Fit a power-law rating curve on the original scale — rc

Fits the power law \(Q = a (h - c)^b\) by nonlinear least squares on the original scale of discharge, with the scatter of the gaugings modelled as set by variance. The curve estimates the mean discharge at each stage.

Usage

rc_powerlaw(
  discharge,
  stage,
  ...,
  data = NULL,
  variance = var_none(),
  offset = NULL,
  control = stats::nls.control(maxiter = 1000, tol = 1e-06)
)

Arguments

discharge: <data-masking> Discharge: a vector, or an expression evaluated in data, such as a column name.
stage: <data-masking> Stage: a vector, or an expression evaluated in data, such as a column name.
...: Must be empty. Present so that every argument after it has to be named in full.
data: Optional data frame in which discharge and stage are evaluated.
variance: How the variance of the gaugings about the curve is modelled: var_none() (or "none", the default), var_prop() (or "prop"), var_power() (or "power"), or var_spec() with the variances. See variance.
offset: The offset, \(c\) in the formula: NULL (the default) to estimate it, or a known value, below every gauged stage, to hold it fixed.
control: Settings for stats::nls(), as from stats::nls.control().

Value

An rc_powerlaw object; see rating_curve for its contents.

The offset

\(c\) is the offset: for a single power law, the stage at which the flow would stop. It is estimated unless it is given, through offset, say from a survey of the control. A given offset is held fixed, so the limits from predict() carry no uncertainty in it. Fixing it at the value estimated by a first fit treats an estimate as known: the curve is the same, but the limits are narrower than they should be.

Examples

fit <- rc_powerlaw(discharge, stage, data = thompson)
predict(fit, new_stage = c(1, 3, 6), conflev = 0.95)
#> # A tibble: 3 × 4
#>   stage   fit ci_lwr ci_upr
#>   <dbl> <dbl>  <dbl>  <dbl>
#> 1     1  243.   222.   264.
#> 2     3  832.   814.   850.
#> 3     6 2209.  2188.  2230.

# constant coefficient of variation instead of constant variance
fit_prop <- rc_powerlaw(discharge, stage, data = thompson, variance = "prop")
predict(fit_prop, new_stage = c(1, 3, 6), conflev = 0.95)
#> # A tibble: 3 × 4
#>   stage   fit ci_lwr ci_upr
#>   <dbl> <dbl>  <dbl>  <dbl>
#> 1     1  252.   249.   255.
#> 2     3  816.   806.   826.
#> 3     6 2205.  2178.  2232.

# the offset known, say from a survey of the control
rc_powerlaw(discharge, stage, data = thompson, variance = "prop", offset = -1.3)
#> Rating curve model.
#> - Method: rc_powerlaw