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NASA Contractor Report 178292
Development of Response Models for the Earth
Radiation Budget Experiment (ERBE) Sensors:
Part I - Dynamic Models and Computer
Simulations for the ERBE Nonscanner,
Scanner and Solar Monitor Sensors
(NASA-CR-178292) DEVELOPRENT OF RESPONSE
MODELS FOR THE EARTH RADIATION BUDGET
EXPERIMENT 'ERBE) SENSORS. PART I: D¥_K_IC
MODELS AND COMPUTER SI_IULATIONS FOR TLE E_S Unclas
NONSCANNER, [Inforj_ation. and Co.tirol Syste=s) G3/35 0111320
Nesim Halyo, Sang H. Choi, Dan A. Chrisman, Jr.and Richard W. Samms
N88-13557
Information & Control Systems, IncorporatedHampton, Virginia 23666
Contract NAS1-16130
March 20, 1987
N/ ANallonal Aeronaulics andSpace Administration
Langley Research CenterHampton, Virginia 23665-5225
https://ntrs.nasa.gov/search.jsp?R=19880004175 2020-04-19T22:42:07+00:00Z
FOREWORD
This report entitled "Development of Response Models for the Earth Radiation
Budget Experiment (ERBE) Sensors" consists of the following four parts.
This is Part I, NASA CR-178292, entitled "Dynamic Models and Computer
Simulations for the ERBE Nonscanner, Scanner and Solar Monitor Sensors".
Part II, NASA CR-178293, is entitled "Analysis of the ERBE Integrating
Sphere Ground Calibration".
Part III, NASA CR-178294, is entitled "ERBE Scanner Measurement Accuracy
Analysis Due to Reduced Housekeeping Data".
Part IV, NASA CR-178295, is entitled "Preliminary Nonscanner Models and
Count Conversion Algorithms".
TABLEOFCONTENTS
page
FOREWORD .............................. i
LIST OF FIGURES ........................... iv
LIST OF SYMBOLS ........................... vl
LIST OF ACRONYMS .......................... x111
i. INTRODUCTION .......................... 1
2. SENSOR MODELING ......................... 5
2.1 GENERAL DESCRIPTION OF THE ERBE SENSOR MODELS ....... 6
2.1.1 MODELS ....................... 6
2.1.2 THERMAL CONDUCTIVE AND RADIATIVE INTERACTIONS .... 14
2.1.3 COUPLING ENCLOSURES ................. 21
2.1.4 GOVERNING EQUATION ................. 27
2.1.5 ELECTRONIC CONTROL SYSTEM MODEL FOR THE
NONSCANNER ..................... 28
2.1.6 SIMULATION OF THE NONSCANNER MODEL ......... 32
2.1.6.A PARAMETER AND COEFFICIENT DETERMINATION . . 34
2.1.6.B INSTRUMENT SIMULATION ........... 35
2.1.7 COUNT CONVERSION FOR NONSCANNER ........... 36
2.1.7.A NONSCANNER TOTAL CHANNEL .......... 36
2.1.7.B NONSCANNER FILTERED CHANNEL FOR
SHORTWAVE ................. 38
2.1.8 SOLAR MONITOR MODEL ANALYSIS ............ 40
2.1.9 SOLAR MONITOR COUNT CONVERSION ALGORITHM ...... 50
2.1.10 SCANNER CONTROL ELECTRONICS ............. 51
2.1.11 SCANNER STEADY-STATE ANALYSIS ............ 54
11
TABLEOFCONTENTS(CONCLUDED)
page
2.1.12 SCANNERCOUNTCONVERSION.............. 58
2.1.13 ACCURACYANDDIAGNOSTICCHECKS............ 61
3. INSTRUMENTPERFORMANCEANDCOUNTCONVERSIONTEST ......... 63
3.1 DESCRIPTIONOFROUTINES................... 68
3.1.1 ROUTINEI ....................... 68
3.1.2 ROUTINEII ...................... 69
3.1.3 ROUTINEIII ...................... 69
3.1.4 ROUTINEIV ..................... 70
4. DISCUSSIONOFPARAMETERESTIMATION................ 71
REFERENCES.............................. 76
APPENDIXA: PRELIMINARYRESULTSONERBENONSCANNERINSTRUMENTMODELING,
COUNTCONVERSIONANDSENSITIVITYANALYSIS ........ 78
APPENDIXB: ERBESCANNERSIMULATIONS................. 166
APPENDIXC: ERBENONSCANNERTHERMALMODEL............. 184
APPENDIXD: ERBESCANNERTHERMALMODEL................ 202
APPENDIXE: ERBESOLARMONITORTHERMALMODEL............. 250
iii
FIGURE I.
FIGURE 2.
FIGURE 3.
FIGURE 4.
FIGURE 5.
FIGURE 6.
FIGURE 7.
FIGURE 8.
FIGURE 9.
FIGURE i0.
FIGURE ll.
FIGURE 12.
FIGURE 13.
FIGURE 14.
FIGURE 15.
FIGURE 16.
FIGURE 17.
FIGURE 18.
LIST OF FIGURES
NONSCANNER MEDIUM FIELD-OF-VIEW TOTAL (MFOVT) ......
NONSCANNER MEDIUM FIELD-OF-VIEW SHORTWAVE (MFOVSW) . . .
NONSCANNER WIDE FIELD-OF-VIEW TOTAL (WFOVT) .......
page
7
8
9
NONSCANNER WIDE FIELD-OF-VIEW SHORTWAVE (WFOVSW) .... I0
SCANNER NARROW FIELD-OF-VIEW TOTAL (NFOVT) ........ ii
SOLAR MONITOR ...................... 12
A SIMPLE THERMAL CONTACT RESISTANCE MODEL ........ 15
RADIATION FLUX INTERCHANGE ............... 18
INTERACTIVE ENCLOSURE MODEL ............... 23
ACTIVE CAVITY RADIOMETER (ACR) CONTROL SYSTEM MODEL . . . 29
NONSCANNER HEAT SINK HEATER CONTROL SYSTEM MODEL .... 32
FLOW DIAGRAM INDICATING FILE AND PROGRAM USAGE IN
SIMULATION ....................... 33
THE SHUTTER MODEL .................... 41
SHUTTER TEMPERATURE AT x = 1.5 cm WITH 20% ABSORPTION
OF SOLAR FLUX Qs = 0.142 w/cm 2 ............. 43
SHUTTER TEMPERATURE AT i00 SECONDS WITH 20% ABSORPTION
OF SOLAR FLUX Qs = 0.142 w/cm 2 .............. 44
A SIMPLIFIED CONFIGURATION OF SOLAR MONITOR BARREL . . . 45
BARREL TEMPERATURE WITH RESPECT TO TIME AT x = 3 cm
FROM THE SECONDARY APERTURE ...... _ ........ 47
BARREL TEMPERATURE ALONG THE AXIAL DIRECTION AT THE
TIME OF 200 SECONDS ................... 48
iv
LIST OF FIGURES (CONCLUDED)
page
FIGURE 19. SCANNER ELECTRONIC CONTROL MODEL ............ 53
FIGURE 20. INSTRUMENT PERFORMANCE AND COUNT CONVERSION TEST .... 65
FIGURE 21. ERBE INSTRUMENT VALIDATION PLAN ............. 66
FIGURE 22. EXPLANATIONS ...................... 74
v
LIST OF SYMBOLS
Symbol
Ac
Ar
A i
Ar
Ac
Definition
Area of conductive transfer in an element
Area of radiative transfer in an element
Contacting area between elements k and i
Radius of a contacting point, e.g. Fig. 7
thInside surface area for i surface
Constant defined in Eq. 14
Area of radiative interaction
Area of nodes covered by a heater
Thermal wave velocity, e.g. Eq. 58
b
B
BQ
bE
Bk
Bias term, e.g. Eq. 25
Vector of coefficients for irradiance from the source
Coefficient which describes heating contributions
from the control systems
Electronic bias
Constants related with the irradiance at a FOV
limiter
Ck
C
AC B
Specific heat capacity of element k
Constant defined in Eq. 15
Number of counts
Baseband signal
D Enclosure coupling factor through the interactiveboundaries
vi
LIST OF SYMBOLS(CONTINUED)
STmbol
ek
ETM
n
mn
e
AEos
El(t)
ei(t)
EBi(T B)
EBx(T B)
Fk- i
Fk_ i
Definition
Energy in element k
Rate of incoming radiant energy to surface n inenclosure m
Rate of incoming radiant energy to non-opaque
boundary surface n in enclosure m from a neighboringsurface
Noise term, Eq. 25
Total incident radiation esitmate
Baseband signal
Spectral irradiance at the instrument due to radia-
tion from the footprint at time t
Combined error of the measurement and count con-
version accuracy
Output of count conversion when viewing the black-
body at temperature TB
Spectral irradiance from the blackbody at TB
Configuration factor between elements k and i
Exchange factor between elements k and i
F
FE(S)
System boundary condi_tions imposed by surrounding_
Control electronics transfer function.
hk- i
Hz
Thermal conductance of the contact between elements
k and i
Hertz
vii
LIST OF SYMBOLS (CONTI_._UED)
Symbol
I
i
Definition
Identity Matrix
Calibrat ion sequence
Number of measurements in the scanner mode.
kk
K.l
kfo
Thermal conductivity of element k
Electronic gain for component i
Post amplification gain
i
Lk- i
L
I
Longwave
Shortest distance between the center of mass of
element k and the contacting point of element k
to its neighboring element i
Length of a solar monitor shutter
Exposure length to solar radiation
M_.l
M°
l
ms
m
Opaque radios[ty - the rate of outgoing radiant
energy from surface i from diffuse and specular
reflectance
Total radiosity - opaque radiosity plus energy flux
transmitted through surface i
Milliseconds
Meter
n
n i
Number of contacts per unit area
Noise from source i
viii
LIST OFSYMBOLS(CONTINUED)
S)nnbol
,l!
qc
Definition
Dynamic heat transfer due to behavior of element k
as a heat source or sink
Heat flux by conduction through the contacting area
between two elements
R i
Ro
Thermal resistance through the contacting area be-
tween elements k and i
Inside radius of solar monitor barrel
Outside radius of solar monitor barrel
Smn
Heat flux arriving at the system boundary at surface
n in enclosure m
Shortwave
T
T k
Tk
T(i)
Total radiation channel
Temperature (vector) of element k
Governing equation for a radiometer with a dynamic
response to an incoming radiance at element k
Temperature change for thermal node i or voltage
change for electronic node i
T
T b
At
tk
t
Ambient temperature
Current shutter temperature in solar monitor
Total scan period
Time at end of space-look
Sampling instant
ix
LIST OFSYMBOLS(CONTINUED)
Symbo I
T B
Definition
Temperature of a blackbody
u(x) Unit step function
V k
v(t), v(t)
_(tk)
V(tki)
VB(t)
Vd(t k)
VBi(T B)
Volume of element k
Instrument output (in volts) at time t
Space clamp value at time tk, i.e. at the end ofspace look
Instrument output (in volts) when viewing space
at time tk
Detector bridge bias voltage at time t or most re-
cent value (v)
Drift balance DAC voltage measurement at time t
or most recent value (v)
.thSteady-state output of the i
blackbody temperature T B
channel when viewing
W Watts
x
x
x
A_ i
Distance from end of barrel in a solar monitor
Vector consisting of electrical signal processing
variables, e.g. Eq. 73
Vector consisting of the offset temperature and
radiative constant, e.g. Eq. 109
Lower frequency signal portion of Ax for component
i, e.g. Eq. 79
Noise and irradiance portion of Ax for component
i, e.g. Eq. 79
x
LIST OF SYMBOI,.q (CONTINUED)
Greek
S_ mbol
Yk
1
_k
n k
_k
Ok
P
o
2
° k
"rk
T.1
T
i(k)
Definition
Density of element k
Surface roughness of surface 1
Thickness of solar monitor shutter
Estimate of unaccounted drift during k th scan
period (v)
Emissivity of element k
Exchange factors between specular surfaces kand i
Attenuation factor for element k
Wavelength k
Reflectivity of surface k
Density of solar monitor shutter
Stefan-Boltzmann constant
Noise variance estimate during space look
Transmissivity of surface k
Time constant for electronic element i
Average time lag (in seconds)
Transmittance of ith channel
Radiative energy supplied to surface k
Superscript
d
S
m
Diffuse
Specular
Number of an enclosure in a system
xi
LIST OFSYMBOLS(CONCLUDED)
Superscript
M
W
P
Definition
Medium field-of-view
Wide field-of-vlew
Photo-diode
Subscript
s
1
s
k-i
n
H
A
F
B
A
E
v
B
o
s
H
Shortwave band
Longwave band
Surface, Eq. 6
between k and i elements
Number of an enclosure's surface in a system
Heater, Eq. 35
Precision aperture
Field-of-vlew llmlter
Bridge
Amplifier
Electronic
Volatage
Blackbody
Initial
Solar, Eq. 58
Heat sink, Eq. 99
xii
LIST OFACRONYMS
Acronym
ACR
A1 Heat Sink
A/D
B.C.
CCA
Cu Heat Sink
DAC
ERBE
EXP
FOV
FOVL
G.E.
HK
HKD
IBB
I/0
I.C.
LW
MFOV
MFVOT
MFOVSW
MAM
MRBB
Definition
Active Cavity Radiometer
Aluminum IIeat Sink
Analog-to-Digital (conversion)
Boundary Conditions
Count Conversion Algorithm
Copper Heat Sink
Digital-to-Analog Conversion
Earth Radiation Budget Experiment
Exponential
Field-of-view
Field-of-view limter
Governing Equations
Housekeeping
Housekeeping data
Internal blackbody
Input/Output (data processing)
Initial Conditions
Longwave
Medium Field-of-View
Medium Field-of-View Total
Medium Field-of-View Shortwave
Mirror Attenuator Mosaic
Master Reference Blackbody
xiii
LIST OF ACRONYMS (CONCLUDED)
Acronym
ms
NS
NFOV
RHS
RMS
SW
SWICS
SiPD
T
WFOV
WFOVT
WFOVSW
WRT
Def inlt ion
Millisecond
Nonscanner
Narrow Field-of-View
Right hand side
Root mean square
Shortwave
Shortwave Internal Calibration Source
Silicon Photo-Diode
Total radiation channel
Wide Field-of-View
Wide Field-of-View Total
Wide Field-of-Vlew Shortwave
With respect to
xiv
i. INTRODUCTION
The radlometrlc measurement system for the Earth Radiation Budget Experi-
ment (ERBE) is designed to measure the earth's radiation fluxes which result
from the earth and its atmosphere's emmltted energy as well as from reflected
energy, i.e., the earth's albedo. A general overview of the concept behind the
three-satellite ERBE radiometric system and its instruments has been described
in many internal NASA documents, such as the Science Team Minutes and Contractor
Status Reports, as well as in journal puhlications, (e.g., Ref. i).
The radiometer package on each ERBE satellite consists of eight instruments
distinguished by their mode of operation, field-of-view and spectral response.
Four of the eight instruments are nonscanner (NS) type radiometers, two with
a Medium Field-of-View (MFOV) and two a Wide Field-of-View (WFOV). Two of the
nonscanner radiometers, one MFOV and one WFOV, have a suprasil filter which
eliminates the instrument's response to longwave radiative effects so that they
only remain sensitive to shortwave radiative effects. Shortwave (SW) radiative
effects are defined here as having wavelengths between 0.2 and 5 _m and long-
wave radiative effects as having wavelengths between 5 and 50 _m. Since radio-
meters without filters measure total radiation, they are designated "Total" (T).
The four nonscanner radiometers are therefore the Medium Field-of-View Total
(MFOVT), Medium Field-of-View Shortwave (MFOVSW), Wide Field-of-View Total
(WFOVT) and Wide Field-of-View Shortwave (WFOVSW). Three of the remaining in-
struments or "channels" are Narrow Field-of-View (NFOV) scanning radiometers,
two of which have filters. One scanner radiometer has a filter which eliminates
longwave radiative effects and another has a filter which eliminates shortwave
radiative effects. Hence the three scanner radiometers are the Narrow Field-of-
View Shortwave (NFOVSW), the Narrow Field-of-View Longwave (NFOVLW) and the
Narrow Field-of-View Total (NFOVT). The eighth instrument in the ERBE radio-
metric package is the solar monitor radiometer.
After the ERBE radiometers perform their earth, space and/or sun view and
calibration measurements, the measurement results undergo an analog-to-digital
(A/D) conversion and the data now in counts is transmitted to the earth through
the spacecraft telemetry system. Counts for the nonscanners are measured
every 0.8 seconds. Heat sink temperatures are read every 8 seconds and other
housekeeping (HK) temperatures every 16 seconds. Therefore, in 16 seconds
the data record for one nonscanner consists of 20 radiometric count readings,
2 heat sink temperature readings and I reading for each of the other HK
temperatures.
The scanners perform a measurement cycle once cv=Ly _ seconds. During a
normal, earth-viewing scan each scanner performs 8 space-look measurements, 62
earth-view measurements and 4 calibration measurements. The Longwave and Total
NFOV scanners utilize the Internal Blackbody (IBB) for calibration measurements,
whereas the Shortwave NFOV scanner utilizes the Shortwave Internal Calibration
Source (SWICS). During a Mirror Attenuator Mosaic (NAM) scan cycle, the NFOVT
and NFOVSW scanners measure solar flux performing 8 space-look measurements,
46 MAM measurements and 4 calibration measurements. Housekeeping (HK) data is
sampled once per scan.
The solar monitor uses the same type of Active Cavity Radiometer (ACR) that
is used by the nonscanners. The three major differences between these ACR's
are as follows:
a) the solar monitor has a shutter to cut off the solar flux every
32 seconds,
b) the configuration of the solar monitor housing is totally different
and
c) the solar monitor has no heat sink heater.
Solar Monitor count measurementsare madeevery 0.033 seconds and HKmeasurements
are madeevery 4 seconds.
The radiometric instruments described above will have inherently different
behavior and responses to arbitrary input signals since no instrument can be
exactly duplicated or subjected to exactly the samecalibration or operating
environment. Evenwhenviewing the problems of sensor performance from th_s
macroscopic point of view, there are many things that have to be dealt with and
accounted for. For example, environmental contamination and aging effects in-
fluence the material degradation of the sensors. Sensor behavior may also be
questioned when the thermal load and leakage exceed the design level. One ex-
ample of abnormal thermal load is in the sun-blip case, where the sensor is
subjected to a sunrise and sunset during its orbit.
In attempting to account for such problems found in the sensor's radiometric
performance and capability, it is therefore necessary to develop a radiometer
sensor model. A sensor model should describe both a transient and steady-state
sensor response for a given radiative input. It must provide sufficiently simple
analytical expressions for parameters of particular interest and describe the
model's behavior, e.g., the model's sensitivity to changes in the modeling assump-
tions. The model must also be able to incorporate changes resulting from in-
strument design requirements during its construction as well as instrument aging,
i.e., sensor material degradation and unexpected thermal load and leakage during
its long-term orbital operation. The products obtained from the sensor model and
simulation must be in a form that is useful to the users. The sensors' response
must therefore be validated by updating model parameters and converted into data
with useful engineering units.
This report entitled "Development of Response Models for tile Earth Radiation
Budget Experiment (ERBE) Sensors" consists of tlle following four parts.
Part I, NASA CR-178292, is entitled "Dynamic Models and Computer Simulations
for the ERBE Nonscanner, Scanner and Solar Monitor Sensors".
Part II, NASA CR-178293, is entitled "Analysis of the ERBE Integrating
Sphere Ground Calibration".
Part III, NASA CR-178294, is entitled "ERBE Scanner Measurement Accuracy
Analysis Due to Reduced Housekeeping Data".
Part IV, NASA CR-178295, is entitled "Preliminary Nonscanner Models and
Count Conversion Algorithms".
4
2. SENSOR MODELING
The following ERBE radiometer sensor models were developed by Information &
Control Systems, Incorporated (ICS) to consider thermal and radiative exchange
effects. These models consider the surface specularity and spectral dependence
of a filter as well as the thermal conductive and radiative interactions among
the nodes in an enclosure of a radiometer. An enclosure is a domain in which
an instrument's geometry is divided into appropriate elements. These enclosures
in an instrument's model are set up using real or fictitious boundaries to ef-
fectively alleviate the computational load for defining configuration factors.
The instrument's model also includes the electronic thermal control sytems which
are coupled with the energy equations. A general form of the energy equations
can be used for all the sensors, despite differences in the mode of operation,
field-of-view and spectral response, as long as the radiative enclosures and
thermal nodes in the sytem's sensor configuration comply with the first law of
thermodynamics.
A sensor model can be tested and evaluated by comparing its behavior with
the actual results obtained from the sensor's ground and inflight calibrations.
The sensor model includes additional uncertainty since it incorporates outputs
from the calibration source models. Once the sensor and calibration source models
prod,lce sufficiently accurate results, they can be used to define the parameters
for the count conversion procedure.
The characteristics of a sensor'_ _imulation model are such that it has
a dynamic response to the time dependence of an arbitrary input signal. The
model also has a spectral response which is due to the spectral dependence of
a sensor's filter and its exposed surfaces. Thermal effects for the model can
be determined from the conductive and radiative properties of the sensor. _le
model's measurementerror can therefore be determined after its time response,
spectral response and thermal effects have been obtained. Themodel can simu-
late the sensor's response to the temporal incident radiation received through
the field-of-view limiter in both the transient and steady-states.
The purpose of model_ngand simulating is therefore I) to understand sensor
performance, 2) to improve measurementaccuracy by updating the model parameters
and 3) to provide appropriate constants necessary for the count conversion
algorithms.
2.1 General Description of the ERBE Sensor Models
2.1.1 Models
A sensor's geometry can be appropriately divided by considering its ther-
mal and radiative interactions as a consequence of its design. Figures 1
through 6 provide diagrams of how sensor elements have been geometrically
partitioned for the nonscanner, scanner and solar monitor. Each element has
a time resonse to the energy flowing to and from its neighboring elements.
One might assume that maximizing the number of sensor nodes would maximize
the accuracy of the model's prediction of energy from a target source. However,
increasing the number of nodes would introduce additional sources of error
since the large set of equations required would incorporate additional para-
meter approximations. Such equation approximations would need to be made to
calculate constants such as thermal resistance, exchange factors and surface
spectral and specular characteristics. There is an accuracy tradeoff between
increasing the number of nodes and approximating the sensor's size, material
properties, geometry, enclosure surface characteristics, temporal incident
NONSCANNER MEDIUM FIELD-OF-VIEW TOTAL (MFOVT)
5
H
4 \ E¥,
2
4
• 2 P
!
H
1 ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 Al HEAT SINK
5 BASE PLATE
6 FOV LIMITER
7 Al HEAT SINK
8 APERTURE
FIGURE I. H HEATER
P TEMPERATURE PROBE
NONSCANNER MEDIUM FIELD-OF-VIEW SHORTWAVE (MFOVSW)
S4
H 2
I
|,
|
• 2 P
i!
H
FIGURE 2.
1 ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 AI HEAT SINK
5 BASE PLATE
6 FOV LIMITER
7 AI HEAT SINK
8 APERTURE
9 SW FILTER DOME
H HEATER
P TEMPERATURE PROBE
NONSCANNEIt ",VIDE FIELD-OF-VIEW TOTAL (WFOVT)
/7
PI
'lS
_._ 8 8!
i I
M
4
m
l
2
1
AA P
A P
H
i
i
• 2 P I.I
l
7
I ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 Al HEAT SINK
5 BASE PLATE
| 6 FOV LIMITER
7 AI HEAT SINK
8 APERTURE
FIGURE 3.
H HEATER
P TEMPERATURE PROBE
NONSCANNER WIDE FIELD-OF-VIEW SHORTWAVE (WFOVSW)
/'!.J
H
•, .y
4 EV.
2
7 "
r
m
4
• 2 P
[ !
FIGURE 4.
H
5.r-- [
1 ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 AI HEAT SINK
5 BASE PLATE
6 FOV LIMITER
7 Al HEAT SINK
8 APERTURE
9 SW FILTER DOME
H HEATER
P TEMPERATURE PROBE
lO
o
_o
ZZ
!
I
11
SOLAR MONITOR
'+-h
III
-.-4-I!IIIIIII
11IIIIIIII
IJ
11
12 13
II j ' |
,oP
-j
+,III
IItIIIIII
1,
6-_, __-_-,,f /-6 ,.I
1
I..,,....
2
|9 19
14
f
|. ACTIVE CAVITY2. REFERENCrCAVITY
3. BUTTON.DUPLEXCONECAVITIES
4. HEAT SINK._.AFERTURE.PRIMARY6, SHIELD. APERTURE1. BODY IREAR BARRELI8. BODY IFRON! blARRELt9, APERTURE.SE.CO_DARY
OR FOV LI&ll IER t|0. h_OTORHOUSIhG||. CHOPPERgLADE|2, CHOPPERPENDLU/_,I13. MOTOR. CHOPPER|4, FOVLIMITERiS. FRONTTOP HOUSING COVER|6. REARTOPHOUSING COVER17. FRONTHOUSING|8. REARHOUSING19. HOLDER. HEAT SINK
?0 FICTIIIOUS SURFACE
p. TI'_4PERATURE PROBES
FICUR_ 6.
12
energy flux intensity and electronic control system. If the temperature
gradients within a node and between nodes are small enough, a model approx-
imation with only a few nodes would not Jeopardize the accuracy required to
perform the parameter estimates and error analysis. Factors which would de-
crease these temperature gradients within a sensor include uniform thermal
properties, simple and symmetrical geometry, uniformity in directional and
spectral properties of an enclosure surface and small temporal change in in-
cident energy flux.
A nonscanner sensor has 9 nodes if it has a suprasil filter dome (Figures
2 and 4) and 8 nodes if it does not (Figures i and 3). The sensor's geometry
is divided into nodes based on material homogeneity, overall sensor size, thermal
conductivity of materials, thermal resistance between materials and surface
radiative properties. The 9 nodes of the _OVSW model (Figure 2) are there-
fore, as numbered: i) the active cavity with a heating and sensor coil 2) the
copper heat sink with a heater, 3) the reference cavity with a sensor coil, 4)
the aluminum heat sink, _lich is partly covered by a heater, 5) the base plate
or substrate, 6) the field-of-view limiter, 7) the aluminum heat sink which
houses the reference cabity, which is partly covered by a heater, 8) the pre-
cision aperture and 9) the filter dome for shortwave channels.
Similarly, the scanner model (Figure 5) is divided into 12 nodes for the
total channel, 13 nodes for the longwave channel and 14 nodes for the short-
wave channel. The solar monitor is divided into 19 nodes (Figure 6).
In addition to their nodes, these sensor models incorporate fictitious sur-
faces to define and simplify enclosure geometries- Since the specular prop-
erty of each surface in an enclosure is taken into consideration, the simple
geometry facilitates the exchange factor calculation, alleviating the compu-
tational work load for determining the exchange factors. A net radiation
13
analysis is performed for each enclosure which has surfaces with specular and
spectral characteristics. This analysis appropriately deals with any partial
specular and diffuse characteristics of an enclosure surface (Ref. 2). The
model can also easily include transient response and thermal conduction. This
net radiation method utilizes the wavebandapproximation to an enclosure's
surface which has a specific spectral dependence.
2.1.2 Thermal Conductive and Radiative Interactions
A volumetrically defined node, or control volume, complies with the first
law of thermodynamics as in tile following equation
de
d--_= Ac qc'"+ Ar _ (I)
The first and second terms of the right hand side of Equation i represent the
conductive transfer through contacting surfaces and radiative transfer from
surrounding elements of an enclosure, respectively. The rate of energy change
in a single node, therefore, depends on the difference between the ingoing and
outgoing energy fluxes through a control surface or "boundary". This rate of
energy gain or loss is composed of the rate of energy stored in a node as well
as the rate of energy source or sink as indicated in the following equation
de = VyC dTd--_ _- _ (2)
where V is the volume of a node, y the density of a node, C the specific heat,
T the temperature and Q the rate of energy, or heat, at a source or sink.
Combining Equations 1 and 2 yields
dT •,,VyC _-_ = Ac qc + Ar _ + _ (3)
14
Equation 3 represents an energy balance in a single node. In the geometrically
partitioned model, the conduction term can be written in the following form
A _ T. - T.c _'' = j i
c/i-j Ri-j
(4)
where Ai_ j is the contacting area of i element to j element, qc the heat flux
by conduction through the contacting area between i and j elements. Ri_ j is the
thermal resistance between i and j elements. The thermal resistance between nodes
is defined by the following realationship
L 1 Li-j .I-!
Ri-j = k + +i Ai-j hi_ j Ai_ j k.3 AJ -i
(s)
where Li_ j is the distance from the thermal center of a node to the center of the
contacting surface, k the thermal conductivity and h the thermal conductance of
the contact between i and j elements. The thermal center is normally coincident
with the geometrical center if a node material has homogeneous and isotropic ther-
mal properties. The contact resistance (1/hA) is mainly dependent on the surface
roughness and the contacting pressure. The well-machined surface has a regular
and uniform roughness. In such a case, the contact resistance can be determined
by the following empirical relations (Ref. 3) as functions of the surface rough-
ness and the thermal conductivities of two contacting materials.
Figure 7.
//
J\
\\\
---------m
A Simple Thermal Contact Resistance Model
15
Let surfaces contact each other with surface roughness 61 and _2 and thermal
conductivities kI and k2, respectively. Assume the contacting area per periodic
2 i
contact is _a . Then the number of contacts per unit area n = _ and the
Atotal number of contacts for two surfaces with total contacting area A is _ .
_a-
Conductance can therefore be calculate,d:
h=T +n (6)
where
ks
6 = 61 + 62 (= 261 or 262 )
n = 2.13_ _ , where q is a parameter for roughness
a+c, where c is the half-width of the surface cavity
1n = _ = number of contacts per unit area
kl,k 2 ffithermal conductivity of nodes i and 2
61,82 = root mean square surface roughness of nodes 1 and 2
A = total node interface area
a ffiradius of a contacting point
For the radiative interchanges in an enclosure including both specular
and diffuse surfaces, a radiative property such as a reflectance, 0, can be
divided into a diffuse and a specular component as well as shortwave and long-
wave bands.
That is, for shortwave bands:
16
d sPs -- Ps + Ps
and for longwave bands:
d s
p£ -- p£ + P£
where the superscripts, d and s, signify the diffuse and specular, and the sub-
scripts the short and longwave bands. For a semi-transparent or an absorbing
medium, the radiative properties have the following relationship among the
reflectance 0, emittance or absorptance _:, and transmittance, T.
_s + Ps + Ts = i for shortwave
EZ + 0£ + T£ = i for longwave
The energy fluxes incident upon and leaving a typical seml-transparent
surface of the enclosure are described in Figure 8. Consider the ith inside
surface area, Ai, of the enclosure which neighbors with other enclosures. The
quantities E i and M_ are the rates of incoming and outgoing radiant energy
while considering diffuse and specular reflectance of the surface per unit in-
side area, respectively. The quantity M_ does not include the rate of trans-
mitted incident radiant energy from the other neighboring enclosures. The
quantity _i is the rate of incoming radiant energy to the non-opaque boundary
surface from the neighboring enclosure. The quanltity #i is the energy flux
supplied by some external means to the surface to make up for the net radiative
loss and thereby maintain the specified surface temperature. A heat balance
at the surface provides the relation
_i = El - Mi (7)
A second equation results from the fact that the energy flux leaving the
17
_th
SURFACE
z_.EiIIIII
Ei
FIGURE 8. Radiation Flux Interchange
18
surface having diffuse and specular characteristics is composed of directly
emitted plus reflectcd energy. Th_s gives
4 d sM_ = Ei O T i + Oi Ei + Pi Ei (8)
Equation 8 is called an opaque radiosity. The total radioslty, considering
the rate of transmitted radiant energy which is originally coming from the
neighboring enclosure, is composed of the opaque radiosity M_ plus the energy
flux transmitted through the surface with transmissivlty, T i. This gives
M i = M_I + Ti _i (9)
The diffuse total radiosity considering only a diffuse factor in Eq. 9
and excluding the specular reflection term is described as the following
d 4 dM i = E i 0 T i + O i E i + Ti _i (I0)
The contribution to Ei from the other diffusely reflecting surfaces in
an enclosure can be represented in accordance with the relationship defined by
the configuration factors in an enclosure with diffuse surfaces, except that
now the configuration factors are replaced by the exchange factors, or
= E - M_ (11)Ei j Fij S
where F.. is the exchange factor defined between i element and j element through13
j's images on the element i. Equations 7 through ii can be represensted by the
short and longwaves, respectively.
To obtain the heat balance (Eq. 7) as functions of surface temperatures Jn
an enclosure and the irradiance, or incident radiation, from the target source at
the field-of-view limiter of the radiometer models, first Eq. 9 is substituted
into Eq. 7, yielding
19
d sin which Pi + Oi = Pi where Oiis the total reflectance of the surface and
_=OT 4.
Thus
¢i = (i - pl) EI - gl _I - _i _i (12)
Then, combining Eqs. i0 and Ii and rearranging, yields the following equation
El = _ (_iJ - Fij 0_ )-I FIj(¢J _j + Tj _j). (13)
For brevity, we introduce new variables from Eq. 13, that is
AIj = (_lJ - Fij P_ )-I FiJ Ej-
Thus, Eq. 12 with the above constants can be written
(14)
(15)
EI = E Gj + Rj (16)J A±j Clj
Equation 16 has an important role to couple an enclosure with the neighboring
enclosures through a semi-transparent or transparent boundary surface.
The above relation (Eq. 16) can be used for the cases when an enclosure
is partly or totally surrounded by the neighboring enclosures and also _len
part or all of'the surfaces of the enclosure are non-opaque, i.e., seml-trans-
parent or transparent. To include all the configurations of the enclosures and
to couple them together through interactive boundary surfaces, the above rela-
tion (Eq. 16) can be generalized in the matrix form, that is,
E (17)
20
where A and C denote Eqs. 14 and 15, respectively. The bars above the notation
A and C are to distinguish Eq. 14 and 15 from the notation for the area and
the conduction constant of governing equation to be discussed in the Section 2.1.4.
2.1.3 Coupling Enclosures
In the above Eq. 17, E can be directly related to E by defining the geo-
metrical coupling factor. The relationship between E and E is apparently de-
pendent on the geometrical formation of enclosures through the interactive
(non-opaque) surfaces as in the following example. Consider the system shown
in Fig. 9 which consists of nine enclosures and interacts at the boundaries with
the incident radiation from the environment. The incoming radiation E can be
E =
written
"]17
21]
(18)
•,m l
L -
where the superscript m denotes the number of enclosures in the prescribed
system• Each enclosure of the system has a certain number of interacting sur-
faces. In such a case, the incident radiation upon the surfaces of the ith
enclosure can be described by
Ei J" ill=!EI
1,
E 2
E trl
and 1 _ i K m (19)
21
where the subscript n denotes the number of a surface in an enclosure of the
i 2
system. As shown in Fig. 9, E4 and E 2 define the incident radiation on an in-
teractive boundary surface between the first and second enclosures. The sub-
scripts 4 and 2 here, respectively denote the surface numbers as seen or counted
from each enclosure. These surfaces designated by 4 and 2 in the enclosure 1 and
2, respectively, are supposed to be the interactive boundary surface co-owned by
enclosures 1 and 2. On the contrary, incident radiation E coming from the op-
posite, or neighboring, enclosure upon the boundary surface is defined by ~2E2 and
~1E4, respectively. Accordingly, observing enclosures i and 2 with respect to the
incident radiation on the interactive boundary surface, we find the following
coincidences:
1 2 2 1E4 -- _2 and E2 = _4
Equation 20 tells us that the incident radiation
(20)
1E4, upon the interactive
boundary surface of the first enclosure is virtually regarded as the opposite of
2incident radiation _2' on the back surface of the same boundary, whether the
second enclosure is considered to be formulated or vice versa. Thls coincidence
on each interactive boundary surface is uniquely determined by the formation
of the enclosure in a given system.
To generalize the above relationship, Eq. 20, for the enclosure which
has interactive boundaries and also system boundaries, let us consider the
first enclosure shown in Figure 9. The first enclosure has not only two inter-
active boundaries at 3rd and 4th surfaces, but also two system boundaries at
ist and 2nd surfaces (Fig. 9). Extending the above relationship (Eq. 20) to
include the system boundaries and describing it in a matrix form, we have
22
$2
7
$2
l2 ~i(= E_ = E l) s_
_tE_
4t
2
E_i _Jt
E_ ~_Es
7
E_#
E_ = _S
E s = El
5 ~_ q'
E_,
EiS = _S
E8 = _7 #
_ g
E._
, _
'tE_
_3
E_ = E_
@
E_ _E9 = _.8 4.
2 c,_ @ E_
3
S_A
6
S_
9
7
S, 3 3
FIGURE 9. Interactive Enclosure Model
23
E 1 =
0 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0
E2 +
"0 0 0 O"
0 0 0 0
1 0 0 0
0 0 0 0
E4 +
1 0 0 O"
0 i 0 0
0 0 0 "0
0 0 0 0
s i21)
Likewise, the other enclosures have the same expression as depicted for the
first enclosure. Thus, introducing a new varlable.D for the enclosure coupling
factor, the relationship (Eq. 21) can be written
= DE + FS (22)
where D is a matrix representing the enclosure coupling factors through the
interactive boundaries, F is a matrix representing the system boundary conditions
imposed by surroundings, and S is a column vector of the heat fluxes arriving
at the system boundaries. Now substituting Eq. 22 into Eq. 17 and collecting
the same variables, we have, with I being an Identity matrix
E = (I - CD) -I _Q + (I - CD) -I _FS (23)
Equation 23 illustrates that tile incident radiation E upon a surface of the
enclosure accounts for all the effects of radiation from the surfaces with
different temperatures not only in its own enclosure, but also in neighboring
enclosures, and the heat flux at the system boundaries.
The energy balance at a node or a surface in the enclosure due to the
incident radiation is then described by inserting Eqs. 22 and 23 into Eq. 12.
Then, by eliminating E and _ from Eq. 12, the radiative energy balance is
obtained as only functions of the temperatures of enclosure surfaces and the
irradiance from the target source or any other kinds of heat flux which inter-
acts with the system boundaries. That is,
24
(24)
When the radiative energy balance described by Eq. 24 is applied to the
enclosures obtained by properly dividing the geometry of the system, the
number of the radiative energy balance equations, _, obtained from Eq. 24 is
more than the number of nodal points because consideration is given twice to
the interactive boundary between two enclosures. Accordingly, in the multiple-
enclosure system the radiative energy balance is calculated for each surface
of enclosures. Then, for the overlapped boundary surface, the radiative energy
balances from both enclosures are added together. The radiative energy balance
(Eq. 24)is then plugged into the energy equation (Eq. 3) of the system.
The enclosure coupling technique developed here has many potential
applications and advantages over the conventional methods currently used.
Among the advantages, this technique can be used for any kind of geometry
by only setting up enclosures via dividing the geometry and defining the D
and F matrices in Eq. 22. The enclosure space with irregular geometry can be
properly divided into many smaller regular sub-spaces to adopt the enclosure
coupling technique. For an enclosure with complicated geometry, the division
of the space geometry into well-deflned sub-spaces facilitates the computation
of the configuration factor or the exchange factor, especially for an enclosure
with specular surfaces which is totally dependent on the geometry.
This technique can also be extended to treat radiation interactions in-
cluding the scattering and absorption with optically thin or thick participating
media in an enclosure. Even though there are temperature and density gradients
in participating media, by dividing the enclosure space into sub-spaces which
are small enough to assume the sub-spaces to be isothermal and isotropic, the
25
enclosure theory can be directly applied for the system without loss of gener-
ality.
In short, tile advantages and potential applications of th_ enclosure COul)li,,g
technique are listed below,
ADVANTAGES:
i. It minimizes the effort required to define the exchange factors
among the surfaces in an enclosure with complex geometry, and so
may be able to decrease a degree of possible approximation in the
computation of exchange factors.
2. It may take the wavelength characteristics of materials or participating
media into account in a simple manner.
3. It can be easily coupled with the thermal model of the system so that
the material and thermal properties are considered in the coupled
dynamic system without approximations and assumptions such as those
in the boundary definitions.
4. It enables the system to deal with any imposed initial conditions
and prescribed boundary conditions.
5. It may enhance the accuracy in the computation of a geometrically
partitioned conductive and radiative model.
6. As a consequence, this technique can avoid the length of time for
computation and a cumbersome manipulation which is often revealed in
the other methods such as ray tracing or Monte Carlo methods.
APPLICATIONS:
i. It can be used for any enclosure geometry.
2. It can be used for either the enclosure with a vacuum _pace or the
enclosure with participating media (solid, liquid or gas).
26
3. It can be applied to any optical system's anslyses and design regard-
less of the directional, spectral, and/or temperature dependencesof
its enclosure spaces.
4. Applications of the enclosure coupling technique to optical systems
mayvary from a typical radiometer to any wavegu_desystem.
2.1.4 Governing Equation
The governing equation which describes the performance of a radiometer that
has a dynamic response to an incoming radiance can be obtained by substituting
Eqs. 4 and 24 into Eq. 3 and collecting terms. Then we can separate the in-
coming radiation flux from tlle radiation balance term. The equation can be
written in a general form to describe tile energy balance in a node as fo]low_
= AT + DT 4 + BE + BQQ + b + e
where
i
MR
OAr rI.(I- P-D=--_-- L
D)(I - _m)-1 _ - E]
Ar [ D)(I CD)-I C T F]B ----M--(Z,, p - - -
BQ = _M
b = a bias term
e = a noise term
R = the thermal resistance between nodes (see Eq. 5)
M = VyC
(25)
27
Ar
AC
= the area for the radiative interaction
-- the node area covered by the heat sink heater
Equation 25 represents a general form that consists of a set of equations
for a multinode system. Considering only the thermal model for the nonscanner
total radiometer which is divided into 8 nodes, we have a set of 8 equations
which represent the energy balance in each node separately. The inclusion of the
electronic thermal control system equations will increase the total system by
the number of the transfer functions. The constants of Eq. 25 for the nonscanner,
the scanner, and the solar monitor are described in Appendices C, D, and E.
2.1.5 Electronic Control System Model for the Nonscanner
The block diagramshown in Fig. i0 illustrates the nonscanner ACR control
system model. The control electronics transfer function is of the form (cf.
ERBE Instruments, Monthly Status Report - Vol. i, Nov. 1980, unpublished)
KE(TIS+ i)(T2S+ i)
FE(S) = S(T3S + I)(T4S + i) (26)
To obtain the differential equations describing the electronics transfer
function, we define the state variables TI0, TII , TI2, as follows
• [.. ]TI0 = VB Ka _ Ra(TI) + Rr(T3) - BE + KE nE (27)
R4
Note BE = R3 + R4 is an input constant, where R3 and R4 are the bridge
resistances.
R i(Tj) = A i + B.TIj + CiT_' i = a, r; j = 1,3 (28)
TII" =-_31 (_Tl I + Tl0 + TITI0) (29)
28
l.,¢
_f
I
c_u
o_
T-t-
I
i.,4o
F-4#
,-40o"
uo
o
g
0
_._
o
<_
k__J
go
-r.I
1.1.r,I
IlJ
u
29
Substituting Eq. 27 into Eq. 29 gives
TII = T_ (-TII + TI0) +_3 VB Ka KE Ra(TI) + Rr(T3) +_3 KE nE (30)
• 1 (_TI 2 + + T 2 TII ) (31)TI2 - _3 TII
For full expression, substitute TII in Eq. 30 intoEq. 31.
Note that TI2 is now the output of the electronics, so that the voltage
output is
v = TI2 + nv (327
To convert to counts, C, use the following relation:
C = KcV = Ke TI2 + Kc nv, K = 819.1 counts/volt.C
The heat flow provided to the active cavity to maintain its temperature
at a constant level is given by
Q1 K v 2 K 2 + 2K n + K n 2 (337= a = a TI2 a TI2 v a v
This expression for QI brings to light the major effects that the noise
terms introduce into the operation of the radiometer• Note that due to the
use of square feedback, the linear noise terms acquire a gain of 2K a TI2 ,
while a quadratic noise term (having non-zero mean) is introduced. Coils
which are wrapped around the active and reference cones generate heat and are
expressed by the following terms•
2 Ra (TI)
QJI = VB [Ra(Tl) + Rr(T3)]2
• 2 Rr(T3)
QJ3 = VB IRa(T1) + Rr(T3)]z
30
To obtain the closed-loop system, the expression Eq. 33 is simply
substituted into the equation governing the active cavity temperature.
_ a 2 + a TII vT1 = AI2(T2 TI) + E 4 k 2K n
k Dik Tk + BI E +_i TII M1
2K n 2 R __(TI)
+ a---mY+ [Ra(TI )aM 1 VB + Rr(T3)] z(34)
The block diagram shown in Fig. II illustrates the non-scanner ACR heat
sink heater control system model.
The control electronics transfer function can be approximated in the
form
(THS + i)
FilE(S) = KHE s (35)
The, differential equations describing the heat sink heater controller can be
expressed as
T13 = KH13 KHE (TH - Z2) + KHE nH (36)
VH = T13 + TH KHE KH13 (T H - T2) + T H KHE n H (37)
In Eq. 25 the heat source term, Q, is substituted by Eq. 38 for the aluminum
and copper heat sinks.
2.1.6 Simulation of the Nonscanner Model
The simulation of the ERBE nonscanner sensors is divided into two major
parts; the first determines the coefficients of the system, the second solves
the 3ystems over time and plots the results. Figure 12 is a flow diagram
31
32
I
N_
H
.CX
o 8
o
e-4o
o
.r_c_
b40J
O
OZ
,-"4
c.0
TZ
33
indicating file and program usage. The equation describing the change of temp-
erature in a given node at time i is
DT4(i) Bnn (39)T(i) = AT(i) + + BE + BQQ +
Where A is the conductance coefficient matrix, D is the radiative coefficient
matrix, B is the irradiance source vector and BQ is a vector of coefficients
which describe heating contributions from the control system. T(i) represents
either the temperature change for a thermal node or a voltage change for an
electronic node. The general method of solution is a fourth order Runge-Kutta
algorithm, all of which is discussed in great detail in the following sections.
2.1.6.A Parameter and Coefficient Determination
The coefficient matrices and vectors are determined in the first program
in the sequence ERBEPRM. The data inputted to this is determined from material
specifications and engineering drawings supplied by the manufacturer. The code
is partitioned into 5 major sections, not including the program input/output,
The first of these is the subroutine "COMPUTA". This routine takes the heat
capacities and node to node thermal resistances and returns the conductance
coefficient matrix. The program then enters the routine ADDELC which is designed
to modify the conductance matrix in such a way as to allow the ACR heater con-
trois to use the same equations in the simulation as the node temperatures.
The output from this routine is an enlarged conductance matrix and several elec-
tronic vectors. Once these two routines have been computed COMPUTH calculates
the radiative exchange coefficient between surfaces for each cavity. Due to
the linking of cavities a and b, as defined in Appendix C, dome filter or
fictitious surfaces, the radiative exchange coefficients for these cavities
34
are stored for later use. The results of COMPUTH are then added together to
give the total radiative exchange coefficient matrix _n COMPUTD, COMPUTB and
CMPUTBP. The results are then issued to two files, one formatted for in-
spection and one binary for use by the simulation program.
2.1.6.B Instrument Simulation
The program consists of four logical sections, an Input, the Integration,
the intermediate Output and the Plotting section. The actual solution to the
system of equations that describes each instrument is accomplished by the pro-
gram ERBESIM. The program steps through a determined time interval using an
evenly spaced discrete time step and solves each system using a Fourth Order
Runge-Kutta scheme to evaluate the system at the next step. Each system is
sampled at another rate controlled by the user until 20 samplings are taken
or the end of simulated time is reached. The results are then stored on mag-
netic medium for use by the plotting routines. The final product is the sen-
sor's resposne described by plots and tables of the modeled response of the
instrument, i.e., time-dependent temperatures of the nodes, instrument response
(in counts), electronic voltages (in volts) and input source irradiance (in
W/m2).
The first section of the program reads the coefficient matrices and vectors,
intitial conditions from binary files, and the simulation control variables
from a namelist file. The Namellst file contains variables for controlling
simulation time step size, sampling rates and the combination of instruments
and radiation profiles, etc. which are to be simulated. The simulation is
35
capable of allowing the user to choose either a sawtooth, Fourier series, step
function, constant or ramp radiance profile.
The Runge-Kutta method used was developed by Fehlberg and solves the system
5
Yi,m+l = Y" + h Ajl, m j _0 Kj i'
where
i = i, 2, 3, ... number of nodes
i
Kji = f(t m + C.3 h, Ym + h E_Z_O bjE KEj)
where f is the function describing T(i) from Eq. 39 as functions of time and
temperature, a.,j bjE, and C.3 are constants determined by Fehlberg for solving
first order non-stiff systems of ordinary differential equations.
2.1.7 Count Conversion for Nonscanner
2.1.7.A Nonscanner Total Channel
In the active cavity radiometers, the electronics control the amount of
heat added to the active cavity to maintain the temperature of the cavity
about half a degree above the reference cavity. The control of the heat input
to the active cavity is inversely proportional to the incident radiation en-
tering through the field of view and precision aperture from a given source.
In other words, the control system provides more heat input for the weak incident
radiation and less heat input for the strong incident radiation. Thus, the
voltage driving the active cavity heater provides the sensor output proportional
to the incident radiation. The voltage reading, after A/D conversion, provides
the sensor output in counts. The sensor output in counts is then converted into
engineering units. The algorithms for the count conversion are of the gain/offset
type, and are derived based on the sampled instrument data, e.g., sensor response
(in counts), HK temperatures, etc. and the instruments radiative and conductive
interactions.
36
The following is the count conversion equation for NS total channels.
0 ¸ C,
(40)
The constants in Eq. 40 are defined as follows
B Ka a
A --v B 1
4 3
k_AE,F Dlk TkO
DI8 3
-
DI6 3
A F = - 4 -_i T6o
AI2 bE n Dlk 4" -
2V B
4R B M I B 1
where
V = instrument output (in volts)
(41)
(42)
(43)
(44)
(45)
TH = measured heat sink temperature (OK) at time t
TA = measured precision aperture temperature (OK) at time t
TF = measured FOV limlter temperature (OK) at time t
The remaining quantities, i.e., Ba, Ka, BI, T, Dik, AI2 and bE are con-
stants which are specified for each instrument. These constants for the count
donversion equations are determined for the simulation study of the instrument
model using ground and inflight calibration data. These parameters or constauts
37
may be altered at various points in time, usually after a f] ight calibration,
in order to reflect any changes in the instrument characteristics.
2.1.7.B Nonscanner Filtered Channel for Shortwave
For the filtered channels, the count conversion has a set of two equations
for short and longwaves as follows:
all Es + a12 AT9 = 81 (46)
a21 _s + a22 AT9 = 82 (47)
The coefficients of the terms in the above equations are given by the
following equations,
all ffiBls - BI£(48)
3
a12 = 4DI, 9 T9 (49)O
a21 = B9£ - B9s (50)
a22 = A9, 4 + A9, 8 - 4D9, 9 T_
O 2 ns 23 VB 8r
ns 4 VB ) - 4(k_'6 Dik Tko 16_ M IBI = (AI2 bE - k_ I Dil Tk ° 4R B M 1 - )
(51)
• 33 (TA TA ) _ 4D16 T6 (TF. - TF )* (TH - TH ) - _DI8 T8
0 0 0 0 0
2- B K v -a a s Bie (52)
n! 3ns 3 + A98(TA T9 ) + Ad8(TH T9 ) + 4(k_4 8,9 D9k Tk )_i ffi k--E1D9k Tk - -O O O O
3 3
, (TH - TH ) + 4D98 T 8 (TA - TA ) - 4D16 T6 (TF - T F ) + BI£ E0 0 0 O 0
(53)
38
The filtered channel count converstion is obtained from the solution of
two simultaneous equations (Eqs. 46 and 47). Thus the solution to the equations
is given by
= is all (_22 - a12 a21 (a22 Bl - a12 82) (54)
T9 = I (- a21 81 + all B2) . (55)all e22 - _12 _21
The longwave incident radiation estimate is then defined by subtracting
the shortwave incident radiation from the total incident radiation estimate.
That is,
The constants in Eqs. 46 - 56 are
v = shortwave channel output (in volts)
^ m2E ffishortwave incident radiation estimate (W/ )s
E£ = longwave incident radiation estimate (W/m 2)
= total incident radiation estimate (W/n$)
The temperatures of the heat sink, aperture and FOV limiter, TH, TA, and TF,
are measured as before in total NS channel.
39
,
2.1.8 Solar MonitorModel Analysis
The solar monitor is different from tile nonscanner in size, shape and
design feature as shown in Figure 6. It has a barrel structure ~ 9.5 cm long
between the primary and secondary apertures. The tail of the core body of the
instrument is attached to the housing and includes the primary and secondary
apertures, barrel, heat sink and active and reference cavities. This instru-
ment has a shutter located in front of the secondary aperture. The shutter
operates on a 64 second "on and off" cycle in which it is open for 32 seconds
and then closed for 32 seconds.
Since this instrument deals with a strong intensity of the incident
radiant energy and does not have the heat sink temperature controller, the
heat sink has a larger heat capacity for damping out the effect of large in-
put temperature gradients.
llowever, Jt is desirable to check whether there is any transient effect
due to the periodical on and off mode of the shutter, the strong intensity of
solar radiation, and the excessively long barrel geometry. In other words,
it is desirable to check whether the steady-state approximation is appropriate
to develop the count conversion algorithm for the solar monitor. Thus, we
introduce two simplified geometries here to check how fast the temperature
reaches a stable condition, as every 32 seconds the solar radiation is
viewed.
First, let us assume that the shutter has a rectangular shape of thin
aluminum plate and the shutter is partly exposed to the incident solar radiation
for a 32 second interval of the full 64 second cycle and has a radiation inter-
action with the isothermal neighboring components at any time. Figure 13 shows
4O
the shutter model.
£
!
I
II
I
!
!i
FLGURE i3. The Shutter Model
It is assumed that the ]eft end of tile shutter is insulated and the right
end is attached to a temperature reservoir which has a large heat capacity.
Then the energy balance equation, the initial and boundary conditions govern-
ing the above model can be written as
Dr k _2T _" (57)G.E . PC _ " = pC Bx---_+ 6 -
I.C . T('r = o) = 'rO
B.C • Tx(X = 0, 3) = 0
Tx(X = L, 3) = Tb
The above equation with Neumann and Dirichlet boundary conditions can be
easily solved by introducing a new variable and separating the variables.
Thus the solution to the equation is described by
(-1) n -aAn2T- e
T(x, z) = T b + 2(To-T b) n=O (%nL)
._L2 {+-2k_- [I - (L)2] -4 nZ0
COS(Xnx)
(-1) n -aXn2T }(XnL)_ e COS(Xnx) (58)
41
where T is an initial temperature set by 273.16 OK, L the total length of theO
shutter (6.5 cm), 6 the thickness of the shutter (0.2 cm), k the thermal con-
ductivity (1.557 w/cm°K), _ the exposure length to solar radiation (2.5 cm),
0 the density of the shutter (2.713 g/cm3), c the specific heat of the
shutter (0.887 J/g°K), a the thermal wave velocity (0.647 cm2/sec)
n 2L n = 0, i, 2, ...
"l!
qs is the solar flux (0.142 w/cm 2) x 0.2
= F c O [T__ - T4 ]qo_
'I!
q" = qs'"u(%-x) u(32-z) + qs u(_-x) u(z-64) u(96-T) ... + 2 4"
F the configuration factor between the shutter and the neighboring
components (0.85)
the emissivity of the shutter surface (0.8)
o = the Stephan-Boltzmann constant
To= = the ambient temperature (use T = Tb)
Tb = the reservoir temperature set by 273°K,
T = current shutter temperature, but use T(T-I)
u(x) = the unit step function.
The graphical description of the solution is shown in Figures 14 and 15.
Figure 14 shows that the shutter temperature not only responds to the on and off
mode of the solar flux, but the peak temperatures as well as the minimum temp-
eratures of each cycle (64 sec) also stay at the same ranges. That is, the time
averaged temperature of the shutter does not change sifnificantly. And Figure
14 also shows that the temperatures before reaching minimum and maximum points
become stable. Figure 15 shows the temperatures along the length of the shutter
after i00 seconds passed.
Secondly, the long barrel which is placed in the housing and is between
the pr£mary and secondary apertures is considered by the followlng assumptions:
i) no direct exposure to the solar fl_x
42
t_S
_rm
_a
S
I I I I I I I I I I
TIME( 8EC.t FOR X = 1.5 CH
FIGURE 14. SHUTTER TEMPERATURE AT X ,, 1.5 CM WITH 20% ABSORPTIONt*
OF SOLAR FLUX Qs = 0.142 W/CM 2.
43
I
GOLLI
w
ta.J
b--
o
I I I I I I•6 1.0 ! .5 2.0 2.8 3.0
XICYL_ FORTIME = 100 SEC
FIGURE 15. SHUTTER TEMPERATURE AT I00 SECONDS WITH 20% ABSORPTION
OF SOLAR FLUX Qs = 0.142 W/CM 2.
44
2) radially uniform temperature since the wall thickness is thin
3) no circumferential temperature gradient
4) radiation exchanges only at inside and outside walls
The configuration of the model is shown in Figure 16.
ell
-i...... ..... .....
0 x L --,
FIGURE 16. A SIMPLIFIED CONFIGURATION OF SOLAR MONITOR BARREL
The equations and boundary conditions governing the thermal conditions
imposed upon the above geometry can be written as
aT a2T
G.E. pc-_ = k_-2x + Ro_R i
I.C. T(T=0) = Tb
(59)
B.C. Tx(X=O) = q_'" (set to be constant)
T(x=L) = Tb .
Then the solution to the equation Is obtained as
T ,= T b + 2k(Ro_Rt ) 1 - (L) 2 - _!" L (1 - _)
2k(Ro-R i) el:: 1__.z.)._ _"(L),n) _ ..... L (L_n) CoB (_,nx)l / (60)
45
where
{_11 " |I _ " II= qi qo
43 = FI_IeI_[(T4(x , t-5) - T4(x, t-l)) l
qi'"= ['"-qs "'q_[£2FI-2
q" = Fl_3e3_[T4av(in x) -T4(x=0, t-l)][u(64-T) u(T-32) + u(128-T) u(T-96) + ...]
Tb = 273.16 OK
L -- I0 cm
k -- 1.557 w/cm OK
a = 0.647 cm2/sec
Ro-Ri = 0.5 cm
An = (2n+l_ 7r2L
_s = 0.146 [u(32-T)+ u(T-64)u(96-T)+ ...]
FI_ 1 = 0.75
eI = 0.8
FI_ 2 = 0.4
e2 = 0.08
FI_ 3 = 0.4
e 3 = 0.5
The above values are used in both the shutter and barrel cases. These estimates
were selected to best represent the solor monitor based on the material and
thermal properties of the shutter and barrel. The results of the calculations
described in Figures 17 and 18 demonstrate that the temperatures do not change
very much with respect to time or in the axial direction of the barrel since
they exhibit only about a half degree of temperature change in both cases.
46
FIGURE 17. BARREL TEMPERATURE WRT TIME AT X=3 CM FROM THE SECONDARYAPERTURE.
47
m
Le75.207_
27q.O_r_ _
0
I I I I I I I I I I I I I.77 l._a e.3o s.o7 s.M q.eo s._ s.t_ s.st 7.s_ s.q4 s.et s.y7
XT(CHt FORTIME = 200 SEC
FIGURE 18. BARREL TEMPERATURE ALONG THE AXIAL DIRECTION AT THE TIME OF 200
SECONDS.
48
Therefore, these results show that the steady-state approximation for the
count conversion algorithm does not contain any crucial errors. Therefore,
the open and closed modesof the shutter were considered in the algorithm
development. Namely, the output responds to the direct radiation which falls
into the active cavity according to the shutter's on and off modes. Even
though the instrument has a small time constant and hence it is stabilized
quickly, within the 32 seconds of a half cycle, the output data which is used
for the conversion algorithm has to be picked up at the near-end of a half
cycle just before the shutter closes the aperture.
49
2.1.9 Solor Monitor Count Conversion Al_orlthm
The approximations made here including the steady-state approximation are
the following:
i. The nodal points are selected to be adjacent to where the temperature
probes are so that the temperatures measured by the probes may be
regarded as the temperature of the nodes.
2. T 1 = T 2 + b E where b E is the electronical bias.
3. The temperature of the reference cavity is the same as that of the
heat sink, that is, T 2 = T 3. Accordingly, Tlo = T3o + b E where the
subscript o signifies the initial temperature.
4. T 1 - Tlo can be replaced by T 3 - T3o.
With the above approximations, the equation for count conversion was derived as
v 3 3 - T3o 5 (t) - TSo
+ A 7 [T7(t) - T7o ] + B (61)
where T3, T5, and T 7 are temperatures in OK which are measured in the instrument.
These temperatures are measured and telemetered every 16 seconds. If these
temperatures are measured twice during the open period of the shutter, then the
second measured temperatures can be used for the count conversion algorithm.
Otherwise, as the temperatures are measured arbitrarily without regard to the
shutter on-and-off mode, the temperature measured at or near the time before the
shutter closure must be used. As well as the temperatures measured, the count
has to be picked up at every 40th data point or near the time before the shutter
closure. V(t) in the equation is the instrument output in volts and is tele-
metered at every 0.8 second intervals. The constants in the equations are
obtained from the thermal control electronics and conductive and radiative
heat transfer, all of which interact with the active cavity and the nodes
50
which have temperature probes. In the equation,
Av'- BI
[ 434 DI_IT31o + DI_2T_o + DI_3T3o + DI-4T o ÷ DI-6T6oA3 " - B-_
+ DI_ISTI8 o + DI_x9T319o
---[ + D1-6T6 3 ]4 DI_5T3o oA 5 - BI
= _--- DI_I4TI4 oA7 BI 1-7T o + DI-8T8° DI-9T9° + DI-IIT_I°+ 3
3 + DI_I7T37o]+ DI-15TI56
B _ bE Ek Dlk T4 _ E= BI - BI ko - BI - B_"
(62)
(63)
(64)
(65)
(66)
the quantities such as Ba, Bj, Ka, Kj, V, BI, T, Dik, Tk , AI3 and b E are theo
constants to be specified for the instrument.
2.1.10 Scanner Control Electronics
The control electronics for the scanner have a different feature from
that of the nonscanner. The difference between them is attributed to the
type of sensor in the scanner. The scanner uses thermister blolometers for
both active and reference flakes. When the temperature of the thermister
varies due to the energy exchange, the current through the thermister varies
with the change of the temperature-dependent thermister resistance. In the
scanner instrument, the active flake that is attached to the heat sink in-
directly responds to the incident radiation from the source through its op-
tical arrangement. The reference flake is also attached to the heat sink
and is placed where the space is totally sealed by the surrounding materials
such as the heat sink and housing. Thus it is assumed that the reference flake's
temperarture is equal to that of the heat sink and its housing. Accordingly,
the observed difference in resistances between the active and reference flakes,
is proportional to the incoming radiation.
51
Figure 19 is the block diagram for the scanner control electronics and
their transfer functions.
The thermister resistance as a function of temperature is given by
RI(T ) ffiRi(To) EXP 8i(_i - _) , i = I, 3 (67)O
Using the polynominal expansion of the exponential in Eq. 67, the difference
in resistances between the active and reference flakes is approximated as
Rlo _l R 8 81 833o 3 RIo - R3o
RI(T I) -R3(T 3) ffi_ TI + T_ ° T3 + To(68)
The constant in the bridge is defined by
2VB k_l + k)
K B =-_-- (f + 2k) 2O
The transfer function of the bridge is transformed into a differential
equation as shown in the following
(69)
TI _i + Xl = KB(RI - R3)(70)
where T 1 is a time constant which, for the case of the scanner is less than
I0 ms.
For the preamplifier, we end up with the equation
T2 x2 + x2 = Xl + T3 Xl - Kc CB + (Eos + n) + T3(Eos + *_) (71)
Thus, the following equations describe the energy balance equation (Eq. 72) and
the differential forms of the transfer functions, (Eq. 73) which include general-
ized forms of the amplifier and filter and a model of each scanner channel
= AT + DT 4 + BQ (_8 + BE (72)
52
,,-41
0
i,--I
"1"
÷÷
0
v
i i i i ! I I I i I I I I ! i I I I I
°1°•t" ÷
Co mmml_m
V
I
4-
0
r._
o
,.-4o
l,J
ou
u.r4
0
I.-4 ,sJr._ U
U
!,,.-4
v
m imm _lm m _ mmmo _Qo_m e i_m°|m_m !
53
_: = A x + BT + B (E + n) + B CB + b (73)X OS OS C
_ere T is a 14 component vector consisting of node temperatures, x is a 6
vector consisting of electrical signal processing variables, Q8 iscomponent
the heater heat flow rate, Eos, CB, and b are electrical and flake related vari-
ables, n is the total noise as viewed at the preamplifier input, and E is the
incoming radiant incidence vector. _le measured quantities which are telemetered
to ground are the raw counts, x6, and the heat sink temperature TT, measured by
a thermister.
Ix61y = + v (74)
T 7
where v is the measurement noise including quantization error due to A/D
conversion.
2.1.11 Scanner Steady-State Analysis
Two types of transient behavior will be no¢iceable in the instruments.
One type occurs when the heater Q8 is activated, i.e., when the instrument
is turned on. The transient behavior in this case is rather slow, as the
heat sink time constant is large and the conductivity of the filters (for
SW and LW instruments) is low. Thus, the time period necessary for the
various node temperatures to reach their relatively steady operating temper-
atures, particularly the filters, will be long. The precise period can be
determined by simulation or measurement.
The second and more important transient behavior is due to quick
variations in the input (or observed) radiation incidence, E, after the opera-
ting condition has been reached. The time constants, in this case, are
determined largely by the electronic signal processing parameters and the
54
flake characteristics. After the operating condition has been reached,
the temperature variations of most nodes should be negligible. The active
flake temperature then varies with the incoming radiation. The flake
resistance and other electrical variables also vary with the incoming radiation
and can produce transient behavior which can be observed if the target level
is changed suddenly, as a step function. On the other hand, if the incoming
radiation varies slowly relative to the system time constants, then the system
will operate in quasi-steady state condition. A preliminary analysis indicates
that, for the parameters selected by the designer, the applicable time constants
are in the vicinity of i0 ms. Thus, if the input radiation does not contain
frequencies above i0 Hz, the transient will be negligible, while frequencies
above 17 Hz, will_be attenauted. Whereas the earth scan may have low content
in the high frequencies (_lich are of less interest to ERBE), the transition
from space to Earth may produce a necessarily short period of transient behavior.f
It may be of interest to analyze the magnitude and duration of such transients,
which probably last less than 50 ms.
Since the quasl-steady state condition is of interest to this analysis,
let us first consider one scan period. The scanner looks at space for a period
"sufficient" for the transient effect to die down and obtains an average of the
last few values (to reduce noise and errors) as the "space clamp". Then the
scanner sees the Earth and atmosphere, then looks at two black bodies at ambient,
but known, temperatures as well as the SWICS in the off condition. The k_t
correspond to the kth space clamp, so that T(kAt) and x(kAt) are the temperature
and electronic vectors at the space clamp measurement. Now let
Ax(t) = x(t) - x(kAt), AT(t) = T(t) - T(kAt), kAt < t <_ (k+l)At. (75)
From Eq. 72 and Eq. 73
55
3_AT+ BQA(_8 + BEAT = AAT + 4D Tkt.
A_ = AxAX + BTAT + Ad + BosAn,
(76)
(77)
where it has been assumed that the thermal and radiative properties are in-
cluded in A, D, BQ and B and the time constants for the electronics in Ax, Bos
and B have remained unchanged over the scan period of 4 seconds• Also we canc
define Ad, the drift component of the voltage as,
= + Ab + BC AC BAd AB T T(kAt) + Bos AEos (78)
Considering the amplifier and Bessel filter to be in a quasl-steady
state condition, it can be seen that
Ax6(t) = Kf A--'_2(t)+ Kf _6(t) (79)O o
Q
where Ax2(t) i's the part of Ax2(t) with lower frequency content (below I0 Hz)
consisting of an averaged part of Ax2(t), which may be considered the signal
and _x6(t) is the part of Ax6(t ) consisting largely of noise and irradiancepart,
variations over short distances at the top of the atmosphere with zero mean.
As the filter attenuates high frequencies, Ax6(t ) will have a low amplitude.
The preamp equations are expressed as
T2 A_2 + Ax2 ffiAXl + T3 A_I + AEos + An + T3(AEos + A6), - Kc ACB (80)
Now assume that
T2 _x 2 - T 3 AEos = 0 (81)
then
_x 2 = A-_1 + T3 A--_1 + AEos - Kc ACB + A-_ (82)
56
where E 1
that AEos and AC B are baseband signals.
is the low frequency portion of AXl, and where it has been assumed
Note that
Ax2 = 7x2 + fx2 (83)
where _x 2 contains the higher frequency components of Ax 2 and causes the term
_x 6. The bridge equations are now expressed as
T I ALl AXl TI ATI TI AT3 + Abl + Tl TI+ ffi BTI 1 + BTI 3 ABTII
+ T3 TI ATI +i AT3TIABTI3 + ABTI I + ABTI 3
(84)
It is assumed that the only variable on the RHS containing higher
frequency components is T I, whereas the remaining variables are slow moving
drift components.
_x I = +I(BTII AT I + BTI 3 AT 3 - Ax I)(85)
Combining Eqs. 85, 82, and 79,
_-_2 ffi _-@i AT3) + (+3+I(BTII + BTI 3
+Eos
- +i) +2
- K AC B + Anc(86)
Ax6 Kf ° Tl _i + AT3)= (BTI I BTI3
+ Kf [(T3 - +i ) A--Xl- +2 _2 + Agoso
- K ACB +_'n+ _x6]C
Now consider the thermal equation
N 3 AT k + BI EAT I = AI2(AT2 - AT I) + 4 k_ I Dik T k
N 3 AT k + _IB1E ffi A12(AT 1 - AT2) - 4 k_ 1 Dlk T k
(87)
(88)
(89)
57
1 i
AT1 = Kf T 1 "_x6 AT3 T 1 _Io BTII BTII BTII
(90)
.... K AC B + A_ + A_ 661 (T3 TI) _i T2 _2 + AEos c (91)
Substituting Eq. 88, 90, and 91 into Eq. 89, we have
BIE = (AI2 - 4 DI1 T_)TI Ax 6 - AI2 AT 2 BTI3 (AI2 - 4 DII T_)AT 3
Kf ° BTII BTII
N 3 AT k + A+ I - (AI2 - 4 DI1 T_)
- k_ 2 4 Dik Tk TI BTII 61 (92)
2.1.12 Scanner Count Conversion
The scanner count conversion algorithm below Is based on steady state
approximation. The structure (i.e., the form of the equations) of the algorithm
is the same for each of the three scanner channels. The parameters have
different values for each channel.
The average of scan points during space clamp is given below
V(tk ) 1 n tki)= n i=Zl v( ' nffi 8 (93)
The mean variance of the counts compared with the space clamp during a scan
cycle is obtained by the equation
2 1 n [V(tk) V(tk)]2Ok ffi--ni=_l (94)
The scanner count conversion equation is then defined by the following
equations
t - tk
At
tk. I _ t • tk (95)
58
The coefficients in Eq. 95 are the following
3
AI2 - 4 DII T1 VBo
Av = K Kf zI BTI1 VB(t)o
14 Fl_ 3+ 3+ 3+ 3 3 3
AH = - k _-11 T1 DI2 T2 DI3 T3 DI4 T4 + DIS T5 + DI6 T6
3 + 3 ]- Al2(1+ _) I+ DIST8 DII3TI3J
BTII
A = c
3
AI2 - 4 DII T 1 VBo
K T I BTI I VB(t ) ffic Kfo Av
0
T 1
- _-- (BTII T1o
+BTI3 T3)[VB(tk)- VB(tk_l)]:
+ Kd[Vd(tk)- Vd(tk_l) ]
(96)
(97)
(98)
(99)
tk = tk_ I + At (100)
In the last term of Eq. 95, (t-tk)/At can be replaced by the number of measure-
ments in the scan mode as follows:
t - tk= 0.0135 (5-8) (I01)
At
where J is the number of measurements in the sca_ mode. The constants in the
above equation are the following:
_(tk) = space clamp value at time tk
V(tki ) ffiinstrument output (in volts) when viewing space at tk
2Ok = noise variance estimate during space look
59
v(t) = instrument output sample at time t
TH(t ) = heat sink temperature measurementat t or most recent value (OK)
At = total scan period (4 sec.)
t k = time at end of space look (see.)
t = sampling instant (sec.)
VB(t ) = detector bridge bias voltage measurement at time t or most recent
value (v)
Vd(t k) = drift balance DAC voltage measurement at time t or most recent
value (v)
6k = estimate of unaccounted drift during kth scan period (V)
T = average time lag (sec.)
kf = post amplification gaino
Note that to compute 6k, it is necessary to have the following space clamp
value; i.e., _(tk). If the term with _k in Eq. 95 is neglected, then the count
conversion may be done without using V(tk). The variance o_ is computed for
diagnostic purposes and should be output as described below. The converted
value E.(t - T), i = s,£, T (for shortwave, longwave, and total, respectively)i
can be interpreted as follows
EiCt) = / _i(X) ExCt) d_ + ei(t) (lO2)
th
where Ti(%)__ is the transmittance of the i channel, E%(t) is the spectral
irradlance at the instrument due to radiation from the footprint at time t]
60
and e.(t) is the error in the measurementcombinedwith the count conversioni
process, and can be estimated through analysis of calibration data and model
results.
2.1.13 Accuracy and Diagnostic Checks
During each scan, the instruments view two black bodies at ambient, but
known temperatures. These data can be used to determine if the basic operation
of the instruments have changed as well as provide an approximate accuracy check
of the combined error el(t) in Eq. i01.
i. Store a table of elements
(VBs(TB) , VB_(TB) , VBT(TB) , EBs(TB) , EB_(TB) , EBT(TB)) where T B = TBj,
i _ j K N. The values VBi(T B) represent the steady state output of the ith
channel when viewing the black body at temperature TB, and may be obtained in
calibration tests. The values EBi(T B) represent the output of count conversion
when viewing the black body at temperature TB, given by
EBi(TB) = fTi(X ) EBX(TB) dX (lo3)
where EBx(T B) is the spectral irradiance from the black body at temperature T B.
2. Compute EBi using the measured black body temperature TB(t) as
TB(t)- TBI [EBi(TB2) - EBi(TBI) ] (104)EBi(TB(t), -- EBi + TB2 TBI
where TBI £ TB(t ) _ TB2
3. Compute E.(t) using the count conversion algorithm with the correspondingI
value v.(t).I
4. Plot vi(t) and VBi(TB(t)) on same axes.
61
6. Compute and plot 0_, and
evi,k+l = avi k + (i - a)(vl(t ) - VBi(TB(t)) )
ei,k+l = a eik + (i - a)(Ei(t ) - EBI(TB(t)) )
2 02 [ 1] 2Ox.z,k+l = a vi k + (i - a) (vi(t) - VBi(TB(t)) - _i,k +
m,k+l = a Oik + (i - a) (t) - EBi(TB(t)) - el,k+ 1
(1o5)
(I06)
(107)
(I08)
62
3. INSTRUMENT PERFORMANCE AND COUNT CONVERSION TEST
The instrument models developed for the non-scanner, the scanner, and
the solar monitor are based upon the theoretical approach with a certain
degree of approximation. For instance, the energy equations for the instru-
ments were formulated with the assumptions that the node (or control volume)
has uniform temperature and homogeneous thermal and radiative properties.
However, this lumped geometry of the node, in reality, cannot hold the
uniform temperature through the node geometry unless the temperatures of the
neighbol nodes are tile same, or the node boundaries are perfectly insulated,
or the thermal conductivity of the node material is infinity. Otherwise,
if there is any heat flux allowed into or from the node, a temperature grad-
ient from the center of node to tile boundary will exist.
The radiative properties that are used in the ERBE radiometer instrument
models may be the values picked out of the handbook or measured and averaged
values over a certain band of wavelength. No matter where these values were
obtained, they would introduce some errors when they are used for the compu-
tation of the radiation exchange between the surfaces. The spectral distri-
bution of the emissive energy flux from a source of radiation such as MRBB
may lie on the limited range of wavelength in which the averaged radiative
properties used may be far above or below the real values within the spectral
energy distribution. Therefore, the averaged values may not represent
reality in instrument performance. Also, during long term operation in space,
these properties will be subject to changes due to various reasons such as
thermal and environmental contamination, and aging effect.
The electronic systems are used in the radiometer to measure the temper-
ature difference between the active and reference cavities or flakes, and to
63
digitize the signal, then to transmit it through a telemetry system. The
parts such as resistors of this electronic system are sensitive to a wide
range of temperature changes. For instance, if the resistance of the bridge
slightly changes from the reference point, then the error in Irradiance
measurementdue to this temperature change maybe in error by a few watts
per square meter. Although the instrument models are well-conceptualized and
designed, the real behavior of the radiometers are, as expected, different
from what is was designed to be.
Other factors to be considered, which influence the sensor performance
and measurementaccuracy, are the inflight changes from the states that the
sensor normally operates in and the calibration source variability in its
operation modeor the changes in source characteristics. An environmental
change was expected in the long-term operation of the instrument due to the
orbital changeswhich caused cyclic variations in the heat load of the in-
strument. Another change, was due to the contamination or degradation of
then senosr surfaces due to radiative interactions with other surfaces.
The internal calibration sources such as MRBBand SWICSare also sub-
ject to change. The monitoring of the change in intensity of these sources
by possible variation in power supply and shift of the spectral distribu-
tion in the source output is possible by using the SiPD, although it is also
subject to change, and intercomparing calibration sources.
To account for the possible errors due to the reasons discussed above
a procedure was developed to detect those errors. _,e procedure uses the
algorithm shown in Figs. 20 and 21 for that purpose. This algorithm is
capable of evaluating the sensor model, adjusting the simulation model para-
meters for minimum errors, and establishing confidence intervals for the
64
I|
ll!
|
l
I"
L_I,-I
ORIGINAl] PKGE IS
OF POOR QUALITY.
65
4_
.|
00000 _i
IIIII
III
__ I __ I----_
I
- _N
66
converted data reflecting sensor characteristics and calibration sources.
The sensor simulation model responses, using known calibration sources
as inputs to the model, are checked against the instrument calibration out-
put. Any differences in the slope, offset and curvature between the simula-
tion and calibration results were interpreted as uncertainties of the model
simulating the instrument. These differences, however, generally reflect
the problems discussed earlier. In the algorithm, these differences are
narrowed by checking and adjusting the sensor model, source, and environmental
parameters to minimize the errors. However, the estimated parameters must
stay within the tolerance that can be allowed by the approximations made in
the modeling or by the nature of reasons described earlier.
The algorithm consists of three parts (see Fig. 20):
The first part is a routine to adjust the sensor model parameters by com-
paring sensor simulation outputs such as counts and HK temperatures to the
infllght and ground calibrations obtained for the known sources.
The second part is a procedure to check the errors generated by simulating
a sensor by comparing them to the calibration data for the known sources. It
checks the error generated from the sensor simulation target source, or envrion-
ment at different time intervals for the same known irradlance source as in
the infllght calibration.
The third part consists of routines to search for any changes in the target,
environment and sensor performance due to the degradation of the instrument
surface, vibration, contaminations, and thermal and aging effects on the radia-
tion interactive surfaces.
To do the above job, the algorithm has been designed to have the following
capabilities:
67
i. to adjust the sensor model parameters in comparison with the
calibration data from known sources to minimize the error be-
tween the sensor model and the calibration data,
2. to periodically check any changes in sensor, target and environment
parameters between inflight calibrations by comparing the calibration
data for the same sources at different time intervals (e.g., i or 2
week perlods),
3. to re-define the sensor, target and environment parameters, if
they have been changed,
4. to test and check any differences between calibration data and
responses from the simulated sensor model for the same known
sources,
5. to repeat the routines i through 4 until the errors produced by
step 4 are minimized,
6. to generate the count conversion parameters.
3.1 Description of Routines
3.1.i Routine I
The calibration sources identified during either ground or infllght cali-
brations are numbered by j = i, N at time ti and ti+ 1 where i = I, n. The sensor
simulation model generates temperatures and sensor responses corresponding to the
calibration sources at a time ti. These temperatures and sensor responses, which
are obtained from the model simulation, are compared with measured data to com-
pute errors of the simulation model. Once these errors fall within the pre-
determined error bounds (after repetitive refinement of simulation parameters),
then these errors will be used as criteria to determine the measurement errors
68
against knowntargets on the ground. If not, the parameters for both the tar-
get source and environment effects will be checked; then in turn tile sensor
parameters will be checked and adjusted, _n case when the computedparameters
are not within the acceptable range based on the materials properties, e.g. its
optical properties. Sucha routine will inlcude a decision block to check targets
and environmental effects to avoid unnecessary adjustments of sensor parameters
whenthe errors maybe caused by and comefrom tile target and environment para-
meters.
3.1.2 Routine II
If the errors are within bounds, the sensor responses Vj, j = i, N at dif-
ferent times ti and ti+ 1 will be sorted and evaluated in Routine I to check the
errors from the sensor simulation model for the targets at different times t i
and ti+ 1 .
3.1.3 Routine III
After finishing Routine II, using all the target sources at different times,
ti and ti+l, Routine III checks the sensor respones of the same target at a dif-
ferent time and computes the errors. If the error of the same target at a dif-
ferent time falls within the error bounds, the Routine will check the next tar-
get. If not, the history of target source variation will be studied, and at
the same time the sensor and environmental conditions will be rechecked to deter-
mine the source of errors. This evaluation in Routine I is performed whenever the
error signals the flag. The process continues until the error source is recog-
nlzed and its correction of error is made by adjusting the related parameters in
the simulation model. When this Routine finishes satisfactorily evaluating all
of the target sources, the program will print all the decisions, errors, sensor
69
and calibration source IlK data, sensor parameters and counts and will supply the
final results to the count conversion algorithm.
3.1.4 Routine IV
If the error does not fall within the prescribed error bounds, the target
and environment parameters are checked in order to find whether changes in these
parameters created pseudo-errors in the sensor responses. If there has been a
change in either the target or environmental parameters, they are re-defined
and re-input to Routine I. In spite of repeated checks, if the error cannot be
minimized, then the thorough review of instruments will be required to determine
whether the error bounds are to be reassessed. In this routine, when the target
and environment are checked and relatively free from the erroneous performance
then we would postulate that the instrument's measurement capability might be
deteriorated due partially to the thermal and environmental contamination or by
other means such as aging of the parts and so on.
70
4. DISCUSSION OF PARAMETER ESTIMATION
The simulation code was developed to study the sensor instrument responses
and behavior during the ground and inflight calibration procedures as described
earlier in Section 2.1.6. This simulation code uses the irradiance at the FOV
limiter of the instrument as the input data. All the constants representing
the instrument geometry, material, thermal and radiative properties and the
electronic thermal control systems are parameterized to be used as coefficients
in the system of equations. Then the simulation code generates the counts and
the nodal temperatures of the model corresponding to tile _nput irradiance.
When the simulation code uses the input irradiance which is the same as the
calibration source, the results of counts and housekeeping temperaures from the
simulation are compared to the calibation results. If the differences from these
comparisons for various levels of irradlance are wide, then the simulation model
parameters are adjusted to minimize those differences. This procedure for esti-
mating the model parameters is repeated until the differences can no longer be
reduced. These estimations also continue for various calibration sources until
systematic response patterns for each calibration source are established. All
procedures mentioned above are performed for each individual radiometer sensor.
Even though two instruments of the same type are used in the calibration pro-
cedure, for example, the results may differ due to the inherent differences in
the manyfactors invoZved, e.g. slight differences in the sensor assembly process
of uniform surface roughness, coatings, glue or contact areas.
The following is an example that shows how the parameters for the nonscanner
total channel are selected. Assume that the instrument output derived from
calibration input irradiamces are linear; then the factors which determine the
slope and offset of the cur_e for simulated results would be judged to be the
71
offset temperature between the active and reference cavities, and tile radiative
constant for tlle term of irradlance in Eq. 34, i.e. BI. To determine these two
factors using the measurementdata, the linear form of Eq. 34 can be written
=x -- 8 (I09)
where x is a column vector consisting of the offset temperature and radiative
constantj and a and 8 are defined later• The solution to Eq. 108 is
x = (a' _)-i a' 8
where _' is the transpose of =•
The constant a is a matrix defined by
alj = AI2 + 4 DII(T2) I, E 1
3 E2AI2 + 4 DII(T2) 2,
AI2 + 4 DII(T3)i , Ei
(lZO)
where i is the calibration sequence•
The constant 8 is a column vector
Bi = S1
B2
sll
72
8 as an explicit form is
_i = - Ba Ka V2i - _i - BJ1 QJI
4
_i = kZ Dlk Tki '
(in)
only when k = i, replace Tli by T2i . (112)
The nominal values are used for unmeasured Tki and QJI except T3 = T2 + ST.
The temperature of the reference cavity is a little higher than the heat sink
temperature due to the Joule heating by electric current flowing probe wire
of reference cavity. The temperature difference, _T, due to the Joule heating
cannot be directly measured but can be observed from the simulation results.
The plots of the calibration and simulation results show that they are
not actually linear but could be considered to be linear for estimation (see
for instance, the steady-state response comparison of simulation of calibration
data for the NS WFOVSW sensor on page 100 in Appendix A). Therefore, the above
approach can still adjust the overall slope and offset of the curve by regarding
them as a linear. The linearity and adjustment can be made by adjusting other
parameters. But since almost all parameters are interrelated, the selection
of a parameter that mostly influences the linearity of the curve in a plot of
simulation results must be cautiously taken by testing the linearity with
various parameters•
Since the simulation code was developed, the calibration data for the
nonscanner radiometers using MRBB as a calibration source was the only source
available. Hence, the nonscanner instrument models have been simulated using
MRBB data which was used in the calibration. Figure 22 shows the calibration
data for the MRBB temperatures in the ist column and the count outputs for
the four nonscanner channels in the 6th through 9th columns. The 2nd columm
shows the blackbody radiation corresponding to the temperatures in the ist
73
0_Jv
r.J
t,J
i--i
I
N
_ •
0 I_ C) O_ C) _1 0 '._" ,-I I_ ,'-I I O_ l"_l ,-I
0 (_ C) ,-I 0 ¢'_ 0 "<t 0 u_ (_ ! ,,.0 0 O00 0 0 0 0 0 0 0 0 0 0 0 0
I
O0 U'_ 0 e5 O0 O_ O_
o_ d g d d ,_ ,_
u_
la
r.J
O_f.,.l
6o ._
t_0
_8ZI-I
0
,-I
0
_4
I-4
74
column. The 3rd and 4th columns show the percentages for the shortwave and
longwave portions of the blackbody radiation described in the 5th column,
respectively. These shortwave and longwave portions were calculated by the
integration of Planck's equation for blackbody radiation with respect to the
wavelength from 0 to 5 _m for shortwave and from 5 Mm to i000 _m for longwave.
These portions were used for the MFOV and WFOV shortwave instrument simulations.
The shortwave range selection of 0 to 5 _m was simply made by the shortwave
filter (suprasil) characteristics. In the 6th through 9th columns, the numbers
in the bottom part of the blaock signify the output (in counts) corresponding
to MRBB radiation from the simulation. The 10th and llth columns show the
differences in count between calibration and simulation tests.
75
REFERENCES
i. Barkstrom, B. R., and J. B. Hall, Jr., Earth Radiation Budget Experiment
(ERBE): An Overview Journal of Energy, Vol. 6, No. 2, March - April, 1982,
pp. 141-146.
2. Sparrow, E. M., and R. D. Cess, Radiation Heat Transfer, Hemisphere
Publishing Corporation, 1978.
3. Rohsenow, W. M., Developments in Heat Transfer, The MIT Press, 1964.
76
77
APPENDIX A
PRELIMINARY RESULTS ON ERBE NON-SCANNER INSTRUMENT
MODELING, COUNT CONVERSION AND SENSITIVITY ANALYSIS
, _EC_I, ING PAGE BLANK _O_ __
78,_-. i.,_,_ _L;,NK NOT _$I_11"_1_
GOALS
• DEVELOPA DYrlAMICSENSORMODELFOR EACHINSTRUMENT
• DEVELOPAN ADAPTIVECOUNTCONVERSIONALGORITHMTHAT
CANACCOMMODATECHANGINGCO!IDITIORS
• DETERMIrETHEACCURACYOF THE ERBE INSTRUMENTS
o DETEP_INETHEACCURACYOF THE ERBEMEASUREMENTS
80 .....tNG PAGE BLANK NoT _"II4fED
APPROACHTO ASSESSMENTOF INSTRUMENTACCURACY
• DEVELOPA MATHEMATICALMODEL AND COMPUTERSIMULATIONFOR EACH INSTRUMENT
• COMPAREGROUND CALIBRATIONDATA TO CORRESPONDINGSIMULATIONDATA
• MINIMIZEDIFFERENCEBY ADJUSTINGMODEL PARAMETERS
• PERFORMA SENSITIVITYANALYSIS
• ESTIMATECONFIDENCEINTERVALSFROM GROUNDCALIBRATIONDATA
• REPEATSAME PROCEDUREFOR FLIGHTCALIBRATIONDATA (ERRORANALYSIS)
• ADJUST ESTIMATEOF CONFIDENCEINTERVALSAT EACH FLIGHTCALIBRATION
81
APPROACHTO ADAPTIVECOUNTCONVERSION
• DEVELOPA MATHEMATICALMODEL AND COMPUTERSIMULATIONFOR EACH INSTRUMENT
• COMPAREGROUND CALIBRATIONDATA TO CORRESPONDINGSIMULATIONDATA
• MINIMIZEDIFFERENCEBY FINE TUNINGMODEL PARAMETERS
• DEVELOPA COUNT CONVERSIONALGORITHM(CCA)BASEDON THE MODEL
• SIMULATETHE IrlSTRUMENTAND CCA
• ESTIMATEERROR DUE TO CCA
• REPEATPROCEDUREFOR EACH FLIGHTCALIBRATION
• FINE TUNE MODEL AND CCA PARAMETERS
• OBTAINNEW ERROR ESTIMATES
82
NON-SCANNER MFOV TOTAl.
p
H 2
u
,iV---
"2 P H
1
1 ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 AI HEAT SINK
5 BASE PLATE
6 FOV LIMITER
7 AI HEAT SINK
8 APERTURE
H HEATER
P TEMPERATURE PROBE
83
NO N- SCANNER WFOV SW
m
/7e/,I
-, "--I
r
5
m
4
H 2
I
1"I!
-bm
• 2 P H
7
t
I
1 ACTIVE CAVITY
2 Cu HEAT SINK
3 REFERENCE CAVITY
4 A! HEAT SINK
5 BASE PLATE
6 FOV LIMITER
7 AI HEAT SINK
8 APERTURE
9 SW FILTER DOME
H HEATER
P TEMPERATURE PROBE
84
NON-SCANNERDYNAMICRESPONSE
_OV TOTAL SIMULATION
(D 40"6F
_ 40-'H--.,(
40.3w
L_ 40.21
40.1
4(}.0
40.05
0
Q 40.00
W_C
39.95
<
rv, 39.90
ft.
LU 3g.B51--
39.80
59.75
26.15
(._o 26.10
LIJIZ
26.05I,--.<C
_ 26.0013-
Z 25.g5I---
25.90
25.B50
ACTIVE CAVITY
R£FERENCE CAVITY"
APER'nJRE
CU HEAT 5NK
AL HEAT SINK
BOT /4. HEAT SINK
m
m
BASE PLATEFOV UMITER
I5O
I IIO0 150
TIME (SEC)
I200
I I
86
NON-SCANNERDYN_'IICRESPONSE
WFOV TOTAL SIMULATION
fO .60 --
_n,.52--
_.43 --
h,
;".27--
(_ .18
b_0.10
0
ACT C.4,V - REF CAVACT CAV-OJ HEAT
I5O
I [10o 150
riME(SEC)
I i I200 250 300
87
NON-SCANNERDYIIAMICRESPONSE
WFOV TOTAL SIMULATION
g00N
75O
soo
4so
,v_00
150 m
L_
IRRADIAN_GONVIZRT[D COUNTS
7025
667500
m
v 6-_'25I,.-
5975 -
0 56?.5 --
0u_ 5275 --ZLd
0_n,, 1.67 /n,,I.d 0
z0 -1.67 --
0
RADIATION ID_RORZERO UI_
q
VI I
5O IO0
I I I150 200 250
TIME (SEC)
Iau0
88
NON-SCANNERDYNAMICRESPONSE
WI=0V FILTERED SIMULATION
48,2O(.)Q 46.83t_
.,(
_ 44.10o..
_ 42.73
4.1.37
40.00
ACllYE CAVITYCAVITY
APERTURESW RLIF.R
40.O5
,b 40.00Ld
i_ 3g.g5
L__ 3g.90
3g.75
CU HF.AT SINKTOP At. HEAT SINKBOT AL HEAT
26.10
bJ
_- 25.45
24.800
BASE PLATEFOV UMII_R
............................ •,IqI,
I I IIO0 150 2O0
"hUE(sec)
I2.50
I
89
NON-SCANNERDYt_AMICRESPONSE
W'FOV FILTERED SII_JI_TION
oO .6O
Oci.i
I--.2 7
l,...-uJ(n .18L,I.LI..0 .I0
0
ACT CAV - _ CAVAOT CAV,,-OJ lIE,AT SINK
I.50
I, I I100 150 2OO
TIME (SEC)
I I
90
NON-SCANNERDYNAMICRESPONSE
WFOV FILTERED SIMULATION
30N
_253
2200
<_ 15
<n,- lO
m,
w
m
m
/-
IRRADI,_NCECONVER1ED COUNTS
_. 74ooz"-1735O0o"" 7,..,ooI--:D
¢w0
zIll
7100
5.O0
tw 1.670twtl,,",., 0
.Z0 -1.67
m
m
I
D
n
m
m
m
"' -5.oo I_ I I IO _ 100 150 200
"n_E(SEC)
RADIA_OH ERRQRZERO Lf_
I I
91
NON-SCANNERDYPIAMICRESPONSE
HFOV TOTAL SIHULATION
40.6
o 40.5 -
4O.4 --i--
Ld
-.jL_ 40.21--
40.1 ------
40.0--
40,05 --
o 40.00I.,1.I
_ 3g.g5 --I--
I--
3g.l_ ,,--'---------
3g.7.,+--
26.15
0o 26.10Wn,::) 26.05
26.0On
Z 25.g5I--
25.9O
25.850
m
I5O
ACTIVE CAVITYREFERENCE CAVITYkaERTURE
cu HEAT SINKTOP AL HEATBOT AL HEAT SII,IK
B.,I_£ PLAll[FOY UMITER
I I100 150
TIME (SEO)
I I I__,0
92
NON-SCANNERDYNAMICRESPONSE
HFOV TOTAL SIMULATION
0.60Q
.52
.43n,bJn .35
ILlI-- .27I--I.dO9.18b.h0 .I0
0
B
I5O
ACT _V - REF _VACT CAV.-CU HEAT
I I I100 150 _:uu
_ME (sEe)
I I250
93
NON-SCANIIERDYNAMIC RESPONSE
MFOV TOTAL SIMULATION
_°° B-
_C 450 _-_-
ac: 300 _--_., "'_-
CONVERTED COUNTS
co 8200
0o"-"7900
775O
_7600
7450 -
3.33N
II/-3.$31"--I I II
"-5. I I I I0 50 100 150 200
TIME (SEC)
I560
94
NON-SCANNERDYNAMICRESPONSE
MFOV FILTERED SIMULATION
48,20 ¸_
(D" _.83--bJ
F-
,'v 4.4.10ILl£L
LIJ42.73I.-.
4.1.37
40.00 ..,,.
40.05
(..)o 40.00--LaJn.,:D 39.95 --
_1_2_._ --
Ld 39.85 --I--
39.8O --
3g.75
26.10 --
0" 25.88--_Jn-D 25.67 --I--
2.5.4,5
(1.
,.,25.23 --h-
• 24.80i0
I IIO0
I I150
TIME (SEC)
200
ACTIVE CAVITYI_J_NC[ CAVITYAPERTURESW FII.TER
CU HEAT SINKTOP AL HEAT SNKBOT N,. HEAT SiNK
BASE PLATEFOV UMITER
I25O
I3OO
95
NON-SCANNERDYNAMICRESPONSE
MFOV FILTERED SIMULATION
oO .60
,.i .52n,.
i
m
o,.bJ
L_I--' .27I--t.*J
.18J.
0.100
I5O
I I100 150
TIME (SEO)
ACT CAY - REF' CAVACT CAY-OU HEAT
] I I2OO 25O 3OO
96
flOr|-SCANI'IERDYNAMIC RESPONSE
MFOV FILTERED SIMULATION
2o
oL
IRRAOIANCECONVERTED COUNTS
A
6500 --Z-_ 645O --00_'_6400 --
D_- 6.x50F--D0
6300 --(K0(I) 6250 --zi,iv_ 62oo __
:_ 5"°° F" _
T-5.ooll I
0 5O
RADIA'n_ ERRORZERO LINE
I I I I100 150 200 250
TIME (SEC)
I3OO
97
NOrI-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIOHDATA
WFOV TOTAL
0 CALIBRA_ONO MODEL
7000--
68(X_
6600--
OvI--
6200
5200; ---o loo 200 _oo 4oo soo eoo 700 =.,u
MRBB IRRADIANCE W/M 2
98
NON-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONTO CALIBPATIONDATA
WFOV TOTAL
16
12
I---Z
00 4v
.-I
l.dC:)0"_ 0
I
Z0
_-_n,,[]
-12
-16 I I0 1O0 200
/0,\/ \
/ %
/ \/ \
f %
"'0 / _'0
I I I I3OO 4OO 5O0 600
MRBB IRRADIANCE W/M 2
I I700 BOO
99
NON-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONOF CALIBRATIONDATA
WFOV FILTERED
0 CALIBRATION[] MOOEL
7400
7375
8_v 7325
03 72.75
,<.,7250
7225
720O0
I ,I I ,I I I I ,.I,,,loo 200 _oo 4oo r,oo eoo 700 Boo
MRBB IRRADIANCE W/M =
lOO
NON-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIO_IDATA
WFOV FILTERED
(f)I---Z
0_)v
ili,i
C)
I
Z0
npnl
0
16--
12--
B--
4--
0
--4 I. --
-8--
-12 --
-160
%.
0"- '-- ......-¢
I I I I I II O0 200 300 4.00 _ 600
MRBB IRRADIANCE W/M 2
I I700 BOO
i01
NON-SCANNERSTEADYSTATERESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIONDATA
MFOV TOTAL
0 CALIBRA_ON0 MODEL
102
82oo
81oo
80oo
G"7gO0
_:_7700
_76D0
7,400
73OO
720Oo
I I I I I, I I I1 O0 200 300 4.00 500 600 700 BOO
MRBB IRRADIANCE W/M 2
NON-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIONDATA
MFOV TOTAL
fa3l"-Z
0(0V
..JI,mD0_E
I
Z0
• n,m
(J
16
12--
B--
4--
0
--'4
-12 --
-160
o,,, I
\ /"_ 0 /1
\\ ..s _ i/0 -. /
"'0'
I I I I I I ,1 IX) 200 300 400 500 600
MRBBIRRADIANCEW/M=
I!
II
II
Ix l
I/
X /
//
'o'
I I700 800
103
NON-SCANNERSTEADYSTATE RESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIOIIDATA
MFOV FILTERED
0 CALIBRA_OND MODEL
65OO
620C0 1O0 200 3OO ,$00 5OO 6OO
NIRBB IRRADIANCE W/M z
70O 8OO
104
NON-SCANNERSTEADYSTATERESPONSE
COMPARISONOF SIMULATIONTO CALIBRATIONDATA
IvIFOV FILTERED
0
16_
12_
0,
I--Z
0C) 4_
--JW
0"_ o
I
Z0
n,,m
-1_ _
-110
\\
\
I I1 O0 200
%%
b4t .0
01jill
I I I I300 400 500 600
MRBB IRRADIANCE W/M z
I7OO
IBOO
105
45O
."< Z_.5
O: 150
75
NON-SCANNERDYNAHICRESPONSE
WFOV TOTAL SIMULATION
TIME DELAY (LAG) ESTIMATION
RRARANCE.... _ COtWm
_70_, F
.TSJ-A
4925L
"°°ITll
o__-,..,_IA I
o lOO 200
It_IAlIOH W_10 LIIC
400 500 600
"n_E(sEc)
106
NON-SCANNERDYNAMICRESPONSE
I_gOV TOTAL SIMULATION
TIME DELAY (LAG) ESTIMATION
40.6
0o 40.5
_40.4.cO
_3n
40.2
40.1
40.0
m
m
ACTIVE CAVITYREFERENCE CAVI"FYAPERTURE
m I I I!! ill Illlll _ Illl I_ ill.l Ililll _lJllil Ill II i i I i Ii ! i1.1 _ It ._ ..................
40050o 40_bJQC
3g.g5
n, 30.90ILlD.
39.85
39.80
39.75
CU HEAT SINKTOP AL HEAT SINKBOT AL HEAT SINK
BAS_ PLAFOV LJk_t'FER
26'15 F
- 25.,o_'" "--=-_5.a5/ I
0 100I I
200 300
TIME (SEC)
NON-SCANNERDYNAMICRESPONSE
WFOV TOTAL SIMULATION
TIME DELAY (LAG) ESTIMATION
O .60e
bJ .52n.
_._.43rv"W0-.35
u.I
LI.
m
m
.27 u-
.18
.100
ACT CAV - RE:F CAVACT CAV-*CU H[AT
ItOO
l, I I200 300 4.00
TIME (SEC)
l J500 GO0
108
NON-SCANNERDYNAMICRESPONSE
MFOV TOTAL SIMULATION
TIME DELAY (LAG) ESTIMATION
450
_,.375
0
<C 225Q.<
150
75
IRRADIANCECONVERTED COUNTS
(/) 8200 FI-.Z
a&50 _-.--0
D.. 7750I--
0 7600CIC0
LuZCn7450 F_ 73O0 L
RAI_AT_ONERRORZERO LINE
.oolTn A
-5.oolII U ! I 1 iY [ ............... :0 I O0 200 300 400 500 600
TIME (SEC)
NON-SCANNERDYNAMICRESPONSE
MFOV TOTAL SIMULATION
TIME DELAY (LAG) ESTIMATION
40.4
n
_l_j40.2
p
........._ I
1--
40.1 ........
40.t __
40.0S --
0o 40.00--L_Jn":D 39.95 --
_L_j39.90 --
o-
_j 3g.85 --
I-- 39.80 --
39.75 --
ACTIVE CAVITYREFERENCE CAVITYAPERTURE
26.15 --
o" 26.10-bJ
26.05 --I---<C
_ 26._ _-13.
_ 25..5"-.....F.-
25.90 --
25._0
CU HEAT SINKTOP AL HEAT SINKBOT AL HEAT SINK
I I I100 2O0 3OO
TIME (SEC)
I400
I5O0
I6uO
Ii0
NON-SCANNERDYNAMICRESPONSE
HFOV TOTAL SINULATION
TIHE DELAY (LAG) ESTINATION
0.600
.52
F-•< .43OCb.Ia..35 --
WI-- .27
L_i,0.10
0
m
I
ACT CAV - REF CAVACT CAV-CU HEAT SINK
I I2OO 3OO
TIME (SEC)
I I I40O 5OO 6O0
111
NON-SCANNERDYNAMICRESPONSE
_FOV FILTERED SIHULATIOH
TIME DELAY (LAG) ESTIMATION
IRR/_ANCECONVERTED COUNTS
_S.Ol"-
,2.s [I'
N Ii
_ 10.0
-- S0__.. --
(/1 7400I-Z
735O0u
73OOI-
(2. 7250
07200
iv,0o_ 715O1Z
5.00
_ 3.33
1_ 1.67
Ld 0
_-1.87
_ -3.33
-5.000
m
m
RADIA'IIONERROR]a[RO I.JN£
. I I I I I I1O0 200 300 400 500 800
TIME ($EC)
112
NON-SCANNERDYNAMICRESPONSE
WFOV FILTERED SIMULATION
TIME DELAY (LAC) ESTIMATION
48.20
0o 46.83
W0C
2 45.47
_bj 44.10O.
41.37
40.00
m
m
ACTIVE CAVITY
REFERENCE CAVITY
APERTURE
SW FILTER
40.05
Oo 40.00
LOC__9.95I--
Q.
39.80
39.75
m
CU HEAT _NKTOP AL HEAT _NK
BOT AL HEAT _NK
BASE PLAI_FOV UUITER
26.10
(.)o 25.88
ILln,,:::)25.67I---<C
IZ25,45
L_J25,231--
25.02
24.800
B
m
I • I I I I I1O0 200 300 400 500 600
TIME (SEC)
NON-SCANNERDYNAMICRESPONSE
WFOV FILTERED SIMULATION
TIME DELAY (LAG) ESTIMATION
b-•< .43rvtO
.35 --
Lo_- .27 --b-tuo,I.18 --LL.
0! i I
11111
k£T CAV - REF CAVACT CAV-CU HEAT SINK
200 30O
TIME ($EC)
I i I4OO 500 800
114
NON-SCANtIERDYNAMICRESPONSE
MFOV FILTERED SIMULATION
TIME DELAY (LAG) ESTIMATION
15.0
12.5
10.0m
7.5
5.0
2.5
0
RRN)IkNC(CONVERTED COUNTS
_ -_ I ° ___ -.. /_..-
u) 65O0F-Z
(_ 6450ov 6400F-
6350
63O0
Cn 8250ZLU(n 8200
m
m
w
m
m
RADIATIONERRORZERO LJNE
5"00 f!'V" _
3.33
,,,oL___\, / \-1.6"zi-- _.-.-.----._-
I
_"-5.00! I ! J t ,! I0 100 200 300 400 500 600
T_ME(sEc)
IIS
NON-SCANNERDYNAMICRESPONSE
MFOV FILTERED SIMULATION
TIME DELAY (LAG) ESTIMATION
48.2O
(.,)• 48.83t.d
::)45.47
_j44.10
9.
42.73
41.37
40.00
40.05
• 40.00ILln,
_3g.g5.<
I_J39.909.
_ 39.85
3g.80
39.75
26.10
t)o 25.88n,n,v
25.67
_1_j25.459.
n_,125.2.3
m
am
m
.,.. _ .,, ..
m
m
m
u
m
I--
25.02 --
24.8O0
AClNI CA_IYREFERENCE CATTYAPERIU_SW FILIER
w
I , , I I i
CU HEAT SINKTOP AL HEAT SiNKBOT AL HEAT SINK
BASE PLAIIFOV UMI'II[R
"" t..
]100
I I200 3O0
•nM_"(sEc)
I I I .,400 500 600
116
NON-SCANNERDYNAMICRESPONSE
MFOV FILTERED SIMULATION
TIME DELAY (LAG) ESTIMATION
Q
LU .52 --r_Dk-
.43
LIJ_.._--
_ .27 --
F-
i.
0.100
ACT CAV - I_ CA','ACT CAV-CU HEAT SINK
I I I I I JtO0 2OO 3OO
TIME (SEC)
400 500 600
117
SENSITIVITYANALYSISUSINGDYNAMICMODEL
NFOV TOTAL SIMULATION
ACTIVE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .92)
n- 318.67 _-_...,-- ---
'=E
_S200 F
0761_ -
(/) 7450 --
(I) 73001 _
15FT
IL,M_A'IION ERRORZERO UNE
0 50 I00
I150
"riME(sEc)
I2OO
I25O
!
118
SENSITIVITYANALYSISUSlrIGDYNAMICMODEL
DflPOV TOTAL SIMULATION
ACTIVE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .92)
40.6
Oa 40.5Ldn,D 40.4I--.<
_uJ40.311.
uj_40.2
40.0
40.05
O" 40.00tUn-
39.%5<
39.90O-
39.B0
39.7;
26.15Oo 26.10b.IC_
26.O5.<
b._26.00a.
_t_j25.g5
25.8,5O
A
mm_mNmmm. .....
m
m
--mmmm_mm--.
!5O
!IO0
1150
liME (SEC)
I
AC11VECAVITYREFERENCE CAVITfAPERTURE
OU HEAT S_IKTOP AL HEAT SINKBOT N. HEAT SINK
BASE PLATEFOV UMITER
1250
I
119
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY EMISSIVITY CIt_NCED BY .02 (NOMINAL .92)
.60 --
V_
A-WO..35 --
bU.27 --
F-W(_ .IS --b.b.O .I0
0
I I I I50 100 150 200
"riME(SEC)
ACT CAV - _ OAVACT CAV-OU HEAT SINK
I J
120
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)
eSO.OO
_ 791.67
Z 633.330
__475.00,
<:rv, 316.67
158.331
i"T
mADIANCE_TEO COUNTS
f-%
_o 8200
0"-" 7900I--
n 7750I--
O 76OOn,oul 7450zLUu_ 73oo
m
m
D
I_IATION ERRORZERO IJNE
I300
12!
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SI_LATION
REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)
40.6--
0
" 40.5 ___.b_
40.4 --<n- 40.3--bUn
i_.i40.2--F-
40.1 ----
40.0--
40.05 --
0o 40.00tUiv
39.95 --<
_139.90 --n
_ 39.85 --
39.80 --
39.75 _
26.15 --
0" 26.10-wiv"_ 26.05 --
_ 26.00 _n
_ 25.95 --
25.90 --
25.850
CU H(AT SINKTOP N. FEAT SINKBOT N. HEAT SINK
BASE PLA11[FOV UI411T.R
I I I I50 100 150 200
TIME (SEC)
I25O 30O
122
SENSITIVITYANALYSISUSINGDYNAMIC,_IODEL
MFOV TOTAL SIMULATION
REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)
(.) .600
.52
_.43
m
i
v
b.I0...35 _-,_
.27
I--
.18
1_ .100
I
CAV - RE}"cAVACT CAV.-_ HEAT
I I I I I100 150 20O 25O
TIME (SEC)
123
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
DOME FILTER LONC WAVE TRANSMITTANCE. (LEAK) CHANGED BY .005 (NOHINAL .01)
950.00
_791.67
Z 633.33O
<C 475.00
<:IZ 316.67
158.33
!
m
IRRADIANCECONVERTED COUNTS
f-%
ul 8200I--Z
805000
7900I-
a. 775OI--
O76OO
n-OUl 7450ZI.iJ(/) 730O
m
m
RADIATION ERRORZERO LINE
15-
i 10:
5
-I0 --I
'_-,5 _1 I I 1 I I0 50 I O0 150 200 250
TIME ($EC)
I3O0
124
SENSITIVITYANALYSISUSINGDYrIAMICMODEL
MFOV TOTAL SIMULATION
DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)
40.6
0
o 40.5 =-.--.----Wo,
40.4 --
<
o- 40.3 --LOtL
_L_40.2 --
40.1 m_.
40.0 _
40.05 --
0
o 40.00
tvZ3 39.95I--
_bJ39.90 --
_t_j3g.85 --F-
3g.80 --
39.75 --
26.15 --
0o 26.10-
hJo"D 26.05 --I--.<
_1 25.g5 --I.,-
25.g0 --
25.850
ACTIVE CAVITY
REFERENCE CAVITY
APERTURE
CU HEAT SINK
TOP AL HEAT SINK
BOT AL HEAT SINK
BASE PLATEFOV UMITIER
I50
I100
I150
TIME (SEC)
I200 250
J3OO
•. 125
SENSITIVITYANALYSISUSINGDYNAMICMODEL
HFOV TOTAL SIHULATION
DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)
O .600
.52
._.43n-W_.35
WI--- .27I..-WU') o18b-
O .100
I50
ACT CAV- REF CAVACT CAV-CU HEAT SIHK
I100 150
TiME (SEC)
I i I200 250 300
12_
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)
3O
_25
Z 200I-'<15F,.<oc10
IRRADIANCECONVERTED COUNTS
(/) 6500--i--Z
6450-0
6400 --
S6350 --"-
6300 --
0u') 6250-z
6200-
15Fn- 5
oo_c_LO 0
z0 -5I-
-lo -
n.-15
o
RADIATION[RRORZERO LINE
I5O
I llO0 15o
TIME (SEC)
I200
L250
J3GO
.Ci_3'
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)
48.20
0o 46.83
b.I
45.47I--_CC_ 44.10L_JCL
41.37
m
m
m
40.00 ,,,-
40.05
Oo 40.00 _,-ILl
=) .19.95 --I--
OC 3g.90WO."s
39.85 w
39.80
3g.7_ m
26.10 m
0o 25.88WC_:D 25.67
L_j 25.45
[1.
_L_j25.23
ACIWE: CAVITY
REFERENCE CAVITY
APERTURE
SW FILTER
I-"
25.02
24.800
CU HEAT SINKTOP AL HEAT SINK
BOT AL HEAT SINK
BASE PLATE
FOV UMITIER
I5O
1 I !100 150 200
TIME (SEC)
I25O
JY_.O
128
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)
O .60o
W .52Q::
<[; .43
m
m
L_
11..35--Iw
.27 --p-bJ(,/) .18 --I,.i,.h0.10
0 50
ACT CAV - RIEF CAVACT CAV-CU HEAT SINK
100I
150
TIME (SEC)
t t I200 250 3uO
129
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)
950.00N
_E791.67
3=
Z 633.330F--_.< 475.00r-_<rr- 316.67
158.33
I
I%I!
3-
IRRADIM,_ECONVERTm COLt,ITS
B200 --Z
8050 --c.)"_ 7900 I
_ 7750
7600,4--
_7450 -Z
15_
n." 50
n--O_-
Z0 -5--F
-,O-L/I_ -15
0
R_DIA_tC,N£RROFZERO L_
I I_o !_.
t I i150 2bO 25G
TIME (SEC)
J3:>0
130
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
DOHE FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOHINAL .946)
(.)40.6_. _
F
P.,AVII'YREFERF.N_ cAvn'¥/F'ER'I_
£.)40.00
hin-O 39.95 --
b.I0.
39.80--
28.15
rJo 26.10hin-
26.05.<n-"26._1W
_ 25.95p-
25.00
0
CU HEAT SINKTOP ,14.HEAT 9NKBOT hi. HEAT _INK
m
50 lClO
I L 1150 2b_ 250
TIME (SEC)
131
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)
O .60Q
l.d .5?.n--
_.43n-Ld(1- .35 --
L_
I- .27 --
t--tU
.18 --I.Lb_o .I0
o
.P_ mw
ACT CAV - REF C_kVACT CAV-CU HEAT SINK
I IIoo
I150
TIME (SEC)
I I i21)0 250 300
SENSITIVITYANALYSISUSINGDYNAMICMODEL
NFOV FILTERED SIMULATION
DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)
Oo 48.83W
iv 44.10ILlI1.
LU42.73I-"
41.37
40.00
m_ m_m
m
m
o 40.O5 F
_ _._op-
_ 3g.85 _-
:.:L
20.10Uo 25.88bJn_:D 25.67I-<_
_ 25.45D.
LJ 25.23 --1--
25.02 --
24.800 5O 100
I I15o 200
TIME (SEC)
.*t"llVl: _wrlYIt_F[ItEN_ cAvn'Y.IJ_RTIRE_V FILTER
CU N[AT SINKTOP AL Nr.AT SIl_BOT #4. H[AT SNK
I250
i3OO
133
SENSITIVITYANALYSISUSIIIGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)
_ON
Z20OD
n, 10
5
0
D
A
t-I-
I!
.-I-!II
"rI!
IRRAI_ANCr"CON'VERTEDCOUNTS
(.J"-" 6400
_i_
Z
o'_ 6200'__
o 5o
RADIATION ERRO_ZERO LIN£
IOOI
150
TINIE(SEC)
I I I200 250 300
134
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)
0.600
I.,.I0-.35 --_E
.27 -
I--
.18b_
0 .lC0
AC'TCAV - R[F CAV^C7 CA,V-CU HEAT SINK
I II00
I I I J150 2DO 25O _00
TIME (SEC)
i
SENSITIVITYANALYSISUSIflGDYNAMICMODEL
MFOV FILTERED SINULATION
DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)
20
< 1.5
!!
TI
.-L.III
-I-II
IRRADIANCE
CONVERTED COUNTS
65OO--I--7
6450 --O(D
_-_ 6400 --p-
11. 6350
O6300--
n-OCO 6250 --ZLd(Jl 6200 __
-10_-_ -,s I0 50
1150
lIME (SEC)
I200
RADIA'rIDh ERROR
Z][RO _NE
i250
J
136
SENSITIVITYANALYSISUSINGDYNN'IICMODEL
M_'OV FILTERED SIMULATION
DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)
m
4O.O5
0" 40.00LUcf
31)J5<
30,1_
30.75
D
n
m
m
m
CU HEAT SINKT0P N. HEAT SINKBOT AL HEAT
26.100• 25_sLU
m
<ew 25.45 --tuft."stu 25.2.3--k-
25.02 --
24.8O0
I5O
I I I iloo 150 200
TIUE (SEC)
I3OO
137
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV FILTERED SIMULATION
DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)
0.60o
m
m
_KWO._18 -
_.?.7 -
_ .100
ACT CAV - _ CAVACT CAV-CU HEAT _RNI(
I50
I100
I I I2OO 2.5O _00
138
SENSITIVITYANALYSISUSINGDYNAMICMODEL
HFOV TOTAL SIMULATION
SENSOK BRIDGE VOLTAGE CHANGED BY ,01 v (NOMINAL 10 v)
950.00
"_. "/91.67
633,33
_ 475.00
(2C316.67
158.33
m
r
m
ImADUUW_CONVUtlIO COUNTS
Ul 82OOI--
_8050C.)"'79OO
m
300
139
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
SENSOR BRIDGE VOLTAGE CHANGED BY .01 v (NOMINAL 10 v)
40.6 m
vO
* 40.5 _...._t.u
40.4
_j 40.3
40.2 m
40.1 ----
40.0 --
40o0_ m
0• 40.00bJ
39.95 --
_L_j3g.90
_ 39.85
28.15
0o 26.10i.d
_ 26.05
e_ 26.00 "_W
I_ 25.95 _F-
25.90 _
25.85 i0
ACTIVE CAVITYREFERENCE CAVITYk1_rR11.11_
CU HEAT SBIIKTOP AL HEAT'SB'IKBOT AL HEAT SINK
EIASE PLA11[FOV JMITER
I 1 I ISO 100 150 2O0
•nuv(sEc)
I J30()
140
SENSITIVITYANALYSISUSINGDYNAMICMODEL
TOTAL $II_J_TZ_I
SENSOR BRIDGE VOLTAGE CHANGED BY .01 v (NOMINAL I0 v)
O.OO
"_ .52n-
_.43
Cc.lq --
_.27--
k-
.18 --
_.I00
I
CAV - R_ CAVCAV-.CU HEAT SINK
ItOO
I I I I200 250 ,_
141
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMIfl_TION
ACTIVE CAVITY HEATER RESISTANCE CHANGED BY .I fl (NOMINAL 1492.3 fl)
g_00t_
_ 833.33
_ 475.00
tv 316.67
158.33
m
m
B
"1"
m
m
• (:_IVE:Rll_ C0UN_
8200--
(3"'_ 7900 m
::)0-7780I-
O 7600 -iv,Ocn 7450 -zLUV) 7300 _
15
o-I0=,slL/
_AI_N ERRORZIRO UN_
I I5O 100
I
TIME (SEC)
I I J2OO 25O 3OO
142
SENSITIVITYANALYSISUSINGDYNAMICMODEL
14FOV TOTAL SIMULATTON
ACTIVE CAVITY HEATER RESISTANCE CHANCED BY . 1 L'I (NOMINAL 1&92.3 _)
.,C
,'1
40.1
4O.O
m
R
_lliJlli
RE]_"IE_E]_XcA_rY
wiv
39.05
o.
Immm_,_m,m.
n
QU Frr.AT g, iKTGP AL HEAT SINKBOT N. HEAT _gNK
r,_ 26"15 r
I50
I100
I I I200 25O _mO
143
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY HEATER RESISTANCE CHANGED BY .I fl (NC_INAL 1492.3 fl)
b4
WP'.27 --
_.lS --
.10o
I
ACT CAV- _ O.AVACT CAV--CU HEAT _NK
I100
I1.50
"hUE(s_C)
I I I2._ 30O
144
SENSITIVITYANALYSISUSINGDYNARICMODEL
MFOV TOTAL SIMUlaTION
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANCED BY .01 i_ (NOMINAL 2319.1 f_ e 40"C)
IImAOL4_[¢ON_TI_ COUNTS
_ B200 --
_OOm
7eO0 -
_7450 -
_730C _
. 15
-I0
n- -15
RABIAIION ERRORZERO LINE
0 50I
tOOI I I
150 2DO 2.50
_.E (SEC)
!
i43
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .01 _ (NOMINAL 2319.1 _ @ 40eC)
O 40.6_....___
'oE.......
ADTI_ CAVII_REFERENCE C_kVIT*_APERTURE
4O.O5
U6 40.00w
m
mmmm.m.m..
NEAT SINKTOP ,'41.HE,AT SNIp,BOT N. HEAT SII,_
26.1,5 --
U• 26.10w
w
I-
_._0--
0I
5O tOO 150
TIME ($EC)
i2OO
l2_
i3OO
146
SENSITIVITyANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANCED BY .01 _ (NOHINAL 2319.1 l_ @ 40"C)
OAV - REF OAr'CAV-¢IJ HEAT _,_K
.27
_"'1-
.,ol I I I I , I I50 tO0 150 _ _ 3_
"riME(SEC)0
i47
SENSITIVITYANALYSISUSINGDYNAMICMODEL
HFOV TOTAL S_TION
REFERENCE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .01 _ (N{_4IR_ 2327.1 _ _ 40"C)
I_O.00
_,79! .67
Z 433..3.1O
r_ 316.6"/
1_S.33
m
i
_ _ _ _ ..... immmw_wm,amm,,m,em_
i
IRR._N_C(CONM[RTm COUNTS
7900 I
7eO0 -
_ 74,50 --ZL_(n 73o(
15-
0:: ___00£n"t_ 0 "_
Rb/)IA'TIONERRORZERO L_r
.L I I150 200 2.5O
_E (SEC)
148
SENSITIVITYANALYSISUSINGDYN_IC MODEL
TOTAL SI_[JI_TZOI¢
_Cg CAVITY S_SO& _ RESISTANCE CHA_ED BY .01 fl (_ONI_,L 232_.1 fl _ 400C)
40.6_._.
::r-................
RErlD_NCE CATTY
,U'EI_TURE
40_5
°4OO0
W
30.8O
3g.75
20,15
rj• 2_.1oL:.In..
,<
e" 28.1 _W
,-,25.95
0
m
m
ra-
m
m
m
CU HEAT SINK
TOP AL NEAT SINK
BOT N. I.fAT gNK
PL,ATEF'OV UldnT.R
ISO
ItOO
_ME (SEC)
!2OO
1I ,
149
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REFERENCE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .0l _ (NOMINAL 2327.1 _ @ 40"C)
0.60O
LU ,52£E
_.43,wWO..15
LU(nb.
_S mbmma
m
m
.27--
.16 --
.10 I0 _o
CAV-CU l,,(A'r._.N_,_
I I I I I1oo 150 290 25o
TIME (SEC)
1.50.
SENSITIVITYANALYSISUSINGDYNAMICMODEL
IO_V TOTAL SIHULATION
SENSOR BRIDCE RESISTANCE CHANGED BY .01 it (NOHINAL 2210 it)
9riOJ30N
79! .67
_ 316.a7
IRRADIANC[CONMgTII_ COUNTS
(Jv 7000
N 15
Ld 0
-10-15
-J
I ,I I50 tOO
R.N_^'ll_N ERRORZ_RO LIH£
l I I200 25O 3OO
151
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SII_LATION
SENSOR BRIDGE RESISTANCE CHANGED BY .01 i_ (NOMINAL 2210 i_)
Jm
0'.01.--...............
(.Io 40.00h3
_39_
_30_0
(I.
38_0
30.75
2_.15Ue 28.10bJ
,<r,- 26.OObJ0.
m
m
m
m
m
m
minE,
ACTIVE CAVITYREFE)_D4C( CAVITYAPERIURE
CU H£kT SINKTOP AL HF.AT 9NKBOT #4. Hr_T 5NK
I I I I I5O 100 150 200 250
_.E (sEc)0
I
152
SENSITIVITYANALYSISUSINGDYriAMICMODEL
MFOV TOTAL SIMULATION
SENSOR BRIDGE RESISTANCE CHANCED BY .01 _ (NOMINAL 2210 _)
,,vLU
I-I._(r1.16b.b.0,10
0
I 1 I I_ME(SEC)
CAV - _ CAVAG"TOA¥-CU I,,ll_T SINK
I I
153
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)
. 95O.OO
Tgl.e7
316.47
1M.._
ItV"- ..................T
m
(n s2oo
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0 ----
15
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1_-10
r,, -15
m
m *,
t0 50
ZERO UI_
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_ME(sEC)300
154
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
ACTIVE CAVITY SENSOR WIRE RESZSTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)
40.6--
Ld _ ....
40.4 --
n'40 _ --LU0.=i
40.1 m ........
4_Om
• 40.00L_
.......
26.15 --
O• 26.10-LUn,
26.05 --,<
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ak_E PI.ATEFOV IJMITER
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TIME (SEC)
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155
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SI_..ATION
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)
ACT OAV- REF C,AVACT CAV-,.CUFEAT
.,o_ I I0 BO 100
I150
TIME (SEC)
I2OO
I . I
156
SENSITIVITYANALYSISUSINGDYNAMICMODEL
9,50.00
'_791.8"/'
_I:_475.00
n. 316.67
158.33
iI
7-II
.4II
J
HFOV TOTAL SIHUTATION
REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2
IRRN)Ik_ICE(:OI_[RTED C'_JNT_
!
I t_
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; !3
Z8O5O --
O_.3_' 7900 --i=..
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5
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l
I
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_ _ _i___.......
j t/ , ,200 300 400
TIME (SEC)
I_ATI_ ERRORZI:RO
5OO 6O0
157
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REDUCED CONDUCTANCE BETWEEN NODES I AND 2
40.6
Q 4_LU
n
40.1
4O.O
40.05
O* 4_00LU
_._
(I..
LU 39.85F-
39.80
_.75
F'-V
AC'I_ C.AVITYmc[ cAvrr_APL_'n_
m
m
n
m
D
CU I._ATTOP AI. IEATBOT AL HF.AT
m
m
m ,v
BASE PLA11[FOV LIIml[R
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Oa 26.10-LU,w
26.05 --
r,, 26.00Ld(3.
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2-_gO--
"OJ
1ooI I I
200 3O0 4O0
•n_E(sEc)
I J5OO 600
158
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REDUCED CONDUCTANCE BETWEEN NODES I AND 2
O .60
ivLOU..35
b.O
.IS
.100
ACT CAV - REF CAVACT CAV-CU IEAT SNK
m
I
I I I I100 200 30O 400
TIME (SEC)
I I5OO SO0
159
SENSITIVITYANALYSISUSINGDYN_IC MODEL
MFOV TOTAL SD4ULATION
REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 f_ (NOMINAL 2319.1 f_)
I_0or7"3
N,it'
" i-rl ;/158.33 _- i uoU_
f
M;llll3[CIIIVER'IED
" 8200 i --
ao,sol
7gOOi --I-,'-
0. 775O --
O7600 -
(z:O
7,k_i -Zu,I
RADIATION ERROR
I
"hUE(SEC)
I I I6_
160
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REDUCED CONDUCTANCE BETWEEN NODES I AND 2
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)
40.6
Oe 40.5bJ
4_4
n:E
4Q.I
4O.O
4O.O5
O• 40.00LUrw
_-g'_
(I.
3.9.85
_.80
_.7S
CAVITYREFERENCE CAVITYAPERTURE
m
CU HEAT "INKI"QP AL HEAT _MKBOT AL HEAT _NK
m
m
w
BA._E PLA11[FOV LIMITER
26.15
0• 26.10-Ld
_ 26.05 --.<
w_j26.00 _:_(1.
25"t_ F
0 20OI
T_ME(sEc)
I40O
I5O0
J6O0
161
SENSITIVITYANALYSISUSINGDYNAMICMODEL
MFOV TOTAL SIMULATION
REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2
ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)
twLUD-._5 --=I
I--bJ
.18 --IJ.u.O .10
0I
IO0
ACT CAV - REF CAVACT CAV-CU HEAT _gNIK
I I I2.0O 30O 4OO
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I ISO')
I
i-
162
_GIN._ PXGE
OE'_OR QUALITY
163
CONCLUSIONS
• DYNAMICMODELSHELPTHE UNDERSTANDINGOF THE EP_E INSTRUMENTS
• SIMPLECOUNTCONVERSIONALGORITHMSHAVEBEENDEVELOPED
STEADYSTATESIMULATIOIIAND GROUNDCALIBRATIONRESULTSARE IN
GENERALAGREEMENT
• THE ERRORSDUE TO COUNTCONVERSIONARE BEIMGANALYZEDBY SIMULATION
• INITIALSENSITIVITYAND ERRORANALYSESPROVIDECONSTRUCTIVERESULTS
OVERALLAPPROACHSEEMSWELL-SUITEDTO ASSESSINSTRUMENTPERFORMANCE,
ACCURACYAND VALIDATION
_lNo, PAGeet,aNIINO'I'
165
APPENDIX B
El{BE SCANNER SIMULATIONS
.¢
_cm_o P_o_SL_Ir_ rR4m_•
167
These graphical displays were produced during development of the simula-
tion model. Please note that the plot of "OUTPUT (CNTS)" versus "TIME (SEC)"
in the graph on pages 169, 173, 174, 177 and ig0 does not display quantita-
tively absolute values for counts generated from the simulation model due to
uncertain gain/offset parameters in the electronics.
168
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183
APPENDIX C
ERBE NON-SCANNER THERMAL MODEL
B,.
PRBCED_ PAGE BLANK NOq' FILMED
185
_J
NON-SCANNER MODEL
The energy balance equation
n T. - Tkr J
Mk Tk -- j-g-i _j+ _ _k + Qsource
(i)
The second term in the right-hand side of tile above equation is the
radiation interaction of the kth element to the other elements in an en-
closure. This variables are defined by the following approach.
For the radiant interchange in an enclosure having both specular and
diffuse surfaces, the surface properties are defined as the followings:
The reflectivity is
d s
Ps = Ps + Ps
d s
Ps = Pl + Pl
where subscripts, s and l, signify short and long waves and superscripts,
d and s, denote diffuse and specular.
For diffuse and gray approximation, the following realtionships are
obtained
+_ = iPs + ks s
Pl c._ _[ = I.
From now, the subscripts for short and longwaves are omitted for brevity.
The radiosoty at the kth surface is
d _ d + TkEkMk = E k_T + PkEk(2)
The incident radiant flux at the kth surface is
- d
Ek = _ Fk-jM jJ
(3)
186
Combhlin>_the above two equations yields
Md Z(6." d- -i (( + T.E.)j = . 31- 0jFj-i) k_i I I1
From (3) and (4),
Ek = Z(_iOi + TiEi) _ Fk-j Pj-ii ]
where
_- (_.. d- -iPj-i 31- pjFj-i)
= OT4.
(4)
(5)
The radiant engergy balance equation is the difference between the
incident energy and the radiosoty. That is,
_k = Ek - Mk
s= Ek - Ek _k - Ok Ek - Ok Ek - _k Ek
Substituting (5) into (6)we have
(6)
_k = Z [(i - pk ) c i Z Fk-j P" - (k _ki ] _"• j-i Ii j
+ Z [(i - pk ) T i _ Fk-j Pj-ii 3
- Tk Ski] E i (7)
LetHk_ i = Z Fk-j Pj-i
J
Then, for the cavity a, the equation (5) can be
expanded as
EIO = Hlo-i el _I
+ [HIo-8 E8 _+ HIO-9 (9] _8
a _I0+ HI0-10 _I0
a _ (A)+ HIO_I 0 TIO EIO
b - cavity __
i0
187
ii
b-cavity
EFov
El0
For the cavity, b
~ b f_10El0 -- HI0-5 _5 _5 + HIO-6 e6 _6 + HI0-10 ci0
• bb +
+ HI0_I0 TI0 El0 HI0-11 _ii EFOV(B)
From (A) and (B),
EIO = G _i HI0-1 _i
a n5+ G _5 TI0 _0-5 HI0-10
a
+ G c6 TI0 HIO_6 HI0_I 0 _6
+ G[HI0-8 _8 + HI0-9 _9 ] _8
a a b
+ G _Io[HIo_I0 + TI0 HI0_I 0 HI0_I0] _QI0
a
+ G TI0 TII HI0_I 0 HI0_I 1 EFO v
2 a .b )-i.where G = (i - TI0 HI0_I 0 HI0_I 0
~
EIO = G b nli TI0 HIO-10 HI0-1
+ G 45 HI0_5 _i
+ G _6 HI0-6 _6
b [I[10-8 _8 + _ ] d8+ G TI0 HI0_I 0 HI0-9 9
b Ha+ G _I0 HI0-10[I + TIO i0-I0 ] _i0
b
+ G TII HI0_I I EFO V
188
$i = _[(I - pi ) Hij Ej - _ij ej] _j + Z[(I. - pi ) Hij T.j - _ij Tj]Ejj J
In cavity a:
a 2 G b -i] _i$i = Ei[(i - of)HI_ I + (i - OI) TI0 HI_f0 HI0_I0 HI0_ I
+ (i - OI) (5 TIO G HI_f0 HI0_5 _5
+ (i - PI) (6 TI0 G HI_I0 HI0_6 _"6
2 b
+ (i - OI) {((8 HI-8 + e9 HI-9) + TI0 G HI_I0 HI0_10 ((8 B10-8
+ HI0-9 E9 )} _8b a
+ (i - pz) (to _-zo [I + Tzo c _zo_zo(l + Zlo _1o_zo)] _zo
+ (1 - O1 ) "_i0 "_11 G 1-11_10 I-Ii0_11 EFOV
A 8 A 9
a : ¢8 + _ ¢9$8 A(8+9 ) A(8+9 )
2 b nl¢8 = (i - p8)[_ 1 H8_ I + eI _i0 G H8_10 Hl0_l 0 Hl0_ I]
+ (i - P8 ) e5 %10 G H8_I0 H10_5 _5
+ (i - p8) e6 "rlO G HS_I0 1-110_6 _Q6
2 b
+ {(8[(i - p8 ) H8_ 8 - i] + (]-- p8) TI0 G H8-].0 HI.0-10 (HlO-8 (
+ H10_ 9 (9) + (i- 8) e9 H8-9} _8
b (i + a+ (I .- P8 ) el0 H8_I0 [i + TI0 G HIO_I 0 TIO HI0_I0)] RI0
+ (i- 08) %10 TII G HS_I0 HI0_I I EFO V
189
2 b¢9 = (i - 09) el[H9_I + TI0 G H9_I0 HI0_I0 HI0_I ] _i
+ (i - 09 ) _5 _i0 G H9_I0 HI0_5 _5
+ (I - p9 ) (6 TI0 G H9_I0 HI0_6 _6
+ { (i - p9 )
2 b
9 H9-8 + c9[(I - 09) H9-9 - i] + (i - 09 ) TI0 G H9_I0 HI0_I 0
(HI0_ 8 E8 + HIO_ 9 _9)}f_8
b (i + a+ (I - 09 ) el0 H9_I0[I + TI0 G HI0_I 0 TI0 HI0-10)] _i0
+ (i - 09 ) TIO TII G H9_IO HI0_I 1 EFO v
In Cavity b:
b_5 = (I - 05 ) _I TI0 G H5_I0 H10_l _i
2 a
+ e5 {[(I - 05 ) H5_ 5 - i] + (i - 05) TI0 G.H5_I0 HI0_5 HI0_I 0} _5
2 a
+ (I - 05 ) E6[Hs_ 6 + TI0 G H5_I0 HI0_6 HI0_I 0] _6
+ (i - 05 ) TI0 G H5_I0 (e8 HI0_8 + e9 HI0_9) _8
+ (i 05 ) el0 H5_I0[I + TI0 G a _b- HI0_I 0 .(i + TI0 HI0_I0)] _QI0
2 G a+ (i - 05 ) TI0 TII H5_I0 HI0_I 0 HI0_I I EFO v
b¢6 = el(l - 06) TI0 G H5_I0 HI0_I _i
2 G H_0-10] n5+ (i - 06 ) e6[H6_5 + TI0 H6_I0 HI0_5
2 a
+ E6{[(I - p6 ) H6_ 6 - i] + (i - 06) TID G H6_I0 HI0_5 HI0_10} _6
+ (i - 06) TIO G H6_I0 (_8 HIO-8 + _9 H'O._I-9 ) fl'8
a b
+ (i - 06 ) el0 H5_I0 [i + TI0 G HI0_I 0 (i + TI0 HI0_I 0)] _i0
2 G a+ (i - 06 ) TI0 TII H5_I0 HI0_I 0 HI0_I I EFO v
190
b G[(i ) b - I] _i_i0 -- _I _i0 - 010 HI0-10 HI0-1
2 G[(i 010) b - i] a+ (5 HI0-5 {(i - 010) + TIO HI0-10 H]O-10} _5
2 G[ (i Ol0 ) b - i] a+ _6 HI0-6 { (i - 010 ) + TI0 - HI0_10 HI0-10} R6
b -I] (_ +_+ TI0 G[(I - 010 ) HI0_10 8 _0-8 9 _0-9 ) _8
b -i] {i+ a a b+ (I0 [(I - 010) HI0-10 TI0 G[HI0-10 + TI0 HI0-10 HIO-IO]} _I0
2 G [(i 010 ) b a+ TI0 TII - HI0_I 0 - i] HI0_I 0 HI0_I 1 EFO V
_Oa a Al0b h
_i0 filter - Al0(a+b ) _i0 + AlO(a+b ) _i0
Cavity C:
¢i=.c(i[(i. - Pl ) HI_ I - 1] R1 + (I - pl ) (2 H1-2 _2
+ (I - pl ) (4 HI-4 _4
C
_2 = (I - 02) _i H2-1 _i + E2[(I - 02) H2-2 - i] n2
+ (I - 02) _4 H2-4 _4
C
_4 = (i - 04 ) (i H4-1 nl + (i - 04 ) E2 H4_ 2 _2
+ c4[(i - 04) H4_ 4 - i] _4 "
c c
where _ and 0 are e and 0
Cavity d=
d
_2 = _2 [(l - 02) H2-2 - i] _2 + (i - 02) _3 H2-3 _3
+ (I - 02 ) (7 H2-7 _7
191
r_ •
_3d ffi (1 - 03 ) (2 H3-2 _2 + c3 [(1 - P3 ) H3_ 3 -
+ (1 - P3 ) e7 H3-7 _7
d_7
I] n 3
= (1 - 07 ) ¢2 H7-2 _2 + (1 - 07 ) _3 H7-3 _3
+ c7[( 1 - p7 ) H7_7 - 1] _7
Cavity e:
_le ffi _l[(1 - pl) HI_1 - l] _]
+ (I - P I) _3 HI-3 _3
+ (I - Pl ) ¢2 HI-2 _2
+zeffi (i- 02) _I_2-i_{ +_2[(I- o2)
+ (1 - P2 ) _3 H2-3 _3
e$3 _(1 - 03 ) (1 H3-1 _Q1 + (1 P 3 )
2 H3-2 _2
+ e3[( 1 _ p3 ) H3_ 3 -11 _Q3
Cavity f :
s f ffic3[(i - 03 ) H3_ 3 - i] _3 + (l - P3 ) c 7 H3-7 _73
_b7f = (1 - 07 ) _3 H7-3 _3 + c7[(l - 07) H7-7 - 1] _7
The energY balance equation (i) is written for each node as the
followings :
T2-T I c c e e a a QHI}TI i { + AI $i + AI $I + AI $i +
= _ii RI-2
192
- T4-T2 T3-T2 T7-T2 e e,-7-i{TI-T2 + -- + -- + -- + A2 _2 + Q2 }
T2 = n 2 R2_ I R2- 4 R2- 3 R2- 7
i T2-T3 e e d d f f
T3 = M 3 { _ + A3 _3 + A3 _3 + A3 _3 + QH3 }
i T2-T4 T5-T4 T8-T4 c c= -- + -- + _ + A4 _4 + Q4 }
T4 M4 { R4-2 R4-5 R4-8
- T6-T5 TI2-T5 b _}i {T4 T5 +__+ __+
T5 = M5 R5_ 4 R5_ 6 R5-12 A5
i Ts-T6 b b
= -- + _6 }T6 M6 { R6-5 A6
i T2-T7 d d f f
= M7 --R2-7 + A7 + + Q7}-T7 { _7 A7 _7
I T4-T8 T5-T8 TI0-T8 a _a}9= __+ __+ __+ a+ aT8 M8 { R8-4 R8-5 R8-10 A8 _b8 A9
• i {T4-TI0 + T8-TI0 + a a + _0 b.... ¢lo }TIO MIO R10_4 RIO_8 AI0 _lO
Where M = V X C. In addition there are three more equations which were
derived from the electronics model. These equations are already discussed
in Section II.
In the above equations, the _s can be splitted as
_k = (_k)shortwave + (_k)longwave"
Substituting _k into the above equations and rearranging the constants
yeild a following type of equation
-- . . T4 + Qk + BkTk Z Akj Tj + Z Dkj 3 BQk EFOV3 J
193
where
%j1
kRk_j
Dkj = constants including all radiative parameters
Bk = constants related with the irradiance at FOV limlter of
radiometers.
The above constants are described in the following pages.
i 1
{AI_ I = MIRI_ 2 ' AI_ 2 = _RI_ 2
i
A2_ I - M2R2_I 'i i
A2_ 3 - M2R2_3 ' A2_ 4 - M2R2_4
=__(--_--L + ! + ! +_)A2_ 2 _z R2_ I R2- 4 R2- 3 R-27
i
M2R2_ 7
I I
{A3_ 2 - , A3_ 3 =M3R3_ 2 M3R3- 2
i
1A4_ 2 = M4R4_2
i
A4- 5 = M4R4_ 5
=_z---(--L+ ! +_ l )A4_ 4 M4 R4_ 2 R4- 5 R4- 8
i
A4_ 8 -M4R4_ 8
i
A5_ = M5R5_4 '.i
_A5-6 MsR5_------_ '
= - _!--- ( i + I + I.__.____)A5- 5 M5 R5- 4 Rs- 6 R5-12
A5-12
i
M5R5-12
194
{A6_5 = i , A6_6 = iM6R6_ 5
{A7_ 2 _ iM7R7_ 2
IAs_ -. IMsRs_ 4
iAs_10_ iMsR8_I0
A7_ 7 =M7RT_ 2
i
A8_ 5 = MsRs_5 ' A8_ 5 -z (._k_.z+ __.i_z + z__k_)M 8 R8_ 4 R8_ 5 R8_I0
i i
{AI0_4 = MIoRI0_4 ' AI0_ 8 = MIoRI0_8 ' AI0_I0 =I (-k +__i )
M10 R10- 4 RI0- 8
{AI a a a a 2 a b a= _ _ Ill_f0 HI0_I0 HI0-1DI_ l _ _1[(1 0I) HI_ 1 + (1 Ol ) ri0 G - i]
C c c c - i] + e e e e - I]}+ A 1 _i[( 1 - 01) HI_ 1 A1 _1[( 1 - 01) HI_ 1
= _ {A I (i - 0I) Ec c e e e eDI-2 M 1 2 HI-2 + AI (i - 01) e2 HI_ 2}
O e e e e
= _ HI_ 3DI-3 _ii AI (i 01) e3
a a b
DI-5 _II 5 TI0 HI-10 HI0-5}
= _ c (i - 01) e4 cDI-4 MI AI HI-4
a a a a Hb= - HI-10 10-6DI_ 6 _ A I (i 0 1 ) _6 Tl0 G
195
o a a a a + a a 2 a " b=- HI_I0 HI0-10DI_ 8 M1 A I (i - pl ) {(e 8 HI_ 8 9 HI-9) + TI0 G
a a a a(_8 HI0-8 + e9 HI0-9)}
= o a a a G b a
DI-10 _ii AI el0 HI-10[I + TI0 HI0-10(1 + TI0 HI0-10) ]
i a a G a b
B1 = M-_ AI (I - 01) TI0 TII HI_I0 HI0_11
O C C c c + e e e e
D2_ 1 = _2-2[A2 (i - p2 ) eI H2_ 1 A 2 (i - 02) cI H2_ I]
= O___{A2[(I P2 ) c - I] c + A2d d d d - I]D2-2 M 2 - H2-2 2 2 [(I - 02) H2_ 2
e e e e+ A2 _2[(i - 02) H2_ 2 - I]}
Ad d d d e e e e= _ { (i - P2 ) _3 H2-3 A2 H2-3}D2_ 3 M2 + (i - p2 ) _3
= _ {A 2 (i - p2)e4 H2-4 }D2-4 M 2
d d d d
= _ H2_ 7D2-7 _22 A2 (I 02) e7
(7 e e e e
D3-1 = M--3A3 (i - 03) E1 H3_ 1
o d3 d d d e e e e= _ H3_ 2 A3 H3_ 2 }D3_ 2 _ {A (i 03) e2 + (I- 03) E2
= __ d d[(l- 0_ ) d _ i] + e e eD3_ 3 M3 {A e3 H3_ 3 A 3 e3[(l - 03)
f f f f
+ A 3 e3[(l - 03) H3_ 3 - i]}
196
c) Ad d (I - 0d) d + f (i- 0_) c f fD3-7 = _33 { c7 H3-7 A3 7 H3-7}
= _ C C C C
D4_ I M4 A4 (i - 04) _i H4-1
---- O" C C C C
D4-2 _44 A4 (i - 04) e2 H4-2
c c c c c -i]D4_ 4 = M4 A4 E4 [(I - 04) H4_ 4
O {Ab (I_ f)b)eI G b aD5-I-- H5-I0HI0-I}
2 b b aO b b b) Hb - i] + (I - 0b) TI0 G H5_I0 HIO_5 HI0_I0]}
D5-5 = M'-5{A5[_5[ (i - 05 5-5
c b b b b 2 b b= - [Hs_ 6 TI0 HS-10 HI0-6 Hal0_10]
D5-6 _55 A5 (I 05) e5 + G
c b b) G b a a + eD5-8 = M-55A5 (i - 05 TI0 H5-10 (_8 HI0-8 9 HI0-9)
o b b b b [I + G a bD5-10 = M--5A5(I - 05) el0 HS-IO TI0 HI0-10(I + TI0 HI0-10)]
1 b b) 2 b aB5 = _55 A5 (i - 05 TII[TIO G HS_I0 HI0_I 0 HI0_I I + H5_II]
=c_ b a b G b a
D6-1 _66 A6 E1 (i - 06) TI0 H5_IO HI0_I
o b b) b b 2 G b b a ]D6_ 5 = _66 A6 (i - 06 e6[H6_ 5 + TI0 H6_I0 HI0_5 HI0_I 0
2 G b aD6_6 = M__A6_ b E6b {[(i - 0_) H6_6b _ i] + (i - 0_) TI0 Hl0_ 5 HI0_I0}
197
---- ab b G b (a a +
D6_ 8 _ A 6 (i - 06 ) TI0 H6_I0 8 HI0-8 9 HI0-9)
b b b b G a Hb
D6_I0 =_6 A6 (i - 06) el0 H6-10[I + TI0 HI0_10 (i - TI0 I0_i0 )]
i b b 2 G b a
B 6 = _ A 6 (i - 06) TII[H6_II + TI0 H6_I0 HI0_I 0 HI0_I 0]
__O d (i - 07 ) d dD7-2 M_7 A7 H7-2
=_ Ad d d d + f f f fDT_ 3 _ { (I - 07) _3 H7-3 A 7 (I - 07) c3 HT_ 3}
_ {AT d d d -i] + f efT[(l Pf7) f - i]}D7-7 = M_ c7[(i - p7 ) HT_ 7 A 7 - HT_ 7
D8_ I
a 2 a b ao {A8 (i 08 ) a a +_ G
= M_ Aa - [el HS-I i TI0 H8-10 HI0-10 HI0-1](8+9)
a
A 9 a a + 2 G a b a ]} A+ (i - 09) El[H9_ I TI0 H9-10 HI0-10 HI0-1 (8+9)
Aa(8+9)
b a b a a b a b
= _ {A8 (i - 08) _5 G + (i- 09) _5 GM- TI0 H8-10 HI0-5 A 9 TI0 H9-10 Hlo- 5}D8_ 5--8
a b + a (i- 09) Eb a b° " o-o oD8-6 = M_
D8- 8 O a a a a - i) + (i 0_) 2 G a b a=-- {A8[_8((I - 08) H8_ 8 - TI0 H8-10.HI0-10 (HIo_ 8M 8
aa+ a a a a a a +E8 Hi0-9 _9 ) + (I - 08) _9 H8-9] + A [(I - 09) _9 H9-8 9
((i- 09) H9_ 9 - i)2 G b a
+ (i - 09) TI0 H9_I0 HI0_I0(HI0_8a + a a
_8 HID-9 c9)]}
198
{A8 (i - p_) a a b aDS-Z0 M8 10 H8-10[1 + TZ0 G (i + ]=-- HI0_10 rl0 HI0_10)
a a [i + G b a+ A9 (i - p_) (i0 H9_I0 TI0 HI0_I 0 (i + rl0 HI0_I0)]}
1 a
B8 = _88 {AS (i - p8 ) TI0 rllG a b
}{8-10 HI0-11 + A9 (i - p9 ) TI0 TII G a9-10b }
HI0-11
C a a a a 2 a a b
=_ 010) TI0 010 ) HI0_10 HI0_10]DI0-1 M10 {AI0 'i HI0-1[(1 - + G((I - - I)
a G[ (i b b - i] a1 rl0 - Pl0 ) HI0-10 HI0_ I}
G.q_ a b G[(I a a bDZ0-S = MZ0 {At0 '5 TZ0 - 0Z0) HZ0_Zo - Z U_0_5
b b b 2 b b+ AI0 c5 HI0_5[(1 - + G (iPI0 ) rl0 - Pl0 ) HI0_10
a
z) azo_zo] }
O b a a bD10-6 M10 {Ac0 '6 TI0 G[ (1 Pl0) - i: _ - HZ0-10 "10-6
b b b b 2 G((I b b a+ AI0 (6 HI0-6[(1 - P 0 ) + TI0 Pl0 ) 50_10 i ]}- _ HI0_10
c a a a a a a 2 a aM10 {Al0(e 8 HI0_8 + L9 HI0_9) [(i - + G((IPlo) "qo - Ozo) a_o_zo -z)
b b G[(I b b - i]( a a a aHI0-10] + AI0 TI0 - PI0) HI0-10 8 H10-8 + _9 HI0-9)}
O a
:-- Pzo) _zo-lo - z] [z + Tzo G (1 + ]DI0_I0 MI0 {A10 el0[(l _ a a b aHI0-10 TI0 HI0_I0)
b b b Hb+ AI0 (i0[(i - Of 0 ) i0-i0 - i] [i + TI0 G a bHI0_I 0 (i + TI0 HI0_I0)]}
199
l_!- a G[(I a a - i] b
BI0 = Ml0 {AI0 TI0 _Ii - 010 ) lIl0-10 HI0-11
b G[(Z pzo) b - z] a b+ AI0 TI0 TII - HI0_10 HI0-10 HI0-11
b (i b b b+ AZO - OiO) zlZ HZO-Zl}
b a G[(I a a - I]1 T1 HI0-11 {AI0 Zl0 - 010 ) HI0-10
BI0 = MI0.
b (T_0 G a [(i b b - i] + i - b+ AI0 HI0-10 - 010 ) HI0-10 010 )}
_gOED_G PAGE PL#2_K NOT FDLM_hb
201
APPENDIX D
ERBE SCANNER THERMAL MODEL EQUATIONS
_I-¢,]_EDING PAGE BLANK NOT FILMF._
203
SCANNER MODEL
The scanner model was developed by the same methods which were used
for the non-scanner model development. The Equations (i) through (7) in
the Appendix C are used to generate the constants of the thermal model
Equation (24) in the Section 2.1.3.
To define the coefficients in the themal model, the enclosure coupling
technique was used as another alternative approach. The following pages
contain those coefficients defined by using the enclosure coupling method.
The attached figure shows the scanner model which has node description
and enclosure descriptions. The enclosure has the number which is circled
around. The numbers describing enclosure surfaces are underlined. In the
illustration of the coefficients, the word that explains the procedure of
derivation was omitted because they were already shown in Appendix C.
b ,jJ
204
V V
/._.1LI.J
C)
LI.I
2O5
i i
AI, i = MIRI_ 2 ' AI, 2 = MIRI_ 2
AI,3 AI,4 = AI, 5 = AI, 6 = AI, 7 = AI, 8 = AI, 9 = AI,10 = AI,II = AI,12 =
AI,I 3 = AI,14 =0
i , A2,2 _ (_2)(__i +____i +---_i +---_i +---_i + I-¢A2, I = MIR2_I = R2_I R2- 3 R2-4 R2- 7 R2- 8 R2-13
i , A2, 4 iA2, 3 = M2R2_3 = M2R2_4
A2, 5 = A2, 6 = 0
l I i
A2, 7 = M2R2_7 ' A2, 8 = M2R2_ 8 ' A2,13 = M2R2_I3
_A2, 9 = A2,10 = A2,11 = A2,12 = A2,14 = 0
i i
= 0 , A 3 = + M3R3_2 = M3R3_2
3,4 A3,5 = A3,6 = A3,7 = A3,8 = A3,9 = A3,10 = A3,11 = A3,12 = A3,13 = A3,14 = 0
• 1 =0
A4, I = 0 , A4, 2 = M4R4_2 , A4, 3
: _ (i (__!__i+.___!__z+ ___!_i)A4,4 _4"4) R4_ 2 R4-6 R4- 8
1 = 0A, : , A4, 7_,6 M4R4_ 6
A4"' = 0' 5
' A4 8 .=9
_4,9 = A4,10 = A4,11 = A4,12 = A4,13 = A4,14 = 0
M4R4-8
206
A5,I = A5,2 = AS,3 = AS,4 = 0
i i= , A5,6 =A5,5 MsRs_6 MsRs-6
A5,7 = A5,8 = A5,9 = AS,10 = AS,If = A5,12 = A5,13 = A5,14 = 0
_A6, I = A6,2 = A6,3 = 0
i I- , A6, 5 = M6R6_5A6,4 M6R6_ 4
A6,6 R6_ 4 R6_ 5 R6_ 8
i
A6,7 =" 0 ' A6,8 " M6R6_ 8
_A6, 9 = A6,10 = A6,11 = A6,12 = A6_13 = A6,14 = 0
• = 0 A7,2 = 1A7, I , M7R7_ 2
A7, 3 = AT, 4 = A7, 5 = A7, 6 = 0
A7,7 = _ (M_)( i_.___+ i ) , = iR7_ 2 R7- 8 ' A7, 8 M7R7_------_
_7,9 = AT,10 = A7,11 = A7,12 = A7,13 = A7,14 = 0
207
i i_'A = 0 A8 -- 0 A8 =8,1 ' ,2 = MsR8_2 ' A8,3 ' ,4 M8R8_4
i iAS,5 = 0 , AS,6 = MsRs_6 , AS,7 = MsRs_------_
=_ i___+ i__+!_+A8,8
R8_ 2 R8- 4 R8_ 6 R8_ 7 R8-10 R8-11
i i
- , A 8 -A8, 9 = 0 , AS,10 MsRs_I 0 ,ii MsRs_II
_A8,12 = A8,13 - A8,14 = 0
A9,1 = A9,2 = A9, 3 = A9, 4 = A9, 5 = A9, 6 = A9, 7 = A9, 8 = 0
i , A9,10 = 0 , A9,11 = 0 , A9,12A9, 9 = M9R9_I2
i
M9R9_I2
A9,13 = A9,14 = 0
"AI0, I = AI0,2 = AI0,3 = AI0,4 = AI0,5 = AI0,6 = AI0,7 = 0
i
AI0,8 - MIoRI0_8 _ , AI0_9 = 0
AI0_I 0 = _ (_i)(i+
M1o RI0-8
i i--+ )RI0-11 RI0-14
I =0 =0
_A10_I 1 - M10R10_I 1 A10-12 A10,13 A10,14
1
MIoRI0-1.4
208
All, I = All, 2 = All, 3 = AII,4 = AII,5 = AII,6 = AII,7 = 0
= i =0 i
AII,8 MIIRII_8 ' AII,9 ' AII,I 0 - MIIRII_I0' AII,I 2
_0
= AII,I 4 = 0 , AII,I I = - (.i)( l___!___'+ ___!___l)_ii,13 KII-8 RII-10
AI2,1 = A12,2 ffiA12,3 ffiA12,4 = A12,5 = A12,6 = A12,7 ffiA12,8 = 0
= i , AI2,10 = AI2,11 = 0 A12,12 _ iA12,9 MI2RI2_9 ' MI2RI2_9
AI2,13 ffi0 , AI2,14 = 0
i
AI3,1 ffi0 , A13,2 - MI3RI3_2 , A13,3 = A13,4 = A13,5 = A13,6
"0
A13,7 = A13,8 ffiA13,9 ffiAI3,10 = AI3,11 = AI3,12 = 0
i
_13,13 - MI3RI3_2 ' A13,14 = 0
AI4-1 = A14,2 = A14,3 = A14,4 = A14,5 = A14,6 = A14,7 = A14,8 = A14,9 ffi0
_ i _0 =0 =0
AI4 ,i0 MI4RI4_I0 ' AI4 ,ii ' AI4 ,12 ' AI4 ,13
Al4ml 4 -MI4RI4-10
209
For the total channel, the filter glasses are removed, Accordingly,
the (DI-_--) related with node number of filters have to set zero.
i-3
BQ = [0, 0, 0, 0, 0, 0, 0, M_' 0, 0, 0, 0, 0, 0]
Scanner:
Let
(Hkji)A = (_.k-j PJ-i)A '3
where subscript A denotes the enclosure A.
Then let
2 -i
G 1 = [i- TI3 (HI3jI3) D (H13m13) F]
2 GI )FSI = (H17mlT) F + TI3 (Hl7ml 3 (HI3jI3)D (HI3mI7)F
2 -i
G2 = [I- TI7 SI(HI7jlT) C]
2 SIS2 = (HI6jI6) C + TI7 G2(HI7jI6) C (HI6jI7)C
2 $2 )B]-IG 3 = [I - TI6 (Hl6ml 6
2
S3 = (Hl4ml4) B + TI6 S2 G3(HI4mI6) B (H16m14) B
2 -i
G4 = [I- TI4 S3 (HI4jI4) A]
210
XI = FI,I_ I + FI,2_ 2 + FI,4_ 4 + FI,5_ 5 + FI,6_ 6 + FI,8_ 8 + FI,9_ 9
+ FI,13_13
where
FI,I
FI,2
FI,4
= TI3 _i GI G2(HITmI3)F (HI3jl)D
2 GI )F= _2 G2 {(HI7m2)F + TI3 GI(HITmI3)F (HI3J2)D + TI3 (H17m13
(HI3jI3) D (H13m2) F}
2
= _4 G2 {(HI7m4)F + TI3 GI(HI7mI3)F (HI3mI3)D (HI3m4)F}
FI,5 = TIT _5 Sl G2(HITJ5)C
FI,6 = TI7 c6 Sl G2(HITj6)C
FI,8 = TI7 _8 Sl G2(HI7j8)C
FI, 9 = TI7 c 9 SI G2(HI7j9) C
FI,13 = 2 GI_13 G2 {(HITmI3)F + TI3 GI(HI7mI3)F (HI3jI3)D + TI3 (HITmI3)F
(HI3jI3) D (H13m13) F}
X 2 = F2,1_ I + F2,2_ 2 + F2,4_ 4 + F2,5_ 5 + F2,6_6 + F2,8_ 8 + F2,9_ 9 + F2,11_II
F2,12_12 + F2,13_13 + F2,14_14
211
where
= (HI6jlT) c (HITmI3)F (HI3jl) DF2,1 TI3 TIT GI G2 G3
F2,2 = TIT 2 G2 G3(HI6jI7)C {(HlTm2) F + TI3 GI(HI7mI3) F (HI3j2) D 2 GI+ TI3
(H17m13) F (HI3jI3)D (H13m2) F}
2 GIF2,4 = TI7 _4 G2 G3(HI6jI7)C {(HITm4)F + TI3 (HI7mI3)F (HI3mI3)D (HI3m4)F}
2 Sl 5)C}F2,5 = e5 G3 {(HI6j5)C + _17 Gz(HI6JI7)C (HI7j
F2,6 = _6 G3 {(HI6j6)C
F2,8 = (8 G3 {(HI6J8)C
F2, 9 = c9 G 3 {(HI6j9) C
2 SI (H1 6)C}+ TIT G2 (HI6JI7)C 7j
2 Sl 8)C}+ TIT G2(HI6jlT) C (HI7 j
2 SI G2 )C }+ TI6 S2(HI6m9)B + _17 (HI6jlT)C (HITj9 "
F2,11 = TI6 _ii S2 G3(HI6mlI)B
F2,12 = TI6 c12 S2 G3(HI6mI2) B
F2,132 GI= TIT El3 G2 G3(HI6jI7) C (H17m13) F {i + TI3 GI(HI3jI3) D + TI3
(HI3jI3) D (H13m13) F}
F2,14 = TI6 14 $2 G3(HI6mI4)B
212
v
El4 = UI,I_ I + UI,2_ 2 + UI,4R 3 + UI,5_ 5 + UI,6_ 6 + UI,8_ 8 + UI,9_ 9 + Ul,10_10
+ UI,II_II + UI,12_12 + UI,13_13 + UI,14_14 + UI,15E
Ul, I ffiTI6 G4(HI4mI6) B F2,1
UI, 2 = TI6 G4(HI4mI6) B F2,2
UI, 4 ffiTI6 G4(HI4mI6) B F2, 4
Ul,5 = _16 G4(HI4mI6)B F2,5
Ul, 6 ffi TI6 G4 (H14m16) B F2,6
ffi (H14m16) B F2,8U1,8 T16 G4
Ul,9 -= G4{YI6(HI4mI6) B F2, 9 + _9(HI4m9)B}
Ul,10 = TI0 S3 G 4 _Io(HI4jI0)A
Ul,ll = G4 {Cll(Hl4mll) B + TI6(HI4mI6) B F2,11}
Ul,12 = G4 {el2(Hl4ml2)B + TI4 c12
/
Ul,13 ffiTI6 G4(HI4mI6) B F2,13
S3(HI4jI2) A + TI6(HI4mI6)B F2,12}
Ul,14 = G4 {c 14(HI4mI4)B + T14 c14 S3(HI4jI4) A + TI6(HI4mI6) B F2,14}
Ul,15 ffiTI5 G4(HI4jI5) A
2i3
El4 = U2,1_ I + U2,2_ 2 + U2,4_ 4 + U2,5_ 5 + U2,6_ 6 + U2,8_ 8 + U2,9_ 9 + U2,10_I0
+ U2,11_II + U2,12_12 + U2,13_13 + U2,14_14 + U2,15 E
U2,1 = TI4 TI6 G4(HI4jI4) A (H14m16) B F2,1
U2, 2 = TI4 TI6 G4(HI4hI4) A (H14m16) B F2,2
U2,4 = TI4 YI6 G4(HI4JI4)A (HI4mI6)B F2,4
U2, 5 = TI4 TI6 G4(HI4jI4) A (H14m16) B F2,5
U2,6 = TI4 Y16 G4(HI4JI4)A (HI4mI6)B F2,6
U2,8 = TI4 z16 G4(HI4JI4)A (HI4mI6)B F2,8
U2, 9 = G4(HI4jI4) A {TI4 TI6(HI4mI6) B F2, 9 + T14 c9(H14m9) B}
= 2 S3 G4U2,10 _I0(HI4jI0)A {i + TI4 (HI4jI4) A}
U2,11 = G4(HI4jI4) A {TI4 TI6(HI4mI6) B F2,11 + TI4 Cil(Hl4mll) B}
U2,12
2
= TI4 TI6 G4(HI4jI4) A (H14m16) B F2,12 + _I2(HI4jI2)A[I + TI4 S3 G4
(HI4jI4)A] + TI4 _12 G4(HI4jI4)A (HI4mI2)B
U2,13 = TI4 YI6 G4(HI4jI4)A (HI4mI6)B F2,13
214
U2,14 -- (HI4JI4)A {TI4 el4 G4(HI4mI4)B + TI4 TI6 G4(HI4mI6)B F2,14 + _14
2 S3 G4 }[I + TI4 (HI4jI4) A]
U2,15 = TI5(HI4jI5) A {i + TI4 G4(HI4jI4) A}
A
El6 = U3,1f_ I + U3,2f22 + U3,4D,4 + U3,5_ 5 + U3,6f_ 6 + U3,Sf_8 + U3,9_ 9 + U3,10_I0
+ U3,11_II + U3,12_12 + U3,13_13 + U3,14_14 + U3,15E
U3,1 : TI4 TI6 S2 G3(HI6mI4) B U2,1 + TI3 TIT G1 G 2 G 3 _I(HI6jI7)C (H17m13) F
(HI3jl) D
U3,2 = TI4 TI6 $2 G3(HI6mI4)B U2,2 + TI3 TI7 _2 GI G2 G3(HI6jI7)C (HI7mI3)F
2
(HI3jI3)D + TI7 _2 G2 G3(HI6JI7)C (HITm2)F + TI3 TI7 c2 GI G2 G3
(HI6JI7) C (HITmI3)F (HI3JI3) D * (H13m2) F
U3,4 = TI4 TI6 $2 G3(HI6mI4)B U2,4 + TI7 _4 G2 G3(HI6jI7)C (Hl7m4)F
2
+ TI3 TI7 _4 GI G2 G3(HI6jI7)C (HI7nI3)F (HI3jI3)D (HI3m4)F
2 SI G2U3, 5 = TI4 TI6 S2 G3(HI6mI4) B U2,5 + G 3 c5(H16j5) C + TIT e 5 (HI6jI7) C
(HI7j5) C
2
U3,6 = TI4 TI6 G3(HI6mI4)B U2,6 + _6 G3(HI6j6)C + TI7 _6 Sl G2 G3 (HI6jI7)C
(HITj6) C
215
2U3,8 = TI4 r16 S2 G3(HI6mI4)BU2,8 + _8 G3(HI6jS)C + TIT c8 SI G2 G3(HI6JI7)C
(HI7j8)C
U3,9 = TI4 TI6 S2 G3(HI6mI4)B U2,9 + 2 _8 SI G29 G3(HI6j9)C + _17 G3(HI6JI7)C
(HITj9)C + YI6 _9 $2 G3(HI7j9)C
U3,10 = TI4 TI0 S2 G3(HI6mI4)B U2,10
U3,11 = YI4 TI6 S2 G3(HI6mI4)BU2,11 + TI6 Ell $2 G3(HI7JlI)C
= (H16m14 ,12 TI6 c12 G3(HI7jI2)CU3,12 TI4 TI6 $2 G3 )B U2 + $2
U3,13 = TI4 YI6 S2 G3(HI6mI4)B U2,13 + TI7 _13 G3(HI6JI7)C (HI7mI3)F
+ TI3 TI7 _13 GI G2 G3(HI6JlT)C (HI3jI3)D (HI7mI3)F[I + _13(H13 ml3)F]
= (H16m14) B U2,14 TI6 c14 G3(HITjI4) CU3,14 TI4 TI6 S2 G3 + S2
= (Hl6mi4) B U2,15U3,15 TI4 YI6 $2 G3
v
El6 = U 4,1_I + U4,2f_2 + U4,4_4 + U4,5_5 + U 4,6_Q6 + U4,8_Q8 + U4,9f_9 + U 4,10_-I0
+ U4,11_II + U4,12_12 + U4,13_13 + U4,14_14 + U4,15E
2 S2 G 3 + TU4, I = T14(Hl6mi4)B[l + TI6 (H16m16) B] U2,1 16 TI7 G3(HI6mI6)B
FI(HI6jI7)C ,i
216
2 S2 G3 U2 + G3U4,2 = TI4(HI6mI4)B [I + TI6 (HI6mI6)B] ,'2 TI6 TI7 (HI6mI6)B
(HI6j 17)C FI,2
2 $2 )B] U2 +U4,4 = TI4(HI6mI4)B [I + TI6 G3(HI6mI6 ,4 TI6 TI7 G3(HI6mI6)B
(HI6jI7) C FI,4
2 $2 B] U2 +U4,5 = TI4(HI6mI4)B[I + TI6 G3(HI6mI6) ,5 TI6 TIT G3(HI6mI6)B
(HI6j 17)C FI,5 + TI6 _5 G3(H]_6mI6)B(HI6j5)C
2 $2 B] +U4,8 = TI4(HI6mI4)B [I + TI6 G3(HI6mI6) U2,8 _16 TIT G3(HI6mI6)B
(HI6jI7)C FI,8 + TI6 _8 G3(HI6mI6)B (HI6j8)C
2 $2 G3 + G3U4,9 = TI4(HI6mI4)B[I + TI6 (H16m16)B] U2,9 TI6 TI7 (H16m16)B
(HI6jI7)C FI,9 + _9(HI6m9)B+ TI6 _9 G3(HI6mI6)B (HI6j9)C
2+ TI6 9 $2 G3(HI6mI6)B (HI6m9)B
2 $2 G3 U2U4,10 = TI4(HI6mI4)B[I + TI6 (HI6mI6)B] ,i0
U4,11 2 $2 )B] U2= TI4(HI6mI4)B[I + TI6 G3(HI6mI6 ,ii
2 $2+ TI6 ell G3(HI6mI6)B (Hl6mll) B
+ ell(Hl6mll)B
U4,12 = TI4(HI6mI4)B[I + T2 S2 G3 U216 (HI6mI6)B] ,12
2 $2 G3 )B+ TI6 C12 (H16m16 (Hl6mi2) B
+ _I2(HI6mI2)B
217
2 S2 G3 B] + r1.14,13= TI4(HI6mI4)B[I + TI6 (Hl6ml6) U2,13 16 TI7 C3(HI6mI6)B
(HI6jI7)C FI,13
22 $2 B] U2, + T _ S2 G3U4,14 = _I4(HI6mI4)B [I + r16 G3(HI6mI6) 14 16 14 (HI6mI6)B
(H16m14)B + (14(HI6mI4)B
2U4,15 = TI4(HI6mI4)B[I + TI6 S2 G3(HI6mI6)B] U2,15
El7 = U5,1_I + U5,2_2 + U5,4_4 + U5,5_5 + U5,6_6 + U5,8_8 + U5,9_9 + U5,10_I0
+ U5,11_II U5,12_12 + U5,13_13 + U5,14_14 + U 5,15E
= _ (HI7jI6)C ,iU5,1 i TI3 G1 G2(HI7mI3)F (HI3jl)D + TI6 TI7 SI G2 U4
= _ c (HI3jI3) DU5,2 2 G2(HI7m2)F+ 2 TI3 G1 G2(HI7mI3)F[(HI3j2)D + (H13m2)F
TI3] + TI6 TI7 SI G2(HI7jI6) C U4,2
U5,4 = _4 G2(HI7m4)F+ _2
4 TI3 G1 G2(HI7mI3)F (HI3jI3)D (H13m4)F + TI6 TI7
SI G2(HI7jI6) C U4,4
= _ (HI7jI6 ,5U5,5 5 TI7 SI G2(HI7j5)C + TI6 TI7 SI G2 )C U4
U5,6 = (6 TI7 SI G2(HITj6)C + TI6 TI7 SI G2(HITjI6)C U4,6
U5,8 = _8 TI7 Sl G2(HI7j8)C + TI6 TI7 Sl G2(HI7jI6)C U4,8
218
U5,9 = _9 TI7 Sl G2(HI7j9)C + "[16 "[17 SI G2(HI7jI6)C U4,9
U5,10 = TI6 TI7 SI G2(HITjI6) C U4,10
U5,11 = _16 TI7 SI G2(HI7jI6)C U4,11
U5,12 = TI6 TI7 S1 G2(HI7jI6) C U4,12
= (H17m13)FU5,13 el3 G2 + _13 TI3 GI G2(HI7mI3)F (HI3jI3)D[I + TI3(HI3mI3 )F]
+ TI6 TI7 SI G2(HI7jI6) C U4,13
U5,14 = TI6 TI7 SI G2(HI7jI6) C U4,14
U5,15 = TI6 TI7 SI G2(HITjI6) C U4,15
A
El7 --U6,1f21 + U6,2f_2 + U6,4f_4 + U6,5_ 5 + U6,6f_ 6 + U6,8f_ 8 + U6,9_ 9 + U6,10f_10
+ U6,11_II + U6,12_12 + U6,13_13 + U6,14_14 + U6,15E
U6,1 = _i TI3 TI7 GI G2(HI7jI7)C (HI7mI3)F (HI3jl)D + _16 G2(HI7jI6)C U4,1
U6,2 = _2 TI7 G2(HI7jI7)C (HI7m2)F + e2 TI3 TI7 GI G2(HI7j 17)C (HI7mI3)F
(HI7jI6 U4,2[(HI3j2) D + TI3(HI3jI3)D (HI3m2)F] + TI6 G2 )C
2
U6,4 = _4 TI7 G2(HI7jI7)C (HI7m4(F + _4 TI3 TI7 GI G2(HI7jI7)C (H17m13) F
(HI3jI3) D (H13m4) F + TI6 G2(HI7jI6) C U4,4
219
U6,5 = _5 G2(HI7j5)C + TI6 G2(HI7jI6)C U4,5
U6,6 -- E6 G2(HI7j6) c + TI6 G2(HI7jI6) C U4,6
= _ G2(HI + G2 ) U4U6,8 8 7j8)C TI6 (HI7jI6 C ,8
U6,9 = _9 G2(HI7j9)C + TI6 G2(HI7jI6)C U4,9
U6,10 --TI6 G2(HI7jI6) C U4,10
U6,11 -- TI6 G2(HI7jI6) C U4,11
U6,12 = TI6 G2(HI7jI6) C U4,12
U6,13 = _13 TI3 TI7 GI G2(HI7jI7)C (HI7mI3)F (HI3jI3)D[I + TI3(HI3mI3)F]
+ _13 TI7 G2(HI7jI7)C (HI7nI3)F + TI6 G2(HI7jI6)C U4,13
U6,14 = TI6 G2(HI7jI6) C U4,14
U6,15 = TI6 G2(HI7jI6) C U4,15
El3 = U7,1_ 1 + U7,2_ 2 + U7,4_ 4 + U7.5_ 5 + U7,6_ 6 + U7,8oQ 8 + U7,9_ 9 + U7,10_10
+ U7,11_II + U7,12_12 + U7,13_13 + U7,14_14 + U7,15E
U7,1 = _i GI(HI3jl)D + TI3 TI7 GI(HI3jI3)D (HI3mI7)F U6,1
220
U7,2 = _2 GI(HI3j2)D + ¢2 TI3 GI(HI3jI3)D (HI3m2)F + "[13 TI7 GI(HI3jI3)D
(H13m17) F U6,2
U7,4 = _4 TI3 GI(HI3jI3)D (HI3m4)F + YI3 TI7 GI(HI3jI3)D (HI3mlT)F U6,4
U7, 5 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,5
U7,6 = "[13 TI7 GI(HI3jI3)D (HI3mI7)F U6,6
U7, 8 = TI3 TIT GI(HI3jI3)D (H13m17) F U6,8
U7, 9 = TI3 TI7 G1 (HI3jI3) D (H13mlT) F U6,9
U7,10 = TI3 TI7 GI(HI3jI3) D (HI3mI7)F U6,10
U7,11 : TI3 TI7 GI(H13jI3) D (14k3mlT) F U6,11
U7,12 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,12
U7,13
U •
7,14
el3 GI(HI3jI3)D + _13 TI3 GI(HI3jI3)D (HI3mI3)F + TI3 TIT GI
(HI3jI3) D (H13m17) F U6,13
TI3 TI7 GI(HI3jI3) D (HI3mI7)F U6,14
U7,15 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,15
221
El3 = U8,1_I + U8,2_2 + U8,4_4 + U8,5_5 + U8,6_6 + U8,8_8 + U8,9_9 + U8,10_I0
+ U8,11_II + U8,12_12+ U8,13_13 + U8_1414 + U8,15E
U8,1 = _i TI3 GI!HI3mI3)F (HI3jl)D + TI7 GI(HI3mlT)F U6,1
U8,2 = E2 TI3 GI(HI3mI3)F (H13i2)D + 2 G2(Hlqm2_F+ TI7 GI(HI3mlT)F U6,2
U8,4 = c4 GI(HI3m4)F + TI7 GI(HI3mlT) F U6,4
U8,5 = TI7 GI(HI3mI7)F U6,5
U8,6 = TI7. GI(HI3mI7)F U6,6
= (H13m17 ,8U8,8 TI7 GI )F U6
U8,9 = TI7 GI(HI3mI7)F U6,9
U8,10 = TI7 GI(HI3mI7)F U6,10
= (H13m17)F U6,11U8,11 TI7 GI
U8,12 = TI7 G!(HI3mI7)F U6,12
= _ _ (H13m13)F TI7 GI(HI3mI7)F U6,13U8,13 13 TI3 GI(HI3mI3(F (HI3jI3)D + 13 G1 +
U8,14 = r17 GI(HI3mI7)F U6,14
U8,15 = r17 GI(HI3mI7)F U6,15
222
OAI_- (-_) [(i-_i)
DI,I_i (HIJl)D
- E + (I-Q I) T13 (HljI3)D US,I]S.L1
oAl + (1-pI) (Hlj 2]DI,2 = (__i)[(I_01) _2(HIj2)D "[13 13)D US, S,L
=0DI,3
_A I
DI, 4 = (-_i)[(I-0 I) TI3(HIjI3)D US,4]S,L
,_(_AI _I3(HIjI3) D U 8,5]S,L
DI, 5 = _--_;[(i-0 I)
°A I U 8
DI, 6 = (-_I)[(I-Pl) TI3(HIjI3)D ,6]S,L
=0DI,7
oAf _I3(HIjI3) D U8 _8]S9L
DI, 8 = (-MI)[(I-O I)
oA I
DI, 9 = (-_i)[(I-0 I) _I3(HIjI3)D US,9]S,L
___AI_I3(HIjI3)D U8,10]S,L
DI,10 = t-_l)[(l-o I)
aA 1
nl,ll = (-_i)[(I-0 I) TI3(HIjI3)D US,II]S,L
°At U8 ]S.= (__i)[(I-01) _I3(HIjI3)D ,12 LDI,12
(_AI + (I-0 I)
DI,13 = (-_i)[(I-0 I) _I3(HIjI3)D UI,13
<_AI
DI,14 = (--_I)[(I-01) _I3(HIjI3)D U8'I4]S'L
13(HIjI3)D]S,L
223
D2,1 = (_2) {A2D(I-02)[eI(H2jl)D + TI3(H2jI3) D U8, I] + A2F(I-P2)[TI3
(H2m13) F UT,I + TI7(H2mI7)F U6,1]}S, L
D2, 2 -- (_2) {A2E[(I'o 2) _2(H2j2)E- e2 ] + A2D[(I-o 2) _2(H2j2 ) -c 2
+ (1-02) TI3(H2jI3)D U8,2] + A2F|(I-o2) c2(H2m2)F - _2 + (1-02)
+ (1-02) T13 *(H2m13) F U7, 2 + (1-02 ) TI7(H2mI7) F U6,2]}S, L
D2, 3 = (_2)[A2E(I-02) e3(H2j3)E]S,L
D2, 4 = (_2) {A2D(J-o 2) TI3(H2jI3) D U8, 4 # A2F[(I-02)_4 (H2m4)F + (1-02) TI3
(H2m13) F U7,4 + (1-02) TIT(H2mI7)F U6,4]}S, L
D2,5 = (_2){A2D (1-02) TI3 (H2jI3) D U8,5 + A2F(I-P2) [TI3(H2mI3)F U7, 5 + TIT
(H2m17) F U6,5]}S,L
D2, 6 = (_2) {A2D(I-o 2) TI3(H2jI3) D U8,6 + A2F(I-02)[TI3(H2mI3)F U7, 6 + TI7
(H2m17) F U6,6]}S, L
D2, 7 = (_2)[A2E(I-O 2) e7(H2j7)E]S,L
D2, 8 = ( ) {A2D(I-P2 ) TI3(H2jI3) D U8,8 + A2F(I-02)[YI3(H2mI3) F UT, 8 + TI7
(H2m17) F U6,8]}S, L
224
oD2,9 = (_-_2){A2D(I-o 2) Ti3(H2jI3) D U8,9 + A2F(I-O2)[TI3(H2mI3) F U7,9
+ TI7(H2mI7)F U6,9]}S,L
D2,10 = (_2) {A2D(I-P2) TI3(H2jI3) D U8,10 + A2F(I-o2)[TI3(H2mI3) F U7,10
+ TIT(H2mI7)F U6,10]}S,L
D2,11 = (-_2) {A2D(I-P2) TI3(H2jI3)D U8,11 + A2F(I-P2)[TI3(H2mI3) F UT,ll
+ TI7(H2mlT)F U6,11]}S,L
D2,12 = (_22) {AzD(I-P2) TI3(HzJ 13)D U8,12 + A2F(I-P2)[TI3(H2mI3)F U7,12
+ TI7(H2mI7) F U6,12]}S,L
D2,13 = (-_2) {A2D(I-P2)[eI3(H2jI3)D + TI3(H2jI3)D'U8,13] + A2F(I-P2)[_I3
(H2m13) F + TI3(H2mI3) F U7,13 + 17(H2mI7)F U6,13]}S,L
D2,14 = (_2) {A2D(I-o 2) TI3(H2jI3) D U8,14 + A2F(!-O2)[TI3(H2mI3) F U7,14
+ TI7(H2mI7) F U6,14]}S,L
D3, I = 0
D3, 2 = (_._3)A3[(I-D 3) _2(H3j2)E]S,L
225
D3,3
D3,4 = D3, 5
A3[(I-03) , 3(H3j3)E - _3]S,L
=0= D3, 6
D3,7
D3,8
A3[(1-03 ) ¢7(H3j7)E]S,L
= D3,10 = D3,11 = D3,12 = D3,13 = D3,14 = 0
D4,1
D4,2
oA 4
oA 4
--
{(I-04)[TI3(H4mI3)F U7,1 + TI7(H4mI7) F U6,1]}S,L
+ TI7(H4mI7) F U6,2]}S,L{ (1-04) [E2(H4mZ)F + _I3(H4mI3)F U 7,2
D4,3
D4,4
--0
oA 4
(--_4) { (I-04) _ 4(H4m4)F
(H4m17) F U6,4}S,L
- _4 + (1-04) TI3 (H4m13) F U7,4 + (l-p 4) TI7
D4,5
D4,6
oA 4 + _ U 6 5 ]}
(__4) {(I_04)[TI3(H4mI3)F U7, 5 17(H4mI7)F , S,L
(_4){ U7 U6,6]}(i-04) [TI3 (H4m13) F ,6 + TI7 (H4m17) F S,L
=0D4,7
D4,8
oA 4-- { (1-04 ) [_I3(H4mI3)F U7,8
= (M4)
+ TI7(H4,17) F U6,S]}S,L
226
D4,9 =
D4,10
D4,11
D4,12 =
D4,13 =
D4,14 =
OA4 U7(___44){ 41-04) [_13(H4mI3)F ,9
+ TI7(H4mI7)F U6,9]}S,L
41-04) [_13(H4m13)F ,i0(_4){ u7
(_A4
(__4) { (1-P4) [_I3(H4mI3)F U7 ,ii
(1-P4) [_13(H4m13 ) U7,12(_4){ .F
[_I3(H4mI3)F + _I3(H4mI3)F U7,13
(_A4 U 7
(__4) { (!-p4) [_Z3(H4mI3).F ,14
+ rl7(H4ml7)F U6,10]}S,L
+ _I7(H4mI7)F U6,11]}S,L
+ TI7(H4mI7)F U6,12]}S,L
+ TI7(H4mI7)F U6,13] (I-P4)}S,L
+ _17(ll4ml7)F U6,14]}S,L
D5,1
D5,2
_- { (l-P5) [TI6 (l_15j16) c U4,1
°A 5 U 4
= (__5) { (i-05) [T16 (H5j 16) ¢ ,2
= _lT(H5j17)c U5,1]}S,L
+ _17(H5j17)c U5,2]}S,L
D5,3 = 0
+ _IT(H5jI7) c U(_A5 U 4
D5,4 = (___5){(l-05)[_16(H5j16)c ,4
D5,5
(_A5 U 4
= (___5){ (i-05) [T16 (H5j 16) c ,5
+ _17(H5]17 )
(H5j5) c - _5]}S,L
oA 5 + ,6
D5, 6 = (-_5){(I-P5)[_6(H5j6)c TI6(H5jI6)c U4
5,4]}S,L
c u5,5] + [(t-Ps) _5
+ _17(H5j17)c U5,6]}B,L
227
=0D5,7
°A5 cD5, 8 -- (--_5){(I-05)[Es(H5j8) + _16(H5j16)c U4,8
+ TI7 (H5jI7) c U5 ,8]}S,L
D5,9= (i-05) [_9 (H5j 9) c r16 (H5jl6)c U4,9
+ _17(H5j17)c U5,9]}S,L
_- °A5 +D5,10 (--_5){(I-05)[TI6(H5j 16)c U4,10 TIT(H5jI7)c U5,10]}S,L
D5,11
oA 5
= (--_5){(i-05) [_16(H5j16)c U4,11+ TI7(H5jI7) c U5,11]}S,L
oA 5 +
D5,12 = (-_5){(I-05)[TI6(H5jI6)c U4,12 TI7(H5jI7)c U5,12]}S,L
D5,13 (O--AM_){ c= . (1-05) [TI6(H5jI6) U4,13 + TI7(H5jI7) c U5,13]}S,L
DS, 14
_A 5
= (--_5){(I-05)[II6(HDjI6)c U4,14+ TI7(H5jI7) c U 5,14])S,L
D6,1°A6 U 4
= (--_6)((i-06) [TI6 (H6j 16) c ,i+ TI7(H6jI7) c U5,1]}S,L
$6,2
oA 6
= (-_6){(I-06)[TI6(H6jI6)c U4,2+ TI7(H6jI6) c U5,2]}S,L
=0D6,3
D6,4= (1-06)[TI6(H6jI6) c U4,4 + %17(H6jlT)c U5,4]}S,L
228
D6,5 (o_) { c= (i-06)[_5(H6j5)c + TI6(H6jI6)c U4,5 + TI7(H6j17) U5,5]}S,L
oA6 c -
(-_6){(I-06)[¢6(H6j6) c + TI6(H6jI6) c U4,6 + _17(}{6j17 ) U5,6 C6}S,L
=0D6, 7
aA6 + +
D6, 8 = (--_6){(I-06)[_8(H6j8)c _I6(H6jI6)c U4,8 TiT(H6jI7) U5,8]}S,L
D6,9 (°_--66){ +_- (1-06) [E9(H6jg) e Tl6(H6jl6)c U4,9+ _17(H6j17)c U5,9]}S,L
aA 6 +
D6,10 = (7-_6){(I-06)[TI6(H6hI6) e U4,10 _17(H6j17)c U5,10]}S,L
D6,11
oA 6
= (-_6)( (1-06 ) [TI6(H6jI6) c U4,11
+ TI7(H6jI7)c U5,11]}S,L
oA6 cD6,12 = (-_6){(I-06)[TI6(H6jI6) U4,12 + TI7(H6jlT) c U5,12]}S,L
D6,13 (_6){= (1-06) [_16(H6j16)c U4,13+ YI7(H6jI7)c U5,13]}S,L
D6,14
oA 6
= (-_6){(I-P6) [TI6(H6jI6)c U4,14+ TI7(H6jI7) c U5,15]]S,L
=0D7,1
_A 7
D7,2 = (-_7)E{(1-07) _2(H7j2)E}S'L
229
aA 7
D7, 3 = (-_7)E{(I-07) 3(H7j3 )E}S,L
= 0= D 7
D7, 4 = D7, 5 ,6
oA 7
D7,7 = (-_7) f (l-P7)E
D7, 8 = D7, 9 = D7,10
- _7,}7(H7j7)E
= D 7= D7 ,ii ,12
S,L
= D7,13 = D7,14
=0
D8,1
D8,2
: (1_08) [TI6 (HsjI6) c ,i(_s){ u4
oA 8
: (__8) { (1_08) [TI6(H8jI6) c U4,2
+ "[17(Hsj17)c U5,1]}S,L
+ TI7(HsjI7)c U5,2]}S,L
D8,3
D8,4
D8,5
D8,6
=0
: (l_P8) [-[16 (H8jI6)c ,4(_s){ u4
oA 8 +
: (-_S) { (L-_8) [_ 5(HSj 5)c 16(Hsj 16)
8j17) u 41}s+ TI7( H c 5, ,L
U 4c ,5
oA8 (H8j 6) + TI6(HSjI6)c U4,6
(-_8) { (I-P 8) [¢ 6 c
+ TI7(Hsjl7)c U5,5]}S,L
+ TI7(HBjI7)c U5,6]}S,L
=0D8,7
_A 8 + .
D8,8 = (__8){(l_P8.)ie8(H8j8) c TI6(HSjI6)c U4,8
+ TI7 (H8jI7)c U5,8] - _8}$'L
230
oA8D8,9 = (--_8)
{ (l-f) 8) [e9(I18j9)¢ + T16(liSj16) c U4,9 + _17(H8j17)c U5,9]}S,L
OA8 + cDS,10 = (--_8){(l-08)[T16(li8j16)¢ U4,10 _I7(H8jI7 ) U5,10]}S,L
D8,11 (_s){= (i-08) ['[16(H8j16)c U4 ,ii+ _lT(Hsj17)c U5,11]}S,L
D8,12 - (_8){ (1-08 ) [_16(H8j16)c U4,12
+ "[17(H8jlT)c U5,12]}S,L
gAs c +D8,13 = (-_8){(I-08)[_I6(H8jI6) U4,13 "_I7(H8JlT)c U5,13]}S,L
D8,14
OA 8
-- (--_8){ (i-08) [Tl6(liSjl6)c U4,14
+ _17 (H8j17) c U5,14]}S,L
D9,1
D9,2
+ T16(l'19m16)B U3,1] + A9c(i-09) c
= (_9){A9B(I-P9)B[_I4(H9mI4)B U2,1
+ "[17(H9jlT)c U5,1]}S,L[TI6(H9jI6)c U4,1
-- (_9){A9B(I-P9)B[_I4(H9mI4)B U2,2 + Tl6(li9ml6) B U3,2] + A9c(I-P9) c
+ TI7(H9jI7)c U5,2]}S,L[_16 (H9jI6)c U4,2
D9,3
D9,4
--0
+ "[16(H9mI6)B U3,4] + A9c(I-P9) c
(_9){A9B(I-o9B[TI4 (H9mI4)B U2,4
+ _[17(H9j17)c U5,4]}S,L['[16(H9j 16) c U4,4
231
D9,5 (_9){A9B(I-O9)B[TI4(H9mI4)B U2,5 + TI6(H9mI6)B U3,5] + A9¢ (l-P9) c
[E5(H9j5)c + TI6(H9jI6) c U4,5 + TI7(H9jI7)c U5,5]}S,L
D9,6 (M9__)o{A9B(I-P9)B[TI4(H9mI4) B U2,6 + TI6(H9mI6)B U3,6] + A9c(l-P9)c
[E6(H9j6)c + TI6(H9jI6) c U4,6 ÷ TI7(H9jlT) c U5,6]}S,L
D9,7 = 0
D9,8 (_9){A9B(!-P9)B[TI4(H9mI4)B U2,8 + TI6(H9mI6)B U3,8] + A9c(l-P9)c
[_8(}{9j6)c + T16(H9jl6)c U4,8 + TI7(H9jI7) c U5,8]}S,L
D9,9 = (%){A9B(I-09)B[_9(H9m9)B + TI4(H9mI4)B U4,9 + TI6(H9jI6)c U5,9] - A9B_9
+ A9c(i-O9)c[C9(H9j 9) + c U4 + U5 ] -c TI6 (H9jI6) ,9 TI7 (H9jI7)c ,9 A9c 9}S,L
D9,10 (_9){A9B(I-P9)B[TI4(H9mI4)B U2,10 + TI6(H9mI6)B U3,10]
(H9jI6) c U4,10 + TI7(H9jI7) c U5,10]}S,L
+ A9c[TI6
D9,11 = (%){A9B(I-O9)B[ClI(H9mlI)B = TI4(H9mI4)B U2,11 + TI6(H9mI6)B U3,11]
+ A9c(l-09)c[_16(H9jl6)c U4,11 + TI7(H9jI7) c U5,11]}S,L
D9,12 (_9) {A9B(1-09) B[e12(H9m12)+ U2B TI4 (H9m14)B ,12 + TI6(H9mI6)B U3,12]
+ A9c(I-O9)c[TI6(H9jI6) c U4,12 + TI7(H9jI7) c U5,12]}S,L
232
D9,13
D9,14
(_9){A9B(I-09)B[TI4(H9mI4)B U2,3 + "c16(H9m16) m
[TI6(H9jI6)c U4,13 + TIT(H9jI7) c U5,13]}S,L
U3,13 ! + A9c(I-09) c
+ TI6(FI9mI6) B U3,14]
= (_9){A9B(I-09)B[_ 14(H9mI4)B + TI4(H9mI4) B U2,14
+ A9c(I-09)c[TI6(H9jI6) c U4,14 + TI7(H9jI7) c U5,14]}S,L
DI0,1
DI0,2
oAI0
--{(i-010)= ( MZ0 )
oaf0
: (Tzo){(z-Ozo)
TI4 (HI0jI4)A UI,I}S,L
TI4(HIoJ14 )A UI,2 }S,L
=0DI0,3
DI0,4
DI0,5
DIO,6
oAI0
: (Tzo){(z-Ozo)
OAI0
-- {(I-010): ( MI0 )
_AI0
: (--_i0){(1-010)
TI4 (HI0jI4)A UI,4}S,L
"[14 (HI0jI4)A UI, 5} S,L
TI4 (HI0j I4)A UI,6}S,L
=0DI0,7
DI0,8
DI0,9
°Al0
: (TlO){ (1-olO)
OAl0
: (_i0 ){ (i-010)
TI4 (HI0j 14) A UI,8}S,L
TI4(HI0jI4)A UI,9}S,L
233
(_AI0 _ +DI0,10 = (-_I0)[(I-PI0) _I0(HI0jI0)A el0 (I-P10) TI4(HI0jI4)A UI,10}S,L
oAI0DI0,11 = (-_I0){(I-PI0) _I4(HIojI4)A UI,II}
(_AI0{(i-01 O) TI4(HI0jI4)A UI }SDI0,12 = (.MIo) [eI2(HI0jI2)A + ,12] ,L
°AloDI0,13 = (-_i0){(i-010) TI4(HIojI4)A UI,13}S,L
(_AI0DI0,14 = (-_i0){ cI4(HIOjI4)A
+ _I4(HIojI4)A UI,14}
DII,I_AII { (i-011) U2
= (MII--) [TI4(HIImI4)B ,i+ TI6(HIImI6) B U3,1]}S,L
oAII +DII,2 = (--_II){(I-PlI)[_I4(HIImI4)B U2,2 YI6(HIImI6)B U3,2]}S,L
DII,3 = 0
oAII += (Hllml4) BDII,4 (--_II){(I-oII)[TI4 U2,4 _I6(HIImI6)B U3,4]}S,L
°AllDII,5 = (-_II){(I-PlI)[TI4(HIImI4)B U2,5 + 16(HIImI6)B U3,5]}S,L
DII,6°All U2-- { (l-Pil) IT14(Hllml4) B ,6
= ( MII )+ TI6(HIImI6) B U3,6]}S,L
=0DII,7
DII,8OAII U 2-- {(I-PlI)[TI4(HIImI4)B ,8
= ( M11)+ _I6(HIImI6)B U3,8]}S,L
234
°All + +DII,9 " (-_II){E9(HIIm9)B YI4(HIImI4)B U2,9 TI6(HIImI6)B U3,9]}S,L
OAll= (IMI-){_I4(HIImI4)B U2'I0DII,10
+ TI6(HIImI6) B U3,10]}S,L
°All= (--Mll){(l-Pll) ell(Hllmll)B - _II + TI4(HIImI4)B U2,11DII, ii
+ TI6(HIImI6) B U3,11}S,L
°All= -- { (I-P11) [_12(HIImI2)B + TI4(HIImI4) B U2,12
Dli, 12 ( Mil )
+ TI6(HIImI6) B U3,12]}S,L
DII,13
DII,14
°All-- {(i-01 I)[_I4(HIImI4)B U2,13 + _[16(HIImI6)B U3,13]}S,L
= ( MII )
OAII
= (--_II){(I-011)[_I4(HIImI4)B + TI4(HIImI4)B U2,14
+ TI6(HIImI6) B U3,14]}S,L
DI2,1
D12,2
o {AI2A(I_oI2) A _14 (HI2j 14 )= (MI2)
+ TI6(HI2mI6) B U3,1]}S,L
A UI,I
= (M_2){AI2A(I-PI2) A TI4
+ TI6(HI2mI6) B U3,2]}S,L
(HI2j 14)A UI,2
+ AI2B(1-OI2)B[_I4(HI2mI4)B U2,1
+ AI2B(I-012)B[TI4(HI2mI4) B U2,2
=0D12,3
235
D12,4 = (M_2){AI2A(I-012) A TI4(HI2JI4)A UI,
+ TI6(HI2mI6)B U3,4]}S,L
4 + AI2B (I-012) B[TI4 (HI2mI4)B U2'4
D12,5 = (M_2) AI2A(I-012)A _I4(HI2JI4)A UI,5
+ _I6(HI2mI6)B U3,5]}S,L
+ AI2 B (1-012) B[TI4 (H12m14) B U2,5
D12,6 = (M_2){AI2A(I-012) A rI4(HI2jI4)A UI
_+ TI6(HI2mI6) B U3,6]}S,L
,6 + AI2B(I-012)B[TI4(HI2mI4)B U2,6
D12,7 = 0
D12,8 = (M_2){AI2A(I-oI2) A TI4(HI2JI4)A UI,8
+ _I6(HI2mI6)B U3,S]}S,L
+ AI2B(I-oI2)B[TI4 (HI2mI4)B U2,8
D12,9
DI2,10
(_2) {AI2A(I-012) ATI4 (HI2JI4)A UI,9 + AI2B(I-012) B [E9 (H12m9) B
+ TI4(HI2mI4) B U2,9 + TI6(HI2mI6)B U3,9]}S,L
-- (M_2){AI2A(I-012)A[_I0(HI2jI0) A + TI2(HI2jI4) A UI,10] + AI2B(I-oi2) B
[TI4(HI2mI4) B U2,10 + TI6(HI2mI6)B U3,10]}S,L
DI2,11 = (M_2){AI2A(I-012) A TI4(HI2jI4)A UI,II + AI2B(I-012)B[¢II(HI2mlI) B
+TI4(HI2mI4)B U2,11 + TI6(HI2mI6) B U3,11]}S,L
236
D12,12 = (M_2){AI2A(I-oI2)A[_I2(HI2jI2) A + TI4(HI2jI4)A Ul,12] - AI2A El2
++ AI2B(I-PI2)B[_I2(HI2mI2) B _I4(HI2jI4)B U2,12 + TI6(HI2mI6B
U3,12] - AI2B eI2}S,L
D12,13 = (M--_2){AI2A(I-PI2)A TI4(HI2jI4)A UI,13
+ TI6(HI2mI6) B U3,13]}S,L
+ AI2B(I-PI2)B[TI4(HI2mI4) B U2,13
DI2,14 (M_2){AI2A(I-012)[_I2(HI2JI4)A + YI4(HI2jI4)A Ul,14 + AI2B(I-oI2) B
[_I4(HI2mI4)B + TI4(HI2mI4) B U2,4 + TI6(HI2mI6) B U3,14]}S,L
DI3,1 = (_3){AI3F(I-PI3)F[TI7(HI3mlT)F U6,1 + TI3(HI3mI3) F U7,1] - AI3 F U7,1
+ AI3D(I-oI3)D[_I(HI3jl) D + TI3(HI3jI3) D U8,1] - AI3 D TI3 U8,1}S,L
D13,2 = (M---_3){AI3F(I-013)F[TI7(HI3mlT) F U6,2 + TI3(HI3mI3) F U7,2
+ Ez(HI3m2) F] - AI3 F TI3 U7,2 + AI3D(I-013)D[C2(HI3j2) D
+ TI3(HI3jI3) D U8,2] - AI3 D TI3 U8,2}S,L
D13,3 = 0
D14,4 -- (M-_3){AI3F(I-PI3)F[TI7(HI3mI7) F U6,4 + TI3(HI3mI3) F + e4(Hl3m4)F]
_13 U7,4 + AI3D[(I-PI3)D TI3(HI3jI3)D - TI3] U8,4}S,L
- AI3 F
237
D13,5 (M-_3){AI3F(I-013)F[TI7(HI3mI7)F U6,5 + TI3(HI3mI3) F U7,5] - AI3F
%13U7,5 + AI3D[(I-oI3)D TI3(HI3j 13)D - %13] U8,5}S,L
D13,6 (M-_3){AI3F(I-PI3)F[TI7(HI3mI7)F U6,6 + %I3(HI3mI3)F U7,6] - AI3F
%13U7,6 + AI3D[(I-013)D %I3(HI3j 13)D - %13] U8,6}S'L
D13,7 = 0
D13,8 U6,8 + 8](M--_3){AI3F(I-PI3)F[%I7(HI3mI7)F %I3(HI3mI3)F U7,
%13U7,8 + AI3D[(I-QI3)D TI3(HI3j 13)D - %13] U8,8}S,L
- AI3F
D13,9 = (¢?{AI3F(I-013)F[TI7(HI3mI7) F U6,9 + YI3(HI3mI3)F U7,9] - AI3F
%13U7,9 + Ai3D[(I-PI3)D %I3(HI3j 13)D - TI3]U8,9}S,L
DI3,10 (M--_3){AI3F(I-oI3)F[YI7(HI3mI7)F U6,10 + %I3(HI3mI3)F U7,10] - AI3F
%13 U7,10 + AI3D[(I-PI3)D %I3(HI3j 13)D - %13] U8,Z0}S'L
DI3,11 (M_3){AI3F(I-PI3)F[TIT(HI3mI7)F U6,11 + %I3(HI3mI3)F U7,11]
%13 U7,11 + AI3D[(I-013)D %I3(HI3j 13)D - TI3] U8,11}S'L
- AI3 F
D13,12 (M-_3){AI3F(I-PI3)F[TI7(HI3mI7)F U6,12 + %I3(HI3mI3)F U7,12]
Y13 U7,12 + AI3D[(I-013)D YI3(HI3j 13)D - TI3] U8,12}S'L
- AI3 F
238
D13,13 = (M_3){AI3F(I-oI3)F[TI7(HI3mI7)F U6,13 + TI3(HI3mI3)F U7,13 + _13
(H13m13)F] - AI3F(eI3 + r13 U7,13) + AI3D(I-oI3)D[_I3(HI3jI3) D
+ TI3(HI3jI3)D U8,13] - AI3D (el3 + TI3 U8,13)}S,L
DI3,14 = (M_3){AI3F(I-oI3)F[TIT(HI3mI7) F U6,14 + TI3(HI3mI3)F U7,14] - AI3F
TI3 U7,14 + AI3D[(I-013) D 13(HI3jI3)D - TI3] U8,14}S,L
DI4,1 = (#4){AI4A TI4[(I-014)(HI4jI4)A - i] Ul, I + AI4B[(I-014) TI4
(H14m14) B - r14] U2, I + AI4 B r16(l-014)(H14m16)B U3,1}S, L
D14,2 = (M--_4){AI4A rl4[(l-Pl4)(Hl4jl4)A - i] UI, 2 + AI4B[(I-014) TI4
(H14m14) B - TI4] U2, 2 + AI4 B T16(I-014 )(H14m16) B U3,2}S,L
D14,3 = 0
D14,4 = (M--_4){AI4A TI4[(I-oI4)(HI4jI4) A - i] UI, 4 + AI4 B T14[(I-014)
(H14m14) B - i] U2, 4 + AI4 B TI6(I-014)(HI4mI6) B U3,4}S, L
D14,5 = (M_4){AI4 A TI4[(1-014)(HI4jI4) A - l] Ul, 5 + AI4 B T14[(1-014)
(H14m14) B - l]U2, 5 + AI4 B TI6(I-014)(HI4mI6) B U3,5}S, L
239
D14,6 (M_4){AI4A TI4[(1-014)(HI4jI4) A - I] UI, 6 + AI4B T14[(1-014)
(H14m14)B - i] U2,6 + AI4B TI6(I-014)(HI4mI6)B U3,6}S,L
D14,7 = 0
D14,8 (M_4){AI4A -[14[ (1-014)(H14h14)A - i] UI, 8 + AI4B TI4[ (I-P14)
(H14m14)B - i] U2,8 + AI4B TI6(I-014)(HI4mI6)B U3,8}S,L
DI4,9 (M_4){AI4A _I4[(I-oI4)(HI4j 14)A - i] UI, 9 + AI4B(I-014) B[E9(H14m9)B
+ TI4(HI4mI4) B U2,9 + TI6(HI4mI6)B U3,9] - AI4B TI4 U2,9}S,L
DI4, i0 ffi (M_4){AI4A(I-014)[_Io(HI4jI0)A + TI4(HI4jI4) A UI,IO]
+ AI4B(I-PI4)[TI4(HI4mI4) B U2,10 + TI6(HI4mI6)B U3,10]
- AI4A TI4 UI,IO
- AI4B TI4 U2,10}S,L
DI4,11 = (M_4){AI4A TI4[(I-0i4)(HI4jI4)A - i] UI,II + AI4B(I-014)[ClI(HI4mlI) B
+ TI4(HI4mI4) B U2,11 + TI6(HI4mI6) B U3,11] - AI4B TI4 U2,11}S,L
D14,12 = (M_4){AI4A(I-Pi4)[eI2(HI4jI2) A + TI4(HI4jI4)A Ul,12] - AI4A TI4 UI,2
+ AI4B(I-014)[cI2(HI4mI2) B + TI4(HI4mI4)B U2,12 + TI6(HI4mI6)B U3,12]
- AI4B TI4 U2,12}S,L
D14,13 = (M_4){AI4A TI4[(I-oI4)(HI4jI4) A - i] UI,13 + AI4B(I-oI4)[TI4(HI4mI4) B
U2,13 - TI6(HI4mI6)B U3,13] - AI4B TI4 U2,13}S,L
240
DI4,14 = (M-_4){AI4A[(I-014)(HI4jI4) A - i](c14 + TI4 UI,14) + AI4B(I-014)
[_ )B + ) U2 + U314(H14m14 TI4(HI4mI4 B ,14 TI6(HI4mI6)B ,14] - AI4B
(_14 + TI4 U2,14)}S,L
(BI)s, L = (M_)[AI(I-0 I) TI3(HIjI3) D U8,15]S, L
(B2)s, L = (M_)[A2D(I-02) TI3(H2jI3) D U8,15 + A2F(I-02){TI3(H2mI3) F U7,15
+ TI7(H2mI7)F U6,15}]S, L
(B4)s,L =(M_)[A4(I-o4){_I3(H4mI3)F U7,15 + TI7(H4mlT)F U6,15}] S,L
(B5)s, L = (M_)[A5(I-05) {_16(H5j16)c U4,15 + _17(H5j17)c U5,15}]S,L
(B6)s, L = (M_)[A6(I-o6){TI6(H6jI6) c U4,15 + TI7(H6jI7) c U5,15}]S,L
(B8)s, L = (M_)[As(I-O8){TI6(H8jI6) c U4,15 + TI7(H8jI7) c U5,15}]S, L
(B9)s, L = (M_)[A9B(I-o9)(TI4(H9mI4) B U2,15 + TI6(H9mI6)B U3,15} + A9c(l-09)
{TI6(H9jI6)c U4,15 + TI7(H9jI7)c U5,15}] S,L
(BI0)S, L = (M-_0)[AI0(I-010){TI4(HI0jI4) A UI,15 + TI5(HI0jI5)A}]S, L
(BII)S, L = (MI-_)[AII(I-011){TI4(HIImI4) B U2,15 + TI6(HIImI6) B U3,15}]S,L
241
(BI2)S, L = (M_2)[AI2A(I-012){TI4(HI2jI4) A UI,15 + TI5(HI2jI5) A} + AI2B(I-012)
{TI4(HI2mI4)B U2,15 + rl6(Hl2ml6)B U3,15}]S, L
(BI3)S, L = (..i)[AI3 D TI3 { - I}U8, + { )m13 (1-013) (HI3jI3)D 15 AI3F TI3 (i-013
(H13m13)F - i}U7,15 + AI3F(I-PI3) TI7(HI3mI7) F U6,15]S,L
(BI4)S, L (MI_)[AI4A YI4{ ) )A I}UI, += (1-014 (HI4jI4 - 5 AI4A(I-oI4) YI5
(HI4jI5) A + AI4 B TI4((I-014)(HI4m14) B - I}U2,15 + AI4 B TI6
(l-P14) (H14m16) B U3,15 ]S, L
For the total channel,
(Bi)s, L B.1
TI3 = TI4 = i ,
:" c = 0 .13 14
Scanner Model Equations
Total Channel
" AijT j 4 + BQiQ + B.E + b + eT.1 = + Dij Tj I
SW/LW Channels
Ti = A._jT.3 +D..T4_3 j + BQ iQ + (Bi)sEs + (Bi)LEL + b + e
242
Scanner (using Enclosure Coupling Method)
E
o
:i
i2
_3
_4
_5
6where m = 6
For each cavity (or enclosure in the scanner model)
EE_I i
ETI E ETI E
. -- _I
E E
L _
v
i
Es= EI!
E°f
i i i i 1 _ 1 i
where E1 = El5 , E2 = El0 , E3 = EI__ , E4 = El2
2 2 2 2 2 2 2 2 2
E1 = El4 , E 2 = Ell , E3 = Ek6 , E4 = E9 , E5 = El2
3 3 _ 3 3 ,3 3 3 3 3 3 3
E1 = EIO ' E2 = E8 ' E3 = '6 ' E4 = E5 ' E5 = El7 ' E6 = E9
4 4 E_ 4 4 _4 E4 4E1 = El7 ' _ = E4 ' E3 = '2 ' 4 = El3
5 5 , E_ 5 5 5E1 = El3 _ =oE 2 , E3 = E 1
6 6 6 6 6 6
E1 = E3 , E2 = E2 , E3 = E 7
243
E1 100000_ I E2= 000
L ooooj000
E2=0010
0000
0000
0000
0000
E1 +0000000000
i00000 I000000
000000J
E3
_010_
;_= °°°° I
iOOOO
_0000
0000
0000liO 00il E4
E2+ 00
O O00
[o°°00
Ioooll= 000000000000
_o_E3+ 00
U °
E5
= 00
00
_4 _6:o9
Thus D matrix is formed by the above Es.
244
. °
0010
0000
0000
I0000
0000
0000
0000
I0000
_0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
O000i
00001
00001
0000
0000
0000
0000
0000
0000
OlO0
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
000o
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
tO00
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
oooo
0000
0000
000'0
0000
O0
O0
O0
O0
000000
000000
000000
000000
000000_
000000
000000
000000
000000
001000
000000
I00000
000000
000000
000000
O0000t
0ooooo
000000
O0
O0
O0
000
000
000
000
000
000
000
000
000
000
000
000
000
000
I00
000
ooo
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
ooo
000
o
sl
m
10000 ..........
00000 .......... 0
00000 .......... 0
00000 .......... 0
o.°,.°°,°.°...,
00000 .......... 01
245
0
x14
0
T14
0
1
0
0
1
0
0
0
1
0
1
0
0
x13
x13
0
0
0
0
246
TD= O0 0 0
O0 0 0
00 0 Oi
00 0 0
00 TI40
O0 0 0
O0 0 0
O0 0 0
'00 0 0
00 0 0
O0 0 0
00 0 0
O0 0 0
00 0 0
00 0 0
O0 0 0
00 0 0
O0 0 0
00 0 0
O0 0 0
00 0 0
00 0 0
O0 0 0
O0 0 0
00 0 0
0 0000
0 0000
TI40000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0100
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000
0 0000)
0 0000
0 0000
0 0000
0 0000
0 0000
000000
000000
000000
000000
i000000
000000
I.00000)
000000
000000
0000_0
000000
000000
000000
000000
000000
000010
000000
000000
000000
000000
000000
000000
000000
000000
000000
000
000
000
000
000 0
000 0
000 0
000 0
000 0
000 0
000 0
000 0
000 0
i00 0
000 0
000 0
000 0
000 0
000 0
000"r13
lO00 0
_000 0
000 0
000 0
000 0
0
0
0
0
0
0
0
0
0
0
0
0
0
'l"13
0
0
0
0
0
0
0 O0
0 O0
0 00
0 00
0 00000
0 00000
0 00000
0 00000
0 00000
00000
00000
00000
00000
00000
001000
00000
00000
00000
00000
00000
00000
00000
00
00
O0
247
A
1
IAI| AI2 AI3 At4
A21 A22 A23 A24
A31 A32 A33 A34
A41 A42 A43 A44
4
All A12 AI3 A14 AI5
A21 A22 A23 A24 A25
A31 A32 A33 A34 A35
IA41 A42 A43 A44 A45
A51 A52 A53 A54 A55
!All AI2 A13 A14 AI5 A16
A21 A22 A23 A24 A25 A26
A31 A32 A33 A34 A35 A36
A4! A42 A43 A44 A45 A46
A51 A52 A53 A54 A55 A56
A6! A62 A63 A64 A65 A66
All A12 A13 A14
A21 A22 _23 A24
A31 A32 A33 A34
A4I A42 A43 A44
All AI2 AI3
A21 A22 A23
A31 A32 A33
6
AI] AI2 AI31
A21 A22 A231
A31 A32 A33J
A = (I - dE)-I E
/_'_EDING PAGE BLANK NOT FILMED
249
APPENDIX E
ERBE SOLAR MONITOR THERMAL MODEL
_bl2qG PAGE BLANK NOT FILMI_)
251
SOLAR MONITOR MODEL
As described in the non-scanner and scanner modeling efforts, the same
notations are used for constants and variables in developing the energy
balance equation of the solar monitor.
To avoid the burden to write the same equations as used in developing
the other models, the same variable name in the equation such as Rij is
here simply used.
A general form of the equation for the solar monitor which is divided
into 19 nodes is presented here as
T = AT + DT 4 +BQQ + BE + b.
Now the computational algorithm for the constants of the above equation
has been developed by using the enclosure coupling technique. It is avail-
able and can be used for all 8 non-scanner, scanner and solar monitor models.
The following constants, Aij, represent the coefficients of themal
conduction terms in the energy balance equations•
All = - I/(RI3MI )
AI2 = 0
AI3 = I/(RI3M I)
A21 = 0
A22 = - I/(R23M_)
A23 = I/(R23M 2)
252
C
!
CII C12 Ct3 C14
C21 C22 C23 C24
C31 C32 C33 C34
C41 C42 C43 C44
CII C12 C13 C14 C15 i
C2I C22 C23 C24 C25
C31 C32 C33 C34 C35
C41 C42 C43 C44 C45
CSI C52 C53 C54 C55
CI 1 C12 C13 C14 C15 C16
IC2] C22 C23 C24 C25 C26,
C31 C32 C33 C34 C35 C36
C4I C42 C43 C44 C45 C461
C51 C52 C53 C54 C55 C56
C6! C62 C63 C64 C65 C66
4 5 6
CI! CI2 el3 C14
iC2l C22 C23 C24
C31 C32 C33 C34
C41 C42 C43 C44
CI1 C12 C]3
C21 C22 C23
C31 C32 C33
Cll C12 C13
C21 C22 C23
C31 C32 C33
C = (I - 0dE) -I E T"
= [(I -0- TD)(I - CD) I- A - _]_ + [(i -0- TD)(I - CD)-
C-T] F S
253
A31 = i/(R3_3)
A32 = l/(R32M3 )
= _(i__ + _: + _i )/._A33 R31 R32 K34 J
A34 ffi1/(R34H3 )
A43 ffiI/(R43M4 )
--!__)/M4: -( i_ + _ + i + R4A44 R43 R45 R47
A45 ffiI/(R45M4 )
A47 ffiI/(R47M4 )
A4_I 9 ffii/(R4_I9M4 )
A54 = I/(R54M5 )
= _(i_!_ + Z----)/M5A55 _54 R56
A56 = I/(R56M 5)
A65 = I/(R65M6 )
= -(i____+ l___)/M 6A66 R65 R67
A67 : I/(R67M6 )
254
W
®
II'
.4-,--
I"!
!
II
!III
III!
II!
SOLAR MONITOR MODEL
i. Active Cavity
2. Reference Cavity
3. Button, Duplex Cone
Cavities
4. Heat Sink
5. Aperture, Primary
6. Shield, Aperture
7. Body (Rear Barrel)
8. Body (Front Barrel)
9. Aperture, Secondary
(or FOV Limiter)
i0. Motor Housing
11. Chopper Blade
12. Chopper Pendulum
13. Motor, Chopper
14. FOV Limiter
15. Front Top Housing Cover
16. Rear Top Housing Cover
17. Front Housing
18. Rear Housing
19. Holder, Heat Sink
20. Fictitious Surface
p. Temperature Probe
E. Total Incident Radiatio_
E.. Irradiance in enclosure a
E. Irradiance in enclosure b
Rate of Incident Radiation
From a Neighboring Surface
Cavities Within The
Sensor, i.e., Enclosures
or Spaces Between Sensor
Nodes
255
A74 = I/(R74M 7)
A76 = I/(R76M 7)
= _(i_.i__+ _ + i )IM7A77 R74 R76 R78
A78 = 1/(R78M 7)
A87 -- 1/(Rs7M 8)
ASS- I(i--!--+ i___+ I____)/MsR87 R89 RS-10
A89 = I/(R87M8 )
As_j0= l/(R8_IOM8)
A9_ 8 = i/(Rg_sM 9)
I
A9_ 9 = R9_8M9
Ai0_8 = I/(Ri0_8Mi0)
AI0_I0 = - i/(RI0_8MI0 )
AI0_I2 = i/(RI0_I2MI0 )
AI0_I 3 1 I/(RI0_I3MI0 )
- I/(RI0_I2MI0) - i/(RIo_I3MI0 )
AII_I 2 = I/(RII_I2MII )
AII_I I = - I/(RII_I2MII)
256
AI2_I I = I/(RI2_IIMI2 )
AI2_I 2 = -(I/RI2_I I + I/RI2_I3)/MI2
A12_13 _ i/(RI2_IIMI2 )
A13_12 -- i/(RI3_I2MI2)
AI3_I 3 = -(I/RI3_I 2 + I/RI3_I0)/MI2
A 3_Io --l!(R13_1o is)
A14-14
A14-15
A14-17
= -(I/RI4_I 5 + I/RI4_I7)/MI4
= I/(RI4_I5MI4 )
= i/(RI4_ITMI4 )
A15_15 = - I/(RIs_I6MI5) - I/(RI5_I4RI5 )
A15_16 = i/(RI5_I6MI5)
A15_14 = i/(RI5_I4MI5 )
A16_15 -- 1/(RI6_I5MI6 )
AI6_I 6 = -(I/RI6_I 5 + I/RI6_I9)/MI6
A16_19 = I/(RI6_I9MI6 )
257
AI7_I 4 = I/(RI7_I4MI7)
Al7_l 7 = -(I/RI7_I 4 + I/RI7_I8)/MI7
AI7_I 8 = I/(RI7_I8MI7 )
AIS_I7 = I/(RIs_I7MI8)
AI8_I 8 = -(I/RI8_I 7 + I/RI8_I9)/MI8
_8-19 = I/(RI8-19MI8)
A19_4= I/(RI9_4MI9)
AIg_I 6 = I/(19_16M19)
AI9_I 8 = I/(RI9_I8MI9 )
AI9_I 9 = -(I/R19_4 + I/RI9_I 6 + I/RI9_I8)/MI9
For the radiative interaction term, considering the specular reflection,
d s (1)p=p +p
E +p+T =i (2)
d _ d + Tk_kM k = (kOT + PkEk(3)
=. M dEk Z Ek_ j 33
(4)
The energy balance equation is now
_k = Ek = Mk(5)
258
where
s + dMk = kEk Mk •
Thus
d s _ Yk_k<k = Ek = ¢k_k - OkEk - okEk(6)
where
flk = °T4
From (3) and (4),
n dE ,-I, fi + r._.)M_ = _ (_ij - Oj j_l _ ( i i i l
i=l
(7)
From (7) through (4)
Ek = _ Ek- j _ Pj-i (e + -j i_i TiE i)
n
[(I_ ifli + _ "m )i i Ek-j Pj-i± j--1
(8)
Equation (6) becomes
_k = (! - Ok )Ek - _k_qk - _kEk
= I [(1 - Ok)_ i ! Ek-jPj-i - 'k_ki ] _ii ]
+ _[(i - pk)'ri _ Ek_jPj_ i - rk_ki ] Eii O
(9)
Let
Hkj i j Ek-j Pj-i
=Hnm £ L En_mPm-£
m
259
From the above drawing, the enclosures, a, b and c, are found. For
each enclosure, the incoming radiance to a certain surface can be described
by Equation (8).
Choose the enclosure b and c and couple them together, then one can
define the transmitted radiance through the filtering surface.
First,
enclosure b
enclosure c
elements 6,7,8,9,11,20
elements 1,5,20
Index
i,j,k
_,m_n
From Equation (8)
l 3 i j
where in this equation i, j and k are dummy index
or
Ek = [ e o Hk j + _ T E Hkji i_i i _ ii i
For the enclosure c, Equation (8) becomes
A
E20 = (I _£H2om£) c + T20 E20(H2om20)c£
(i0)
where £, m and n = i, 5 and 20.
260
Likewise, for the enclosure b,
A
E20 = (I ¢i_iH20ji)b + T2oE20(H20j20)b + TIIEII(H20jlI)bi
(ii)
where i and j = 6, 7, 8, 9, ii and 20
Combining (i0) and (ii) to obtain E20 and E20 as a function of Eli, we
have
E20 = GI[( _El_fH2omZ) c + T20(H20m20)c (! c i_iH2oji)b
I
+ TII T20(H20m20)c (H20jll)b Eli ](12)
and
*%
E20 = GI[(_ _i_i H2oji) n + T20(H20j20)b (_ E£_£H20m£)c1
+ TII(H20jlI)b Eli ](13)
where
2 c]-IGI = [i - T20(H20j20)b (g20m20)
The radiation energy balances in the enclosures b and c are
: [[(i - p k) _ -_klb i i Hkji k_ki ] _i
+ I[(1 - pk)T i Hkj ii
- Tk6ki ] (Ei=20 or Ei=ll)(14)
261
and
= I[(1 - On) eIHnml - en6n£] _££
A
+ I[(i - On) TlHnm l - Tn_n£ ] El£
(15)
Second,
enclosure d
enclosure b
elements 11,14,21
elements 6,7,8,9,11,20
Index
l,m,n
i,j,k
From Equation (8)
D
Eli = (I _i_iHllji)b + TII(HIIjlI)bEII + T20(HIIj20)bE20i
(16)
and
A
Eli = (_" _£Hllml)a + Tll(Hllmll)a Eli + T£
21(Hllm21)a E(17)
Substituting (12) into (16), Equation (16) becomes
A
Eli = ([ _i_iHllji)b + T20 GI(HIIj20) b (£_ _£H20m£)ci
2 GI c .+ T20 (HIIj20)D (H20m20) (_. _i_iH20ji)bl
2 GI (H2 J ] -+[TII(HIIjlI)b + TII _20 (Hllj20)b (H20m20)c - ll)b Eli(18)
Substituting (18) into (17)
Eli = G2[(_ _£_£ Hnml)a+ T21(HIIm21)aE + Tll(Hllmll)a (_. Ei_iHllji)b
l
+ TII T20 GI(HIImlI) a (Hllj20)b (_ _£_£ H20m£) c
2 GI c .+ TII T20 (Hllmll) a (HIIj20) b (H20m20) (_ Ei_iH20ji) b]I
(19)
262
where2 -i
G2 -- [i -TII SI(HIImlI) a]
2 GISI = (HIIjlI) b + T20 (HIIj20) b (H20m20)c (H2ojlI) b
On the other hand, substitution (17) into (18)
A
Ell -- G2[(_._i_iHllji) b + T20 GI(HIIj20) b (_ _I_£ H20ml) c11.
2
+ T20 GI(HIIj20) b (H20m20) c (_ ci_iH20ji)b
+rll SI( _ _l_l Hllm£) a + TIIT21 SI(HIIm21) a E](2O)
For the enclosure a: nodes: 11,14,21
10 for open position
= [(i - 011 ) Cll(_llmll) a - Cll]_ll + (i- 014 ) _14(_im14)a _14
+ [I - 01 I) T21(Hllm21)a E + [(i - 011 ) S(Hllmll)a. - TII]EII
0 for closed position
0 for open position
I - -- E= (i - 0i4 ) l(Hl4mll)a _iI + [(! 014 ) Gl4(Hl4ml4)a 14 ] _14
A
+ (i - 014 ) Tll(Hl4mll)a Ell + (i - 014 ) T21(Hl4m21)aE
263
For the enclosure b: nodes: 6,7,8,9,11,20
_6 b = [(i - 06) _6(H6j6)b - _6] _6 + (i - 06) E7(H6j7)b _7
0 for open position
+ (i - 06) _8(H6j8)b _8 + (i - 06) c9(H6j9) b _9 + (i - 06) _ll(H6jll)b _ii
_0 for closed position ~+ (i - 06) _II(H6jlI)D Eli + (i - 06) T20(H6j20)B E20
_7Ib = (i - 07) _6(H7j6)b _6 +[(i - 07) c
+ (i - 07) _8(H7j8)b _8 + (L - 07)
7(H7j7)b - _7] fZ7
0 for open position
9(H7jg)b _9 + (i - 07) ]II(H7jlI)B _ii
/ -+ (i - 07 ) ll(H7jll)b Ell + (i - 07) T20(HTj20) b E20
0 for closed position
= (1 - o8) 6(H8j6)b _6 + (i - 08) 7(H8j7)b _7 + [(i - 08) ¢8
(H8js)b - _8 ] _8
0 for open position
+ (! - 08 ) _9(H8j9)b _9 ÷ (i - 08) _ll(H8jll)b _ii
+ (! - 08)/ll(Hsjll)b Eli + (i - 08) T20(H7j20)b E20
0 for closed position
264
I = (i - 09) E6(H9j6)b _6 + (i - 09) E7(H9jT)b _7 + (L - 09) _8(H9j8)b _8_9 b
0 for open position
+ [(i - 09) (9(H9j9)b - (9 ] _9 + (i - 09) /ll(H9jll)b _ii
7 -+ (i - 09) ll(H9jll)b Eli + (i - 07) T20(H9j20) b E20
0 for closed position
b: (i - Pll ) (6(H11j6)b f_6 + (1 - Pll ) E
(HllJ 8)b _8
7 (Hllj7)b _7 + (1 - Pll )
+(i - Pll ) f9(Hllj9)b _9 + [(i - PlI)SI(HlljlI) b - (11 ] _ii
0 for open position
+ (L - Pll ) T20(HIIj20) b E20 + [(1 -Pll ) 1(Hzlj11)b - "[II ] Ell
0 for closed position
@20 b = (i - P20 ) E6(H20j6)b _6 + (I - P20 ) E7(H20j7) b _7
O_r open position
+ (I- 020 ) _8(H20j8)D _8 + (i - P20 ) _9(H20j9)b _9 + (I - P20)_l
(H2ojll)b _ii
+ (i - P20) TII(H2ojlI) b Eli + [(i - P20 ) _20(H20j20)b - T20] E20
265
Enclosure c: Nodes: 1,5,20
= [(i - 01) _l(Hljl)c - -el] _I + (L 01) 5(Hlj5)c _5
A
(1 - 01) r20(Hlj20)c E20
_51 c = (i - 05 ) _l(H5jl)c _i + [(i - 05 ) e5(H5h5)c -
+ (1 - 05 ) "r20(tt5j20) c E20
I - _5_20 = (i - 020 ) EI(H2ojl) c _i + (I 020 ) f5(H2oh5)c
c
A
+ [(i - 020 ) T20(H20j20) - T20] E20
For the enclosure d: Nodes: i,j,k = 1,3,4,5
_I Id = [(I-01) ei(Hlil)d" -_I] _I + (i - 01) _3(Hlj3)d _3
+ (i - 01) e4(Hlj4)d _4 + (i - 01) _5(Hij5)d _5
I : (I - 03) _l(H3jl)d f_l + [(i - 03) _3(H3j3)d - E3] f_3_3 d
+ (i - 03 ) _4(H3j4)d _4 + (i - 03) E3(H3j5) d _Q5
4)4 d : (i - 04 ) EI(H4jl) d _i + (I - 04) _3(H4j3)d f_3
+ [(i - 04) ¢4(H4j4)d - E4] _Q4 + (i - 04) _5(H4j5)d _5
266
_5Id = (1-05) Cl(H5jl)d _i + (1-05) _3 (H5j3)d _3
+ (1-05) E4 (H5j4)d _.4+[(1-05) _5 (H5j5)d - _5] _5
For the enclosure e: Nodes: 4,16,18,19
= [(1-04) c4 (H4j4) e - E4] _4 + (1-04) _16 (H4jl6)e _16
+ (1-04) f18 (H4jl8)e f_18+ (I-P4) el9 (H4jl9)e _219
_161e = (1-016) c4 (Hl6j4)e _4 + [(1-016) _16 (Hl6jl6)e - _16] _16
+ (1-016) _18 (Hl6jl8)e _18 + (1-016) _19 (Hl6jl9)e _19
= (I-P18 16 (HI8jI6 n16_18 e (I-P18) £4 (Hl8j4)e _4 + ) _ )e
+ [(I-PI8) _18 (Hl8jl8)e - _18 ] _18 + (I-P18) _19 (Hl8jl9)e _19
_191e = (1-019) _4 (Hl9j4)e _4 + (!-0i9) _16 (Hl9jl6)e _16
+ ) )e + [ (1-019) _(I-P19 ¢18 (H19j18 _218 19 (H19j19)e - _19 ] 219
For the enclosure f: Nodes: 4,7,8,14,15,16,17,18
_4 If = [(1-04) e4 (H4j4)f - _4 ] _4 + (I-P4) _7 (H4j7)f _7
+ (1-P4) _.8 (H4j8)f _8 + (1-04) ci4 (H4jl4).f _14 + (1-04) _15 (H4jl5)-f _15
+ (1-04) _16 (H4jl6)f _16 + (1-P4) _17 (H4jl7)f _17 + (1-04) El8 (H4jl8)f _18
267
-- (I-P7) '4(H7j4)f _4 + [(I-P 7) _
(H7j8)j _28
7 (H7j7)f - _7 ] _7 + (I-P 7) c 8
+ (1-07) (14 (H7jl4)f _14 + (I-0 7) _15 (H7jl5)f _215 + (1-0 7) _16
(H7jl6)f f_16
+ (l-P7) (17 (H7jI7) f f217 + (l-P7) el8 (H7jI8) f _218
¢81 --"(I-P8) E4 Q4 + (f (H8j4)f (I-P 8)
(Hsj 8)f - (8 ] f_8
7 (H8j 7) f n7 + [ (1-ps) (s
+ (I-P8) (14 (Hsjl4)f _14 + (I-P 8) (15 (H8jI5) f _15 + (I-P 8) (16
(HsjI6) f n16
+ (I-P8) (17 (Hsjl7)f _217 + (I-P 8) (18 (Hsjl8)f f_18
l:_14 f (I-P14) _4 (Hl4jl4)f f_4 + (I-P14) _7 7) _7 + ) _(HI4 j f (1 -pl 4
(m14js)f_8
+ [(I-PI4) (14 (Hl4jl4)f (14] fZl4 +- (l-Pl 4) f
(I-P14) (16 (Hl4jl6)f f_16
15 (Hl4jl5)f _15 +
+ (1-P14) (17 (H14ji7)f _17+ (1-P14) _18 (H14j18)f _18
268
= (1-015) 4 (Hl5j4)f f24 + ) ((1-015
(H15j8) f f_8
7 (Hl5j7)f _7 + .((l-P15) 8
+ (1-015) (14 (Hl5jl4)f _214 + [(1-015) _15 (Hl5Jl5)f - _15 ] _15 +
(1-015) _16 (Hl5jl6)f f_16
+ (1-015) _17 (Hl5jl7)f f_17 + (i -015 ) _18 (Hl5jl8)f P'18
= (1-016) 4 (Hl6j4)f f_4 + (1-016) (7 7) _7 +e (HI6 j f (1-016) E
(HI6j8) f f28
+ (I-P16) e14 (H16j14)f _14 + (I-P16) (15 (Hl6jl5)f _15 + [(1-P16) (
(Hl6jl6)f - _16 ] f_16
+ (1-016) (17 (Hl6jl7)f f217 + (I-PI7) el8 (HI6jI8) f _18
16
-- (I-P17) e4 (HlTj4)f f_4 + (I-P17) e
(HI7j S) f _8
7 (Hl7j7)f _7 + ) ((I-P17
+ (I-P17) 14 (Hl7jl4)f DI4 + (I-PI7) _15 (Hl7jl5)f _15 + ) (( (l-Pl 7
(H17j16)f _16
16
+ [(1-P17)c17 (Hl7jlT)f - _17 ] _17 + (I-P17) el8 (HI7jI8) f ni8
269
_181f = (I-P18) _4 (Hl8j4)f _f + (1-018) c7 (HI8j7) f _7 + (1-018) ¢8
(Hlsjl8)f _8
+ (l-Pi8) _14 (Hl8jl4)f _14 + (I-@18) '15 (Hl8jl5)f n15 + (1-018)
el6 (Hl8jl6)f _16
+ )_(I-P18 17 (Hl8jl7)f _17 + [(1-018) _18 (Hl8jl8)f - _18 ] ill8
For the enclosure g: Nodes: 8,9,14
= [(1-P8) E8 (H8j8)g - _8 ] _8 + (l-Q8) E
+ (1-08) ¢14 (H8jl4)g _14
9 (H8j9)f _9
_9 Ig = (1-P9) E8 (H9j8)g _8 + [(i-09) _
+ (1-09) _14 (H9jl4)g _14
9 (H9j9)g-_9 ] _9
_14 Ig = (1-014) _8 (Iil4j8)g _8 + (1-014) ¢
+ [(I-P14) el4 (Hl4jl4)g - _14 ] _14
9 (Hl4j9)g n9
For the enclosure h: Nodes: 8,9,14,17 - i,j,k
_8 lh = [(I-P8) _8 (H8j8)h - E8] _8 + (1-08) _9 (H8j9)h _9
+ (I-P8) _14 (H8jl4)h _QI4 + (i-08) _17 (H8jl7)h QI7
27O
-( 9] _9
= (I-P 9) (8 (I{9_8)h _8 + t(I-P9) _9 (B9J 9)h
+ (I-P 9) _14 (H9314)h _14 +(l-Q9) _17 (_9317)h _17
ffi(l-Pi4) B (I_1418)h _B + (l-Of4) E9 (B14]9)h _9 n17) (-17 (_14JlT)h
+ [(1-P14 ) (14 (1414_14)h- _14] _14 + (1-P14
qblTlh (I-P17) (8 (_II7_B)B _IB + (i-017) _9 (H17_g)h %39
+ (l-Pl7) ¢14 (BlT_14)h _14 + [(1-p17) _17 (HlT_17)h 17 ] _17
For the enclOSUre l: NodeS:
¢2I_ L(i-o2) 2 (Bziz) i
+ (1_o2) 44 (_2_4)i _4
2,3,4 - i'_'k
¢2 ] _2 + (1-02) _3 (B233) i_3
= (1-P 3) _ 2 (_3_ 2) t e2 + [(l-P3)
¢3/i
3 (B3_3) - (3 ] _3
+ L(i-p_) 4 (B434) i
The following are the constants for the radiation transfer terms in
the energy balance equation.
_- 2GI [i- T20 (H20j20)b (H20m20)c]-I
2 GISI = (HIIjlI) b + T20 (Hllj20)b (H20m20)c(H20jll)b
2 sl ]-iG 2 = [i - rll (Hllmll) a
2 GIUk = G 2 ek [(Hllhk) b + T20 (Hllj20)b (H20m20)c (H20jk) b]
2 GI G2Vk = GI _k T20 [(H20j20)b (H20mk)c + TII (H20jll)b (Hllmll)a (HIIJ 20)b
(H20mk) c ]
2 2 GI G2Yk = GI _k [(H20mk)c + Ill T20 (H20m20)c (H20jlI)B (Hllmll)a (HIIJ 20)b
(H20mk) c ]
2
W k --G I [(H20jk) b ek + TII (Hllmll) a (H20jll)b U k]
DII = (_i) {Alc[(l-0 I) eI (Hljl) c -e l] + Alc (i-0 I) T20 (HIj20) c V I
+ AId[(l-01) _i (Hljl)d - el ]}
DI2 = 0
DI3 = (_i) Aid (i-01) _3 (Hlj3)d
DI4 = (_i) Aid (I-01) _4 (Hlj4)d
272
DI5 = (_i) {Alc[(l-Pl) _5 (Hlj5)c + (i-01) T20 (Hlj20)c V5] + Aid (i-01)
_5 (HljD)d}
DI6 : (_i) Alc (Z-01) T20 (HIj20) c W6
DI7 = (_i) Alc (I-0 I) T20 (HIj20) c W7
DIS = (_i--) Alc (I-0 I) T20 (HIj20) c W8
DI9 = (_I) Alc (i-0 I) T20 (HIj20) c W9
DI_I0 = 0
DI-II = (_I) Alc (i-01) T20 (Hlj20)c [WII + G1 G2 TII ell (H20jll)b (Hllmll)a]
DI_I2 = DI_I3 = 0
DI_I4 = (_i) Alc (i-0 I) T20 (HIj20) c TII
DI_I5 = DI_I6 = DI_I7 = DI_I8 = DI_I9 = 0
14 GI G2 (H20jll)b (Hllml4)c
= (Alc) aBI M I TII r20 T21 (i-0 I) G I G 2 (HIj20) c (H20jlI) b (Hllm2 I)
D2_ I = 0
D2_ 2 = (_2) A2i[(i-02)- c 2 (H2j2) i - c2]
273
D2-3 ffi (_i) A2i (1-02) _3 (H2j3)i
D2-4 = (_2) Azi (I-P2) _4 (H2j4)i
D2_5 = D2_ 6 ffiD2_ 7 = D2_ 8 = D2_ 9 = D2_I0 = D2_II ffiD2_12 = D2_13 ffiD2_14
= D2_15 = D2_16 ffiD2_17 = D2_18 = D2_19 = 0
D3_ 1 : (_3) A3d (1-P 3) eI (H3jl) d
D3_ 2 : (_3) A3d (1-0 3 ) E2 (H3j2) i
D3-3 ffi (_3) {.A3d[(1-P3) _3 (H3j3)d- c3] + A3i[(1-P3) _3 (H3j3)i- _3 ]}
D3-4 ffi (_3) {A3d (1-03) c4 (H3j4)d + A3i (1-P3) _4 (H3j4)i}
D3-5 = (_3) A3d (1-03) _5 (H3j5)d
D3_ 6 = D3_ 7 ffi D3_ 8 = D3_ 9 = D3_10 ffi D3_ll ffi D3_12 = D3_13 = D3_14 " D3_15
= D3_16 = D3_17 = D3_18 "= D3_19 = 0
ffi eI (H4j 1D4-1 (_4) A4d (1-04) )d
D4_ 2 = ([_4) A4i (1-04 ) (2 (H4j 2) i
D4-3 = (_4) {A4d (1-04) c3 (H4j3)d + A4i (1-04) _3 (H4h3)i}
274
D4_4 -- (_4) {A4d[(l-o 4) 4 (H4j4)d - _4] + A4e[(l-04) _4 (H4j4)e - _4]
+ A4f[(l-04) c4 (H4j4)f -E4] + A4i[(i-04) _4 (H4j4)i- _4]}
D4-5 =(_44) {A4d (1-04) _5 (H4j5)d}
D4_6 = 0
D4-7 = (_4) A4f (i-P4) e7 (H4j7)f
D4-8 _ (_4) A4f (1-04) ¢8 (H4h8)f
D4_ 9 =" D4_IO = D4=11 = D4_12 = D4_13 = 0
D4-14 = (_4) A4f (1-04) _14 (H4jl4)f
D4-15 = (_4) A4f (1-04) _15 (H4jlS)f
D4_16 = (_4) {A4e (1-04 ) E16 (H4j16) + A4f (1-04) Cl 6 (H4jl6)f }
D4-17 = (_4) A4f (I-04) _17 (H4jlT)f
D4-18 = (_4) {A4e (i-04) f18 (H4jlS)e + A4f (1-D 4 ) _18 (H4jl8)f}
D4-19 = (_4) A4e (1-04) ¢19 (H4jl9)e
D5_ I = (_5) {A5c (1-05)[_ 1 (H5jl) c +
(H5j i)d }
r20 (H5j 20)c Vz] + A5d (I-P5) el
275
D5_2 = 0
D5_3 = (_5) A5d (i-o 5) c3 (H5j3)d
D5_4 -- (_5) A5d (i-05) c4 (HDj4)d
D5-5 = (_5) {ADd[(l-05) _5 (HDh5)c- _5
+ A5d[(1-05) _5 (H5j5)d - _5]d }
D5_6 -- (_5) A5c (1-05 ) T20 (H5j20) c W6
D5_7 = (_5) A5c (1-05 ) T20 (H5j20) c W7
D5_8 = (_5) A5c (1-05 ) T20 (H5j20) c W8
D5_9 = (_5) A5c (1-05 ) T20 (H5j20) c W9
D5_I0 = 0
+ (1-05) T20 (H5j20) c V5]
D5-11 = (_5) A5c (1-05) T20 (H5j20)c[WII + GI G2 TII _ii (H20jll)b
(Hllmll) a _14]
D5_12= D5_13= 0
D5_14= (_) A5c TII T20 GI G2"(l-05)(H5j20)c (H20jll)b (Hllml4)a _14
D5_I5 = D5_16= D5_17= D5_18= D5_19= 0
276
B5 (A5c) (1-05) GI G2 a= M5 TII T20 _21 (H5j20)c (H20jll)b (Hllm21)
D6-1-- (M6--)A6b (I-P6)[_211 T20 _i GI G2 (H6jll)b (Hllmll)a (Hllj20)b
(H20ml)c + T20 (Hgj20)b YI ]
D6_2 -- 0 , D6_3 -- 0 , D6_4 = 0
c (1_P6)[ 2D6-5-- (M-_)A6b TII "[20
+ T20 (H6j20)b Y5]
(5 GI G2 (H6jll)b (Hllmll)a (HIIJ20) c
D6-6 = (-_6) A6b{(I-P6) (6 (H6j6)b - (
2+ T20 (H6j20) b (H20m20) c W6]}
6 + (I-P6)['r121 (H6jll)b (Hllm11)a U6
_6 2 )a U7D6_ 7 = ( ) A6b (1-06 ) {E 7 (H6j7) b + TII (H6jll)b (Hllml I
2
+ T20 (H6j20) b (H20m20) c W 7}
_6 2D6L8 = ( ) A6b (1-06) {(8 (H6j8)b + TII (H6jll)b
2
+ T20 (H6j20) b (H20m20) c W8}
(Hllmll) a U 8
_6 2D6-9 = ( ) A6b (1-06) {E9 (H6j9)b + TII (H6jll)b (Hllmll)a U9
2
+ T20 (H6j20) b (H20m20) c W9}
D6_I0 = 0
277
= _ (1-P6) {( )b +D6-11 (M6) A6b ii (H6jll TZl (H6jll)b (Hllmll)a
2(TII Ull + _ii G2) + T20 (H6j20)b (H20m20)c[Wll + GI G2 +ii TII
(H20jlI) b (Hllmll) a}
D6_12= D6_13= 0
_6 2 GI G2D6-14 = ( ) A6b (I-P6)[TII el4 G2 (H6jll)b (Hllm14) + TII T20 (14
(H6j20)b (H20m20)c (H20jlI) b (Hllmll) a]
D5_I5 = D6_16= D6_17= D6_18= D6_19= 0
A_.
B6 = (M--_)(1-P6) TII T21 G2[(H6jlI)b (Hllm21)a
(H20jlI)b (Hllm21)a]
2
+ GI T20 (H6j20)b (H20m20) c
D7_ I = (_7) A7b (I-P7){T_I T20 GI G2 (H7jlI) b (Hllmll)a (Hllj20)b (H20ml) c e1
+ _20 (H7j20)b YI }
DT_ 2 = D7_ 3 = 0
D7_ 4 = (_i) A7f (1-P 7) E4 (H7j4) f.
D7-5 = (_7) A7b (1-07) T20 {T_I _5 GI G2 (H7jll)b (Hllmll)a (Hllj20)b
(H20m5) c + (H7j20) b Y5 }
D7-6 = (_7) A7b (1-07) {(6 (H7j6)b2 (H7j U6 + 2+ TII ll)b (Hllmll) a T20
(HTj20) b (H20m20) c W6}
278
D7_7 = (_7) {A7b[(1-07) c7 (H7j7)b - _7 + (1-07)[T_I (H7jll)b (Hllmll)a U7
2 (H7j )c W7]] + [ _7 - ]}+ T20 20)b (H20m20 A7f (1-07) (H7jT) f E7
(_7 2D7_8 -- ) {A7b (l-P7)[c 8 (H7j8)b + TII (H7jlI) b (Hllmll) a Us
2+ T20 (H7j20)b (H20m20)cW8] + A7f (l-P7) _8 (H7j8)f}
D7_9 -- (_7) {A7b (l-P7)[_ 9 (H7j9)b
2+ T20 (H7j20)b (H20m20)c W9]}
2+ TII (H7jlI) b (Hllmll) a U9
DT_I0 = 0
D7_11-- (_7) A7b
(Hl-ml i )a
(I-P7) {eli (H7jll)b + TII (H7jll)b (TII UII + ell G2)
2+ T20 (HTj20)b (H20m20)c[Wll + G1 G2 ell TII (H20jlI) b
(Hllmll)a]}
D7_12= D7_13= 0
_7) + 2 GID7-14 = ( {ATb (1-07) TII G2 _14[(H7jll)b (Hllml4)a _20 (H7j20)b
(H20m20)c(H20jll)b (Hllml4)a] + A7f (1-07) _14 (H7jl4)f}
D7_15= (_7) A7f (1-07 )
D7_16= (_7) A7f (1-07)
15 (H7jl5)f
16 (H7jl6)f
279
D7-17 = (_7) A7f (1-07) _17 (H7jl7)f
D7_18 = (_7) A7f (1-07 ) el8 (HTjI8) f
D7_19 = 0
= (ATb) G2 {T21 aB7 M_-- (1-07) TII (H7jll)b (Hllm21)
2
+ T20 T21 G I (H7j20) b
(H20m20) c (H20jll)b (Hllm21) a}
D8_ I = (_8) A8b (I-08){T_I T20 eI G I G 2 (H8jlI) b (Hllmll) a (HIIj20) b (H20ml) c
+ T20 (H8j20) b YI }
DS_ 2 = DS_ 3 = 0
D8-4 = (_8) A8f (1-08) _4 (H8j4)f
D8-5-- (_8) A8b (I-08){T211 T20 _5 GI G2 (H8jll)b (Hllmll)a (Hllj20)b
(II20m5) c + T20 (H8j20) b Y5 }
2 )a U6D8_ 6 = (_8) A8b (1-08 ) {E6 (H8j6) b + TII (H8jlI) b (Hllmll
2
+ T20 (H8j20)b (H20m20)c W 6}
D8_ 7 = (_8) {Asb (I-08)[c7 (H8j7)b + T211 (H8jll)b
2
(Hllmll) a U7 + T20
(H8j20)b (H20m20) W7] + A8f (1-08) _7 (H8j7)f}
28O
{A8b[ (l-P8) _ (H8j - E8 + (1-08) 2D8-8 = (M--8) 8 8)b [_ii (H8jll)b
2 )c W8]] + [(1-08) _8 c+ T20 (H8j20)b (}120m20 A8f (H8js) f -
[(1-°8) _8 (H8j8)g
(Hllmll) a U8
- _8 ] + A8h[(1-08 )
8] + A8g
_8 (Hsjs)h - _8 ]}
2 2
= _ {A8b (I-08)[E9 (H8j9)b + YII (H8jll)b (Hllmll)a U9 + T20D8- 9 (M8)
(H8j20) b (H20m20)c W9] + A8g (1-08) _9 (H8j9)g + A8h (1-08) _
(H8j9) h}
D8_I0 -- (%) x 0 -- 0
D8-11 = (_88) A8b (I-08) {¢ii (H8jll)b + _Ii (H8jll)b (llllmll)a (TII UII
2
+ ell G2) + T20 (H8j20)b (H20m20)c[Wll + TII ell G I G 2 (H20jlI) b
(Hllmll)a]}
D8_12 = D8_13 = 0
2 GID8-14 = (_8) {A8b (i-08) TII _14 G2[(H8jlI)B (Hllml4)a - T20 (H8j 20)b
(H20m20) c (H20jll)b (Hllml4)a] + A8f (1-08) _14 (H8jl4)f
+ A8g (1-08 ) ,_14 (H8jI4) +g A8h (I-P 8) 14 (H8jl4)h}
D8-15 = (_8) A8f (1-08) _15 (Hsjl5)f
D8_16 -- (_8) A8f (i- 8) _16 (Hsjl6)f
281
D8-17 = (_8)[A8f (1-08) _17 (H8jl7)f + A8h (1-08) _17 (H9jl7)h]
D8_19 = 0
B8 (A8b) (1-08) G2 { a + 2 G1 a= M8 TII T21 (Hllm2 I) T20 (H8j20) b (H20m20) b (Hllm2 I) }
2
D9_ I = (%) A9b (1-09 ) T20{TII GI G 2 (H9jlI) b (Hllmll) a (HIIj20) b (H20ml)
c (H9j20) bi + YI }
D9_ 2 = 0 , D9_ 3 = 0 , D9_ 4 -- 0
2 GI G2 (H9jD9- 5 = ( ) A9b (I-P9) T20{_II _5 ll)b (Hllmll)a (HllJ20)b
(H20m5) c + (H9j20) b Y5 }
2 2
D9-6 = (_9) A9b (1-09) {_6 (H9j6)b + Tll (H9jll)b (Hllmll)a U6 + T20
(H9j20) b (H20m20)c W 6}
D9_7 = (_9) A9b (1-09) {E2 U7 + 27 (H9jT)b + TII (H9jll)b (Hllmll)a T20
(H9j20) b (H20m20)c W 7}
= _ {A9b (i-09)[_D9- 8 (M9)
2 2
8 (H9j8)b + TII (H9jll)b (Hllmll)a U8 + T20
(H9j20)b (H20m20)c W8 + A9g (1-09) _8 (H9j8)g + A9h (1-09) _8
(H9j 8) h}
282
= (H9j 9)b (Hgjll)b (Hllmll)D9_ 9 (_99) {A9D[(I-09) c9 - _9 + (I-09)[T211 a U9
2 _ -_9] ++ T20 (H9j20) b (H20m20) c W9]] + A9g[(l-09) 9 (H9j9)g A9h
[(1-09) _9 - _9]}(Hgj9) h
D9_I0 = 0
D9-11 = (_99) A9b (1-09) {_Ii (H9jll)b + TII (H9jll)b (Hllmll)a (Yll UII +
2
(iI G2) + T20 (H9j 20)b (H20m20)c[WII + _Ii TII GI G2 (H20Jll)b
(Hllmll)a]}
D9_12 = D9_13 = 0
-_9 2 GID9-14 = ( ) {A9b TII _14 G2[(H9JlI)b (Hllml4)a + Y20 (H9j20)b (H20m20)c
(H20jlI) b (Hllml4) a] + A9g (1-09) _14 (H9jl4)g + A9h (1-09) _14
(H9jI4) h}
D9_I 5 = D9_16 = 0
D9-17 = (%) A9h (1-09) _17 (H9jl9)h
D9-18 = D9-19 = 0
: (Asb) (1-09) c2{B9 M 9 TII T21 (H9jll)b (Hllm21)a
(H20jlI) b (Hllm21)a}
2 G1+ T20 (H9j20)b (H20m20)c
283
DI0_I = DI0_2 = D10_3 = D10_4 =
DI0_I 0 = DI0_I 1 = DI0_I 2 = DI0_I 3
DI0_I 7 = DI0_I 8 -- DI0_I 9 = 0
D11-6
DII-7 = (M_I)[AIIb {(i-011) e7 (Hllj7)b
DIO_5 = DI0_6 = DIO_7 = DIO_8 = DIO_9 =
= DI0_I 4 = DI0_I 5 = DI0_I 6 --
= 2 G I G 2 1 [ - i]DII-I (M_I) [AIIb{TII T20 ¢ (i-011) (HIIjlI) b (Hllmll) a (HIIj20)
(H20ml)c + (i-011) T20 (Hllj20)b Y1 } + Alla TII T20 _i G1 G2[(I-oII)
(Hllmll) a - i] (Hllj20) a (H20ml)c ]
DII_2 = DII_3 = DII_4 = 0
DII_ 5 z (M_I)[AIID{TII2 "['20GI G2 _5[(i-011 ) (Hlljll)n - l](Hllmll) a (HIIj20) b
(H20mS)c + (i-011) T20 (Hllj20)b Y5 } +" Alla _ii Y20 _5 G1 G5
[(l-Pl I) (H_ ) - i]llmll a (Hllj20)b (H20m5)c]
= (M_I)[AIIb {(I-PlI) _6 (Hllj6)b + T_I[(I-oII) (Hlljll)b -i]
2
(Hllmll)a U6 + r20 (i-011) (Hllj20)b (H20m20)c W6} +All a TII
[(I-011) (Hllmll) a - i] U6]
2
+ YII [(I-011) (HIIjlI) 5 - I] U7
2
+ T20 (I-011) (HIIj20) b (H20m20) c W9} + All a TII[(I-011)
(Hllmll) a - i] U7]
284
= 2
DZi-8 (M_I)[Alib {(1-011) _8 (Hlij8)b + Tli[(l-Pll) (Hlljil) b - i]
2
(Hllmll) a U8 + T20 (Z-Pll) (HIIj20) b (H20m20) c W8}+ Ali a TII
[(I-o11) (Hllmll) a - 1] US]
DII-9 ( ) [Aiib { (l-Oil) _ 9 (Hiij9)b + _il [ (i-Pii) (Hiljli)b - i]
2
(Hllmll) a U9 + T20 (1-011) (H11j20) b (H20m20) e W9}+ All a Tll
[ (1-011) (Hllmll) a - i] U9]
D : 011-i0
DII-11 = (M#I)[Allb {(1-011) Eli (H11jI1)b - _11 + %11[(1-011) (Hlljll) b -i]
2
(Hllmll) a (TII Ull + eli G 2) + T20 (1-0il) (HIIj20) b (Hl120m20) c
[WII + ii TII GI G2 )a]} + { )E (H20jII) b (Hllml I Alla (i_011 ell
(Hllmll)a - _ii + %11[(i-011 ) (Hllmll) a - l][Ull + TII ell G2
S1 )a]}](Hllml I
DII_I 2 : DII_I 3 : 0
DII_I4 = (M_I)[AIIb{%11 c14[(1-011) (HlljlI) b _ 1]G2 (Hllml4)a + (1-011) TII
2
T20 el4 G1 G2 (Hllj20)b (H20m20)c(H20jll) b (Hllml4) a} + Ali a (I-011)
El4 SI G2[(I-011) (Hllmll)a}]
285
DII-15 = DII-16 = Dli_17 = DII_I 8 DII_I 9 = 0
BII = (M_I)[AIIb{TII r21 G2[(I-DII) (HIIjlI) 5 - I] (Hllm21) a + (l_Pll) T11
2
T20 T21 G1 G2 (HIIj20) 5 (H20m20) c (H20jlI) b (Hllm2k)} + All a {(l-Pll)
2
Y21 (Hllm21)a + TII T21 S1 G2[(I-Ol I) (Hllmll) a - i] (Hllm21)a} ]
DI2_ 1 DI2_ 2 = D12_3 = D12_4 = D12_5 = D12_6 = D12_7 = D12_8 = D12_9
DI2_I 0 = DI2_I 1 = DI2_I 2 = DI2_I 3 = DI2_I 4 = DI2_I 5 = DI2_I 6 =
D12-17 = D12_18 = D12_19 = 0
DI3_I = D13_2 = D13_3 = D13_4 = D13_5 = D13_6 = D13_7 = D13_8 = D13_9 "
DI3_I 0 = DI3_II = D13_12 = DI3_I 3 = DI3_I 4 = DI3_I 5 = DI3_I 6
D13_17 = D13_18 = DI3_I 9 = 0
DI4-1 = D14-2 = D14-3 = D14-4 = DI4_ 5 = D14_6 = D14_7 = 0
D14-8 = (M_4)[AI4g (1-014) E8 )g + _8(HI4j8 AI4 h (1-014) (HI4jS) h ]
D14-9 = (M_4) [Al4g (l-P14) _9 (Hl4j9)g + AI4 h (l-P14) e9 (Hl4jg)h]
DI4-10 = DI4_II = D14_12 = DI4_I 3 = 0
DI4_I 4 = (M_ 4) {Al4g
D14_15 = DI4_I 6 = 0
14[(I-P14) (Hl4jl4)g - i] + AI4 h _14[(i_P14) (H14j14) h _ i]}
DI4_I7 = (M_4) Al4h (1-014) el7 (l_l14jlT)h
D14_18= D14_19= 0
DIS_ I = DI5_ 2 = DIS_ 3 = 0
D14_4 = (M_ 5) AI5 f (1-015) _4 (Hl5j4)f
DI5_ 5 -_ D15_6 = 0
D15_7 = (M_5) Al5f (1-015) E 7 (HlSj7)f
D15_8 = (M_5) AI5 f (1-015) _8 (Hl5j8)f
D15-9 = DIS-10 = DIS_If = D15_12 = D15_13 = 0
DI5_I 4 = (M_5) AI5 f (1-015) _14 (H15314)f
DIS_I 5 = (M_5) A15f[(l-015) _15 (Hl5jl5)f- _15]
_) (1-015) (HISjI6) fD15_16 = (Mi 5 AI5 f _16
DI5_I 7 = (M_ 5) Al5f (1-015) c17
DI5_I 8 = (M#5) AI5 f (1-015) _18
D15_19 = 0
(HI5jI7) f
(HI5jis)f
DI6_ I = DI6_ 2 = DI6_ 3 = 0
287
: (M_. + Al6f(l-Pl6) _4 (Hl6j4)f}D16-4 ) {AI6e(I-PI6) (4 (Hl6j4)e
D16_5 : D16_6 : 0
D16-7 : (M_6) A16f(I-PI6) (7 (Hl6jT)f
D16-8 : (M_6) Al6f(l-Pl6) (8 (Hl6j8)f
DI6_ 9 : DI6_I0 : D16_11 : DI6_I 2 : DI6_I 3 = 0
D16-14 : (M_6) A16f(l-P16) (14 (Hl6jl4)f
D16-15 : (#6) AI6f(I-PI6) _15 (Hl6jl5)f
D16_16 : (M--_6) {A16e[(l-P16)(HZ6j16) e - i] + AI6 f (16[(I-o16)(HI6jI6) f _ 1]}
D16-17 : (M_6) AI6f(I-PI6) (17 (Hl6jl7)f
D16-18 : (M_6) {AI6e(I-PI6)_18 (HI6jI8) + A16f(1-PI6)_18 (HI6j 18)f}
D16-19 : (M_6) Al6f(l-Pl6) _19 (Hl6jl9)e
DI7_I = D17_2 : D17_3 = 0
D17-4 : (M--_7)Al7f(l-Pl7) (4 (Hl7j4)f
D17_5 = D17_6 = 0
D17-7 : (M--_7) Al7f(l-Pl7) _7 (H17j7)f
288
D17-8 = (M--_7){AlTf(l-Pl7) c8 (HlTjS)f + Al7h(l-Pl7) _8 (HlTjS)h}
D17-9-- (M--_7)A17h(l-017) _9 (Hl7j9)h
DI7_I0 = DI7_I I = DIT_I2 = DIT=I3 = 0
DI7_I4 = (M-_7){Ai7f(l-Pl7) 14 (Hl7jl4)f + Al7h(l-Pl7) _14 (HlTjl4)h}
D17-15 = (M_7) AI7f(I-PI7) (15 (Hl7jl5)f
D17-16 = (¢7) AI7f(I-PI7) _16 (Hl7jl6)f
DI7_I7 -- (M--_7){AI7 f (17[(l-P17)(H17j17) f - i] + AI7h _I7[(I-oI7)(HI7jI7) h - I]}
DIT_I8 = D17_19= 0
DIS_I = D18_2= D18_3= 0
D18-4 = (M--_8){Al8e(l-Pl8) e4 (Hl8j4)e + Alsf(l-PlS) E4 (Hl8j4)h}
D18_5= D18_6= 0
D18_7= (M--_8)AI8f(I-PI8) E7 (HI8j7) f
D18-8 = (M-_8)AI8f(I-PI8) _8 (Hl8jS)f
D18_9= DI8_I0 = DI8_I 1 = DI8_I2 = DIS_I3 = 0
D18-14 = (M--_8)AI8f(I-PI8) _14 (Hl8jl4)f
289
DI8_I5 = (M_--_)Alsf(l-Pl8) c15 (HI8jI5) f18
D18-16 = (M-_8) {AI8e(I-PI8) _16 (HIsjI6) + AI8f(I-PI8) _16 (Hl8jl6)f}
D18-17 = (M_8) Al8f(l-Pl8) _17 (Hl8jl7)f
DI8_I 8 = (M_ 8) {AI8 e _18 [(I-pls)(HI8jI8) e - i] + AI8 f _18 [(l-p18)(H18j18) f -I]}
M_8 •DI8_I 9 = ( ) AI8e(I'-PI8) El9 (HI8jI9) e
DI9_I = D19_2 = D19_3 = 0
D19-4 = (M--_9)Al9e(l-Pl9) _4 (Hl9j4)e
D19_5 = D19_6 = D19_7 = D19_8 = D19_9 -- DI9_I 0 = DI9_I 1 = DI9_I 2 -- DI9_I 3 "
DIg_I 4 = DIg_I 5 = 0
DI9_I 6 - (M--_9)AI9e(I-PI9) El6 (HI9jI6) e
DI9_I 7 = 0
D19-18 = (M--_8)Al9e(l-Pl9) _18 (HI9jI8)
D19_19 = (M--_8)AI9 e EI9[(1-PI9)(HI9jI9) e - i]
290
Standard Bibliographic Page
I. Report No.
NASA CR-1782922. Government Accession No.
4.Title_dSubtitle Development of Response Models for the
Earth Radiation Budget Experiment (ERBE) Sensors:
Partl - Dynamic Models and Computer Simulations for th
EP_E Nonscanner, Scanner and Solar Monitor Sensors
7. Author(s)
Nesim Halyo, Sang H. Choi,
Dan A. Chrisman, Jr., Richard W. Sannns
9. Per_rming Org_ization Name and Address
Information & Control Systems, Incorporated
28 Research Drive
Hampton, VA 23666
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Langley Research CenterR_mnf on VA 2_665
15. Supplementary Notes
3. Recipient's Catalog No.
5. Report Date
March 20, 1987_B. Performing Organization Code
8. Performing Organisation Report No.
FR 687103
10. Work Unit No.
665-45-30-01
11. Contract or Grant No.
NASI-16130
13. Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
Langley Technical Monitor: Robert J. Keynton
Final Report
16. A_tract
Dynamic models and computer simulations were developed for the radiometric
sensors utilized in the Earth Radiation Budget Experiment (ERBE). The models were
developed to understand sensor performance, improve measurement accuracy by updatin@
model parameters and provide the constants needed for the count conversion algor-
ithms. Model simulations were compared with the sensor's actual responses demon-
strated in the ground and inflight calibrations. The models consider thermal and
radiative exchange effects, surface specularity, spectral dependence of a filter,
radiative interactions among an enclosure's nodes, partial specular and diffuse
enclosure surface characteristics and steady-state and transient sensor responses.
Relatively few sensor nodes were chosen for the models since there is an accuracy
tradeoff between increasing the number of nodes and approximating parameters such
as the sensor's size, material properties, geometry, enclosure surface character-
istics, etc. Given that the temperature gradients within a node and between nodes
are small enough, approximating with only a few nodes does not jeopardize the
accuracy required to perform the parameter estimates and error analyses.
17.Key Words(Suggestedby Authors(s))
Dynamic Models, Computer Simulations,
Radiometric Sensors, Count Conversion
Algorithms, Ground and Inflight Calibra-
tions, Thermal and Radiative Exchange
Effects, Surface Specularity,
Spectral Dependence, Enclosure Nodes
18. Distribution St_ement
Unclassified - Unlimited
Subject Category 35
19. Security Classif.(of this report)
Unclassified 20. Secu_ty Classif.(ofthispage) 21. No. ofPages122. P_ceUnclassified 305 AI4
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