Precision of the MDI Continuum Intensity

J. Todd Hoeksema
SOI-TN-132: 23 February 1996

The Ic computation in the MDI IP is actually:

 Ic =	{ [ 2*Si + Id) >> 1 ] + Sd>>2 } << 15  

		then extract the upper 16 of the lower 32 bits in the

		accumulator and finally subtract Cadc from that result.
This closely resembles Sd/8 + Id/4 + Si/2 - Cadc.

Where

F0-F4 are filtergrams at tuning positions 0 (cont) - 4.

	Si is the sum of F0

	Sd is the sum of F1+F2+F3+F4

	Sa is the difference F1-F3

	Sc is the difference F2-F4

	Id is sqrt(2*(Sa**2 + Sc**2))

	Cadc is the offset value of the CCD A/D converter.
There is a complete sequence of F0-4 each 15 seconds.

The shot noise level of a typical 3000 DN filtergram is 5.48 DN.

** NOTE: The Cadc value subtracted in the Ic computation must be properly set for various repeats, it must be different for 15, 30, or 60 second (1,2, or 4 repeat) observations.

We can rewrite Ic in terms of filtergrams as:

Ic  =		Sd/8	+	       Id/4 			     + Si/2 

Ic  =  	(F1+F2+F3+F4)/8 + (1/4) * sqrt{2*[(F1-F3)**2 + (F2-F4)**2]}  + F0/2
For the purposes of determining the noise, if |F1-F3| is about equal to |F2-F4| (the 0 velocity, well-tuned case) we can rewrite as follows 

	sqrt{2*[(F1-F3)**2 + (F2-F4)**2]} Ý F1-F3 + F4-F2
Then writing out the whole expression and combining terms, Ic becomes just a weighted sum of the 5 filtergrams, approximately

	Ic Ý	1/8 * [ 4*F0 + 3*F1 + 3*F4 - F2 - F3 ] - Cadc
If the noise is about the same in each filtergram and we just RMS the errors we find that Ic has an error related to the filtergram noise level by 5.48*6/8 or 4.11 DN in 15 seconds. Typical values would be about 3125 DN (assuming F0=F1=F4=3000; F2=F3=2500).

Observing for multiples of 15 seconds will increase the value linearly but the noise by the sqrt of the number of 15 second intervals. So a 60-second Ic should be about 12500 DN with an error of 8.22 DN; 30 seconds (2 passes) gives a signal of 6250 DN with error 5.81 DN.

This agrees pretty well with the sqrt compressed values Dnyanesh reported:

1100 (= 10*sqrt(12100)) (though with the incorrect ID lookup table). It also compares favorably with Phil's MDI model computation that for 0 velocity gave:

	O velocity	Ic	Ic_rms	V_rms

	60 Second	11600	7.8	9.4

	30 Second	 5810	5.7	12.4
For completeness I'll just note the expected signals and shot noise for various quantities:

		15s: Signal; noise 	30s: S;N	60s: S;N


	Ic	  3K;  	4 DN	 	6K;  6 DN	12K;  8 DN

	Si	  3K; 	5.5 DN		6K;  7.75 DN	12K;  11 DN

	Sd	 12K; 	11 DN	       24K;  15.5 DN	48K;  22 DN

	Sa	0.5K; 	7.75 DN		1K;  11 DN	 2K;  15.5 DN

	Sc	0.5K; 	7.75 DN		1K;  11 DN	 2K;  15.5 DN

	Id	  1K; 	11 DN		2K;  15.5 DN	 4K;  22 DN
As noted above, the actual calculation of Ic is:

 Ic =   { [ 2*Si + Id) >> 1 ] + Sd>>2 } << 15

		then extract the upper 16 of the lower 32 bits in the

		accumulator and finally subtract Cadc from that result.
Two bits are lost from Sd, so the precision is 4 DN;
This is less than the noise in Sd;

One bit is lost from the sum of 2*Si+Id, so the precision is 2 DN;
 This is less than the noise of that quantity.

So the computation is carried out with sufficient precision.

As currently calculated a 60s Ic computation has a signal that is about 1500 times the noise level. The ratio is lower for shorter integrations. Thus 11 bits are sufficient for Ic.

Last Modified: 04:21pm PST, March 11, 1996