LASCO C1 Emission Line Signal - Three Image Method
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Derivation of the Emission Signal Extraction Equation

The 3-image method of extracting emission signal from LASCO C1 images is based on a model for the stray light in an image which has been found to be accurate to within a few percent for all cases. The following 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.

For each pixel we have:

	S(w) = R I(w) + E(w) + L                        Eq(1)

where S(w) is the signal as a function of wavelength w (corrected for offset bias and exposure time), I(w) is the disk-averaged solar "Fraunhofer" irradiance, E(w) is the emission signal, and R and L are constants with respect to wavelength. All variables except I(w) may vary from pixel to pixel.

I(w) is Fraunhofer radiation which is largely present due to scattering of the solar disk radiation from the first mirror in the instrument, which is subjected to the full un-occulted view of the sun. To a lesser extent, the F-Corona also contributes to this portion of the signal. The L variable is a white-light background which is practically constant with respect to wavelength, and will account for white stray light as well as the K-corona which is essentially constant over the bandwidth of a typical C1 scan.

The emission images are reduced using both "open door" and "closed door" scans. The open door scans are direct images of the corona, with the solar disc occulted. The closed door scans have a translucent screen at the front of the telescope which nominally illuminates the detector with a uniform light intensity from the entire solar disc.

It is assumed that the constants R and L may be different for open and closed door scans. If we choose two off-line wavelengths (w₁ and w₂), for which we know the emission signal will be zero, and one on-line wavelength (w_x) at which we wish to measure the emission signal, we can form three open-door equations:

	S1 = R I1 + L             Eq (2a)
        S2 = R I2 + L             Eq (2b)
        Sx = R Ix + L + E         Eq (2c)

where S₁ = = S(w₁) etc. and E is the emission line signal at wavelength w_x. We may also form three closed door equations:

	Sc1 = Rc I1 + Lc           Eq (3a)
        Sc2 = Rc I2 + Lc           Eq (3b)
        Scx = Rc Ix + Lc           Eq (3c)

where subscript "c" indicates a closed-door image. Note that there is no emission signal for the closed door image S_cx because, unlike the open door image in which the disk of the sun is occulted, the closed door image is illuminated by the full disk of the sun, which effectively overpowers the relatively weak emission line signal.

Given only the six measured images S₁, S₂, S_x, S_c1, S_c2, and S_cx, the above six equations can be solved for the emission signal by simple algebra:

	E = (Sx-S2) - (S1-S2)(Scx-Sc2)/(Sc1-Sc2)         Eq (4)

Fig 1 below is a raw signal image which has only been corrected for offset bias and exposure. Fig. 2 is the same image after all extraneous light except the FeXIV line signal has been eliminated using the above procedure.

           

Considerations in Choosing the Reference Wavelengths

It can be seen immediately that the denominator in the second term (S_c1-S_c2) of Equation (4) should never be zero, and in fact should be as large as possible in order to maximize the signal-to-noise ratio. Therefore, when looking for suitable off line wavelengths, we will seek a "high signal" wavelength and a "low signal" wavelength which will serve as a reliable measure of the intensity of the Fraunhofer radiation. The low signal wavelength will generally be near the minimum of a Fraunhofer absorption line. Consideration must be made of the fact that the transmitted wavelength of the FP varies over the image. The low signal wavelength should be chosen so that it gives a relatively low signal for all important points on the image.

The model equation (1) is an approximation to the true situation. The most serious error in this equation can be described as a function which is linear in wavelength which gives the signal versus wavelength function a "tilted" appearance. This effect is especially strong in the "ghost" image in the lower right quadrant of an image. In order to minimize the effect of this tilt, it will be desireable to have the two off-line wavelengths and the on-line wavelength to be as close to each other as possible.

Noise Considerations

A typical closed door image has a lower signal level than the interesting areas of the corresponding open door image. Much of the noise in an image is photon noise. The standard deviation of photon noise is the square root of the number of photons collected by a given pixel. The total noise may be written:

        D(E) = Sqrt[ D2(S) + D2(Sc)(S/Sc)2 ]            Eq(5)

where D(S) is the noise in an open door image:

        D2(S) = Q S/(X g)                              Eq(6)

and D(S_c) is the noise in a closed door image:

        D2(Sc) = Q Sc/(Xc g)         Eq(7)

where S is a typical open door signal, S_c a typical closed door signal, X and X_c are the respective exposure times, g is the photon sensitivity of the detector (about 13 photons/DN), and Q is a constant.

In order that the closed door scans contribute noise in an amount equal to that of the open door scans, we would need to collect an equal number of photons for each scan. To completely eliminate noise in the closed door scan, we would need to collect an infinite number of photons, but practically speaking, collecting two or three times the number of photons as found in the open door scan will be sufficient. For example, the signal in an open door FeXIV scan will be about 200 DN/pixel/sec near the occulter, while the closed door scan will have a signal level of about 12 DN/pixel/second. To obtain the same amount of noise in the closed door scan as in the open door scan, we would need 200/12 = 17 closed door images (taken at the same exposure time as the open door images) in order to give a contribution to the noise equal to that of the open door images. Using double or triple that number would give a significant yet practical reduction in the total noise.

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PR Apr 16, 1998