SOI-TN-132: 23 February 1996

The Ic computation in the MDI IP is actually:

Ic = { [ 2*Si + Id) >> 1 ] + Sd>>2 } << 15This closely resembles Sd/8 + Id/4 + Si/2 - Cadc.then extract the upper 16 of the lower 32 bits in the

accumulator and finally subtract Cadc from that result.

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

Si is the sum of F0There is a complete sequence of F0-4 each 15 seconds.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.

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/2For 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 followsIc = (F1+F2+F3+F4)/8 + (1/4) * sqrt{2*[(F1-F3)**2 + (F2-F4)**2]} + F0/2

sqrt{2*[(F1-F3)**2 + (F2-F4)**2]} Ý F1-F3 + F4-F2Then 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 ] - CadcIf 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_rmsFor completeness I'll just note the expected signals and shot noise for various quantities:60 Second 11600 7.8 9.4

30 Second 5810 5.7 12.4

15s: Signal; noise 30s: S;N 60s: S;NAs noted above, the actual calculation of Ic is: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

Ic = { [ 2*Si + Id) >> 1 ] + Sd>>2 } << 15Two bits are lost from Sd, so the precision is 4 DN;then extract the upper 16 of the lower 32 bits in the

accumulator and finally subtract Cadc from that result.

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.