37 Evaluation of Air Quality Maps Using Cross-Validation …
239
Fig. 37.1 Variance of observation-minus-analysis residuals of PM 2.5 for both active and independent observations
dash line. A series of analyses were performed with different observation weight, as
controlled by the ratio σ
2
o /σ
2
b while maintaining the sum of observation and model
error variance constant and equal to the var (O-B) [5, 7].
To illustrate the different diagnostics from which an estimate of analysis error
variance can be obtained, we use a geometrical representation of random variables,
where uncorrelated variables are orthogonal, and the norm (or length) of a side represents the standard deviation. In Fig. 37.2 is displayed the truth T, O the observations
used for the analysis, B the background or model value, and O c the independent
observations used to evaluate the analysis. Since the analysis is a linear combination
of the observations and the model (background) the analysis A lies on the line joining
O and B. As usual, we assume that observation error, i.e. the vector in the direction
(T, O), is uncorrelated with the model error, i.e. vector in the direction (T, B). Thus
the triangle OTB is a right triangle. The analysis is optimal when the analysis error,
i.e. the norm of the vector (T, A), is minimal, which actually occur when (T, A) is
orthogonal to (O, B).
For an optimal analysis, the triangle OAT and TAB are right triangles. Assuming that the independent observations O c have uncorrelated errors with active observation errors (T, O) and have also uncorrelated errors with the background error
(T, B), the independent observation O c thus lie perpendicular to the analysis plane
formed by T, O, B, and A. It is then evident that the var(O c -A) which is plotted in
Fig. 37.1, which is the length of (O c , A) reaches a minimum when the analysis A is
closest to T, that is when the analysis is optimal. Different points along the line (O, B)
correspond to different ratio σ
2
o /σ
2
b , so we conclude that the minimum observed in
239
Fig. 37.1 Variance of observation-minus-analysis residuals of PM 2.5 for both active and independent observations
dash line. A series of analyses were performed with different observation weight, as
controlled by the ratio σ
2
o /σ
2
b while maintaining the sum of observation and model
error variance constant and equal to the var (O-B) [5, 7].
To illustrate the different diagnostics from which an estimate of analysis error
variance can be obtained, we use a geometrical representation of random variables,
where uncorrelated variables are orthogonal, and the norm (or length) of a side represents the standard deviation. In Fig. 37.2 is displayed the truth T, O the observations
used for the analysis, B the background or model value, and O c the independent
observations used to evaluate the analysis. Since the analysis is a linear combination
of the observations and the model (background) the analysis A lies on the line joining
O and B. As usual, we assume that observation error, i.e. the vector in the direction
(T, O), is uncorrelated with the model error, i.e. vector in the direction (T, B). Thus
the triangle OTB is a right triangle. The analysis is optimal when the analysis error,
i.e. the norm of the vector (T, A), is minimal, which actually occur when (T, A) is
orthogonal to (O, B).
For an optimal analysis, the triangle OAT and TAB are right triangles. Assuming that the independent observations O c have uncorrelated errors with active observation errors (T, O) and have also uncorrelated errors with the background error
(T, B), the independent observation O c thus lie perpendicular to the analysis plane
formed by T, O, B, and A. It is then evident that the var(O c -A) which is plotted in
Fig. 37.1, which is the length of (O c , A) reaches a minimum when the analysis A is
closest to T, that is when the analysis is optimal. Different points along the line (O, B)
correspond to different ratio σ
2
o /σ
2
b , so we conclude that the minimum observed in