John L. Sabo and David M. Post. 2008. Quantifying periodic, stochastic, and catastrophic environmental variation. Ecological Monographs 78:19–40.

Appendix C. Simulation validation of the slope as an estimator of the standard deviation of residual discharge events.

From Eq. 15 we have a linear relationship between the squared residual discharge values (M_r) and the ln-transformed number of observations of events of this size (o_r):

	(C.1)
where,
	(C.2a)
and,
.	(C.2b)

These equations lead us to the definition of the standard deviation of residual discharge events via the intercept (A) and slope (B) viz:

,	(C.3a)
or,
.	(C.3b)

In this appendix we use simulated data to measure the accuracy of estimates of as estimated via Eq. 17b using the slope.

First we generate a series of random (independent) variates with a mean of zero and a range of values for the standard deviation:

.

(C.4)

In this analysis we used values between 0.1 and 10 at increments of 0.1 for , reflecting the range of values measured in streams (Appendix D) and drew 20 years of daily variates (i.e., N = 7300) reflecting the size of datasets analyzed by Fourier (Appendix B). We then extracted positive residuals (e.g., high-flow events) from this dataset (F~3650), counted observations within five equally spaced bins and estimated the slope and intercept of this relationship (via Eq. C.1). Finally, we compared our estimates of to the actual standard deviation , used to generate residual discharge values in Eq. C.4. Estimation of via the slope was nearly unbiased and the relationship was almost exactly 1:1 (Fig. C1). Moreover, precision (e.g., gauged by the range of the 95% confidence interval) is also high for all but very high values for the true standard deviation. Thus, we use the slope, B to estimate using Eqs. 17b and C.3b.