Exponential store with zero flows and thresholded loss

An exponential store (linear transfer function) which has a loss term, produces no flow when the store drops below a level, and can therefore model longer-term disconnection of a store from streamflow.

leakyExpStore.sim(x, tau, loss, thres, init = 0, return_components = FALSE)

Arguments

x: input time series.
tau: Time constant for flow from exponential store.
loss: Constant loss that occurs while the value of the store is greater than thres. See details.
thres: Trigger level to turn off loss, which can be interpreted as a store capacity.
init: Initial value of exponential store
return_components: whether to return store value (G) as well as flow (Q)

Value

the flow time series with the same dimensions and time windows as the input x. If return_components = TRUE, it will have multiple columns named G and Q.

Details

Storage is increased by effective rainfall and decreased by flow and losses. $$G[k] = G[k-1] + U[k] - Q[k] - L[k]$$

Flow is proportional to storage $$Q[k] = a G[k], G[k]>0$$ $$Q[k] = 0, otherwise$$

Loss switches off at some threshold, as a piece-wise continuous function. $$L[k] = L, G[k]>T+L (with T<=0)$$ $$L[k] = G[k] - T, T+L > G[k] > T$$ $$L[k] = 0, G[k]<T$$

Author

Joseph Guillaume joseph.guillaume@anu.edu.au with advice from Barry Croke

Examples


U <- ts(c(1, rep(0, 10), 1, rep(0, 20)))

## Without a loss, equivalent to expuh
##  Loss threshold has no effect
all.equal(leakyExpStore.sim(U, 5, loss = 0, thres = 0), expuh.sim(U, tau_s = 5))
#> [1] TRUE

## With losses stopping when flow stops, equivalent to expuh with a loss
all.equal(
  leakyExpStore.sim(U, 5, loss = 0.1, thres = 0),
  expuh.sim(U, tau_s = 5, loss = 0.1)
)
#> [1] TRUE

## Plot of unit hydrographs
xyplot(cbind(
  "U" = U,
  "loss=0" = expuh.sim(U, tau_s = 5),
  "loss=0.1 & thres=0" = expuh.sim(U, tau_s = 5, loss = 0.1),
  "loss=0.1 & thres=-0.3" = leakyExpStore.sim(U, 5, loss = 0.1, thres = -0.3),
  "loss=0.1 & thres=-Inf" = leakyExpStore.sim(U, 5, loss = 0.1, thres = -Inf)
), superpose = TRUE)


## Time series plot of value of store
xyplot(cbind(
  "thres=0" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = 0,
    return_components = TRUE
  )[, "G"],
  "thres=-0.3" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = -0.3,
    return_components = TRUE
  )[, "G"],
  "thres=-Inf" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = -Inf,
    return_components = TRUE
  )[, "G"]
), superpose = TRUE)


## Time series of loss
xyplot(cbind(
  "thres=0" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = 0,
    return_components = TRUE
  )[, "L"],
  "thres=-0.3" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = -0.3,
    return_components = TRUE
  )[, "L"],
  "thres=-Inf" = leakyExpStore.sim(U, 5,
    loss = 0.1, thres = -Inf,
    return_components = TRUE
  )[, "L"]
), superpose = TRUE)