Distribution of Forecast Error Doubling Times Using Lorenz’s Quadratic Model

Abstract One way to quantify the quality of a forecast is to estimate the forecast error doubling time—the longer the error doubling time, the better is the forecast. Lorenz in 1982 using the well-known logistic model obtained a point estimate of the error doubling time for the growth of root-mean-square (RMS) errors in the numerical model prediction of 500-hPa height by following divergence of the analog pairs over a 100-day period covering 1 December 1980–10 March 1981. While this approach to the point estimate of error doubling time gained widespread acceptance, little reflection reveals that the RMS error itself is subjected to non-Gaussian noise resulting from the errors in the initial conditions used by the numerical model to create the forecast of the 500-hPa height field. By postulating that the noise corrupting the RMS error follows the chi-squared distribution and combining it with the forward sensitivity method (FSM) for dynamic data assimilation, we develop a statistical approach to obtaining the distribution of error doubling time as opposed to a point estimate. Significance Statement Lorenz developed a method for estimating a single-point value of forecast error doubling time using data from ECMWF’s operational forecast model in 1980–81. The uniqueness of his method was the use of analog pairs, in particular the divergence of pairs, fitted to the logistic equation. The method was widely used by numerical weather prediction (NWP) researchers in the meteorology community, but it was found that the same data, the divergence of analog pairs, used with models other than the logistic equation, produced different single-point values of the error doubling time. Is there a single value for estimating the forecast error doubling time using Lorenz’s method? We have answered this question by considering two different approaches: a deterministic approach and a stochastic approach. Knowledge gained from numerical experiments with these two approaches are discussed along with their value to forecasting practice.