| expuh {hydromad} | R Documentation |
A unit hydrograph (linear transfer function) defined as a system of exponentially receding components. Each component is defined by its time constant and fractional volume, and if there are multiple (up to 3) such components they may be in a parallel and/or series configuration.
expuh.sim(U, delay = 0,
tau_s = 0, tau_q = 0, tau_3 = 0,
v_s = 1, v_q = NA, v_3 = 0,
series = 0, loss = 0,
Xs_0 = 0, Xq_0 = 0, X3_0 = 0,
pars = NULL,
return_components = FALSE,
na.action = na.pass,
epsilon = hydromad.getOption("sim.epsilon"))
ssg.expuh(theta)
normalise.expuh(theta)
U |
input time series. |
delay |
lag (dead time) between input and response, in time steps. |
tau_s, tau_q, tau_3 |
time constants (τ) for the exponential components. |
v_s, v_q, v_3 |
fractional volumes (v) for the exponential components. |
series |
defines the configuration of exponential components, as being in parallel and/or series (for second or third order models). See Details. |
loss |
a constant loss (or gain, if negative) term subtracted from the
slow ( |
Xs_0, Xq_0, X3_0 |
initial values of the exponential components. |
pars |
the parameters as a named vector. If this is given, it will over-ride the named parmameter arguments. |
return_components |
whether to return all component time series. |
na.action |
function to remove missing values,
e.g. |
epsilon |
values smaller than this in the output will be set to zero. |
theta |
the parameters as a named vector. |
The expuh model is a transfer function translating an
input time series U into an output series X.
It describes a configuration of exponentially
decaying components, each defined by a recession rate α and
peak response β. However, in hydrology these parameters are more
easily interpreted in terms of time constants τ (number of time
steps to reduce to a fraction 1/e, 37%) and fractional volumes
v. These are directly related as:
τ = -1 / \log(α)
v = β / (1 - α)
If there are two components in parallel, these are conventionally called slow (s) and quick (q) flow components. The total simulated flow X is the sum of these; X[t] = X_s[t] + X_q[t], and:
X_s[t] = α_s X_s[t-1] + β_s U[t]
X_q[t] = α_q X_q[t-1] + β_q U[t]
Two components might also be arranged in series rather than parallel, in which case:
X_s[t] = α_s X_s[t-1] + β_s U[t]
X[t] = α_q X[t-1] + β_q X_s[t]
This configuration is specified by the argument series = 1. The
default series = 0 specifies all components to be in parallel.
In the case of three components, with corresponding time constants
τ_s, τ_q and tau_3 (tau_s, tau_q,
tau_3), there are four possible types of configuration:
series = 0all 3 components in parallel, i.e. independent flows:
X = s + q + 3. In this case v_q defaults to
1 - v_s - v_3 in order to ensure that the total volume is
1.
series = 1one component in parallel with two in series: the q
component is in series with the 3 component, and the
s component is in parallel:
X = (q * 3) + s. In this case v_q defaults to 1.
series = 2two components in parallel with one in series: the s
and q components are in parallel and the 3 component
is in series:
X = 3 * (s + q). In this case v_q defaults to
1 - v_s in order to ensure that the total volume of the
parallel component is 1. The total volume will be 1 if v_3
is also 1.
series = 3all 3 components in series: X = s * q * 3.
In this case v_q defaults to 1. The total volume will be 1
if v_s and v_3 are also 1.
the model output as a ts object,
with the same dimensions and time window as the input U.
If return_components = TRUE, it will have multiple
columns named
Xs, Xq and, if relevant, X3.
Felix Andrews felix@nfrac.org
Jakeman, A.J., I.G. Littlewood, and P.G. Whitehead (1990), Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments, Journal of Hydrology, 117: 275-300.
data(HydroTestData)
mod1 <- hydromad(HydroTestData, routing = "expuh",
tau_s = 30, tau_q = 5, v_s = 0.5)
flowcomps <- predict(mod1, return_components = TRUE)
xyplot(cbind(`Slow component` = flowcomps[,"Xs"],
`Total flow` = flowcomps[,1] + flowcomps[,2]),
superpose = TRUE) +
layer(panel.refline(h = 0))
U <- ts(c(1, rep(0, 30)))
xyplot(cbind("tau_s = 10" = expuh.sim(U, tau_s = 10),
"& tau_q = 1" = expuh.sim(U, tau_s = 10, tau_q = 1, v_s = 0.5),
"&& v_s = 0.9" = expuh.sim(U, tau_s = 10, tau_q = 1, v_s = 0.9)),
superpose = TRUE)