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Fig. 8 | Earth, Planets and Space

Fig. 8

From: Improving the estimation of thermospheric neutral density via two-step assimilation of in situ neutral density into a numerical model

Fig. 8

This height profile illustrates the average ratio of the analyzed (\(y-H(x^a)\)) and forecasted (\(y-H(x^f)\)) observational residuals for the first nine analysis steps. For easier interpretation we subtract the ratio from one: \(\frac{H(x^a)-H(x^f)}{y-H(x^f)} = 1-\frac{y-H(x^a)}{y-H(x^f)}\). The numerator contains the differences between the analyzed and forecasted observations and the denominator is the innovation. We subdivide the x axis into three intervals: \([-\infty ,0)\), [0, 1], and \((1,\infty ]\). For each interval we provide exemplary illustrations for the ratio. In the first interval (negative ratio), the analyzed state (transformed to the observation space) is pushed away from the observations, in the third interval (ratio greater than one) the correction applied to the forecast overshoots the innovation. A value of zero means that observations have no impact and the analyzed observations are equal to the forecasted observations \(H(x^a)=H(x^f)\). A value of one means that the full innovation is adopted \(H(x^a)=y\). Ideally, the ratio is in the interval [0, 1], which means that the analyzed observations are between the real observations and the modeled observations. The solid bold line is the median computed over all layers of the data grid at the corresponding color coded analysis step. The shaded area marks the interval between the 25th and 75th percentile. Here \(x^a\) is the analyzed state after applying constraints

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