Abstract

The so-called True Skill Statistic (TSS) and the Heidke Skill Score (S), as used in the context of the contingency, table approach to forecast verification, are compared. It is shown that the TSS approaches the Probability of Detection (POD) whenever the forecasting is dominated by correct forecasts of non-occurrence, i.e., forecasting rare events like severe local storms. This means that the TSS is vulnerable to “hedging” in rare event forecasting. The S-statistic is shown to be superior to the TSS in this situation, accounting for correct forecasts of null events in a controlled fashion. It turns out that the TSS and S values are related in a subtle way, becoming identical when the expected values (due to chance in a k × k contingency table) remain unchanged when comparing the actual forecast table to that of a hypothetical perfect set of forecasts. Examples of the behavior of the TSS and S values in different situations are provided which support the recommendation that S be used in preference to TSS for rare event forecasting. A geometrical interpretation is also given for certain aspects of the 2 × 2 contingency table and this is generalized to the k × l case. Using this geometrical interpretation, it is shown to be possible to apply dichotomous verification techniques in polychotomous situations, thus allowing a direct comparison between dichotomous and polychotomous forecasting.

Abstract

Although considerable understanding of mountain waves has been gained with the aid of the steady-state assumption, it is clear that mountain waves evolve over time. Group velocity arguments indicate that this evolution can occur in less than 1 h. This study uses observations of trapped lee waves to measure the rate at which their horizontal wavelengths change, including a detailed analysis of two events in which such changes are clearly documented. In one case, Doppler lidar observations show a steady increase in horizontal wavelength of 6% h⁻¹ over 4 h and clearly illustrates the relationship between the wave clouds and wave motions. In a second case, visible satellite imagery reveals an increase in wavelength of 14% h⁻¹, which is related to temporal changes in vertical air motions measured by wind profilers within the wave field. Hourly vertical profiles of wind and virtual temperature measured by radio acoustic sounding systems (RASS) and wind profilers reveal important changes in the wave environment. These data were used to initialize a two-dimensional nonlinear nonhydrostatic numerical model with soundings representing five times over 8 h. Each simulation produced trapped lee waves. The simulations support the conclusion that the observed increase in wavelength resulted from changes in the wave environment. Uncertainty in the predicted wavelength due to using measurements within the trapped lee waves as initial conditions is shown to be small in this case. The wind profiler and RASS measurement accuracies are adequate to measure changes leading to trapped lee wave nonstationarity.

The results from these two case studies are combined with evidence of nonstationarity found in earlier papers and additional events documented here using visible satellite imagery. These 24 observations of nonstationarity indicate that the horizontal wavelength of trapped lee waves can change by as much as 30% h⁻¹. The average of all events, most of which occurred at midday, is a 9% h⁻¹ temporal increase. It is suggested that the deepening of the mixed layer thins the elevated stable layer that is a key part of the waveguide and that this thinning causes the systematic temporal increase of the horizontal wavelength.

While this study focused on trapped lee waves, it can be inferred that vertically propagating waves can also change significantly over a few hours. Because aircraft measurements of vertical momentum flux profiles typically require 2–5 h to complete, and stationarity is required over that time, it is recommended that steadiness should be measured rather than assumed in such studies.