powuh {hydromad}R Documentation

Power law transfer function models

Description

A power-law form of unit hydrograph (transfer function).

Usage

powuh.sim(U, delay = 0,
          a, b = 1, c = 1, init = 0, uhsteps = 100,
	  na.action = na.pass,
          epsilon = hydromad.getOption("sim.epsilon"))

Arguments

U

input time series.

delay

lag (dead time) between input and response, in time steps.

a

the time for flow to drop by half after a peak, if c = 1. See Details.

b

persistence of the flow response; defines the recession curve tail.

c

curvature at half-peak point.

init

initial flow value(s) used in convolution filter.

uhsteps

number of time steps to use in approximating the unit hydrograph convolution filter.

na.action

function to remove missing values, e.g. na.omit.

epsilon

values smaller than this will be set to zero.

Details

The power law form of the unit hydrograph is:

H = 1 / (1 + (t/a)^{b/c}) ^ c

where H is the fraction of peak flow, t is the time since peak, and a, b and c are parameters.

From Croke (2006):

Parameter a is the value of t (time since peak) at which the ordinate of the asymptote (t/a)^(-b) has a value of 1, b determines the persistence of the flow response and c defines the shape of the response curve near its peak. The c parameter appears twice in order to reduce interaction between the b and c parameters (in this form, the c parameter only influences the curvature near t = a, and doesn't influence the asymptote, which is determined solely by the b parameter). The time for H to decrease to 0.5 is a(2^(1/c) - 1)^(c/b). While this is a three parameter model, for t >> a only the b parameter is significant. Since the value of the a parameter is typically significantly less than one (see Table 1) the recession curve can be written as

H = (t_r / t)^b

where t_r is some reference time (t_r >> a) at which the hydrograph profile has been normalized. Thus the remaining two parameters (a and c) only influence the response curve near the event peak, and [the equation above] can be taken as a single parameter recession model.

Value

the model output as a ts object, with the same dimensions and time window as the input U.

Author(s)

Felix Andrews felix@nfrac.org

References

Croke, B.F.W. (2006). A technique for deriving an average event unit hydrograph from streamflow-only data for ephemeral quick-flow-dominant catchments. Advances in Water Resources 29, pp. 493–502.

See Also

expuh armax

Examples

U  <- ts(c(1, rep(0, 99)))
xyplot(cbind("a = 5" = powuh.sim(U, a = 5),
             "& b = 2" = powuh.sim(U, a = 5, b = 2),
             "& c = 2" = powuh.sim(U, a = 5, c = 2)),
       superpose = TRUE)
[Package hydromad version 0.9-18 Index]