Deriving Nonlinearity Coefficients from Changing Background
Transcript of Deriving Nonlinearity Coefficients from Changing Background
Deriving Nonlinearity Coefficients from Changing Background
Mark Esplin, Deron Scott, Bryce Walker, Ben EsplinUtah State University Space Dynamics Laboratory
1
Contentsā¢ Quadric Nonlinearityā¢ Nonlinearity Model and Sensitivity Studiesā¢ Application to CrIS Sensorā¢ Check Method with Stepped Blackbody Dataā¢ Conclusions
2
Quadratic Nonlinearity Equations
3
An interferometer with quadratic nonlinearity can be expressed as:š¼š¼ā² = (š¼š¼ + šš) ā šš2 š¼š¼ + šš 2
where: I = original undistorted interferogramIā = interferogram distorted with quadratic nonlinearityV = offset voltagea2 = quadratic nonlinearity coefficient
āšš2(šš ā šš) + 1 ā 2šš2šš šš
)Iā² = āšš2š¼š¼2 + 1 ā 2šš2šš š¼š¼ + (šš ā šš2šš2Rearranging the terms yields:
This term is just a constant and can be removed before taking the Fourier transform. After taking the Fourier transform the result is:
where: S is the Fourier transform of I, and convolution is indicated by ā.If the spectrum is band limited, the first term can be ignored, resulting in:
1 ā 2šš2šš šš
This is the desired spectrum, S, multiplied by a gain term
0
40
80
120
0 40 80 120 160
40
80
120
80 120 160
Graphical Representation of Nonlinearity
V -- Voltage Offseta2 -- Quadratic Nonlinearity CoefficientI -- Linear Input InterferogramIā -- Nonlinear Interferogram
š¼š¼ā² = š¼š¼ + šš ā šš2 š¼š¼ + šš 2
ā¢ CrISās photons to voltage relationship is nonlinearā¢ The magnitude of the nonlinearity has been
exaggerated for clarity
Iā
I
V
Nonlinear Spectrum
ā¢ When S is band limited as shown, the (šš ā šš) component does not overlap spectrally and can readily be removed
ā¢ The nonlinearity corrected spectra is then:
5
šššš ā šš
Sā = āšš2(šš ā šš) + 1 ā 2šš2šš šš
V -- Voltage OffsetS -- Correct SpectrumSā -- Nonlinear Spectruma2 -- Quadratic Nonlinearity Coefficient
S = ššā²/ 1 ā 2šš2šš
Nonlinear Spectrum
Nor
mal
ized
Resp
onse
Nyquist Frequency
Standard Nonlinearity Correction Methodā¢ Measure source with known output radiances
ā Need at least two different radiance levelsā¢ Derive nonlinearity coefficients that yield known radiancesā¢ Advantage:
ā Simple intuitive methodā¢ Disadvantage:
ā Radiance of source needs to be known very accurately at different radiance levels
6
Alternative Nonlinearity Methodā¢ Observe external source (ECT) held at a constant output radiance
ā Although required to be stable, the actual value of radiance is unimportant
ā¢ Make measurements with different sensor temperaturesā¢ Internal calibration target (ICT) temperature floats with sensor
temperatureā¢ Reprocess the data with different nonlinearity coefficientsā¢ Optimize nonlinearity coefficient by minimizing spread in calculated
radiancesā¢ Why it works:
ā Varying background sensor temperature moves up and down the system gain curve
ā Nonlinearity errors cause calibrated radiance to change even though the radiance from source has not changed
7
Sensor Simulation Modelā¢ Built model to study sensitivities of the methodā¢ Derived nonlinearity a2 coefficients from simulated dataā¢ Model included:
ā Internal calibration with Deep space (DS) and Internal Calibration Target (ICT)
ā ICT temperature errorsā Time dependent temperature changes for ICT and ECTā Warm instrument background effectsā Noise
ā¢ Results show low sensitivity to all parameters except for Earth Calibration Target (ECT) radiance changes correlated with sensor/ICT variations
ā¢ ECT radiance can vary as long as it is not correlated with sensor temperature
8
Nonlinearity Correction Method Applied to CrIS
ā¢ Cross-Track Infrared Sounder (CrIS)ā¢ The CrIS sensor is an infrared
Fourier transform spectrometerā¢ One CrIS instrument is currently
flying on the Suomi National Polar-orbiting Partnership (NPP) spacecraft
ā¢ Another CrIS (J1) is being preparing for launch in 2017
http://www.jpss.noaa.gov/images/media-gallery/gallery-jpss1_1.jpg
ā¢ 8-second scans ā¢ 30 Earth locations
ā 9 FOVs per locationā 3 spectral bands per FOV
ā¢ LWIR 650-1095 cm-1, resolution: 0.625 cm-1
ā¢ MWIR 1210-1750 cm-1, resolution: 1.25 cm-1
ā¢ SWIR 2155-2550 cm-1, resolution: 2.5 cm-1
ā¢ Two calibration views per scanā Internal calibration target (ICT) ā Warm (ambient temperature)ā Deep space (DS) ā Coldā Separate calibration for two interferometer scan directions
ā¢ Two redundant electronic sides
CrIS Sensor Includes Self Contained Radiance Calibration
30 Earth Scenes Sampling 9 FOVs
DS(Cold)
ICT(Warm)
8 sec
Nonlinearity Correction Method Applied to CrISā¢ CrIS model J1 has completed thermal vacuum (TVAC) testingā¢ Long Term Repeatability (LTR) data taken during TVACā¢ Provides opportunity to verify method with real sensor dataā¢ LTR data consists of 18 collects each 2 hours long taken over a
period of a month ā¢ The LTR gives a metric for the stability of the sensorā¢ A small LTR means a stable sensorā¢ LTR is defined as:
ā Average together the 2 hours of spectra in each setā Convert to a percentage of a 287 K blackbodyā Find the standard deviation at each wavenumber
ā¢ CrIS J1 meets spec with large margin even without a nonlinearity correction
11
Time History of LWIR LTR Data
ā¢ Temperatures for several sensor elements shown on leftā SSM (scene select mirror), ICT (internal calibration target), OMA
(interferometer or opto-mechanical assembly) ā¢ BT error shown on right
ā Measured brightness temperature minus 287 Kā¢ BT error tracks sensor temperature changes
12
Sensor Temperatures BT - 287 K
Derive Nonlinearity a2s from LTR dataā¢ Vary a2s to minimize LTR using an iterative technique
ā Start with an initial set of a2sā Calculate the LTRā Vary a2s until minimum LTR is found
ā¢ Derivation of a2s uses no explicit temperature knowledgeā ICT and DS temperatures used by Science Data Record (SDR)
algorithmā¢ Only consider more central regions of each band
ā LWIR 680 ā 1020 cm-1, MWIR 1220 ā 1600 cm-1, SWIR 2160 ā 2400 cm-1
ā¢ LTR histories created by concatenated measurements into single time series
ā¢ Analysis performed for both electronic sides
13
LWIR LTR Before and After Nonlinearity Optimization
ā¢ LW FOV5 most nonlinear and shows the largest LTRā¢ After optimization of a2s, LTR is much lower ā¢ Excellent agreement between FOVs
14
No Nonlinearity Correction Optimized a2s
Magnitude of LTR with Different a2s
ā¢ LTR calculated with a factor times preliminary Exelis a2sā¢ Calculated at a number of points then spline interpolatedā¢ Solved for factor that minimizes LTR for each FOV
15
MWIR LTR Before and After Optimization
ā¢ Only MW FOV9 has significant nonlinearityā¢ Only FOV9 a2 optimized, all other MW a2s set to zero
16
MWIR LTR
ā¢ Only MW FOV9 a2 varied, other a2s set to zeroā¢ MWIR side 1 data
17
LTR Before Nonlinearity Optimization
ā¢ LW FOV5 and MW FOV9 brightness temperature differences on order of 50 mK
ā¢ Radiance jump at center of test apparent with nonlinear FOVs
18
LWIR MWIR
LTR After Nonlinearity Optimization
ā¢ Excellent agreement between LWIR and MWIR time historiesā¢ ECT gradients rows FOVs 1-2-3 to 7-8-9 clearly visibleā¢ Remaining structure probably ECT temperature instability
ā Equal magnitude for all bands
19
LWIR MWIR
LTR After Nonlinearity Optimization
ā¢ Excellent agreement between LWIR and SWIRā¢ SWIR has higher noise ā¢ No nonlinearity correction for SWIR
20
LWIR SWIR
Derived a2s
ā¢ Side1: a2s derived from side 1 LTR dataā¢ Side2: a2s derived from side 2 LTR dataā¢ Results consistent with a2s generated with conventional method
21
LW 1 LW 2 LW 3 LW 4 LW 5 LW 6 LW 7 LW 8 LW 9 MW 9Preliminary 0.0116 0.0128 0.012 0.0119 0.0386 0.0085 0.0093 0.0121 0.0064 0.0513Flight 0.0141 0.0145 0.0152 0.0159 0.0355 0.011 0.0095 0.0157 0.0085 0.0507S1 0.0131 0.0147 0.0127 0.018 0.0359 0.0113 0.0127 0.0169 0.0086 0.0446S2 0.0139 0.0137 0.0128 0.0158 0.0324 0.0094 0.0076 0.0107 0.0005 0.0321
0
0.01
0.02
0.03
0.04
0.05
0.06
COEF
FICI
ENT
VALU
E
FOV #
a2 Comparisons
Check of LTR Derived Nonlinearity a2sā¢ Conventional method of determining nonlinearity uses the ECT
stepped through a range of temperaturesā¢ Plot stepped blackbody brightness temperatures similar to LTR
time histories ā¢ Combine data into a time series subtract ECT set point
temperatureā¢ Again integrate over spectral content
ā LWIR 680 ā 1020 cm-1
ā MWIR 1220 ā 1600 cm-1
ā SWIR 2160 ā 2400 cm-1
ā¢ No correction for ECT gradients or temperature errorsā¢ Side 1 a2s consistent with stepped blackbody data
22
Stepped Blackbody No Nonlinearity Correction
ā¢ No nonlinearity correctionā¢ Nonlinearity of LW FOV5 and MW FOV9 clearly visibleā¢ Mission Nominal Temperature (MN) stepped blackbody, side 1
23
LWIR
200K233K
260K
278K
299K
310K
200K 233K 260K 278K
299K 310K
MWIR
With Nonlinearity Correction
ā¢ Note the scale change from the preceding slideā¢ Good agreement between LW and MWā¢ a2s derived from side 1 LTR
24
LWIR MWIR
MWIR Compared to SWIR
ā¢ Good agreement with SWIRā¢ High noise for cold temperatures with MWIR and SWIR
25
SWIRMWIR
Discussionā¢ Method works well even though sensor temperature difference was
only on the order of 1.5 Kā¢ Larger change in sensor temperature expected to produce more
accurate a2sā¢ Variations in observed brightness temperature used to derive a2s
smaller than ECT gradientsā¢ With quadratic nonlinearity large temperature differences are not
neededā¢ However, side 2 LTR has less temperature variation and derived
a2s not as consistent with stepped blackbodyā¢ Method relies on relative differences not absolute differencesā¢ Method can still work with ECT temperature variation if they are
not correlated with the sensor temperature variations
26
Conclusionā¢ a2 nonlinearity coefficients can be derived using alternate method
ā Potential on-orbit application with uniform temperature sceneā¢ Derived a2s consistent with stepped blackbodyā¢ Side 1 LTR data showed larger brightness temperature differences
and resulted in better a2sā¢ Potentially higher accuracy since method can partially compensates
for inaccuracies of other coefficients and modelā¢ Probably only practical with system like CrIS with self contained
calibration
27