CRiSP1.6 Theory & Calibration Manual: II.7 - Stochastic Processes

II.7 - Stochastic Processes

CRiSP.1 provides the ability to vary parameters over a run. This allows a representation of random factors. The randomness is incorporated in different ways for flow, dam passage, reservoir mortality and travel time. The approach is to describe specific parameters as having a deterministic part and a stochastic part. A deterministic part may change with the independent variables that determine the parameter but the value obtained does not change from one model run to another if all factors are the same. The stochastic part changes each time it is calculated in CRiSP.1 or between model runs. The value of the stochastic part is obtained from a random number distribution function using a broken-stick distribution function. This is described along with deterministic and stochastic parts of the parameters in the following sections.

II.7.1 - Stochastic Parameter Probability Density

Variation in many of the stochastic rate parameters is described by a broken-stick probability distribution function (pdf). This is a simple function based on a piecewise linear distribution. The probability density function and the cumulative density function are illustrated in Fig. 54. It is described using the 0, 50 and 100% cumulative probability levels.

Fig. 54 Probability function (pdf) and cumulative function of the broken-stick probability distribution

Random deviates for this broken-stick density distribution are obtained from the following transformation formula:

(135)

where

• x = unit uniform random deviate range 0 < x < 1
• y_l = lower limit of the distribution range
• y_m = distribution of the median value
• y_u = upper limit of the distribution range.

Although the distribution uses the median, the broken-stick input windows in CRiSP.1 use the mean value since most data reports include a mean in addition to the minimum and maximum values. The median is estimated from these three measures as:

(136)

assuming the mean of the distribution is equal to the average of the mean of the lowest 50% of the distribution and the highest 50%. These are simply the average of the minimum and median, and maximum and median, respectively.

Note that in a skewed distribution the mean and median are different. The result is that the mean specified by the user must fall in the middle two quartiles of the distribution, i.e. if the user specifies a minimum of 0 and a maximum of 100 for some distribution, the mean must lie between 25 and 75, inclusive. If the user specifies a distribution outside this range, CRiSP.1 will post a message to that effect in the message window and will direct the user to choose a mean that lies in the acceptable range.

II.7.2 - Stochastic Parameters

Migration

Variability in the migration rate is determined by the equation:

(137)

where

• r(t) = determined from eq (48)
• V(i) = variance factor which is different for each release i.

The term V(i) is drawn from the broken-stick distribution. The mean value is set at 100%, representing the deterministic r(t) and the upper and lower values are set with sliders in the Migration Rate Variance window in the Behavior menu.

The variance factor assumes that variability in migration velocity relative to water velocity is associated with a particular stock of fish. Studies of travel time support this assumption since particular stocks exhibit their own unique relationship with flow.

Flow

In the Scenario Mode, daily flow variations are described by a random process in headwater flow. Details of this process are described in the Headwater Modulation.

Dam Passage

Variability in dam passage parameters is applied in each dam time slice, typically 6 hours. The variability is generated from the broken-stick distribution and is applied to the following variables:

• bypass mortality
• spill mortality
• turbine mortality
• transportation mortality
• day / night FGE
• spill efficiency.

II.7.3 - Scales of Stochastic Variability

The scales over which stochastic variability are applied is given in Table 45.

Table 45 Model probability density functions

Process	Equation	pdf	Scale
Migration rate variance	eq (52)	broken-stick	release group
Flow in Scenario	eq (16)	Normal	12 hrs
FGE & dam mortality	eq (127)	broken-stick	6 hrs
Spill efficiency	eq (128)	Normal	6 hrs

CRiSP1.6 Theory & Calibration Manual: II.7 - Stochastic Processes