Experimental censorship of bed load particle motions, and bias correction of the associated frequency distributions

Knowledge of the statistical distributions of particle hop properties (distances, travel, and rest times) enables a deeper understanding of bed load sediment transport. However, the measurement of particle hops is prone to censorship: Since many hops cross the boundaries of a spatial-temporal observation window, one knows that they exist but does not know how long they are. An option is to build particle hop samples considering only the hops that are completely observed and excluding (censoring) those observed only partially. Such a choice, however, biases the frequency distributions of the hop properties. Moreover, censorship acts in both space and time, and a hop censored in time will also not contribute to a sample of hop lengths, and vice versa. Time censorship similarly applies to particle rest times. This paper presents a theoretical formulation of censorship that leads to nonparametric bias corrections recovering estimates of values of the underlying distributions of hop distance, travel time, and rest time up to sampling window dimensions. We illustrate the occurrence and consequences of experimental censorship, and the benefit of applying the bias corrections, for both synthetic and laboratory samples of particle hops. The corrections reasonably recover the relative proportions of frequency distributions represented by the data up to the sampling dimensions and improve the estimates of the first two moments of particle hop properties. Recommendations are given regarding how the size of an observation window may be chosen to reduce the bias to below some prescribed value, if the forms of the underlying distributions are known.