pastas.rfunc.Hantush.get_tmax_approximation#

pastas.rfunc.Hantush.get_tmax_approximation(p: pastas.typing.ArrayLike, cutoff: float | None = None) → float#

Approximates the time (tmax) when the step response reaches a specified cutoff.

This analytical approximation is derived by evaluating the tail of the impulse response integral. The derivation relies on the following steps: 1. The tail integral is mapped to dimensionless time x = t/a. 2. Deep in the tail, the b/t term in the exponent is assumed negligible,

reducing the integral to the standard Exponential Integral, E1(x).

  1. E1(x) is approximated by its leading asymptotic term: exp(-x) / x.

  2. Equating this to the remaining area (1 - cutoff) yields an equation of the form x * exp(x) = z, which is classically solved using the Lambert W function: x = W(z).

To prevent crashing the calculation when rho is large (causing k0 to underflow and z to overflow), the math is translated into log-space. The exponentially scaled Bessel function (k0e) and the Wright Omega function (the exact analytical solution to y + ln(y) = log_z) are used. This provides a globally continuous, unbreakable equivalent to lambertw(z).

Parameters:

  • p (array_like) – Response function parameters [A, a, b].

  • cutoff (float, optional) – The fraction of the total step response area reached at tmax. Must be strictly between 0 and 1. Defaults to self.cutoff if cutoff is None.

Returns:

The approximated tmax value.

Return type:

float