Technical Note: A measure of watershed nonlinearity II: re-introducing an IFP inverse fractional power transform for streamflow recession analysis

Abstract. This note illustrates, in the context of Brutsaert–Nieber (1977) model: −dQ/dt = aQ^b, the utility of a newly rediscovered inverse fractional power (IFP) transform of the flow rates. This method of streamflow recession analysis dates back a half-century. The IFP transform Δ_b on an operand Q is defined as Δ_b Q = 1/Q^b-1. Brutsaert–Nieber model by IFP transform thus becomes: Δ_bQ(t) = Δ_bQ(0) + (b−1) at, if b ≠ 1. The IFP transformed recession curve appears as a straight line on a semi-IFP plot. The method has both the advantage of being independent of the size of computational time step, and the disadvantage of being depending on the parameter b value. This is used to calibrate the Brutsaert–Nieber recession flow model in which b is a slope (or shape) parameter, and a is an intercept (or a scale parameter). It is applied to four observed events on the Spoon River in Illinois (4237 km²). The results show that the IFP transform method gives a narrower range of parameter b values than the regression method in a recession plot. Theoretically, an IFP transformed recession curve for large watersheds falls between those performed by the reciprocal of the cubic root (RoCR) transform and the reciprocal of the square root (RoSR) one. In general, the forgotten IFP transform method merits a fresh look, especially for hillslopes and zero-order catchments, the building blocks of a watershed system. In particular, because of its origin in hillslope hydrology, the 1-parameter RoSR transform need be falsified or verified for application to headwater catchments.