Rotational Raman scattering in the Earth's atmosphere ... - SRON

Accurate modeling of spectral fine-structure in Earth radiance spectra measured with GOME 89
vectors v 1 ,...,v M describe the row space of the retrieval. The solution (4.14) is also known as the
solution of minimum length (see e.g. Menke [1989]) and can be written as
x est = K T ( KK T) −1
y , (4.15)
which is useful for any numerical implementation. For GOME measurements x est is strongly affected
by the measurement error e sun in Eq. (4.12). This is caused by those terms in Eq. (4.14) that belong
to small singular values. Here the noise on the measurement y gets amplified and therefore has a
significant contribution to x est . However, owing to the spectral smoothing that is part of the forward
model of the Earth radiance spectrum in Eq. (4.12), these error contributions are smoothed out and
thus have only a minor effect on the corresponding simulations of an Earth radiance spectrum.
The minimum length solution in Eq. (4.14) obviously does not contain information of the solar
spectrum in the null-space components. Since the null-space components describe the fine spectral
structures of the solar spectrum, the retrieved solar spectrum is a smooth version of the highresolution
solar spectrum. This smoothing is expressed by the so-called averaging kernel A of the
retrieval [Rodgers, 2000]. For the minimum length solution the averaging kernel is given by
A =
M∑
v i vi T . (4.16)
i=1
Thus, the relation between the true solar spectrum and the retrieved spectrum is
x = Ax true + e x . (4.17)
Here, e x is the error on the retrieved solar spectrum. The averaging kernel A represents the projection
of the state vector onto the row space (i.e. the non-null-space) of the inversion problem. On the other
hand, (1−A) represents the projection onto the null-space, where 1 is the identity matrix.
Figure 4.4 shows the null-space component and the row space component of a solar spectrum
retrieved from a simulated GOME solar measurement. Here we used the solar spectrum of Chance
and Spurr [1997] as the true high-resolution solar spectrum x true . The figure shows that the null-space
part (1−A)x true represents a significant component of the true solar spectrum. The rms of the nullspace
component, related to the mean solar spectrum, is 23.4%. By definition, this null-space part is
mapped to zero in the forward simulation of the GOME solar irradiance spectrum.
4.4.2 Undersampling error
If the solar spectrum is well sampled, the null-space contribution is mapped to zero for any sampling
of the slit-averaged spectrum. This is also true for a sampling of the Earth radiance spectrum, which
is subject to a similar but slightly stronger spectral smoothing (see Fig. 4.1). However, it is known
that the spectral sampling of GOME is too coarse, which results in a model bias for the simulation of
the Earth’s radiance spectra of GOME. This error is called the undersampling error of GOME and has
to be distinguished from the interpolation error described in Section 4.3.