LASCO C1 Emission Line Signal - Two Image Method
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Introduction

Although a rather exact method for extracting the emission line signal has been described in the "LASCO C1 Emission Line Signal - Three Image Method" article, many of the C1 observations were obtained at just two wavelengths. We describe here a modification of the three-image method of extracting an emission line image which requires only two images, and is a significant improvement over a simple image subtraction. As with the three-image method, this method isolates the signal due to the emission line only. (To convert the signal to radiometric units, see the article on LASCO C1 Radiometry.) An IDL program fpc1_2img.pro is available which implements the procedure described in this article.

We can rewrite the equation for the emission line signal from the Three Image Method article as:

     E = (Sx-S2) - f (Scx-Sc2)    Eq. 1

where the dimensionless ``Fraunhofer ratio'' f is given by

     f = (S1-S2)/(Sc1-Sc2)    Eq. 2 

which is measureable, and may be expected to be rather independent of the wavelengths of the signals used to measure it since the ratio R/R_c is, according to the model, independent of wavelength. It can be seen immediately that the denominator S_c1-S_c2 in Equation 2 should never be zero, and in fact should be as large as possible in order to maximize the signal-to-noise ratio. It is desireable to find a method of estimating the emission image when the second off-line image (S₂) is missing, and which is more accurate than the standard method. Referring to Equation 1 it can be seen that this may be done by estimating the Fraunhofer ratio f, which is the only variable which depends upon the missing image S₂. The parameters used to estimate the value of f should include any parameter which is both available and relevant. This would include the 5 available images S₁, S_x, S_c1, S_c2 and S_cx as well as position in the image, distance of the spacecraft to the Sun etc. An additional reason for using the dimensionless variables f and f_s is that they are independent of pixel response, and so any prediction method involving these variables will not have to deal with variations in pixel response or degradation. See photometry section below)

The 3-Gaussian approximation to f

A good estimator of f that can be obtained without the use of the S₁ signal is:

    fs = S2/Sc2

Since we use the fractional error in the signal rather than the absolute error as a figure of merit, It is better to deal with the logarithm of f. Accordingly we will use the variables:

    x = ln(fs)

and

     z = ln(f/fs)

Figure 1 is a scatter plot of z versus x calculated using Equations \ref{3img-E} and \ref{zdef} from average signals at &lambda₁=5297.6001 Angstroms and λ₁=5309.2343 Angstroms from the data of March 16 through June 17, 1998.

FIGURE 1

caption: Fig. 1 - Scatter plot of to \ln(f/f_s) as a function of \ln(f) It can be seen that the standard approximation (z=0) is not bad, but could be improved by a more accurate functional approximation to z. Accordingly, the z(x) function will be modelled as follows:

     z3g=Σ Ai Gi(x,xi,σi)        Eq. 4

where the sum is from i=1 to 3 and A_i are constants and G_i are Gaussian functions:

     Gi(x,xi,σi)=exp(-(x-xi)2/2σi2)  Eq. 5

This model was chosen instead of a polynomial model because z is essentially a perturbation from the standard approximation of z=0 and it is desireable that the model reduce to z=0 for values outside the nominal range of x. The parameters of the model were determined by a least squares fit to suitably ``averaged'' images. The parameters of the model were determined by a least squares fit to the 3/Mar/98 set of closed door images using the averaging method described in the section above. The open door images were averaged images from the 3-image data taken between 16/Mar/98 and 17/Jun/98. These results are summarized in Figure 2 below.

FIGURE 2

caption: Fig. 2 - 3-Gaussian Approximation to ln(f/f_s) as a function of ln(f).

In Figure 2 the diamonds are values of z averaged over 0.1-wide intervals of x. The red line is a fit of the 3-Gaussian model to these points using Equations 4 and 5 with the following parameter values:

 i     Ai        xi       σi
 0  0.08423  -2.39595  0.14103 
 1  0.11093  -1.47551  0.24021
 2  0.65913   1.47500  1.17664