Weather forecasting

{title2=$n_{\mathrm{forecast}}\approx\lambda^{-1}\log(\Delta/\varepsilon_0)$}

A forecast horizon is the duration over which a prediction meets a specified error tolerance. If an initial uncertainty $\varepsilon_0$ grows approximately as $\varepsilon_0e^{\lambda n}$ with positive <Lyapunov exponent> $\lambda$, then solving for when it reaches a tolerance $\Delta$ gives the displayed estimate. Here $\log$ is the <natural logarithm> and $\lambda$ is measured per step of the <discrete dynamical system>; for a <flow map>, use the exponent per unit <time> instead. The estimate assumes the growth approximation remains valid up to the chosen tolerance and excludes additional model errors.

Reducing $\varepsilon_0$ by a factor ten buys approximately $(\log10)/\lambda$ more steps, rather than multiplying the forecast duration by ten. For typical <logistic map> <orbits> at $r=4$, $\lambda=\log2$, so one extra decimal digit buys roughly $3.32$ steps in this approximation. Better measurements help, but their benefit is logarithmic. A forecast of a <probability distribution> or a stable long-term average asks a different question from predicting each individual state.


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