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Let $t_0$ be the time instant of interest, $t_{-1}$ be some time before $t_0$, and $t_1$ be some instant in time after $t_0$.

Now there is no confusion with forecast - if the present time is $t_0$, a forecast at $t_1$, for example, uses a model that assimilates observations at $t_0$, and then step forward in time to make the forecast at $t_1$.

Suppose now the present time is $t_1$. I'm confused as to what a hindcast at time $t_0$ means. Do we start up the model at $t_1$, then go backward in time to compute the hindcast at $t_0$, or do we start up the model at $t_{-1}$, then run the model forward to get to $t_0$?

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A hindcast, also known as a historical re-forecast, integrates the model forward in time just like with a forecast, so you'd initialise the model at $t_{-1}$ and run through to $t_1$. If you have an assimilation system that can make use of observations at $t_0$, then it would use them in the same way that it would with a forecast.

The point of a hindcast is to do the forecast again using something that wasn't available originally. That new something might be observations (for assimilation or for verification), the assimilation system or the forecast model. They can be used to calibrate the modelling system or just to check that updates to the modelling system do actually improve the forecast. They're often used for cases studies of extreme events or situations that are known to be tricky to forecast; after all, why wait for the next 1 in 30 year event to test your new system when you have one in the archive, probably with lots of verification data accumulated over the years.

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  • $\begingroup$ Thank you Deditos - though now I am unclear as to how the hindcast differs from a reanalysis. Reading the Wikipedia article (en.wikipedia.org/wiki/Backtesting#Hindcast), it is said there "Hindcasting usually refers to a numerical model integration of a historical period where no observations have been assimilated. This distinguishes a hindcast run from a reanalysis." Is this right? Does this mean no assimilation at $t_0$, or no assimilation at $t_1$ (the final time period of interest in your example)? And the entire period in your example, $t_-1$ thru $t_1$, are all in the past, right? $\endgroup$ – user4624937 Jul 2 '15 at 20:42
  • $\begingroup$ First, I'll caveat that different disciplines/applications may use the terms in different ways. But from my atmos perspective, an analysis (or re-analysis) runs the model/assimilation combo only for the observation window, whereas a forecast (or re-forecast) runs the model beyond the observation window. In practice these are two steps in the same forecasting system. For example, using a 09-21 UTC observation window to produce an analysis at 12 UTC, which is then used to initialise a free running forecast out to 7 days. $\endgroup$ – Deditos Jul 3 '15 at 10:24
  • $\begingroup$ Thank you Deditos for the clarifications! If you don't mind, I have another question. Is it possible to "integrate backward" in time? For example, say only observations on Jan. 1 and Feb. 1 are available. The time of interest happens to be Jan. 29. Would one have to use the analysis on Jan. 1 and integrate forward 29 days, or is it possible to somehow make use the observations on Feb. 1 and "go backward" two days? $\endgroup$ – user4624937 Jul 4 '15 at 2:57
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    $\begingroup$ No, you can't integrate models backwards in time. If you have an initial value problem and definitely want to use both Jan 1 and Feb 1 obs, then you'd need an observation window that covers both dates and you'd be finding the optimal initial state for some date on or before Jan 1. $\endgroup$ – Deditos Jul 4 '15 at 15:32

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