A Possible Non-Linear Calibration Scheme

From Figure 4 we see that the DCR counts are tending to approach an asymptotic value as the source strength increases. The gain curve can then be approximated as

$$P_{\rm out}(\rm counts) \sim {1 \over {\epsilon + \beta \left[ P_{\rm in}(\rm K) \right]^{-\delta}} }$$ (6)

or

$$P_{\rm in}(\rm K) \sim {1 \over {B \left[ P_{\rm out}(\rm counts) \right]^{-\gamma} - A } }$$ (7)

where $\epsilon$,$\beta$,$\delta$,$A$,$B$,and $\gamma$ are assumed to be greater than or equal to zero. Normal Sig/Ref observations are reduced - assuming a constant gain - using

$$T_{src}(K) = \left({{G Sig^{cal off}(counts) - G Ref^{cal off}(counts)} \over G Ref^{cal off}(counts)} \right) T_{sys}^{Ref}(K)$$

$$= \left( {{Sig^{cal off}(K) - Ref^{cal off}(K)} \over Ref^{cal off}(K)}\right) T_{sys}^{Ref}(K)$$ (8)

where

$$T_{sys}^{Ref}(K) = {T_{cal} \over 2} \left( { G Ref^{cal on}(counts) + G Ref^{cal off}(counts) \over G Ref^{cal on}(counts) + G Ref^{cal off}(counts) } \right)$$

$$= {T_{cal} \over 2} \left( { Ref^{cal on}(K) + Ref^{cal off}(K) \over Ref^{cal on}(K) + Ref^{cal off}(K) } \right).$$ (9)

Equations 8 and 9 assume $Sig(counts)=G T_{Sig}(K)$ and $Ref(counts)=G T_{Ref}(K)$ with $G=$ constant.

If we use Equation 7 then Equation 8 becomes

$$T_{src}(K) = \left({ { {1 \over {B \left[ Sig^{cal off}(\rm ... ...(\rm counts)\right]^{-\gamma} - A}} } \right) T_{sys}^{Ref}(K)$$ (10)

which with a little algebra can be shown to be

$$T_{src}(K) = \left( { { \left[ Ref^{cal off}(\rm counts)\rig... ...}(\rm counts)\right]^{-\gamma} - C} } \right) T_{sys}^{Ref}(K)$$ (11)

where $C = A / B$. Putting Equation 7 into Equation 9 and doing little algebra results in

$$T_{sys}^{Ref}(K) = \left( {{ \left[Ref^{cal off}(counts)\rig... ...f^{cal on}(counts)\right]^{-\gamma}}}\right) {T_{cal} \over 2}$$ (12)

A fit could be done to Equations 11 and 12 to determine $C$ and $\gamma$ while allowing the source strength to be $T_{src}(K) = T_o \nu^\alpha$.

This scheme has the potential to provide calibrated data on a known temperature scale if $\gamma$ and $C$ are stable or slowly varying. The data could then be further processed using the ``Solomon scheme'' to provide good baselines.

If $\gamma$ and $C$ are stable or slowly varying then it may be possible to periodically observe well known calibration sources - with well defined $T_o \nu^\alpha$ to determine $\gamma$ and $C$. The values for $\gamma$ and $C$ determined from the calibrators could then be ``blindly'' applied to the target observations.