Multiplicative Discrete Random Cascade Model (MDRC)
First read rainfall data on a fine resolution (1hour) and aggregate the data to higher time frequencies: (input Level: 1day; upper Level: 12hrs; middle Level: 6hrs, lower level: 3hrs lowest level: 1.5hrs).
Second disaggregating the data through a mutiplicative Discrete Random Cascade model (MDRC) (1440min->720min->360min->180min->90min), at the end, rainfall on a finer resolution will be simulated using a higher time frequency data.
This MDRC is a Microcanonical model: it conserves volumes in every level.
The first step is to find the weights (W1 and W2) of every level, this is done by finding if the volume (V0) in the upper level (60min) has fallen in the first sub interval (V1=V0.W1) or the second (V2=V0.W2) or in both.
Finding model parameters: For every recorded rainfall in the upper level if volume > threshhold (0.3mm) find W1 = R1/R and (W2 = 1-W1). A sample of W is obtained in every level, plot histogram to find distribution.
The weights represent a probability of how the rainfall volume is distributed, Three possible values for the weights: W1 = 0 means all rainfall fell in 2nd sub-interval P (W=0) W1 = 1 means all rainfall fell in 1st sub-interval P (W=1) 0 < W1 < 1 means part of rainfall fell in 1st sub iterval and part in the 2nd For calculating P01, the relation between the volumes and the weights is modeled through a logistic regression. For calculating the prob P (0<W<1) a beta distribution is assigned and using the maximum likelihood method the parameter ß is estimated for every cascade level and every station. The MDRC baseline model has two parameters P01 and ß per level.
the MDRC unbounded model is introduced and allows relating the probability P01 to the rainfall volume R through a logistic regression function the parameters of the logisticRegression fct: a an b are estimated using the maximum likelihood method. This is done by first identifying where w is 0 or 1 and for these values, find the corresponding rainfall volume R and use log(logisticRegression fct) and where w is between ]0:1[ use log(1-logisticRegression fct), in that way the parameters a and b are estimated using all of the observed weights. This is done for every station and every cascade level, therefore the unbounded model has three parameters per level a, b and beta. the value of beta is the same used in the baseline model
Analysing: Once parameters are found, study the effect of the time and space on them: First divide the events into 4 different boxes: (Isolated: 010, Enclosed: 111, Followed: 011, Preceded: 110 ), plot them Second extract the P01 for every month and plot it