/2017WR020529. This article may be used for non-commercial purposes in

accordance with Wiley Terms and Conditions for Self-Archiving.

DOI link to article:

http://doi.org/10.1002/2017WR020529

Date deposited:

14/06/2017

Embargo release date:

12 May 2018

1

A theoretically consistent stochastic cascade for temporal disaggregation of 1

intermittent rainfall 2

3

F. Lombardo

1

, E. Volpi

1

, D. Koutsoyiannis

2

, and F. Serinaldi

3,4

4

5

1

Dipartimento di Ingegneria, Università degli Studi Roma Tre, Via Vito Volterra, 62 – 00146 6

Rome, Italy. 7

2

Department of Water Resources and Environmental Engineering, National Technical 8

University of Athens, Heroon Polytechneiou 5, GR-157 80 Zographou, Greece. 9

3

School of Civil Engineering and Geosciences, Newcastle University - Newcastle Upon Tyne, 10

NE1 7RU, UK. 11

4

Willis Research Network - 51 Lime St., London, EC3M 7DQ, UK. 12

13

Corresponding author: Federico Lombardo (federico.lombardo@uniroma3.it) 14

15

Key Points: 16

Parsimonious generation of fine-scale time series of intermittent rainfall with prescribed 17

dependence structure 18

Modeling approach of mixed type in stationary setting, with discrete description of 19

intermittency and continuous description of rainfall 20

Analytical formulation of autocorrelation for mixed process without making any 21

assumption about dependence structure or marginal probability 22

23

24

2

Abstract 25

Generating fine-scale time series of intermittent rainfall that are fully consistent with any given 26

coarse-scale totals is a key and open issue in many hydrological problems. We propose a 27

stationary disaggregation method that simulates rainfall time series with given dependence 28

structure, wet/dry probability, and marginal distribution at a target finer (lower-level) time scale, 29

preserving full consistency with variables at a parent coarser (higher-level) time scale. We 30

account for the intermittent character of rainfall at fine time scales by merging a discrete 31

stochastic representation of intermittency and a continuous one of rainfall depths. This approach 32

yields a unique and parsimonious mathematical framework providing general analytical 33

formulations of mean, variance, and autocorrelation function (ACF) for a mixed-type stochastic 34

process in terms of mean, variance, and ACFs of both continuous and discrete components, 35

respectively. To achieve the full consistency between variables at finer and coarser time scales in 36

terms of marginal distribution and coarse-scale totals, the generated lower-level series are 37

adjusted according to a procedure that does not affect the stochastic structure implied by the 38

original model. To assess model performance, we study rainfall process as intermittent with both 39

independent and dependent occurrences, where dependence is quantified by the probability that 40

two consecutive time intervals are dry. In either case, we provide analytical formulations of main 41

statistics of our mixed-type disaggregation model and show their clear accordance with Monte 42

Carlo simulations. An application to rainfall time series from real world is shown as a proof of 43

concept. 44

1 Introduction 45

Rainfall is the main input to most hydrological systems. A wide range of studies 46

concerning floods, water resources and water quality require characterization of rainfall inputs at 47

3

fine time scales [Blöschl and Sivapalan, 1995]. This may be possible using empirical 48

observations, but there is often a need to extend available data in terms of temporal resolution 49

satisfying some additive property (i.e. that the sum of the values of consecutive variables within 50

a period be equal to the corresponding coarse-scale amount) [Berne et al., 2004]. Hence, rainfall 51

disaggregation models are required. Both disaggregation and downscaling models refer to 52

transferring information from a given scale (higher-level) to a smaller scale (lower-level), e.g. 53

they generate consistent rainfall time series at a specific scale given a known precipitation 54

measured or simulated at a certain coarser scale. The two approaches are very similar in nature 55

but not identical to each other. Downscaling aims at producing the finer-scale time series with 56

the required statistics, being statistically consistent with the given variables at the coarser scale, 57

while disaggregation has the additional requirement to produce a finer scale time series that adds 58

up to the given coarse-scale total. 59

Although there is substantial experience in stochastic disaggregation of rainfall to fine 60

time scales, most modeling schemes existing in the literature are ad hoc techniques rather than 61

consistent general methods [see review by Koutsoyiannis, 2003a]. Disaggregation models were 62

introduced in hydrology by the pioneering work of Valencia and Schaake [1973], who proposed 63

a simple linear disaggregation model that is fully general for Gaussian random fields without 64

intermittency. However, the skewed distributions and the intermittent nature of the rainfall 65

process at fine time scales are severe obstacles for the application of a theoretically consistent 66

scheme to rainfall disaggregation [Koutsoyiannis and Langousis, 2011]. This paper reports some 67

progress in this respect. Our model exploits the full generality and theoretical consistency of 68

linear disaggregation schemes proposed by Valencia and Schaake [1973] for Gaussian random 69

4

variables, but it generates intermittent time series with lognormal distribution that are more 70

consistent with the actual rainfall process at fine time scales. 71

The following sections expand on a stochastic approach to rainfall disaggregation in time, 72

with an emphasis on the analytical description of a model of the mixed (discrete-continuous) 73

type. Firstly, we generate lognormal time series of rainfall depths with prescribed mean, variance 74

and autocorrelation function (ACF) based on fractional Gaussian noise (fGn), also known as 75

Hurst-Kolmogorov (HK) process [Mandelbrot and Van Ness, 1968]. Note that the lognormality 76

hypothesis and our specific normalizing transformation (see next section) enable the analytical 77

formulation of the main statistics of the rainfall depth process. Secondly, we obtain the 78

intermittent rainfall process by multiplying the synthetic rainfall depths above by user-specified 79

binary sequences (i.e., rainfall occurrences) with given mean and ACF. The resulting stochastic 80

model is of the mixed type and we derive its summary statistics in closed forms. 81

We propose herein an evolution of the downscaling model by Lombardo et al. [2012], 82

which is upgraded and revised to include both a stochastic model accounting for intermittency 83

and an appropriate strategy to preserve the additive property. The preservation of the additive 84

property distinguishes indeed disaggregation from downscaling. This modification required to 85

set up a disaggregation model produces a more realistic rainfall model that retains its primitive 86

simplicity in association with a parsimonious framework for simulation. In brief, the 87

advancements reported under the following sections include: 88

Background information. A basic review with discussion about some improvements on 89

the model structure is presented in the next section. 90

Intermittency. The main novelty of this paper is the introduction of intermittency in the 91

modeling framework, which is fully general and it can be used when simulating mixed-92