Time-frequency analysis of beach bacteria variations and its implication for recreational water quality modeling

This paper exploited the potential of the wavelet analysis in resolving beach bacteria concentration and candidate explanatory variables across multiple time scales with temporal information preserved. The wavelet transform of E. coli concentration and its explanatory variables observed at Huntington Beach, Ohio in 2006 exhibited well-defined patterns of different time scales, phases, and durations, which cannot be clearly shown in conventional time-domain analyses. If linear regression modeling is to be used for the ease of implementation and interpretation,the wavelet-transformed regression model reveals that low model residual can be realized through matching major patterns and their phase angles between E. coli concentration and its explanatory variables. The property of pattern matching for linear regression models can be adopted as a criterion for choosing useful predictors, while phase matching further explains why intuitively good variables such as wave height and onshore wind speed were excluded from the optimal models by model selection processes in Frick et al. (Environ. Sci. Technol. 2008, 42,4818-4824). The phase angles defined by the wavelet analysis in the time-frequency domain can help identify the physical processes and interactions occurring between bacteria concentration and its explanatory variables. It was deduced, for this particular case, that wind events resulted in elevated E. coli concentration, wave height, and turbidity at the beach with a periodicity of 7-8 days. Wind events also brought about increased beach bacteria concentrations through large-scale current circulations in the lake with a period of 21 days. The time length for linear regression models with statistical robustness can also be deduced from the periods of the major patterns in bacteria concentration and explanatory variables, which explains and supplements the modeling efforts performed in (1).