Rainfall disaggregation methods: theory and applications
Demetris Koutsoyiannis
Noted by Li-Pen Wang
1/ Intro
- Higher-level (coarser) time series and lower-level (finer) time series  how to
generate consistent time series
- Flooding warning: paucity of rainfall data for time-scales of one hour or less 
disaggregation model
- General-purpose models: not specific to certain hydrological process/application,
not always applicable to the rainfall process (especially at fine time scales) 
specialised techniques for the rainfall process
- Synthetic fine-scale series should reproduce the important statistical features of
the related hydrological processes at this time scale
2/ General-purpose stochastic disaggregation models
- Valencia and Schaake (1972, 1973), introducing pioneering work in hydrology,
conducted a multivariate model that generates sequences of lower-level values at
many locations as linear combination of the related higher-level values and
independently generated random components.
- The parameter estimation procedure of this model ensures the resemblance of
variance and covariance properties between historical and generated series.
However, it makes no efforts to preserve covariances of the lower-level variables
belonging to consecutive periods. In fact, this model does not assume any
connect between lower-level variables of different periods.
- Drawbacks:
(1) The additive property, which is one of the main attributes of the original
disaggregation scheme, is lost.
(2) Excessive number of parameters due to the large number of cross
correlations that they attempt to reproduce.  the staged disaggregation
model (conducted in two or more steps)/ the condensed disaggregation
model (reduce the number of parameters)/ the dynamic disaggregation
model (DDM, capable of fixing the inconsistency problem of earlier
disaggregation models)
- Comment: the lower-level time interval should not be finer than a month
because these models cannot handle the skewed distributions and the
intermittent nature of the rainfall process at fine time scales.
3/ Rainfall disaggregation models
- Do not exhibit the generality of the Valencia-Schaake type linear schemes (usually
ad hoc techniques).
- Markov chain model (monthly to daily)/ DDM (can handle intermittent rainfall
process, monthly to event duration to hourly, not straightforwardly applied to
daily -> hourly)/ Koutsoyiannis (1994, section 4) / Koutsoyiannis and Onof
(2000/2001, section 5)/ Multifractal (do not perform well in extrapolation) 
limited to single-site.
- Multiple site rainfall disaggregation: the spatial correlation (cross-correlation
among different sites) must be maintained (section 5 and 6).
4/ a generalised stochastic framework for coupling stochastic modes of different
time scales
- Independently run a lower-level model without referring to higher-level model,
and then modify results to be consistent with a given higher-level time series
without affecting stochastic structure implied by the lower-level model.
- Proportional adjusting procedure/ linear adjusting procedure/ generalised linear
adjusting procedure (a generalised framework for coupling stochastic models of
different time series, Koutsoyiannis 2001: ensure preservation of mean and
variance-covariance matrix of variables, and additive property).
5/ Hyetos: a single variate fine time scale rainfall disaggregation model based on
the Bartlett-Lewis process (single site)
- Implementation of generalised linear adjusting procedure mentioned in section 4
(Koutsoyiannis and Onof, 2000; 2001).
- Features: representing rainfall in continuous time; wet/dry structure can be
generated independently.
- General assumptions of Bartlett-Lewis model: storm origins occur following a
Poisson process; origins of cells of each storm arrive following a Poisson process;
arrivals of each storm terminate after a time exponentially distributed; each cell
has a duration exponentially distributed; each cell has a uniform intensity with a
specified (exponential or gamma) distribution.
- Four-level repetition scheme: Level 0 generates L wet days; Level 1 generates the
intensities of all cells and storms and the resulting daily rain depths; Level 2
adjust sequences; Level 3 splits wet day sequence into sub-sequences.
- Good agreement in dry/wet probabilities, variance, skewness, autocorrelation,
and empirical distributions of maximum hourly rainfall.
6/ MuDRain: A model for multivariate disaggregation of rainfall at a fine time scale
- Increased mathematical complexity: multivariate vs. univariate
- A univariate disaggregation model like Hyetos would generate a synthetic hourly
series, fully consistent with the known daily series and, simultaneously,
statistically consistent with the actual hourly rainfall series. Obviously, however, a
synthetic series obtained by such a manner could not coincide with the actual
one, but would be only a likely realisation. Now let us assume that there exist
hourly rainfall data at a neighbouring raingauge. If this is the case and, in addition,
the cross-correlation among the two raingauges is significant, then we would
utilise the available hourly rainfall information at the neighbouring station to
generate spatially and temporally consistent hourly rainfall series at the
raingauge of interest. In other words, the spatial correlation is turned to
advantage, since, in combination with the available single-site hourly rainfall
information, it enables more realistic generation of the synthesised hyetographs.
- A simple AR(1) model is proven to be able to well describe intermittency and
skewness during rain days.
- Case study (Fytilas, 2002):
(1) (hourly cross-correlation) = (daily cross-correlation) ^ m; 2 < m <3
(2) 6 raingauge stations; 6 years of hourly data and 2 daily series.
(3) Good agreement in cross-correlation, lag one autocorrelation coefficient,
autocorrelation functions, probability distribution function, and hyetographs.
7/ Conclusions and discussion
- General-purpose disaggregation model/