Predicted dynamic electrothermal performance of thermistor

Predicted dynamic electrothermalperformance of thermistor bolometerradiometers for Earth radiation budget applications

Martial P. A. Haeffelin, J. Robert Mahan, and Kory J. Priestley

The Earth Radiation Budget Experiment ~ERBE! and the Clouds and the Earth’s Radiant Energy System~CERES! rely on scanning thermistor bolometer radiometers of a similar design for accomplishing theirmission. High-level dynamic electrothermal models of these instruments have been developed on thebasis of the Monte Carlo ray-trace, finite-difference, and finite-element methods. The models are ca-pable of simulating the end-to-end response of the ERBE and the CERES instruments to simulatedsequences of Earth scenes. Such models will prove useful in the design of future generations of similarinstruments, in defining ground-based and in-flight calibration and data-reduction strategies, in theinterpretation of flight data, and in understanding data anomalies that might arise after the instrumentshave been placed in orbit. Two modules that make up the end-to-end model are presented: theoptical–thermal radiative module and the thermistor bolometer dynamic electrothermal module. Theoptics module is used to determine the point-spread function of the optics, which establishes that theinstrument has sharply defined footprints on the Earth. Results obtained with the thermistor bolometerdynamic electrothermal module provide valuable insights into the details of channel operation andestablish its high level of equivalence. The combination of the two modules allows the point-spreadfunction of the instrument to be determined and reveals the potential of this tool for scanning realisticEarth scenes. © 1997 Optical Society of America

1. Introduction

The last third of the 20th century has been charac-terized by an increased awareness of man’s impact onthe environment.1 Initial interest in the role ofgreenhouse gases2 and more recent interest in therole of cloud forcing3 have led to ambitious programsto monitor the Earth’s global radiative energy budgetfrom platforms orbiting in space. Two of these pro-grams, the Earth Radiation Budget Experiment~ERBE!4,5 initiated in the mid-1980’s and the Cloudsand the Earth’s Radiant Energy System ~CERES!6,7

scheduled for operation in the late 1990’s and beyond,rely on scanning thermistor bolometer radiometers toaccomplish their mission.

M. P. A. Haeffelin and J. R. Mahan are with the Virginia Poly-technic Institute and State University, Blacksburg, Virginia24061-0238. K. J. Priestley is with the NASA Langley ResearchCenter, MS 472, Hampton, Virginia 23681-0001. M. P. A. Haef-felin’s current address is NASA Langley Research Center, MS 420,Hampton, Virginia 23681-0001.

Received 15 January 1997; revised manuscript received 13 May1997.

0003-6935y97y287129-14$10.00y0© 1997 Optical Society of America

Variations in the Earth’s radiative energy budgetcapable of signaling climate changes are expected tobe subtle and to occur over periods of years or evendecades. Instruments intended to sense these vari-ations must therefore be capable of providing long-term, continuous data records relatively free ofinstrument-imposed biases. This requirement hasled to an understandable reluctance to change thebasic instrument design from one monitoring pro-gram to the next. Therefore the ERBE and CERESscanning radiometric channels are quite similar intheir design and operation, and it is likely that thisgeneric type of instrument will remain importantwell into the 21st century.

2. ERBE and CERES Scanning Radiometers

ERBE was started by NASA in the early 1980’s to mon-itor the radiative field of the Earth. Each satellite in themission carried two types of instruments: narrow field-of-view scanning thermistor bolometer radiometers andwide field-of-view nonscanning active-cavity radiome-ters. The description of the nonscanning instrumentscan be found in Refs. 8 and 9. Although the active-cavity instrument is an absolute radiometer of great ac-curacy, it will not be used in the next-generation mission,

1 October 1997 y Vol. 36, No. 28 y APPLIED OPTICS 7129

borrego

Typewritten Text

Optical Society of America. Martial P. A. Haeffelin, J. Robert Mahan, and Kory J. Priestley, "Predicted dynamic electrothermal performance of thermistor bolometer radiometers for Earth radiation budget applications," Appl. Opt. 36, 7129-7142 (1997). doi: 10.1364/ao.36.007129

the CERES. The narrow field-of-view scanning radiom-eters were found to be more suitable for Earth radiationbudget monitoring because of their greater spatial reso-lution. For studying the effects of clouds on the Earthradiation budget, the pixel size must be sufficiently smallto obtain data that are not contaminated by the presenceof clouds.

A. Optics

A scanning thermistor bolometer radiometer typicalof those used in ERBE and CERES is depicted in Fig.1. It consists of a baffled Cassegrain-like telescopethat gathers radiation from a restricted Earth sceneand focuses it on a thick-film thermistor bolometer.Radiation incident upon the aperture nearly parallelto the instrument optical axis is specularly reflectedfrom the primary and the secondary mirrors before itenters the field stop located behind the primary mir-ror. The sensing element is mounted behind thefield stop on a massive aluminum substrate thatserves as a heat sink. Although ERBE radiometersmade use of aspherical mirrors, a primary concaveparaboloid, and a secondary convex hyperboloid—thetrue Cassegrain configuration—the mirrors of the

Fig. 1. View of the CERES scanning thermistor bolometer radi-ometer geometry, courtesy of NASA Langley Research Center.

7130 APPLIED OPTICS y Vol. 36, No. 28 y 1 October 1997

CERES telescope are both spherical and so will bereferred to as a Cassegrain-like geometry.

A true Cassegrain geometry allows the sphericalaberrations to be eliminated. Thus, if the detector islocated correctly, the focus of the image can be good.However, such a geometry increases the off-axis ab-errations. The choice of spherical mirrors forCERES was motivated by the necessity to reduceoff-axis aberrations. However, this configurationdoes increase spherical aberrations for radiation par-allel to the main axis of the instrument. Becausethe instrument is not an imager, this has virtually noimpact on the quality of the radiometric products.

Bongiovi10 showed in his thesis that the opticalpoint-spread function ~PSF!—defined in Section4.A.3—of the CERES radiometer is shaped by thefield stop, while the field of view of the ERBE instru-ment seems to be defined by the telescope optics.This, in part, results from the choice of mirrors.Again the spherical mirrors produce an image with amore uniform aberration throughout all angles of in-cidence, a situation that creates a more sharply de-fined PSF. More details on this topic can be found inRef. 11.

Filters are provided to give the desired spectralsensitivity. Three wavelength intervals are moni-tored in both ERBE and CERES. The three ERBEscanning channels had an approximately 60-km fieldof view at nadir and were sensitive in the short ~0.2–5.0 mm!, long ~5.0–50 mm!, and total wavelengthbands. The three CERES channels will have ap-proximately twice the spatial resolution of ERBE,and the longwave channel will be replaced by anatmospheric window ~8.0–12.0 mm! channel.

Although the designs of ERBE and CERES appearsimilar, the CERES scanning radiometer is a com-pletely new instrument. The geometry of the tele-scope has been fine-tuned to reduce contaminationfrom off-axis radiation. Bongiovi10 compares the op-tical PSF of the ERBE and CERES telescopes andshows that the CERES telescope is far superior to theERBE in the definition of the pixel.

B. Detector

The heart of the scanning radiometric channels inboth ERBE and CERES consists of two nearly iden-tical thick-film thermistor bolometer detectorsmounted on adjacent faces of a thick aluminum sub-strate that acts as a heat sink. Temperaturechanges in the heat sink induced by changes in theinstrument thermal environment can be compen-sated for effectively when the two detectors areplaced in adjacent arms of a two-active-arm deflec-tion bridge. The active and compensating detectorsare 40- to 50-mm-thick wafers, consisting of an ab-sorber layer, a thermistor layer, and a thermal-impedance layer, as shown in Fig. 2. Radiation fromthe Earth scene is directed onto the active thermistorbolometer detector by the Cassegrain telescope op-tics.

Radiation incident upon the absorber layer of theactive detector is converted into sensible heat and

conducted through the thermistor and thermal im-pedance to the heat sink, where the temperature ismaintained within fixed narrow bounds by a servo-controller. The change in temperature in the ther-mistor is highly dependent on the rate at which theheat is conducted away to the heat sink. The ther-mal impedance layer between the thermistor and theheat sink maintains a large temperature gradientthat amplifies the temperature changes in the ther-mistor. The resulting temperature rise in the ther-mistor produces a corresponding resistance changethat is sensed by the deflection bridge circuit. Theamplified bridge voltage may then be calibrated interms of the incident radiation. A schematic dia-gram of the bridge amplifier circuit appears in Fig. 3.

Thermistor bolometer radiometers scan the Earthfrom limb to limb with a scan period of the order of afew seconds, and so the incident-radiation signal ishighly transient. This means that the thermal andelectrical frequency response characteristics inter-vene in determining the instrument transfer func-tion. In practice, however, radiance data aregenerally interpreted as if they were instantaneousrepresentations of the scene being viewed. Also, thescenes viewed by these instruments are not necessar-ily spatially homogeneous; the spatial distribution ofthe energy incident to the radiation detector will gen-erally vary from scene to scene. However, radiancedata are interpreted as if all possible spatial distri-butions of a given amount of energy from an Earthscene produce the same instrument deflection, i.e.,

Fig. 2. Cross section of the CERES thermistor bolometer activeand compensating detectors.

equivalence is assumed. Finally, thermal noise thatis emitted and reflected to the detector from the in-strument structure varies from orbit to orbit and, toa lesser extent, with the scene being viewed. Most ofthe thermal noise sources are adequately sampled atthe ends of each scan cycle when the instrument ob-serves cold space. This will occur every 3.3 s inCERES. Scene-dependent variations in thermalnoise that may occur during the scan cycle are notsampled.

High-level dynamic electrothermal models of scan-ning thermistor bolometer radiometers are being de-veloped in support of ERBE and CERES.9,10,12,13 Thiseffort is motivated by several factors: First, it is in-teresting to know the impact of potential sources of themeasurement error described above, i.e., dynamic re-sponse, nonequivalence, and thermal contamination,on the instrument’s accuracy and precision. Suffi-ciently high-level numerical representations of theseinstruments are fully capable of addressing these is-sues. Dynamic electrothermal models have alsoproved useful in the initial design stage for optimizingphysical dimensions and surface optical treatments,for sizing and selecting electronic components, and forpredicting the resulting instrument PSF. Ground-based and in-flight calibration systems and proceduresas well as data-reduction strategies can be designedand tested before actual hardware is available. Fi-nally, these models have promise for use in the inter-pretation of flight data14 and for understanding anddealing with anomalies that may arise after the instru-ment has been placed in orbit.

3. High-Level Numerical Models of a CERES-likeScanning Thermistor Bolometer Radiometer

Our ultimate goal is to produce a high-level end-to-end dynamic electrothermal model of scanning ther-mistor bolometer radiometers of the type used inERBE and CERES. Such a model would consist ofthree major blocks: an optical–thermal radiativemodule, a detector dynamic electrothermal responsemodule, and a structural heat conduction module.The approach has been to develop these three mod-ules separately and then to integrate them into acoherent end-to-end instrument model. We empha-

Fig. 3. Bridge amplifier circuit for the CERES scanning radiometer protoflight model.


size the optical–thermal radiative module and thedetector dynamic electrothermal module.

A. Optical–Thermal Radiative Module

The optical–thermal radiative module is based on theMonte Carlo ray-trace method. Therefore it is lim-ited to wavelengths where diffraction effects are neg-ligible. The dimensions of the ERBE and theCERES instruments are such that diffraction may bedeemed negligible for all but the longest wavelengthsof interest, i.e., beyond approximately 50 mm. Theerror in the overall radiative analysis associated withneglecting diffraction effects is probably less than thatassociated with uncertainties in the surface radiativeproperties used in the analysis. This optical modulewas developed first by Meekins12 for the ERBE instru-ment. Meekins performed a radiative and opticalanalysis of the ERBE radiometer that gave good in-sight into the characteristics of the instrument. Themodel was later modified by Bongiovi10 to include theCERES geometry. The geometry taken into accountin the model was more complete; the baffle, for in-stance, was added to the model. Bongiovi computedthe radiative and optical characteristics of both theERBE and the CERES instruments with greater con-fidence than before, thanks to the advance of comput-ers, which permitted higher speed and accuracy.Finally the model was modified by Haeffelin to pro-vide input to his dynamic electrothermal model of thedetector.

Applications of the Monte Carlo ray-trace methodin optics and thermal radiation are widely docu-mented in the literature15–21; therefore details aboutthe general method are not included here. Briefly, alarge number of energy bundles, or rays, are allowedto enter the radiometer through the baffle openingwith a desired spatial and spectral distribution.Each ray is traced until it strikes a surface, at whichpoint its fate is determined by the comparison of thelocal values of the surface optical properties ~absorp-tivity and degree of specularity of the reflectivity!with random numbers whose values are uniformlydistributed between zero and unity. For example, ifthe first random number drawn is less than the localabsorptivity, the ray is absorbed; otherwise it is re-flected. In this way the surface optical propertiesare interpreted as probabilities, either that a givenray will be absorbed or reflected or, if reflected, thatthe reflection will be specular or diffuse. If a reflec-tion turns out to be diffuse, the values of a pair ofrandom numbers are used to determine the directionof the diffuse reflection. Rays emitted from surfacesof the radiometer are treated in a manner similar torays entering through the baffle opening.

Typically, a ray-trace analysis must be performedfor every wavelength band of interest, assuming theoptical properties of the enclosure to be constantwithin each wavelength interval. For the purpose ofour study we assume constant optical properties overthe entire spectral domain of interest ~0.2–50.0 mm! tolimit computation time. The gray analysis, which isan option in the code, is perfectly adapted to our study.


The interior surfaces of the radiometer are subdi-vided into surface elements whose size and numberdepend on the spatial resolution required to achieveaccuracy goals. When all the entering and emittedrays either have been traced to a surface elementwhere they are absorbed or have exited the radiom-eter through the baffle opening, it is possible to com-pute a quantity called the radiation distributionfactor Dij, defined as the fraction of radiation emittedfrom surface element i ~or entering through the open-ing! that is absorbed by surface element j ~or exitingthrough the opening!. After a statistically signifi-cant number of rays have been traced, usually of theorder of millions depending on the accuracy desired,the distribution factor can be estimated as

Dij .Nij

Ni, (1)

where Nij is the number of rays emitted diffuselyfrom surface element i that are absorbed by surfaceelement j, and Ni is the total number of rays emittedfrom surface element i.

The distribution factors Dij can be shown to havethe following properties:

(j51

n

Dij 5 1.0, i 5 1, 2, . . . , n, (2)

eiAiDij 5 ejAjDji, (3)

(i51

n

eiAiDij 5 ejAj, j 5 1, 2, . . . , n, (4)

where e and A represent emissivity and surface area,respectively. Equation ~2! is a statement of conser-vation of energy and Eq. ~3! expresses reciprocity.We obtain Eq. ~4! by summing Eq. ~3! over i andapplying Eq. ~2! to the result.

After a statistically valid estimate of the distribu-tion factor has been obtained, it may be used to com-pute the radiation absorbed by any surface j,

Qabs,j 5 (i51

n

AieisTi4Dij~W!. (5)

In this circumstance a given Earth scene can be in-terpreted as an appropriate distribution of blackpoint sources with various temperatures located atinfinity in front of the baffle opening. After the tem-peratures of all of the interior surface elements andthe point sources representing the Earth scene arespecified, Eq. ~5! can be used to compute the distri-bution of absorbed radiation on all internal surfacesof the radiometer, including the active flake.

B. Detector Dynamic Electrothermal Module

The thermal part of the model characterizes the tem-perature changes in both the active detector and thecompensator flakes, as well as in the substrates, asso-ciated with temporal and spatial variations of the ra-diative input. The electrical part of the model is usedto predict the variation of overall resistance of both

flakes resulting from thermal changes and to computethe signal response of the electronic circuit. The elec-trical analysis also is used to predict the Joulean heat-ing in both thermistors and pass it to the thermalmodel. The two parts of the model are fully coupled.The model was developed by Haeffelin and Priestley.

The unsteady three-dimensional temperature fieldT~x, y, z, t! in the detector flake is given by the solu-tion to the unsteady heat conduction equation,

]

]x Skx

]T]xD 1

]

]y Sky

]T]yD 1

]

]z Skz

]T]zD 1 q̇ 5 rC

]T]t

,

(6)

subject to appropriate initial and boundary condi-tions. In Eq. ~6!, kx, ky, and kz are the thermal con-ductivities in the x, y, and z directions; q̇ is avolumetric heat source term; r is the mass density;and C is the specific heat.

Two-dimensional electrical diffusion in the ther-mistor layer is described by the Laplace equation,

]

]x Sse

]V]xD 1

]

]y Sse

]V]yD 5 0, (7)

derived from the continuity equation for the currentdensity field J,

=~J! 5 0. (8)

In Eq. ~7!, V is the electric potential and se is theelectrical conductivity. A two-dimensional model isused for electrical diffusion because the thermistorlayer is electrically insulated in the vertical directionby two layers of epoxy.

Both thermal and electrical diffusion in the detectorflake are represented by a fully implicit finite-difference formulation. The finite-difference methodis conceptually simple and straightforward, and it iswell suited to the rectangular geometry of the detec-tor flake. Because the electrical and thermophysicalproperties of the various layers are different, thefinite-difference analysis is developed initially fromthe point of view of control volumes. Each layer isfirst subdivided into control volumes such that theproperties are uniform within a given control volume.The nodes are located at the centers of the controlvolumes, and each node has six neighbors. Thefinite-difference nodal mesh for the CERES-like de-tector flake of Fig. 2 is shown in Fig. 4. The numberof layers of nodes through the flake is fixed, but thenumber of nodes per layer is variable.

In an unsteady finite-difference formulation, thetemperature of a given node at time t is directly re-lated to the temperatures of its six neighbors and toits own temperature at the previous time ~t 2 dt! byan algebraic approximation to Eq. ~6!. The degree ofapproximation depends on the finite-difference gridspacing and the size of the time steps used as well asthe convergence criteria used for each time step.The details of this process are widely available instandard texts, of which Ref. 22 is an example. Thedetails of the current application are given in Ref. 13.

The detector flake represented in Figs. 2 and 4 isgeometrically three-dimensional. Although we knowthat heat is conducted principally in one direction,the three-dimensionality of the problem is retained tomaintain the flexibility of the model and to samplethe effects of three-dimensionality. Equation ~6! isintegrated over time and over each control volume toobtain a discretized representation of the heat diffu-sion equation. The resulting algebraic differenceequations of the problem are solved by a Gauss–Siedel point-by-point method. The solution startswith an initial temperature, usually 311 K, for theentire grid shown in Fig. 4. An initial temperatureguess is assigned to each node to calculate the nodetemperatures at the first time step. Each grid point,or node, is visited successively to calculate the tem-perature at that point with the discretized heat dif-fusion equation. In this step the temperatures ofthe six neighbors are needed and so the most recentvalues are used. When the entire grid has been vis-ited in this manner, one iteration has been com-pleted. Iterations are performed successively untilthe change in node temperatures from one iterationto the next becomes less than an acceptable tolerance.

The active and compensator detector flakes, both ofwhich are analyzed in this manner, are connected inthe detector bridge circuit as shown in Fig. 3. Ineach flake the current passes through the thermistorlayer connected to the circuit by platinum leads.The resistance provided by a square sheet of ther-mistor is related to its temperature by

R 5 R0 expFBS1T

21T0DG , (9)

where R0 is the known thermistor resistance at tem-perature T0, a is the temperature coefficient of resis-

Fig. 4. Plan ~top! and elevation ~bottom! views of the finite-difference nodal mesh for transient thermal and electrical analysisof the CERES bolometer detector.


tance around T0, and B is the activation energy of thethermistor material. For a rectangular cross-section conductor of thickness d, length L, and widthw, the electrical conductivity se is related to the elec-trical resistance R by

se 5L

Rwd. (10)

Equations ~9! and ~10! are applied to a d 3 L 3 wrectangular volume element to compute the localelectrical conductivity from local temperatures pro-vided by the thermal analysis.

When the active flake receives a radiative heatinput, its temperature distribution changes rapidly,which in turn changes the overall resistance of thethermistor. Equation ~9! is nonlinear in tempera-ture and so cannot be used to relate the averagetemperature of the thermistor to its overall resis-tance. Instead, for the value of electrical potentialdifference across the flake at the beginning of a givencalculation time step, a discrete electric field in thethermistor layer is computed with the two-dimensional electrical diffusion equation, Eq. ~7!, andthe local values of the electrical conductivity are ob-tained with Eqs. ~9! and ~10!. We then compute thetwo-dimensional current density field passingthrough the flake by applying Ohm’s law locally. Fi-nally, the overall electrical resistance of the ther-mistor is computed as the ratio of the appliedpotential difference to the computed total currentflowing out of ~or into! the flake.

When the resistances of the active and compensat-ing detector flakes are known, the signal response ofthe detector bridge amplifier circuit can be computed.At a given time step with given values for each com-ponent of the circuit, resistors, capacitors, and oper-ational amplifiers, the electrical potential Vout in Fig.3 can be computed. Its dynamic behavior is ruled bya differential equation,

dVout

dt1 C1Vout 5 C2, (11)

where C1 and C2 are constants whose values dependon the values of the circuit components shown in Fig.3. Equation ~11!, which is based on successive ap-plications of Kirchhoff ’s current law to all the nodesin Fig. 3, is solved. The operational amplifiers areassumed to be ideal amplifiers, which implies thatthe electrical potential at the output of the amplifieris computed as directly proportional to the differencebetween the electrical potential at the inverting inputand that at the noninverting input,

Vout 5 A~Vin1 2 Vin

2!, (12)

where A is the gain of the amplifier. The signalresponse of the detector circuit, Vout, is then com-puted as

Vout 5 C3 exp~2C1t! 1C2

C1. (13)


We obtained the signal response of the circuit ateach time step by successively invoking the combinedthermal and electrical analyses for the radiative in-put condition prevailing at the beginning of the timestep. With sufficiently small time steps, of the orderof 1 ms, the continuous response of the thermistorbolometer radiometer to a sequence of simulatedEarth scenes can be obtained.

The end-to-end model was also meant to include amodule that characterized the heat diffusion in theinstrument structure. However, a commercial soft-ware package was chosen to perform this task,thereby precluding its integration with the other twomodules. The impact of change in thermal noiseacross a scan has been shown to contribute negligiblyto the signal response of the instrument.23 There-fore integration of the thermal conductive moduleinto the end-to-end model is not critical and is notdiscussed further in this research. The end-to-enddynamic radiative and electrothermal model of theentire radiometric channel is thus considered to becomplete because the optical–thermal radiative mod-ule and the thermistor bolometer dynamic electrother-mal module are fully integrated, and the telescopestructural dynamic thermal module can be used off-line to verify that its integration is unnecessary.

4. Predicted Channel Performance with the Model

The optical–thermal radiative module and the ther-mistor bolometer dynamic electrothermal modulehave been used separately and in combination tostudy the optical performance of a radiometer, includ-ing thermal radiative contamination and the equiv-alence and dynamic response characteristics of athermistor bolometer. Studies with the structuredynamic thermal model have established that theresults presented here are not influenced by heatdiffusion in the radiometer structure.

A. Predicted Optical–Thermal Radiative Performance

Three streams of radiation arrive at the active detec-tor during channel operation: ~1! direct radiationfrom the Earth scene that has followed the theoreti-cal optical path defined by the baffle, telescope optics,filters, and a precision field stop ~mounted just abovethe detector in Fig. 2!; ~2! indirect radiation from thehemispherical space above the baffle opening, some ofwhich is from the intended Earth scene but does notfollow the theoretical optical path and the remainderof which is out-of-field radiation; and ~3! radiationemitted from the radiometer structure, which is po-tential thermal radiative contamination. The MonteCarlo ray-trace method allows the radiation that ar-rives at the active detector to be classified into thesethree categories.

The fidelity of the modeling tool used to predict theperformance has been established by Bongiovi.10

The fidelity of a Monte Carlo ray-trace analysis ismeasured by its degree of convergence and by thestatistics of its internal consistency. Degree of con-vergence refers to the number of energy bundles, orrays, that must be traced in order that the result

obtained is no longer sensitive to the number of raystraced; and statistics of internal consistency refers tothe statistical measure of the degree to which theanalysis is consistent with the fundamental laws ofradiative transfer such as conservation of energy andreciprocity.

For the analysis of results that are reported here,the CERES-like radiometer is divided into 746 sur-face elements. Nominal values of the surface opticalproperties ~absorptivity and degree of specularity ofthe reflectivity! have been used for the various surfacecoatings and treatments ~Chemglaze by HughsonChemicals, copper black, electrodeposited nickel, pol-ished aluminum, and Black Velvet by 3M!. See Ref.10 for the surface optical properties.

1. Discretization of the Field of ViewRecall from Section 3.A that a study by Bongiovi10

showed that the field of view of the CERES scanningradiometer was defined predominantly by the shapeand the dimensions of the field stop. Two ap-proaches are available to characterize the field ofview of the instrument with the optics module of theend-to-end model. To define the field of view and theoptical PSF ~see Section 4.A.3!, Bongiovi10 used areverse Monte Carlo ray-trace method, in which en-ergy bundles are diffusely emitted from the detectorand those leaving the instrument through the aper-ture are accounted for. This method does not allowthe field of view to be discretized and, hence, hetero-geneous scenes cannot be studied. In addition, inBongiovi’s study, the detector was not discretized;hence results obtained from the optical model couldnot be used as input to the electrothermal model.

Because the field of view is shaped by the field stop,it was decided to base its angular discretization ac-cording to the two angles that characterize the fieldstop: the scan and the cross-scan angles. Notethat in the case of a cross-track scanning radiometer,the cross-scan direction corresponds to the along-track direction. The scan and cross-scan angles aremeasured from the instrument optical axis in thescan and cross-scan directions, respectively. In thephysical dimensions of the field stop, the maximumscan angle is 60.65° and the maximum cross-scanangle is 61.30°. The discretization is obtained witha 0.0635° increment and 0.1° increment, respectively.The increment in the scan direction is defined accord-ing to the scan rate of the CERES instrument, whichis approximately 63.5 degys, and the numerical timestep of the electrothermal module, 0.001 s. There-fore, a 0.0635° increment allows the field of view ofthe instrument to be updated at every time step whenthe end-to-end model is used to simulate a scanacross a given scene. The increment in the cross-scan direction is chosen so that the number of angu-lar intervals is similar to the number of intervals inthe scan direction. The angles are not limited to thevalues that correspond to the physical dimensions ofthe field stop because some out-of-field radiationreaches the detector through nonoptical paths.

The unit vector that characterizes the direction of

collimated radiation incident upon the aperture canbe defined by two angles in the spherical coordinatesystem of the instrument, u and f, respectively,called the zenith and azimuth angles. An angularbin in the field of view of the instrument is defined bya scan angle and a cross-scan angle, respectively band z, to which correspond two angles u and f thatdefine the direction of radiation for that bin. Be-cause the angular bin is small, radiation associatedwith one bin is assumed to be collimated. Accordingto Fig. 5, the azimuth angle can be defined as

f 5 arctanStan b

tan zD , (14)

and the zenith angle as

u 5 arctanStan b

sin fD . (15)

The angle u is defined between 0 and p and the anglef is between 0 and 2p.

The maximum zenith angle to be considered forthis application is less than 2°; hence cos u is essen-tially equal to unity. Therefore the solid angle, de-fined as “the area intercepted on a unit sphere by aconical angle originating at the sphere center ~non-dimensionalized by the radius square of thesphere!”24 can be considered as an area interceptedon the plane tangent to the sphere at the interceptwith the cone axis. Hence the solid angle associatedwith each angular bin, DV, is to an acceptable ap-proximation the product of the scan-angle incrementand the cross-scan-angle increment,

DV 5 DbDz. (16)

2. Computation of Distribution FactorsDistribution factors, defined in Section 3.A, are usedto determine the radiative exchange between ele-ments that make up an enclosure. Distribution fac-tors vary according to the conditions under which

Fig. 5. Geometry for conversion from cross-scan ~along-track! andscan ~cross-track! angles to zenith and azimuth angles.


they are developed. Some distribution factors havebeen computed in enclosures that have spectral char-acteristics and can be applied only in specific wave-length intervals. Other distribution factorscorrespond to the energy exchange between two sur-faces, but for only limited angular conditions.

To define the field of view of the scanning radiom-eter, the optics module is used to trace radiation in-cident upon the instrument aperture through thetelescope to the detector. To be able to study heter-ogeneous Earth scenes, radiation associated witheach angular bin is traced independently. For eachcombination of scan and cross-scan angles, the anglesu and f are computed with Eqs. ~14! and ~15!, andcollimated rays are emitted in direction u, f from theimaginary surface of the instrument aperture andtraced through the telescope. Because a fairly largefraction of the radiation incident upon the instru-ment aperture actually reaches the detector, approx-imately 30% for on-axis radiation, only 40,000bundles need to be traced to obtain a good definitionof the spatial distribution of energy on the detector.

On the basis of Eq. ~1!, distribution factors arecomputed for each angular bin as

Deltbin .

Neltbin

Nbin , (17)

where Neltbin is the number of rays emitted in the

collimated direction of a given bin from the instru-ment aperture absorbed by a given surface elementon the detector, and Nbin is the total number of raysemitted from the instrument aperture in the colli-mated direction of the bin.

Figure 6~a! shows the spatial distribution of energyabsorbed in the plane of the detector for collimatedradiation that enters the instrument aperture paral-lel to the optical axis. The three legs supporting thesecondary mirror are clearly visible. The image isnearly circular, which corresponds to the geometry ofthe baffle opening. The image is centered on thedetector and has a 0.1-mm diameter. For collimatedon-axis radiation, the field-stop aperture does not in-tercept any radiation.

The grid shown in Fig. 6~a! is that of the elementsthat define the mesh of the electrothermal model.As described in Section 3.B, the surface of the ab-sorber layer of the detector has been discretized in a16 3 16 grid, or 256 elements, of which only thecentral 16 are visible in Fig. 6~a!. Figure 6~b! showsthe distribution factors that correspond to the spatialdistribution of Fig. 6~a!. Note that, although thesize of the image of Fig. 6~a! is approximately that ofone element, four elements are actually illuminatedbecause the image is centered. The remaining ele-ments receive virtually no radiation.

For each angular bin, 256 distribution factors arecomputed and stored. As the scan and cross-scanangles are incremented, the limit of the field of viewis found when all the distribution factor values arezero. Note that the spatial discretization of the de-tector in 256 elements is necessary only when the


optics module serves as an input to the detector elec-trothermal model. For strictly optical and radiativeconsiderations, the absorbing surface of the activedetector can be considered as one element. In thiscase the distribution factor that represents the frac-tion of energy incident upon the instrument aperturewithin a given angular bin that reaches the activedetector can be computed as

Dbin 5 (elt51

256

Deltbin. (18)

In Eq. ~18!, Deltbin is the distribution factor for each of

the 256 elements of the detector. The sum is there-fore carried out on the 256 elements.

3. Point-Spread Function of the OpticsThe distribution factors derived for collimated radi-ation define the spatial distribution of a distant objectin the plane containing the detector. In other words,given a collimated flux originating from a distantpoint source, the collimated distribution factors de-fine the spatial distribution of the image in the planeof the detector. This is the definition of a PSF. The

Fig. 6. Spatial distribution of energy absorbed in the plane of thedetector for collimated radiation parallel to the optical axis of thetelescope: ~a! energy bundle distribution, ~b! distribution factors.

PSF of the optics—i.e., staring instrument—can alsobe used in combination with the image flux for re-trieving the distribution of the object flux. More de-tails can be found in Ref. 9.

The collimated distribution factors of Eq. ~18! for allconsidered angular bins define the PSF of the opticsbecause the PSF shows the relative contribution of auniform array of point sources to the total radiationincident upon the plane of the detector. Figure 7shows the PSF of the optics. We normalized the PSFby dividing each contribution by the maximum value ofthe distribution. Figure 7~a! gives a one-dimensionalview of the PSF in the scan direction, where each curveis associated with a specific cross-scan angle. Figure7~b! represents a two-dimensional distribution withthe relative contribution of each point source definedby an angular bin. This figure shows that the field ofview of the instrument is very sharply defined. Thevalue of the PSF is greater than 0.9 over approxi-mately 40% of the field of view—red area of Fig. 7~b!.The field of view of this CERES-like instrument isthus shaped by the field stop instead of by the geom-etry of the telescope. This is in contrast with theERBE scanning instrument, whose optical PSF isdominated by the telescope optics. This is an impor-tant characteristic for a scanning radiometer because

Fig. 7. PSF of the optics: ~top! one-dimensional representation,~bottom! two-dimensional representation.

diamond-shaped field stops can reduce both aliasingand blur25 when compared with circular apertures.We verify that the out-of-field radiation is of the orderof only a few percent by integrating the PSF over thephysical dimensions of the field stop. A more accu-rate estimate of the out-of-field radiation requiresmuch greater resolution ~0.01° increments! in the dis-cretization of the field of view.

4. Distribution of Energy on the DetectorTo couple the optical module to the electrothermalmodule, the distribution of radiation that arrives fromthe field on the active detector must be computed.The spatial distribution is discretized arbitrarilythrough a 16 3 16 grid. This provides sufficient spa-tial resolution without requiring excessive computa-tional resources. For each of the 256 elements, theradiative power absorbed in the absorber layer, Pelt-bin, is computed as the product of the radiative energyincident upon the instrument aperture and the cor-responding distribution factor:

Peltbin 5 SbinDelt

bin, (19)

where the subscript elt refers to the elements of thedetector grid, the superscript bin refers to the angu-lar bin defined by the cross-scan and scan angles, Sbin

is the power at the aperture associated with the pointsource for a given angular bin, and Delt

bin is the dis-tribution factor defined in Eq. ~17!. It is importantto note that Pelt

bin is indeed the power absorbed bythe detector element, as opposed to the incidentpower, because the absorption coefficient of the de-tector is a surface property that is taken into accountin the ray-trace analysis of the optics module thatdefines Delt

bin. The total power absorbed by the de-tector is

Pbin 5 (elt

Peltbin. (20)

Because the radiation incident upon the instru-ment within its field of view is defined as a radiance~Wym2sr!, it is convenient to compute an equivalentradiance corresponding to the total power absorbedby the detector. To do that we first compute a radi-ance equivalent to the power incident upon the aper-ture for each angular bin,

Lbin 5Sbin

DVbinAap, (21)

where DVbin is the solid angle defined in Eq. ~16! andAap is the surface area of the instrument aperture.The total equivalent radiance at the aperture is com-puted as a weighted average of the radiances in eachangular bin, where the weighting factor is the distri-


bution factor from the aperture to the detector in thegiven angular bin,

Leq 5

(bin

LbinDbin

(bin

Dbin. (22)

In Eq. ~22!, Dbin is the total distribution factor fromthe aperture to the detector in a given angular bin, asdefined in Eq. ~18!. Note that if the power incidentupon the instrument aperture is uniform, the equiv-alent radiance in Eq. ~22! will be equal to any of thebin radiances defined in Eq. ~21!.

Figure 8 shows three stages—input, transfer func-tion, and output—in Earth scene image processingwith the optics module. Figure 8~a! shows the an-gular distribution of power at the aperture of theradiometer produced from an Earth scene that rep-resents solar radiation reflected from an ocean sur-face under 30% cloud cover. The cloud coverconsists of large, low-altitude clouds, comparable to

Fig. 8. Three stages in Earth scene image processing with the useof the optics module: ~top! angular distribution of power at theaperture of the radiometer, ~middle! PSF of the optics, ~bottom!spatial distribution of radiant flux on the detector.


stratocumulus clouds. The data are obtained from aray-trace model developed by Villeneuve.26 Eachpixel in Fig. 8~a! corresponds to an angular bin. Thevalue associated with each pixel is a power instead ofa radiance because power is a quantity required bythe electrothermal module. However, because of thedirect correspondence between power and radiancedefined in Eq. ~21!, this figure can be thought of as animage of the cloud scene. The equivalent radianceassociated with this scene is approximately 125 Wym2sr. Figure 8~b! represents the PSF of the optics,which has already been depicted in Fig. 7. The datashown in Fig. 8~a! are used as input to the transferfunction of Fig. 8~b! to produce the radiant flux ab-sorbed on the detector surface. The spatial distribu-tion of the absorbed flux discretized on a 16 3 16 gridis depicted in Fig. 8~c! along with the geometricallimits of the field stop. In addition to definingsharply the field of view of the instrument, the fieldstop also effectively limits the illuminated area of thedetector to ensure all the radiation in the field of viewcan be detected. Note that the distribution of energyin the plane of the detector is an inverted image of thecloud scene because the detector is located behind thefocal point of the instrument. This demonstratesthe capability of the optical model to produce a two-dimensional distribution of absorbed energy on thedetector given any heterogeneous Earth scene.

B. Predicted Dynamic Electrothermal Performance

The dimensions of the various layers of the active andthe compensating detectors and their thermophysicalproperties are given in Ref. 27, as are the values ofthe electrical components in Fig. 3. Nominal valuesfor the CERES detector were used to obtain the re-sults reported here.

1. Temperature Distribution through the DetectorThe electrothermal module predicts the transient dis-tribution of temperature throughout the entire detec-tor unit, including both thermistor layers and theirsubstrates. At each time step a vertical tempera-ture profile through the center of the detector unitmay be extracted from the complete temperature dis-tribution. Figure 9~a! shows a temperature profilethrough the center of the active detector that corre-sponds to the detector at equilibrium while cold spaceis viewed. This temperature distribution is used asthe initial condition for most simulations. Figure9~b! shows the evolution with time of the verticaltemperature profile through the center of the activedetector after a 300-Wym2sr step input. Note thatonly the part of the detector above the thermal im-pedance, i.e., the flat part of Fig. 9~a!, is depicted inFig. 9~b! to show temperature changes in the ther-mistor. Hence the bottom curve of Fig. 9~b! is adetail of Fig. 9~a!.

Up to a time of t 5 0 the detector is at equilibriumwhile cold space is viewed. During this period, aportion of the Joulean heating resulting from biascurrent flowing through the thermistor is conductedto the left through a bonding layer and the absorber

layer and, subsequently, radiated into space, whilethe remainder of this self-heating is conducted tothe right through another bonding layer and thethermal impedance layer and into the aluminumsubstrate. These two components of heat flux aremanifested in Fig. 9 by downward-sloping line seg-ments, sloping away from a relatively flat plateaunear the center of the graph. The plateau repre-sents the essentially isothermal thermistor layer. Aconstant irradiance—corresponding to a 300-Wym2 srradiance at the instrument aperture—is then appliedto the detector at t 5 0, and the thermistor tempera-ture ~the flat plateau! begins to rise as less net heat isconducted to the left and radiated from the absorber.Note that the temperature axis covers a range of onlya few hundredths of a degree and the thermistor tem-perature changes only approximately 3 mK.

Figure 10~a! shows the evolution with time of thespatially averaged thermistor temperatures in boththe active and the compensator detectors, corre-sponding to the conditions of Fig. 9. Up to a time oft 5 0, both detectors have similar temperatures. Be-yond this time the temperature in the active detector

Fig. 9. Vertical temperature profile through the center of thedetector unit: ~a! the flake temperature at equilibrium while coldspace is viewed, ~b! evolution with time after a 300 Wym2 sr stepinput.

increases owing to the radiative input. At thispoint, changes in the thermal environment can alsoinduce temperature changes in the active detector.The temperature of the compensator detector alsochanges under the influence of the changing electri-cal conditions in the external circuit ~self-heating!.In a previous study,13 Haeffelin showed the time con-stant of the detector is 8.5 ms when only a radiativestep input is considered. Here the time constants ofthe active and the compensating detectors are, re-spectively, 12 ms and 22 ms, assuming a first-ordertime response. Clearly the temperature changes inFig. 10~a! do not correspond to first-order time re-sponses because the energy inputs are not step in-puts. For the active detector it is a combination of aradiative step input with a slowly drifting internalsource, and for the compensating flake it is simply achanging internal source.

Figure 10~b! shows the evolution with time of thesignal response of the instrument to a 300-Wym2srstep input. The time response can be fitted with a

Fig. 10. Evolution with time after a 300 Wym2 sr step input of ~a!the spatially averaged thermistor temperatures in both the activeand the compensator detector flakes, ~b! the signal response of theinstrument.


first-order function and the resulting time constantbecomes 10 ms, which is in the same range as theexperimentally determined instrument time con-stant of the CERES detectors. Hence the modelshows that thermal changes in the environment,whether caused by self-heating or other sources, thatwould affect the response of the active detector areeffectively compensated by a matching compensatingdetector. This is gratifying because nominal dimen-sions and properties and component values furnishedby the manufacturer were used in the model; butotherwise the model result is completely independentof the experimental result.

2. Study of EquivalenceThe concept of equivalence, or nonequivalence, refersto the effect that spatial distribution of a given radi-ative power input to the instrument might have onthe instrument signal response. The equivalence ofa scanning radiometer is an important concept be-cause Earth scenes and calibration scenes of equiva-lent power are likely to produce images in thedetector plane with different spatial distributions.In a previous study by Haeffelin,13 the author showedthat the hypothesis of equivalence for the ERBEscanning radiometer was a good assumption and thatthe instrument probably is not sensitive to the type ofdifferences in Earth radiative fields that would typi-cally be encountered in practice. However, thestudy did not include the optics module, and theequivalence tests were based on absorbed radiativepower on the detector instead of equivalent radianceat the instrument aperture.

a. Response of Detector to Spot Illuminations: Anumerical experiment was performed to simulate alaser point heating experiment performed by the con-structor. A 10-mW spot heating is applied sequen-tially to each of the 256 surface elements into whichthe absorber layer has been divided. The spot size isapproximately 0.1 mm 3 0.1 mm. For each illumi-nation the detector is allowed to evolve from initialconditions, corresponding to a space look, to steady-state conditions.

Figure 11 shows the predicted normalized steady-state detector response, corresponding to the laserpoint heating numerical experiment. Figure 11~a!depicts a one-dimensional representation of the dis-tribution; Fig. 11~b! is a two-dimensional representa-tion. The predicted instrument sensitivity to spotillumination is quite uniform over a large region inthe center of the detector. The response to spotheating falls off only near the edges to about 70% ofthe center value. The average normalized responseis approximately 0.94. Figure 11~b! also shows thegeometrical shape of the field stop. Note that thespatial distribution of absorbed energy depicted inFig. 8~c! falls within a region in which the normalizedresponsivity of the detector is greater than 0.9.

b. Response of Detector to Nonuniform EarthScenes: The combined optical and electrothermalmodules are used to compute the steady-state signalresponse of the instrument to the nonuniform Earth


scene shown in Fig. 8~a!. This scene representslarge decks of low stratiform clouds over an oceansurface. The input equivalent radiance of this non-uniform scene is 126.68 Wym2 sr. Given the radia-tive input of Fig. 8~a!, the combined optical andelectrothermal modules compute a steady-state out-put of 1.505 V. We can convert the output signalinto an output equivalent radiance by applying avoltage-to-radiance calibration based on viewing adiffuse calibration source.23 This value is 126.29 Wym2 sr, which represents a 0.3% difference from theinput radiance. This predicted difference can be in-terpreted in terms of equivalence between a uniformcalibration scene and a typical nonuniform Earth

Fig. 11. ~a! One-dimensional representation, ~b! two-dimensionalrepresentation of the distribution of the normalized steady-statesignal response to 10-mW spot illuminations.

scene. The instrument reveals a high level of equiv-alence.

3. Instrument Point-Spread FunctionAn important part of the radiometric data analysis isto take into account and compensate for the timeresponse of the detector and the electronic circuit.Compensation involves shifting the pixel location bya given amount with respect to the instantaneouslocation of the optical axis. The shift of the pixel isdefined by the scanning instrument PSF, which canbe computed as the product of the scan rate, thedetector time response, and the optical PSF.28,29 InManalo et al.,28 the optical PSF was taken to be gov-erned by the field stop, and the detector response wasassumed to be first order.

With the CERES-like end-to-end model, we cancompute an instrument-like PSF by scanning acrossa uniform line source of radiation, which is a distri-bution of point sources along a line normal to the scandirection, and recording the transient response of themodel. In the numerical experiment, each pointsource produces an output signal that is independentof that of the other point sources. Note that theelectronic circuit that has been modeled is as shownin Fig. 3 and hence does not include the four-poleBessel filter used in the actual CERES instrument.Therefore the results presented here represent onlythe PSF through the preamplifier electronics. Sig-nificant modulation of the signal can occur, depend-ing on the filter characteristics.

Figure 12 shows an instrument-like PSF as a func-tion of the position in the scan and cross-scan direc-tions. Figure 12~a! shows detailed distributions withscan angles for several values of the cross-scan angle.Figure 12~b! shows a two-dimensional representationof the distribution in which the shift with respect to theoptical axis—0.0° in both scan and cross-scan angles—appears clearly. Because of the thermal and electri-cal inertia of the sensing element and signal-conditioning circuit, the tail of the distribution goes outto approximately 4.0° beyond the optical axis of theinstrument, which corresponds to approximately 63ms because the scan rate is 63.5 degys.

During the processing the radiometric data, theshift of the pixel is taken to be the centroid of the PSFwith respect to the origin.28 The centroid in the scandirection, zcen is defined as the weighted average ofthe scan angles in Fig. 12~b!, where the weightingfunction is the PSF, i.e.,

zcen 5

(i

z~i! (j

PSF~i,j!

(i(

jPSF~i,j!

. (23)

In Eq. ~23!, i and j are the indices in the scan andcross-scan directions, respectively, z~i! is the angularposition in the scan direction, and PSF~i, j! is the i, jvalue of the PSF.

In the current configuration of the optics and theelectrothermal models, the centroid is found to be

approximately 0.63°, or 10 ms, behind the origin.This important result is needed for the data process-ing associated with simulation of Earth scans. Notethat a centroid has been computed only in the scandirection because the scan rate produces a scan ve-locity that is much greater than the velocity of thesatellite. The linear velocity of the satellite is ap-proximately 7 kmys or 7 myms, which is equivalent toan angular rate of 5 3 1024 degyms in the cross-scandirection. During the time it takes to integrate thesignal of the field of view, 100 ms, for example, thecross-scan angle will have moved 0.05°, which istwice the resolution of a cross-scan angular bin.Therefore this motion can be neglected in the processof converting the instrument output signal into radi-ance at the aperture.

5. Conclusions

A high-level end-to-end numerical model of the scan-ning thermistor bolometer radiometers planned foruse in CERES is presented, and two modules aredescribed in some detail. Results from the optical–thermal radiative module establish that the instru-ment’s optical system provides relatively uniformweighting over most of its footprint on the Earthwhile it uniformly distributes gathered radiation onthe active detector. Very little out-of-field radiation

Fig. 12. Instrument PSF: ~top! one-dimensional representationand ~bottom! two-dimensional representation.


reaches the active detector. The thermistor bolome-ter dynamic electrothermal module accurately predictsthe instrument frequency response characteristicswhile it lends considerable insight into its operation.It also provides convincing data, indicating that un-certainties associated with nonequivalence are neg-ligible. The two modules are interconnected so thatthe optical–thermal radiative module can scan agiven scene to present a time series of inputs to thethermistor bolometer dynamic electrothermal mod-ule. This capability is used to define the instru-ment PSF taking into account both the instrumentoptics and the thermal and electronic time re-sponses of the detector unit. The potential for usingthis combination of modules to evaluate ground andon-board calibration procedures and even to validatedata-reduction strategies is obvious.

The authors acknowledge the Radiation SciencesBranch of the Atmospheric Sciences Division atNASA’s Langley Research Center for its support ofthis study under grant NAG-1-1456.

References1. Inadvertent Climate Modification, Final Report of the Study of

Man’s Impact ~MIT, Cambridge, Mass., 1971!.2. B. R. Barkstrom and G. L. Smith, “The Earth radiation budget

experiment: science and implementation,” Rev. Geophys. 24,379–390 ~1986!.

3. V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R.Barkstrom, E. Ahmad, and D. Hartmann, “Cloud radiativeforcing and climate: results from the Earth radiation budgetexperiment,” Science 243, 57–63 ~1989!.

4. B. R. Barkstrom and J. B. Hall Jr., “The Earth radiation bud-get experiment ~ERBE!: an overview,” J. Energy 6, 141–146~1982!.

5. B. R. Barkstrom, “The Earth radiation budget experiment~ERBE!,” Bull. Amer. Meteorol. Soc. 65, 1170–1185 ~1984!.

6. B. A. Wielicki and B. R. Barkstrom, “Clouds and the Earth’sradiant energy system ~CERES!: an Earth observing systemexperiment,” in Proceedings of the Second Symposium onGlobal Change Studies, E. J. Barron, J. R. Mahoney, C. B.Baker, R. J. Gurney, and J. T. Merrill, eds. ~American Meteo-rological Society, Boston, Mass., 1991!, pp. 11–16.

7. S. L. Carman, J. E. Cooper, J. Miller, E. F. Harrison, and B. R.Barkstrom, “Clouds and the Earth’s radiant energy system~CERES!,” Adv. Astrophys. Sci. 76, 693–706 ~1992!.

8. N. E. Tira, “Dynamic simulation of solar calibration of the totalEarth-viewing channel of the Earth Radiation Budget Exper-iment (ERBE),” Master’s thesis ~Virginia Polytechnic Instituteand State University, Blacksburg, Va., 1987!.

9. N. E. Tira, “A study of the thermal and optical characteristicsof radiometric channels for Earth radiation budget applica-tions,” Ph.D. dissertation ~Virginia Polytechnic Institute andState University, Blacksburg, Va., 1991!.

10. R. P. Bongiovi, “A parametric study of the radiative and opticalcharacteristics of a scanning radiometer for Earth radiationbudget applications using the Monte-Carlo method,” Master’sthesis ~Virginia Polytechnic Institute and State University,Blacksburg, Va., 1993!.

11. M. Walkup, “Creation of a Monte-Carlo optical work bench forradiometric imaging,” Master’s thesis ~Virginia PolytechnicInstitute and State University, Blacksburg, Va., 1996!.

12. J. L. Meekins, “Optical analysis of the ERBE scanning ther-mistor bolometer using the Monte-Carlo method,” Master’s


thesis ~Virginia Polytechnic Institute and State University,Blacksburg, Va., 1990!.

13. M. P. A. Haeffelin, “A numerical study of equivalence in scan-ning thermistor bolometer radiometers for Earth radiationbudget applications,” Master’s thesis ~Virginia Polytechnic In-stitute and State University, Blacksburg, Va., 1993!.

14. K. J. Priestley, J. R. Mahan, and R. B. Lee III, “Use of high-level numerical radiometric channel models for Earth radia-tion budget signal processing,” presented at the Sixth AnnualInverse Problems in Engineering Seminar, University of Cin-cinnati, Ohio, 1994.

15. J. R. Howell, “Application of Monte Carlo to heat transferproblems,” in Advances in Heat Transfer ~Academic, NewYork, 1968!, Vol. 5, pp. 1–54.

16. T. S. Chou, “A Monte Carlo approach to optical analysis,” Opt.Eng. 13, 299–302 ~1974!.

17. J. R. Mahan and L. D. Eskin, “Application of the Monte Carlotechnique to transient thermal modeling of cavity radiometershaving diffuse-specular surfaces,” in Proceedings of the FourthConference on Atmospheric Radiation, C. L. Mater, ed. ~Amer-ican Meteorological Society, Boston, Mass., 1981!, pp. 181–186.

18. J. R. Mahan and L. D. Eskin, “The radiation distribution factorits calculation using Monte Carlo and an example of its appli-cation,” in Proceedings of the First UK National Heat TransferConference, H. C. Simpson, ed., Report No. 19 ~European Fed-eration of Chemical Engineering Publication Series, Leeds,England, 1984!, pp. 1001–1012.

19. J. R. Mahan, F. Kowsary, and L. D. Eskin, “Apparent emis-sivities of conical and cylindrical cavities: comparison ofMonte Carlo and exact analytical results,” in Radiation, PhaseChange Heat Transfer, and Thermal Systems, Y. Jaluria, V. P.Carey, W. A. Fiveland, and W. Yuen, eds. ~American Society ofMechanical Engineers ~ASME!, New York, 1987!, pp. 1–8.

20. N. E. Tira, J. R. Mahan, and R. B. Lee III, “Dynamic electro-thermal model for the ERBE nonscanning radiometer withapplication to solar observation and thermal noise,” Opt. Eng.29, 351–358 ~1990!.

21. N. E. Tira, J. R. Mahan, R. B. Lee III, and R. J. Keynton,“Linear-array apertures for in-flight dynamic solar calibrationof radiometric channels for Earth radiation-budget applica-tions,” Appl. Opt. 33, 5617–5627 ~1994!.

22. S. V. Patankar, Numerical Heat Transfer and Fluid Flow~Hemisphere, New York, 1980!.

23. M. P. A. Haeffelin, “A study of Earth radiation budget radio-metric channel performance and data interpretation proto-cols,” Ph.D. thesis ~Virginia Polytechnic Institute and StateUniversity, Blacksburg, Va., 1996!.

24. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer~Hemisphere, Washington, D.C., 1992!.

25. S. J. Katzberg, F. O. Huck, and S. D. Wall, “Photosensor ap-erture shaping to reduce aliasing in optical-mechanical line-scan imaging systems,” Appl. Opt. 12, 1054–1060 ~1973!.

26. P. V. Villeneuve, “A numerical study of the sensitivity ofcloudy-scene bidirectional reflectivity distribution functions tovariations in cloud parameters,” Ph.D. thesis ~Virginia Poly-technic Institute and State University, Blacksburg, Va., 1996!.

27. CERES Drawing Package: Encoder and Sensor Drawings~BEI, Servo, Adcole!, TRW Applied Technology Division, Spaceand Electronics Group, One Space Park, Redondo Beach, Calif.90278, 1993.

28. N. Manalo, G. L. Smith, and B. R. Barkstrom, “Transfer func-tion considerations for the CERES scanning radiometer,” inProceedings of the Tenth Conference on Optoelectronics andMicrowave, H. N. Nasr, ed. ~International Society for OpticalEngineering, Bellingham, Wash., 1991!, pp. 106–116.

29. G. L. Smith, “Effects of time response on the point-spreadfunction of a scanning radiometer,” Appl. Opt. 33, 7031–7037~1994!.

Documents

Predicted dynamic electrothermal performance of thermistor