Page 561 - Sensors and Control Systems in Manufacturing

514
T e n
Cha p te r

The idea is to model the nonlinear response of the system by
using a quadratic function instead of a linear function. The uncali-
brated measurement is now expressed as:
σ
σ
σ
σ
σ
σ
σ
S Measured () = Q()( L Source ()+ M Stray ()) + K()( L Source ()+ M Stray ())
2
(10.46)
where S Measured (σ) = measured complex spectrum (arbitrary)
Q(σ) = nonlinear instrument response function
–1
2
2
(arbitrary cm sr cm /W )
K(σ) = complex instrument response function
2
–1
(arbitrary cm sr cm /W)
–1
2
L Source (σ) = source spectral radiance (W/cm sr cm )
M Stray (σ) = spectral power of the stray radiance
2
(W/cm sr cm )
–1
The radiometric calibration is performed as before but there are
now three unknowns per detector, the nonlinear radiometric gain Q(σ),
the linear radiometric gain K(σ), and the radiometric offset M Stray (σ). This
nonlinear calibration is then applied to an uncalibrated measurement
to obtain a calibrated spectrum.
The characterization of the preceding system requires a minimum
of three different temperature blackbody measurements. If we apply
Eq. (10.27) to the three measured blackbodies, we get the following
system of equations.
1 (
)
)
2
σ
σ
σ
σ
σ
σ
σ
S Measured () = Q() L Source ()+ M Stray () + K() L 1 ( Source ()+ M Stray ()
1
(10.47)
)
2 (
2
σ
σ
σ
σ
σ
σ
σ
S Measured () = Q() L Source ()+ M Stray () + K() L 2 ( Source ()+ M Stray () )
2
(10.48)
)
3 (
2
σ
σ
σ
σ
σ
σ
σ
S Measured () = Q() L Source ()+ M Stray () + K() L 3 ( Source ()+ M Stray () )
3

(10.49)
It is now possible to solve for the three unknowns Q(σ), K(σ), and
M Stray (σ). The calibrated spectrum is then given by:
σ
σ
K σ − 2
σ
L Source () = (( ) Q( ) M Stray ( )) ±
σ
2 Q()
σ
σ
σ
σ
σ
σ
σ
σ
2
4
Q
(2 Q( ) M Strray () + K ()) − Q ()( ()M Stray () + K ()M Stray (σ) − S Measured ( ))
σ
2 Q ()
(10.50)