SCULIB_SKYFUNC

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SCULIB_SKYFUNC_1

calculate f(x) and its Jacobian for NAG non-linear fitting routine E04UPF when fitting ETA_TEL, B and TAU in SKYDIP analysis

Description:

E04UPF is a NAG routine that is used to fit a theoretical sky-dip curve to the measured data by varying ETA_TEL, B and TAU. This routine is called by E04UPF to calculate the M sub-functions fi(x) at the current solution vector X, and/or their Jacobian matrix. See the NAG manual if you need to know more about E04UPF.

The solution vector is composed as follows:-

x(1) = ETA_TEL
x(2) = B
x(3) = TAU

For each of the M measured airmasses the subfunction f(x) is:-

$f (x) = J_{}$ theory (Airmass(i)) - J_meas (Airmass(i))

$J_{}$

noise (Airmass(i)) (74)

where:-

$J_{}$ theory = (1-

$η_{}$

tel) J_tel + $η_{}$

telJ_atm (1-B $e x p (- τ A i r m a s s)) (75)$

$J_{}$ atm = J_amb X_G (76)

$X_{G} = 1 + \frac{h 1 \times h 2}{J_{}}$ amb

$e x p (- τ \times A i r m a s s (i) / X_G c o n s t) (77)$

The corresponding Jacobian has component (i,j):-

$\frac{\partial f (x)}{\partial x (j)} = \frac{\partial}{\partial x (j)} J_{}$ theory(Airmass(i))

$\times \frac{1}{J_{}}$

noise(i) (78)

leading to:

$\frac{\partial f (i)}{\partial x (1)} = - J_{}$ tel+J_atm - B J_tel

$e x p (- τ A (i))$

$J_{}$

noise(i) (79)

$\frac{\partial f (i)}{\partial x (2)} = - η_{}$ tel J_atm

$e x p (- τ A (i))$

$J_{}$

noise(i) (80)

$\frac{\partial f (i)}{\partial x (3)} = η_{}$ tel

$\partial J_{}$

atm $\partial τ - η_{}$

tel B $e x p (- τ A (i)) (\partial J_{})$

atm $\partial τ - J_{}$

atm A(i) $J_{}$

noise(i) (81)

where

$\partial J_{}$ atm

$\partial τ = J_{}$

amb $\partial X_{G}$ $\partial τ$ (82)

and

$\frac{\partial X_{G}}{\partial τ} = - \frac{H_{1} H_{2} A (i)}{J_{}}$ amb X_Gconst

$e x p (- τ A (i) / X_{}$

Gconst) (83)

The array USER is used to get the necessary ancillary information into the routine:-

USER (1) = J_TEL
USER (2) = J_AMB
USER (3) = not used
USER (4:M $+$ 3) = the measured airmasses
USER (M $+$ 4:2M $+$ 3) = the measured sky temperatures
USER (2M $+$ 4:3M $+$ 3) = the errors on the measured sky temperatures

SCULIB_SKYFUNC_1

Description:

Invocation

Arguments

See NAG manual description of OBJFUN parameter in E04UPF.

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