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

Abstract: Generating 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.

## 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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