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The Forward model - the imaging process

``It is no use saying `We are doing our best.' You have got to succeed in doing what is necessary.'' - Winston Churchill

In order to make quantitative data analysis possible it is necessary to know the absolute scale of the instrument.

For single station imaging it is sufficient to calculate the absolute sensitivity, i.e. the number of photons needed inside the field-of-view of the pixel on the front lens in order to create one count in the image. The primary measurable in imaging is raw counts. With knowledge of the number of photons per count, the effective area of the camera and the pixel field-of-view, it is possible to convert the raw counts to a surface brightness, defined as photons per solid angle per area per time $ 1/[ster][m^2][s]$. For aeronomic studies it is of more immediate interest to know the total emission from the atmosphere of the spectral features. This conversion from surface brightness is simple provided that the emission is isotropic and not self absorbed. If $ \Bbb{N}$ photons have been detected in a pixel with a field-of-view $ d\Omega$ and an effective area $ A_L$, an exposure time $ t$ and wavelength band $ d\lambda$, the corresponding surface brightness is:

$\displaystyle \Bbb{I} = \frac{\Bbb{N}}{d\Omega A_L d\lambda t}$ (5.1)

This corresponds to the total number of photons emitted from a column with unit area in the direction of the pixel line-of-sight. Assuming that the emission is isotropic, the total number of photons emitted in all directions is:

$\displaystyle I = 4\pi\Bbb{I}$ (5.2)

In aeronomy the unit for column emission rate has been given the name Rayleigh, $ 10^6/[m^2][s]=1 [R]$ ( Hunten et al., 1956).

For an imaging system intended for tomographic inversion it is necessary to know what fraction of photons emitted in a voxel creates a count in a pixel in the image.

		 {\vspace{2cm}\fbox {\epsfig{file=Figures/,width=30mm,height=10cm}}\vfill}
		 \end{figure} Factors which need to to be taken into account are:

Voxel - pixel field-of-view intersection

For all inverse problems in this work there is a voxel representation of the distribution of emission. To calculate the contribution to the image intensity in one pixel from a voxel, the intersection volume, $ V$, between the pixel field-of-view and the voxel must be calculated. This is described in section 5.1.

Pixel field-of-view

The pixel field-of-view $ (d\Omega)$ is solely dependent on the optical characteristics of the camera. A general and straightforward way to calculate this is described in section 5.2.

Atmospheric absorption

The light from the aurora and airglow is absorbed in the lower atmosphere, mainly in the stratosphere. This absorption depends on both zenith angle, $ \theta_z$, and wavelength, $ \lambda$. This is described in section 5.3.

Effective collecting area

The effective collecting area of the optical system is essentially the size of the front lens as seen from the direction of the voxel. Further the limiting aperture of the optics might change with the angle relative the optical axis. This is described in section 5.4.

Transmission of optics

Variation of the transmission of the optical system with angle relative to the optical axis should be accounted for, as described in section 5.5.

Variation in exposure time

The exposure time varies slightly from pixel to pixel due to the working of the shutters; a first order correction for this is described in section 5.6.

Point spread function

The point spread functions (PSF) must be determined. The PSF is the image of a point source. The procedure for determining the PSF is outlined in section 5.7.

Pixel sensitivity

The sensitivity of individual pixels must be determined; the necessary requirements are outlined in section 5.8.

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Next: Voxel Pixel Up: thesis Previous: Evaluation   Contents

copyright Björn Gustavsson 2000-10-24