Column observation operator

The column observation operator

The calculation of modeled column-averaged mixing ratios follows the approach of Rodgers (2000). The column-averaged mixing ratio is related to the true atmospheric profile as in the following equation:

	ĉ_t	= h^Tx_t
		= h^Tx_a + h^TA(x_t-x_a) + h^TGε
(1)		≈ c_a + ax_t - c'_a

where the error term has been ignored, and the following symbols and abbreviations have been used:

ĉ_t...		Observed column-averaged mixing ratio
h^T ...		Pressure weighting function (transposed)
x_t ...		True atmospheric profile (unknown)
x_a ...		Prior profile
A ...		Averaging kernel matrix
c_a ...	h^Tx_a	Prior column-averaged mixing ratio
c'_a ...	h^TAx_a	Prior column-averaged mixing ratio (smoothed)
a ...	h^TA	Pressure-weighted column averaging kernel

Equation (1) is used to compute the modeled column-averaged mixing ratio ĉ_m by substituting the true profile x_t with the modeled atmospheric profile x_m:

ĉ_m ≈ c_a^sd + ax_m

where the error term has been ignored, and the abbreviation c_a^sd = c_a - c'_a has been used (^sd = "smoothing difference").

File format - Atmospheric column data

Column
1	Time [fractional year]
2	Year
3	Month
4	Day
5	Hour
6	Minute
7	Second
8	Latitude
9	Longitude
10	<unused dummy>
11	Observed column-averaged mixing ratio
12	Sigma
13	Modeled column-averaged mixing ratio
14	Difference between prior column-averaged mixing ratio and smoothed prior column-averaged mixing ratio (c_a^sd)
15 to 14+NL	Pressure axis (hPa)
15+NL to 14+2*NL	Pressure-weighted column averaging kernel

Lines starting with "#" or "!", or empty lines, are ignored.
The column "<unused dummy>" takes the place of "height" from the files for station data, to match the column numbers of subsequent quantities.
All profiles are thought of as layer means. However, the model profile is interpolated linearly to the values given in "Pressure axis". This rather applies to level values instead of layer means, and is therefore an approximation.
"NL" means "Number of layers". The code automatically detects NL from the number of columns in the file. The code can also handle different numbers of layers in one file. In that case, both the "Pressure axis" and the averaging kernels need to be filled with negative values, so that the number of columns stays the same in each line. The number of columns doesn't have to be the same in all input files.
The "Pressure axis" must be either strictly monotonically increasing or strictly monotonically decreasing (automatically detected).

The code is currently unable to handle more than one species. However, if this is implemented in a future version, columns 11-14 are duplicated as follows (example for two species):

11-12	Observed column-averaged mixing ratios
13-14	Sigmas
15-16	Modeled column-averaged mixing ratios
17-18	Differences between prior column-averaged mixing ratios and smoothed prior column-averaged mixing ratios

After that, one common pressure axis follows, and after that, the averaging kernels. The first NL elements are the first averaging kernel, then follows the next kernel, and so on.

References:

Rodgers, C. D. Inverse Methods for Atmospheric Sounding. World Scientific Publishing Co. Pte. Ltd.: 2000.

Contact: Friedemann Reum