The Cascade Model after OLSSON

The basic concept of cascade models, is a process of repeatedly dividing a given dimension into successively smaller units. Each division step redistributes a given magnitude following previously defined parameters. The underlying idea is self-similarity over a certain range of scales. In order to apply cascade processes on temporal rainfall disaggregation, a number of studies was conducted (e.g. Olsson et al., 1993; Olsson, 1995), leading to the result, that rainfall is characterised by multifractal behaviour over a range of scales. “Scaling implies that statistical properties of the process observed at different scales, i.e., resolutions, are governed by the same relationship [. . . ]” (Olsson, 1998). One way to preserve these scaling properties at each resolution, is to use a so-called bounded random cascade. The term “bounded” means a fitting of the scale-dependent behaviour of the cascade parameters with a power function depending on scale (see e.g. Molnar & Burlando, 2005). This is usually achieved by deriving a log-log linear relationship between the cascade parameters and the timescale applied. Another approach is represented by unbounded random cascades, where parameters are independent of scale. Such a model was applied by Olsson (1998), who assessed the scaling behaviour of rainfall time series, and found a range, where statistical

properties arescale-invariant, i.e. do not change significantly. In his study area, he verified

the existence of scale-invariant conditions between approximately 1 hour and 1 week.

4.6.2.1. Basic Concept

The model used for temporal disaggregation of precipitation is a multiplicative microcanonical random cascade with branching number 2, defined after Olsson (1998) and extended by Güntner et al. (2001), which is briefly described in this section. It is characterised by exact conservation of mass as opposed to canonical cascades.

Fig. 4.18 visualizes exemplarily the basic scheme. In order to disaggregate the rainfall data,

at first statistics have to be calculated from empirical data to form the so-called generator,

which controls the disaggregation process. The cascadelevel denotes the time series at a

certain temporal resolution, which is doubled or halved at the transition (called modulation)

to a higher, respectively lower level. A time interval at an arbitrary level is called abox, which

can bedryorwet, depending on its rainfall volumeV. The volumes of the boxes are classified

as

• above mean volumeor

• below mean volume.

A split-up of a box during a modulation, abranching, results in two equally sized boxes in the

higher level at the adjacent time intervalst1 andt2. The original volume is redistributed with

two multiplicativeweights W1 andW2, withW1+W2 = 1. Three weighting possibilities are

available at a modulation to a higher level:

• 1/0-division: the whole rainfall volumeV occurred duringt1, thenV1 = 1andV2 = 0on the higher level

100

70 30

10 60 0 30

0 10 40 20 0 0 30 0

starting enclosed ending isolated

x/x 0/1 x/x x/x 0/1 1/0 position: division: Level (resolution): 3 (80 min) 1 (20 min) 2 (40 min) 0 (10 min)

10 = box with 10 mm rainfall

W1 W2

weight:

Fig. 4.18: Schematics of the cascade process as defined byOlsson(1998)

• x/x-division: if rainfall is divided tot1 andt2 withV1 >0andV2 >0

To improve model performance, thepositionof the box is taken into account, which is referred

as:

• starting: if a precipitation event is beginning, i.e. the preceding box is dry and the succeeding box is wet

• enclosed: if the box observed is enclosed by wet boxes

• ending: an ending precipitation event, inverse to the starting case

• isolated: the box is surrounded by dry boxes

For construction of the generator of the model, a successive aggregation of data at the high- est level available has to be done, until the desired resolution, for which disaggregation shall be carried out, is reached. During these aggregation steps statistics regarding volume, po- sition and division are collected, resulting in frequencies for each combination at each level.

These frequencies are converted to probabilitiesP which define the generator. In this way

the generator defined by Eq. (4.56) has 24 parameters, derived from the combination of 2 volume classes, 4 position classes and 3 division classes.

W1, W2 =        1 and 0 withP(1/0) 0 and 1 withP(0/1) W_x/xand1−Wx/x withP(x/x) (4.56) where0< Wx/x<1andP(0/1) +P(1/0) +P(x/x) = 1.

Probabilities of Wx/x values for ending boxes below mean volume 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 16 38 50 66 90 W1 [% of V] P [0 -1 ]

Fig. 4.19: ExemplaryWx/x-distribution for ending boxes below mean volume of a the generator of a

cascade model

Additionally for the case of a x/x-division, a probability distribution for the volume fractions is

calculated, called aWx/x-distribution. TheWx/x-distribution consists of a number of classes

for the weights W, with their associated probabilities, for each volume and position class, as exemplarily shown in Fig. 4.19.

4.6.2.2. Application of a Cascade Model

The steps from preparing the generator to applying the model for disaggregation are illustrated in this section, to provide an understanding for the modifications made in this study to adapt

the model toAtmoStations, which is subject in Section 4.6.3.

As already mentioned, in order to use a cascade model, its generator first has to be cal- culated and calibrated using high resolution empirical data. The properties of the generator then are used to disaggregate the desired low resolution datasets. The following listing sum- marises the required steps:

1. Aggregation: Aggregation starts at the level with the highest temporal resolution (e.g. level 0 in Fig. 4.18). For calculation of the generator, statistics accounting for rainfall properties are collected the following way:

a) The volumes of two adjacent boxes are added up into a box in the upper level. The temporal resolution of the boxes in the upper level is doubled.

b) The type of division is determined from the two aggregated boxes in the lower

level. In case of a x/x-division, the ratio of the weights W1 and W2 forming the

upper box, is calculated. The classes for volume and position are defined by the new, resulting box.

c) All boxes of the current level are evaluated this way and the information gained, concerning volume, position and division for this level, is stored.

This aggregation procedure (steps 1a–1c) is repeated until the desired temporal reso- lution (e.g. level 3 in Fig. 4.18) is reached.

2. Generator calculation: After the aggregation from step 1 frequencies for each modula- tion are available, which are used to form the generator:

a) In order to form a valid generator, the degree of scale invariance has to be as-

sessed. This can be done by analysing the distribution of the probabilitiesP(1/0),

P(0/1)andP(x/x)and the distribution ofWx/xover the aggregated levels. Scale

invariance can be assumed, if the distributions basically show the same shape. b) When scale-invariant levels are defined, the probabilities and weights can be av-

eraged over these levels. The generator is only applicable within these levels. 3. Disaggregation: Now the previously built generator, as well as a low resolution time

series for disaggregation is available. The process is carried out beginning with the first box at the given temporal resolution of the time series (e.g. 80 minutes at level 3 in Fig. 4.18) in these steps:

a) The position and volume class of the box is determined.

b) A uniformly distributed random number is drawn, and from it the type of division is determined using the probabilities for the given volume and position class.

c) The box is split up according to the type of division. In case of a x/x-division a second random number is generated, which is used to determine the volumes

W1 and W2 with the help of the corresponding Wx/x-distribution of the box. The

division occurs by multiplying the original volume with the weights and setting the resolution of the lower level to the half.

d) Division is repeated this way for all boxes of the level, thus forming the next lower level.

Steps 3a – 3d are repeated until the desired resolution (e.g. 10 minutes in Fig. 4.18) is reached.