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Documenting calibration - #494
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nilsvanvelzen
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I do approve but look at some minor things
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| Calibration starts with the definition of a scalar cost function that measures | ||
| the misfit between model predictions and the available observations over the | ||
| assimilation interval. The cost function is typically defined as the sum of |
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Replaced by weigthed sum of squared differences.
| optimal initial condition is the one that minimizes :math:`J`), :math:`x(k)` is | ||
| the variable at time :math:`t_k`, :math:`H` is the observation operator, | ||
| :math:`y^o(k)` is the observation at time :math:`t_k`, and :math:`R` is the | ||
| representation covariance. For the second cost function, we additionally have |
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Representation covariance? Not covariance of measurement errors?
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| * :math:`J(x_o) = \sum_{k=1}^N (y^o(k)-Hx(k))R^{-1}(y^o(k)-Hx(k))^\top`; and | ||
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| * :math:`J(x_o) = (x_o-x^b)^\top(P^b)^{-1}(x_o-x^b)+\sum_{k=1}^N (y^o(k)-Hx(k))R^{-1}(y^o(k)-Hx(k))^\top`. |
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I suppose this is the weak constraint variant? Probably good idea to call it this way in th text.
| :math:`y^o(k)` is the observation at time :math:`t_k`, and :math:`R` is the | ||
| representation covariance. For the second cost function, we additionally have | ||
| :math:`x^b`, the background or initial estimate of :math:`x_o`, and | ||
| :math:`P^b`, its covariance. This additional component ensures that the |
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mention that P^b is a measure for how sure you are of your initial parameter values
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