A common method for diagnosing and correcting model bias in weather and climate prediction is based on analysis increments: the difference between the analysis and the forecast at each data assimilation cycle. Their time average is used to estimate and correct the bias, and machine learning is increasingly used to map the forecast state to the increments, giving a state-dependent correction. Despite its widespread use, the method lacks theoretical foundations. We first analyse a Kalman filter whose forecast model has a constant bias. We show what the averaged increment measures, and prove that iterating the correction removes all of the bias seen through the observations; for linear dynamics, no method can identify the rest. We then extend the analysis to state-dependent biases. For linear dynamics, each increment records the model error over the filter's memory, as seen through the observations. We use this to determine what the increments measure when averaged over a long period, averaged over periods with similar slowly varying conditions, and used without averaging, as in machine-learned corrections. Numerical experiments confirm the theory for a linear model and suggest that its main conclusions carry over to the nonlinear Lorenz '96 model, including for a correction learned by a neural network.

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