Estimating work-magnitude associations in earth surface processes

Abstract

Magnitude-frequency concepts in earth surface processes have found widespread application following the publication of the well-known paper Wolman and Miller. Of particular interest in such studies is the determination of those event magnitudes which make the most important long-term contributions to the total work of a given process. However, there has been little discussion to date concerning an appropriate estimator of the parameter θ, where θ is the long-term work achieved by events within a specified magnitude range, expressed as a proportion of the long-term work achieved by events of all magnitudes. The estimation of θ is straightforward for the time-independent case where short-duration events occur randomly in time, and event magnitudes are independent random variables from a common distribution. For this model, θ exists as a true parameter which can be estimated by\(\hat \omega \), where\(\hat \omega \) is the sample proportion of work contributed by events within the specified magnitude range. This estimator is biased, but it is almost median-unbiased for large samples. An approximate expression for var (\(\hat \omega \)) can be obtained from standard results. A similar approach to the estimation of θ can be applied to estimating the long-term work contribution of the largest events in consecutiveR-year periods. An example is presented using riverbank erosion data. Within the constraints of the time-independent model, the estimation procedure is quite general and can be applied with or without prior specification of the probability distribution of event magnitudes. In some situations, estimation can also be achieved indirectly by using a sample of the causal events which generate the individual work events. This indirect estimation is particularly simple if work magnitude is a power transformation of causal magnitude, and the distribution of causal event magnitudes can be approximated by a lognormal distribution or a Weibull distribution. The relative work achieved by events within ever-smaller magnitude ranges leads in the limit to the work intensity function,P(y). A plot of this function shows the relative importance ofy—magnitude events with respect to their long-term work contributions. Estimation ofP(y) is carried out by first fitting a probability distribution to a sample of event magnitude data. The functionP(y) is unimodal with respect to the following probability distributions of event magnitudes: lognormal, Weibull, unimodal beta, gamma, and inverse Gaussian. A lognormal distribution of event magnitudes produces the maximum work intensity at the lognormal median. In a strict mathematical sense, the long-term work contribution of very large and very small events is insignificant. However, little can be deduced concerning the pattern of work intensity between these two extremes. In particular, there appears no reason to suppose that the maximum work intensity will coincide with work magnitudes classified as “intermediate.”