Journal ArticleDOI

# A theoretically consistent stochastic cascade for temporal disaggregation of intermittent rainfall

Federico Lombardo
01 Jun 2017-Water Resources Research (Wiley-Blackwell)-Vol. 53, Iss: 6, pp 4586-4605

AbstractGenerating fine-scale time series of intermittent rainfall that are fully consistent with any given coarse-scale totals is a key and open issue in many hydrological problems. We propose a stationary disaggregation method that simulates rainfall time series with given dependence structure, wet/dry probability, and marginal distribution at a target finer (lower-level) time scale, preserving full consistency with variables at a parent coarser (higher-level) time scale. We account for the intermittent character of rainfall at fine time scales by merging a discrete stochastic representation of intermittency and a continuous one of rainfall depths. This approach yields a unique and parsimonious mathematical framework providing general analytical formulations of mean, variance, and autocorrelation function (ACF) for a mixed-type stochastic process in terms of mean, variance, and ACFs of both continuous and discrete components, respectively. To achieve the full consistency between variables at finer and coarser time scales in terms of marginal distribution and coarse-scale totals, the generated lower-level series are adjusted according to a procedure that does not affect the stochastic structure implied by the original model. To assess model performance, we study rainfall process as intermittent with both independent and dependent occurrences, where dependence is quantified by the probability that two consecutive time intervals are dry. In either case, we provide analytical formulations of main statistics of our mixed-type disaggregation model and show their clear accordance with Monte Carlo simulations. An application to rainfall time series from real world is shown as a proof of concept.

Topics: Spatial dependence (59%), Autocorrelation (54%), Stochastic process (54%), Marginal distribution (54%)

## Summary (2 min read)

### 1 Introduction

•  Monte Carlo experiments and comparison to observed data.
• In Section 5 and 6, the authors show respectively some Monte Carlo experiments and a case study in order to test the capability of their model to reproduce the statistical behavior of synthetic and real rainfall time series.
• The authors conclude their work with Section 7, where they give an overview on the key ideas and briefly discuss the applicability aspects of their approach.

### 2 Basic concepts and background

• In summary, their model assumes lognormal rainfall, and then it is reasonable to use a (scale-dependent) logarithmic transformation of variables (eq. 5) and perform disaggregation of transformed variables in a Gaussian domain, thus exploiting the desired properties of the normal distribution for linear disaggregation schemes [Koutsoyiannis, 2003a] .
• Indeed, the authors simulate a fGn in the auxiliary domain whose characteristics are modified (by eq. 5) based on the last disaggregation step of interest k, in order to obtain (by eq. 14) 2 k variables in the lognormal domain with the desired statistical properties given by eqs. ( 15)-( 17).

### 2. Markov chain model.

• The preservation of the additive property is guaranteed by applying eq. ( 36) to the generated series (see next section).
• The intermittent component refers exclusively to the target scale, and is combined with the continuous component at that scale.
• Note that mean and variance in eqs. ( 19) and ( 20) are independent of the specific model, while the ACF in eq. ( 29) relies on the dependence structures of both the continuous and binary components.
• In the following, the authors show how this ACF specializes for intermittent components with Bernoulli and Markov structures.

### 3.2 Markovian occurrences

• Conversely, the proportional procedure always results in positive variables, but it is strictly exact only in some special cases that introduce severe limitations.
• The power adjusting procedure has no limitations and works in any case, but it does not preserve the additive property at once.
• Then, the application of eq. ( 36) must be iterative, until the calculated sum of the lower-level variables equals the given , .
• The power adjusting procedure greatly outperforms the other procedures in terms of accuracy.

### 5 Numerical simulations

• Figs. 6 and 7 show empirical vs. theoretical ACFs of two different mixed-type processes assuming respectively purely random and Markovian occurrences, , , with the same parameters as above (clearly, for random occurrences the authors have , 1 0).
• Note that both figures also depict the case with null probability dry, i.e. , 0, which corresponds to the rainfall depth process, , .
• As expected, both of their occurrence models are generally cause for decorrelation of the intermittent process with respect to the process without intermittency.
• For Markovian occurrences (see Fig. 7 ), the autocorrelation is higher for small time lags than that for random occurrences, while it tends to the random case asymptotically (compare Figs.

### 6 Application to observational data

• In Figs. 10 and 11, the authors compare the historical hyetographs for January 1999 and April 2003 to a typical synthetic hyetograph generated by their model.
• In both cases, the authors can see that their model produces realistic traces of the real world hyetograph.
• Other than similarities in the general shapes, the authors showed that their model provides simulations that preserve the statistical behavior observed in real rainfall time series.

### 1. Input parameters

• Estimating such parameters from rainfall data series is relatively straightforward [see also Koutsoyiannis, 2003b] .
• In addition, it should be emphasized that their model fitting does not require the use of statistical moments of order higher than two, which are difficult to be reliably estimated from data [Lombardo et al., 2014] .

### 3. Disaggregation scheme

• This is based on a dyadic random cascade structure (see e.g. Fig. 2 ) such that each higher-level amount is disaggregated into two lower-level amounts satisfying the additivity constraint in eq. ( 6).
• The generation step is based on eq. ( 7) that can account for correlations with other variables previously generated.
• By eq. ( 14), the authors transform lower-level variables generated in the auxiliary domain back to the target domain, but the additive property is not satisfied anymore.

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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

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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. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.