Identification and Correction of Residual Image in the $\hbox{O}_{2}$ A-Band of the Orbiting Carbon...

$: Identification and Correction of Residual Image in the $\hbox{O}_{2}$ A-Band of the Orbiting Carbon Observatory$
2426 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011

Identification and Correction of Residual Image inthe O2 A-Band of the Orbiting Carbon Observatory

Denis M. O’Brien, Randy Pollock, Igor Polonsky, and Matt Rogers

Abstract—The detector used for the O2 A-band (0.76 μm) ofthe National Aeronautics and Space Administration’s OrbitingCarbon Observatory (OCO) employed a HyViSI Hawaii-1RGsensor, operating at 180 K in a rolling read-out mode. During thethermal vacuum testing of the flight instrument, it was discoveredthat the detector exhibited residual images that lasted for manyseconds and were of sufficient magnitude to compromise themission objectives. Independent testing of flight-spare detectorsrevealed that the problem was common to all and was not simply afault of the flight detector. The residual image was found to dependupon even-order derivatives of the spectrum, and its decay was afunction of the number of frames rather than time. An empiricalmodel was developed, which represented the measured spectrumin terms of the true spectrum and a history of all previous changesin the spectra. On the basis of the model, an algorithm wasdevised to correct spectra for the effects of residual image, usinga time-marching analysis of a history of previous spectra. Thealgorithm was tested with spectra acquired during the secondthermal vacuum test of OCO and was found to reduce the effect ofresidual image to almost the noise level of the detector. Numericalsimulations indicate that residual image has a negligible impacton retrieved concentrations of O2 and CO2 once the spectra havebeen corrected.

Index Terms—Hyperspectral sensors, optical detectors, remotesensing, silicon.

I. INTRODUCTION

THE ORBITING Carbon Observatory (OCO) was a Na-tional Aeronautics and Space Administration (NASA)

Earth System Science Pathfinder mission designed to measureglobal CO2 concentrations twice a month with sufficient pre-cision and accuracy to detect sources and sinks of CO2 onregional scales [1]. It was launched on February 24, 2009, butdid not reach orbit due to a failure of the launch vehicle.

The algorithms to retrieve the dry-air mole fraction of CO2

(denoted XCO2) required very high linearity of the calibrated

spectral radiances. Unfortunately, during the thermal vacuumtesting (TVAC) of the instrument, serious residual image prob-lems were found in the focal plane arrays (FPAs) in boththe O2 A-band (0.76 μm) and the weak CO2 band (1.6 μm).The O2 A-band FPA was a silicon HyViSI Hawaii-1RG from

Manuscript received May 6, 2010; revised October 25, 2010; acceptedOctober 31, 2010. Date of publication January 5, 2011; date of current versionMay 20, 2011. The work of D. M. O’Brien, I. Polonsky, and M. Rogers wassupported by the Jet Propulsion Laboratory under Contract 1280999.

D. M. O’Brien, I. Polonsky, and M. Rogers are with the Colorado StateUniversity, Fort Collins, CO 80523-1375 USA.

R. Pollock is with the NASA Jet Propulsion Laboratory, California Instituteof Technology, Pasadena, CA 91109 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2010.2091683

Teledyne Scientific and Imaging, LLC [2], [3]. While residualimage can be an issue in these devices, particularly if driveninto saturation [4], the interaction of the specifics of the focalplane, the spectral content of the incident radiation, and OCO’srolling read-out scheme created residual images of unusuallylarge amplitude and duration (many seconds). Furthermore, theproblem was not unique to the flight detector but rather provedto be a general feature of these devices.

In this paper, the term “residual image” is used generically,because the cause of the residual image observed in the OCOHyViSI Hawaii-1RG detector is unknown. Some aspects of theeffect, such as its dependence upon the second derivative of thespectrum, suggest that it is related to interpixel capacitance.However, a numerical simulation of the detector, includinginterpixel capacitance and a precise representation of the rollingread-out, was unable to reproduce the slow decays observedin practice. Consequently, a pragmatic approach was adopted.The residual image was characterized while the OCO flightinstrument was in the thermal vacuum chamber, a residualimage correction algorithm (RICA) was posited, and experi-ments were conducted to test the efficacy of the RICA. Theexperiments used both real spectra, acquired when sunlightwas directed into the vacuum chamber, and simulated spectra.Identifying the causes of the residual image remains an openarea of research.

In contrast, the term “charge persistence” (or “image per-sistence”) is reserved for the effect in which an elevated darkcurrent is caused by prior illumination. A causal mechanism forcharge persistence has been proposed in the context of HgCdTeHawaii-1RG detectors [5], [6]. The mechanism involves thetrapping of charge in the depleted region of each photodiode,which shrinks as photocharge accumulates, and the subsequentgradual release of the trapped charge once the depleted regionis returned to its original boundary by a reset. Whether thismechanism applies in HyViSI detectors or not is presentlyunclear, because the bulk of the silicon is fully depleted. This,too, is an interesting open area of research.

The O2 A-band detector was operated at a temperature of180 K, which is chosen to minimize the risk of ice contam-ination on the detector while reducing the dark current to anacceptable level. Subsequent development for OCO-2, the re-flight of OCO, suggests that the residual image effects reportedhere depend strongly on temperature and may be eliminated bycooling the detector to 120 K.

The FPA for the weak CO2 band was a HgCdTeHawaii-1RG, also from Teledyne Scientific and Imaging, LLC.Its residual image problem was determined to be specific to theflight detector and had been missed during part-level screening.

0196-2892/$26.00 © 2011 IEEE

O’BRIEN et al.: IDENTIFICATION AND CORRECTION OF RESIDUAL IMAGE IN O2 A-BAND OF OCO 2427

Fig. 1. Uncalibrated spectrum in the O2 A-band measured with the flightinstrument during TVAC at JPL. The light source was a heliostat on theroof of the laboratory. Wavelength increases from right to left, spanning fromapproximately 0.757 to 0.772 μm.

Fig. 2. Single frame of data from the OCO O2 A-band FPA. The wavelengthincreases from 757.47 to 772.21 nm from left to right. The white stripe is froma row with higher output than the FPA average. On the top and bottom of theimage, the edges of the spectrometer’s entrance slit are visible.

Since this is of less general interest, it will not be discussed inthis paper.

II. INSTRUMENT OVERVIEW

The OCO instrument was to measure sunlight reflected bythe Earth in three narrow bands, i.e., the O2 A-band, shownin Fig. 1, and two CO2 bands, termed weak and strong, atapproximately 1.6 and 2.0 μm, respectively. The spectral rangeof each channel included the complete molecular absorptionband as well as some nearby continuum to provide constraintson the optical properties of the surface and aerosols [7].

The instrument consisted of three imaging spectrometers,one for each band [8]. Light was directed into the spectrometersthrough a common telescope and a series of beam splittersand reimagers. Each spectrometer worked in first order andused a flat holographic grating. At each spectrometer’s focus,an area FPA collected the spectrum. As is typical in imagingspectrometers, one dimension measured field angles along theslit and the other measured different wavelengths (Fig. 2).

To measure the depth of each absorption line with sufficientaccuracy to correct the spectra for the effects of scatteringby clouds and aerosols, both absolute (< 5%) and relative(< 1% band-to-band) radiometric calibrations are essential [9],[10]. For XCO2

retrievals, linearity is even more important thanradiometric calibration; nonlinearity at the 0.1% level causeserror in XCO2

exceeding the mission target.The instrument was to be flown in the afternoon constellation

of Earth-observing satellites (also known as the A-Train) [11].In this orbit, the apparent ground velocity is approximately6.8 km/s. As OCO acquired spectra at a rate of 3 Hz, its spatialfootprint was 2.25 km. At this scale, the albedo of the Earthhas significant spatial structures, which would have led to ahigh frequency modulation of radiance in the continuum. In

order to achieve the mission objective of measuring XCO2with

an accuracy of approximately 1 ppmv, these large changes inabsolute signal levels could not be allowed to create spectrallydependent variations that would mask subtle changes in the at-mospheric state. Therefore, residual image errors were critical;if they could not be avoided, then they had to be characterizedand corrected via postprocessing.

The RICA presented in this paper was developed using alimited set of experiments to characterize the detector. Whilea more comprehensive study, involving wider ranges of solarzenith angles, solar intensities, and atmospheric conditions,would have been ideal, it was not feasible within the projectbudget. Equally, hardware modifications were precluded bythe budget. Nevertheless, as the results will show, the RICAperformed well, and OCO was approved for flight.

III. DETECTORS AND READ-OUT SCHEME

Each of OCO’s spectrometers used a Hawaii-1RG FPA.The two CO2 bands used HgCdTe detectors, while the O2

A-band used a HyViSI silicon detector. Because the opticalrequirements in the three bands were nearly identical, detectorswere selected with a common pixel pitch and the same read-out device and controlling electronics were used for all threebands. This commonality ensured high correlation in the spatialsampling of the Earth’s surface.

To provide the spectral coverage, between 800 and 950columns were required for each FPA. The spatial coverageonly required about 160 rows. In order to capitalize on existingread-out integrated circuits (ROICs), it was decided to use a1024 × 1024 array, but to read only 220 of the 1024 rowswhile reading out all 1024 columns. To reduce noise in theanalog electronics, the Hawaii-1RG ROIC was operated in a16-channel mode, in which 64 columns were routed via ananalog multiplexer through each of the channel amplifiers. Arolling read-out scheme was used to maximize the integrationtime while not undersampling the scene; this scheme providedover 95% duty cycle.

These ROICs have a significantly higher reset noise than readnoise, but do support nondestructive reads. That combinationwas exploited in a correlated double sampling (CDS) techniqueto eliminate the sensitivity to the reset noise (Fig. 3). Duringinstrument testing, it was established that the optimal timingincluded a seven-row delay between the reset and the first read(reset read), as shown in Fig. 4. This was accomplished usingthe advanced clocking option of the Hawaii-1RG ROIC [12].

IV. TVAC

In September 2007, the OCO instrument underwent its firstround of TVAC. The primary objective of this campaign wasto focus the optics onto the FPAs. A secondary objective wasto collect small sets of data with each piece of test equipmentin preparation for the formal calibration and performancetesting that would follow the assembly of the instrument. Thesedata sets provided the first hints of problems with the silicondetector.

The first test to show a nonlinear effect involved view-ing a Spectralon plate illuminated with direct sunlight. The


Fig. 3. Illustration of the CDS read-out used for OCO. The top graph shows anideal system; the rate of the increase of the voltage is proportional to the photonflux on the detector immediately above the unit cell collecting the charge. Thebottom graph shows reality; the charge on any pixel is corrupted by the reset ofneighboring rows. OCO allowed a seven-row delay between the reset and resetread to allow the reset value to settle. Creating a corresponding acceleration ofthe integration reads made matters worse and was not adopted for flight.

Fig. 4. Details of OCO read-out. OCO used the ROIC in 16-channel mode, inwhich each output addressed only 64 of the 1024 columns.

instrument was run in a special test mode in which CDSsubtraction was not performed inside the instrument computers.This permitted the reset and integrated reads of each pixel to berecorded independently, at the cost of subsampling the individ-ual frames. Review of these data showed an alarming patternin the reset levels. Whereas a barely detectable positive imageof the spectrum had been expected from charge persistence,instead, a negative residual image was observed with averageamplitude approximately 8% of the spectrum (Fig. 5). Evenworse, the residual image was not proportional to the positiveimage and depended on the second derivative of the spectrum(Fig. 5 (lower plot) and Fig. 6).

A second test (known as the “matador” test) discovereda more troubling aspect of the residual image. In this test,the instrument was run in its normal operating mode, withCDS subtraction performed onboard at the full frame rate, andthe instrument again observed a Spectralon plate illuminatedby sunlight. However, the light was modulated manually byinserting a blocking screen into the entrance beam every 30 s,so the intensity switched from bright to dark and vice versa. Inan ideal system, the response would drop from the bright to thedark level almost instantaneously. In regions of the spectrumwhere the second derivative was small, this ideal behavior was

observed, as shown in the second row of Fig. 7. However,where the second derivative of the spectrum was positive (forexample, at local minima), there was large overshoot followingthe transitions, as shown in the third row of Fig. 7. Conversely,where the second derivative was negative (for example, at localmaxima), the opposite effect was observed with undershoot,as shown in the fourth row of Fig. 7. The decay time of theovershoot and undershoot was many seconds and, therefore,could have compromised OCO’s spectra over rapidly varyingsurfaces.

V. EMPIRICAL MODEL FOR THE DECAY

It was demonstrated through careful independent modelingof the entire detector array and the read-out process that inter-pixel capacitance alone could not explain the long decay timesthat had been observed. Therefore, a purely empirical modelwas developed from the matador test data to predict the errorterm for any spectral index and its decay over time. The modelhas the form

Ms = a0e−b0s + a1e

−b1sM (2)s + a2e

−b2sM (4)s (1)

where Ms denotes the measured spectrum (in digital counts) atframe s = 1, 2, 3, . . . after the matador’s cape began to blockthe beam and M

(2)s and M

(4)s are the second- and fourth-

order spectral differences. The latter are defined for spectralindex j by

M(2)s,j =Ms,j−1 − 2Ms,j +Ms,j+1

M(4)s,j =M

(2)s,j−1 − 2M

(2)s,j +M

(2)s,j+1.

The parameters ak and bk were derived by fitting the modelto a full spectrum simultaneously; values are given in Table I.This model cannot predict the error for the first frame (s = 1)after the matador’s cape begins to block the beam, because thematador was not synchronized with the instrument clock.

The first term (with coefficient a0) probably is related tothe average illumination level; it represents the type of chargepersistence that was expected. There is no significant termrelated to the intensity just before the light is blocked. Inareas of linear spectral content, this error term is quite small,falling to a few counts within two frames. In regions wherethe second derivative is large, the second term dominates.More importantly, the exponential decay rate is much slower.The fourth-derivative term improves the prediction, but, as thefourth difference is much smaller, so too is the correspondingcontribution to the error. Fig. 8 shows the performance of thesimple model.

While these results build confidence that the residual imagecan be modeled in terms of the second (and higher) derivativesof the spectrum, they do not show how to treat the rapidlyvarying signals that OCO would observe. The changes in thesecond derivative caused by varying topography (for example)would induce residual images, whose effect upon estimatedXCO2

would be hard to unravel. A more robust error modelhad to be developed; this is covered in Section VII.


Fig. 5. (Upper plot) (Lower line) Reset level is forced negative in approximate proportion to the (upper line) signal level. (Lower plot) Ratio of these curvesshows that the reset level is not a linear multiplier of the signal level. This figure is for read mode A3 with a three-row delay between the reset and first read, butother testings showed only small differences between the A3 and A7 modes used for the flight instrument.

Fig. 6. These figures show two attempts to predict the reset level from the signal level. The upper curve includes a simple linear fit. It captures most of thebehavior, but adding a second derivative term significantly reduces the deviation from a line. Higher order even derivatives (e.g., fourth and sixth) provide smalladditional improvements.

VI. TESTING OF ENGINEERING DETECTOR

Once this complex residual image problem had been dis-covered, a choice had to be made between replacing the flightdetector and attempting to correct the spectra in software. Thefirst option rested upon the assumption that the flight detectorwas faulty and that residual image with such long delays wasnot a problem common to all detectors of this type. To testthis assumption, a qualification unit was selected, a test bedwas built, and the timing and analog electronics were simulatedusing a microcontroller-based system. The FPA was placed in

a small vacuum Dewar with a transparent window. To simulatethe high contrast spectral patterns of absorption lines (as shownin Fig. 2), a lens was used to image a bar code (shown inFig. 9) onto the FPA. The sharp transitions between blackand white created sizeable second derivatives in the “spectral”direction. The transitions between light and dark were createdwith a shutter in front of the light bulb. Once this system wasoperational, all timing signals were captured on oscilloscopesand compared with similar measurements from the flight elec-tronics. It quickly became clear that the qualification unit also


Fig. 7. During the matador test, a “cape” was placed quickly in front of the instrument aperture for 30 s at a time. (Second row) Where the second derivative ofthe spectrum is small, the transitions from light to dark are nearly instantaneous, falling to the noise level within a frame or two. (Third row) Where the secondderivative is positive, the signal overshoots the steady-state level and takes 5–8 s (15–24 frames) to fall into the noise. (Fourth row) Conversely, where the secondderivative is negative, the signal undershoots the steady-state level. The vertical lines (marked A, B, and C) indicate the positions of the pixels whose time seriesare plotted in the corresponding rows below. In each panel, a vertical section has been excised from the transitions on the right to emphasize the upper and lowerparts of the waveforms, but the absolute scale of the overshoot and undershoot is represented correctly. The ragged tops of the spectra in the second row are causedby a thin cloud at the time of the measurements.

TABLE ICOEFFICIENTS FOR SIMPLE MODEL IN (1)

exhibited residual image and that the sensitivity to the secondderivative of the spectrum (quantified by the parameters a1 andb1) was comparable with that of the flight detector.

Since the flight instrument had a fixed time per frame (1/3 ofa second), the frame time in the test bed was varied in an at-tempt to determine whether the exponential decay constant wasassociated with time or the number of frames. The first wouldpoint to leakage of stored charge as a possible cause, while thesecond would suggest that additional resets were required todrain the stored charge. The tests showed that the exponentialconstant was associated with the number of frames. It may bethat more elaborate timing schemes employing multiple resetscould reduce the residual image, but the approaches tried onthe test bed (recognizing the capabilities of the flight hardware)


Fig. 8. Upper and middle panels show a point in the spectrum where the second derivative is positive. The lower panel shows the (diamonds) overshoot thatoccurs when the matador cape blocks the light and (circles) the model for the overshoot. Other than the first frame, which the model cannot predict, the agreementis within the noise level.

Fig. 9. Bar code imaged by the qualification detector. The vertical black barswere caused by defunct multiplexers in the read-out circuit. They did not affectthe residual image experiments.

did not yield significant improvements over the timing shownin Fig. 4.

VII. ALGORITHM FOR CONTINUOUSLY VARYING SIGNALS

The basis of the O2 A-band RICA is the assumption that themeasured spectrum M(t) at time t is a synthesis of the true

spectrum F (t) at the same time and a history of all precedingchanges in the spectrum, represented mathematically in theform

M(t) = F (t) +

t∫−∞

dτh(t− τ)F ′(τ) (2)

where h(t) is a kernel to be determined and F ′ denotes the timederivative of F .1 This assumption is supported by analysis ofspectra obtained during TVAC tests of the OCO flight instru-ment, but is not based upon a physical model of the processesoccurring inside the detector. In particular, computer models ofthe read-out process based solely on an assumption of interpixelcapacitance do not produce the long time constants seen in theTVAC tests.

1An alternative hypothesis is that the measured spectrum M(t) depends onthe true spectrum F (t) at time t and the history of previous spectra, in whichcase (2) would be replaced by

M(t) = F (t) +

t∫−∞

dτg(t− τ)F (τ).

This form predicts that the measured and true spectra will differ in a steadystate. Although spectra acquired during the matador tests can be corrected withan algorithm of this form, (2) was adopted for routine processing of OCOspectra.


The earlier emphasis on assumption is important. The resid-ual image of the O2 A-band detector was discovered at the endof the first TVAC campaign. In response, a comprehensive setof tests was planned for the second TVAC campaign, to testthe repeatability of the effect, to experiment with alternativeread-out schemes in an attempt to minimize the residual im-age, and (as a last resort) to characterize the residual imagethrough matador tests in which the beam from the heliostatwas switched between 100% and lower levels of 50%, 25%,12.5%, and 0%. The intermediate levels were obtained using amatador’s cape manufactured from an aluminum sheet drilledwith a pattern of holes to provide partial transmission of light.This method of attenuating the beam had the important propertythat it was spectrally neutral, which means that the intensity wasreduced but the shape of the spectrum was not changed. Themathematical model described in the following is consistentwith all the data from these tests and, in principle, can beapplied to spectra that change continuously. However, it wasnot possible to devise a series of experiments with controlledlight sources during the extremely tight schedule of the secondTVAC campaign. The efficacy of the algorithm for the flight in-strument was to be tested after launch, using carefully selectedground targets. The launch failure ended those plans, but thealgorithm still merits discussion.

This section has three subsections. The first and sec-ond describe the functional form assumed for h(t) and themethod by which the constants of the model are determined.The third presents the technique used to recover F at timet0, given a sequence of measured spectra at earlier timest−h, t−h+1, . . . , t−1, t0.

A. Functional Form of the Kernel

Integration by parts in (2), together with the reasonableassumption that h decays rapidly to zero at infinity, leads tothe representation

M(t) = [1 + h(0)]F (t) +

t∫−∞

dτg(t− τ)F (τ) (3)

with g defined by

g(t) =dh(t)

dt.

Evidence from TVAC suggested that g should depend princi-pally upon the second derivative of the spectrum, a functionalform also suggested by the effects of interpixel capacitance,which originally was thought to be the cause of the residualimage. From observations of the time dependence of light-to-dark and dark-to-light transitions in the matador tests, it appearsthat g(t) decays approximately exponentially. Combining thespectral and temporal dependences, we propose the followingansatz for g(t):

g(t) =m−1∑k=0

(q−1∑p=0

αkpe−βkpt

)Ak (4)

where αkp and βkp are constants to be determined and A is thematrix representing second-order differentiation in the spectraldomain2

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −2 1 0 . . . 0 01 −2 1 0 . . . 0 00 1 −2 1 . . . 0 00 0 1 −2 . . . 0 0...

......

. . .. . .

. . ....

0 0 0 . . . 1 −2 10 0 0 . . . 1 −2 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.

In practice, derivatives up to the sixth order (m = 4) areneeded, although the contribution from the zeroth-order deriva-tive turns out to be so small that it may be omitted. Threeexponential terms (q = 3) are required to achieve high-qualityfits to the data.

B. Numerical Implementation

The integral in (3) must be replaced by a quadrature beforethe process can be implemented numerically. We assume asimple Riemann summation for the quadrature rule. Thus, if Ms

and Fs represent the measured and true spectra at frame s, then

Ms = (1− Γ)Fs +m−1∑k=0

∞∑n=1

gknAkFs−n (5)

where

gkn =

q−1∑p=0

αkpe−βkpn.

The operator Γ is chosen to ensure that the measured and truespectra are identical in a steady state. Thus, if M and F denotethe steady-state values, then the geometric series in (5) can besummed to obtain

M = (1− Γ)F +m−1∑k=0

q−1∑p=0

αkpe−βkp

1− e−βkpAkF.

The requirement that M = F then yields, for the linearoperator Γ, the definition

Γ =

m−1∑k=0

q−1∑p=0

γkpAk (6)

where

γkp = αkpe−βkp

1− e−βkp=

αkp

eβkp − 1. (7)

The procedure to determine the parameters αkp and βkp

using data from the matador tests is as follows.Step 1: We select a matador test in which the light level was

switched from 100% to 0%. In the sequence of recorded frames,

2Note that the first and last rows of matrix A enforce the boundary conditionsthat the second derivatives at the end points are equal to the second derivativesat the adjacent points.


Fig. 10. Dashed line represents the input radiance, based on the assumption ofan instantaneous transition. The dots indicate the measurements at successiveframes. Parameter a is the offset from frame n+

j to the transition.

we identify the transition points from light to dark and viceversa. Thus, the time series is divided into J periods when theaperture was open and J periods when the aperture was closed.Let n+

j denote the frame index at which the jth bright periodended, and let n−

j denote the corresponding index for the jthdark period. Generally, n+

j will be the last bright frame beforethe aperture was blocked, the signal in frame (n+

j + 1) willhave an intermediate value, because the closing of the apertureby the matador’s cape was neither fast nor synchronized withthe OCO frame clock, and in frame (n+

j + 2), the signal willbe low. Even though the solar beam was blocked by a personinserting a sheet of blackened aluminum, an inherently slowprocess, for want of a better model or any indication that theassumption is problematic, we assume that the transition fromlight to dark was instantaneous, occurring at frame (n+

j + a),as shown in Fig. 10. Because the digital count Cn+

j+2 at frame

(n+j + 2) is very much lower than the count Cn+

jat frame n+

j

and because the count recorded at any frame s is the integral ofthe signal from time ts−1 to time ts, we estimate a, the frameoffset to the transition, by

a = Cn+j+1/Cn+

j.

Step 2: The periods of light were sufficiently long that thesignal reached a steady state before the transition from lightto dark.3 Therefore, we average the last u+ frames of thejth bright period to determine the asymptotic value of thesignal before the transition. Let d+j denote this value. A similarprocedure applied to the last u− frames of the jth dark periodgives d−j .4 All J values of d−j are averaged to produce w, whichis taken to represent the dark current

w = J−1J∑

j=1

d−j .

We use the symbol w because the values so obtained constitutethe residual w-pattern5 in the dark when all detector columnsare considered. Of the frames immediately after the transition,

3For the experiment with test tag 7503, this was not strictly true. Theapproach to steady state was close but not complete.

4Typical values for u+ and u− were 5 and 30 frames, respectively.5The w-pattern is an artifact of the read-out multiplexers.

we discard the first, which usually is affected by the closing ofthe aperture, but then retain S successive frames for analysis.We denote by d−j,s the measured signal in the dark at time step(n+

j + s+ 1) for s = 1, 2, . . . , S.Step 3: We assume that the transition was instantaneous,

occurring at frame n+j + a, and that the true signal after n+

j + awas zero. Inserting the ansatz of (4) into (3), we find for framess ≥ 0 that

Ms =

m−1∑k=0

q−1∑p=0

∞∑n=s+1

αkpe−βkpnAkFs−n (8)

where

Fs =

{0, if s ≥ 0F, if s < 0.

Here, F is the static spectrum prior to the transition. Summationof the geometric series in (8) yields

Ms =

m−1∑k=0

q−1∑p=0

γkpe−βkpsAkF. (9)

Step 4: Our aim is to determine the parameters αkp andβkp in (4). To do so, we identify F in (9) with d+j definedfor period jafter correction for w-pattern. Thus, we make theidentification for the jth bright period that

F ↔ D+j = d+j − w.

Similarly, we identify the measured spectrum Ms at frame s ≥1 after the transition in the jth dark period with d−j,s, again aftercorrection for w-pattern

Ms ↔ D−j,s = d−j,s − w, s = 1, 2, . . . , S.

The parameters are estimated using nonlinear least squaresoptimization. We define differences between model and data

Φj,s(αkp, βkp) = Ms −D−j,s, s = 1, 2, . . . , S

and the cost function

Φ(αkp, βkp) =J∑

j=1

S∑s=1

Φ2j,s(αkp, βkp)

where the summation over s explicitly excludes the first pointafter the transition, that point usually being affected by the rateof closing of the aperture. Thus, the cost function measures thediscrepancy between model and data over J cycles of light anddark for S observations after each bright-to-dark transition. Theoptimization is repeated for all sounding positions recorded inthe TVAC matador test. In practice, the algorithm optimizes γkpand βkp, and then, αkp is recovered using (7). The optimizationalgorithm is that published by [13].

The optimization algorithm requires the partial derivativesof the cost function with respect to γkp and βkp. These follow


Fig. 11. Upper panel shows the (cross symbols) measured and (plus symbols)modeled signals after the bright-to-dark transitions in the training data set.Measurements for 64 pixels (513–576) from the P-branch of the O2 A-bandwere used to define the cost functions. The lower panel is a scatter plot of themeasured and modeled data.

easily from the corresponding derivatives of Ms in (9)

∂Ms

∂γkp= e−βkpsAkF

∂Ms

∂βkp= −sγkpe

−βkpsAkF.

The upper panel of Fig. 11 shows the measured (crosssymbols) and modeled (plus symbols) signals after the bright-to-dark transitions in the training data set. Measurements for64 pixels (513–576) from the P-branch of the O2 A-band wereused to define the cost functions. The model captures mostof the variability in the data, except perhaps long after eachtransition when the signal is dominated by noise. The lowerpanel of Fig. 11 is similar, except that it presents the measuredand modeled signals in a scatter plot.

C. Recovery of the True Spectrum by Deconvolution

The discretized version of the evolution (5) may be writtenin the compact form

Ms = (1− Γ)Fs +

∞∑n=1

GnFs−n (10)

where Ms and Fs represent the measured and true spectra atframe s and

Gn =

m−1∑k=0

gknAk.

The summation on the right-hand side of (10) represents thecontribution to the measured spectrum at frame s from the truespectra at all preceding frames. Because Gn decays exponen-tially to zero as n increases, the contributions to the historysummation from frames more than h frames before the current

frame s may be neglected.6 A simple deconvolution algorithmthat provides an approximation F̂s to the true spectrum Fs maybe formulated as follows. At the starting frame (s = 0), assumethat the measured spectrum is a reasonable approximation tothe true spectrum

F̂0 = M0.

The error incurred in this approximation will be damped out ex-ponentially as time progresses. At subsequent frames, construct

F̂1 =Λ[M1 −G1F̂0]

F̂2 =Λ[M2 −G1F̂1 −G2F̂0]

F̂3 =Λ[M3 −G1F̂2 −G2F̂1 −G3F̂0]. . .

F̂h =Λ[Mh −G1F̂h−1 −G2F̂h−2 − · · · −GhF̂0]

where Λ denotes the operator

Λ = (1− Γ)−1 (11)

which may be precomputed and stored. After h steps of thisform, in which the “history tail” grows steadily, further stepsmay be truncated at h terms, because contributions from earlierframes will be negligible. Thus, once s > h

F̂s = Λ[Ms −G1F̂s−1 −G2F̂s−2 − · · · −GhF̂s−h]

or more succinctly

F̂s = Λ

[Ms −

h∑n=1

GnF̂s−n

]. (12)

We emphasize that this time-marching deconvolution algo-rithm is not the only tactic that might be used. Indeed, a dis-advantage of the time-marching algorithm is the time requiredto initialize the history buffers, time that would be used betterrecovering data. Furthermore, if there is an interruption to theflow of data, then additional time will be lost as the time-marching algorithm reinitializes. One alternative approach is torecognize that the solution of

M(t) = F (t) +

t∫−∞

dτg(t− τ)F (τ)

may be written in the closed form

F (t) = M(t) +

t∫−∞

dτf(t− τ)M(τ)

where the deconvolution kernel f(t) satisfies the integralequation

f(t) = −g(t)−t∫

0

dτg(t− τ)f(τ).

6In practice, h was assigned a large value, typically 80. With some exper-imentation, it should be possible to reduce this value without affecting theaccuracy of the deconvolution significantly.


Fig. 12. Right-hand panel in the top row shows the (solid) raw and (dashed) corrected time series for a spectral point where the second-derivative is small,indicated by the vertical line in the left-hand panel. The second and third rows are similar, except that they show spectral points where the second derivative ispositive and negative, respectively.

The kernel f(t) could be precomputed, thereby reducing thecost of deconvolution and ensuring that reinitialization of thealgorithm would require only the history of measured spectraand not the reconstructed history of true spectra. A similarsolution is possible for the gradient formulation in (2).

VIII. PERFORMANCE OF THE ALGORITHM

The performance of the algorithm on data used in its trainingis very good (not surprisingly); undershoot and overshoot arereduced almost to the level of the detector noise. Results of amore stringent test with independent data, not seen during thetraining, are shown in Fig. 12. In this test, the intensity of theinput beam was reduced from 100% to 25%; the matador’s capecontained holes. The first row of Fig. 12 is for a spectral pointwhere the second derivative of the spectrum is small; there isalmost no residual image, and the algorithm does not alter thespectrum. The second row of Fig. 12 is for a local minimumof the spectrum, where the second derivative is positive. In thiscase, the raw data overshoots, but the corrected spectrum makesmuch cleaner transitions, except for the frame closest to thetransition, when the uncertainty in timing cannot be eliminated.Importantly, the corrected signal in the O2 A-band respondsrapidly after each transition, returning to the equilibrium valueusually within two frames. The third row of Fig. 12 is for alocal maximum, where the second derivative is negative; the

Fig. 13. Cross and plus symbols respectively denote surface pressures re-trieved from O2 A-band spectra with and without a correction for residualimage. The offset between the retrievals with the cape inserted and removed iscaused by a small error in radiometric calibration. Subsequently, the calibrationerror was fixed.

raw data undershoots after each transition, while the magnitudeof the undershoot is much reduced in the corrected data.

While the correction is not perfect, the errors that remainafter correction for residual image are unlikely to compromisethe retrieval of either the surface pressure or XCO2

. In supportof this conclusion, Fig. 13 shows the surface pressure retrievedfrom solar spectra in the O2 A-band acquired with the heliostat.In this experiment, the cape used by the matador was analuminum mesh whose holes covered approximately 50% ofits area, thereby transmitting 50% of the incident solar intensity


without imposing a spectral signature upon the radiation. Thecape was inserted and removed at intervals of approximately30 s. The variable plotted on the vertical axis is the surface pres-sure inferred from analysis of the sunlight entering the vacuumchamber in the O2 A-band. Over the course of the experiment,the surface pressure barely changed, so the retrieved pressureshould be constant. The cross symbols in Fig. 13 denote thepressure retrieved from the spectra without any correction forresidual image. The large spikes occur at the transitions whenthe cape was inserted or removed. The amplitude of the upwardspikes is approximately 8 hPa, grossly exceeding the missiontarget of 1 hPa. Superimposed on the retrieved pressure is astep of approximately 3 hPa, caused by incorrect radiometriccalibration. Later work corrected the calibration, and the offsetwas reduced to less than 1 hPa [9]. The plus symbols in Fig. 13show the retrieved surface pressure after the spectra had beencorrected for residual image. In the first cycle of the cape,the correction algorithm acquires the history needed for latercorrections. Thereafter, the correction algorithm eliminates thesharp spikes, and the errors meet the mission target.

In the constrained context of the thermal vacuum chambertests, it was not practical to experiment with continuously vary-ing input intensity, as would be experienced by the instrumentin flight observing sunlight reflected from natural surfaces.Therefore, we resorted to numerical simulations to test theefficacy of the RICA. A simulated instrument was flown overa simulated world. The instrument model included the lineshape function, spectral dispersion, expected noise, and signal-to-noise ratio of the real OCO instrument. It followed the OCOorbit, as shown in the upper panel of Fig. 14, and simulatedthe OCO scan modes. The simulated world included cloudsand aerosols as observed by NASA’s CloudSat and CALIPSOsatellites, surface properties (with polarization) from MODISand POLDER, meteorology from the European Centre forMedium Range Weather Forecasting, and trace gases from achemical transport model. The spectra in the O2 A-band wereperturbed with residual image, and then, noise typical of theinstrument was added in all three spectral bands to producea set of corrupted spectra. Next the RICA was applied to thecorrupted spectra in the O2 A-band to produce a set of correctedspectra. Finally, surface pressure and XCO2

were retrieved fromthe original spectra, the corrupted spectra, and the correctedspectra via simultaneous analysis of all three OCO bands. Thedifferences between the values retrieved from the corruptedand original spectra indicate the deleterious effect of residualimage. Those differences for surface pressure and XCO2

areshown by cross symbols in the middle and lower panels ofFig. 14. The residual image causes unacceptably large errors,well in excess of the mission targets. In contrast, the differencesbetween the values of surface pressure and XCO2

retrieved fromthe corrected and original spectra, shown in Fig. 14 by plussymbols, show that the RICA has reduced the impact of residualimage significantly. The means and biases over the orbit areacceptable at 0.03 and 0.64 hPa for surface pressure and 0.01and 0.20 ppmv for XCO2

.Unlike the analysis of the solar spectra from the heliostat to

recover surface pressure, there is a degree of circularity in thesimulations, because the residual image was simulated using

Fig. 14. Top panel shows the orbit track. Retrievals were not made over theocean, which is too dark in the OCO bands, or when the solar zenith angleexceeded 70◦. The cross symbols in the middle and lower panels show thechanges in the surface pressure and XCO2 when these variables are retrievedfrom spectra corrupted by residual image in the O2 A-band rather than theoriginal uncorrupted spectra. The plus symbols show the errors after the RICAhas been applied.

the initial model described by (1). However, the RICA usedparameters derived from different matador tests, and the spectrawere corrupted further with noise. Despite the limitations, theresults suggest that the RICA should be able to cope with con-tinuously varying intensities. In particular, there is no evidenceof noise amplification or instability.

IX. CONCLUSION

The residual image observed in the O2 A-band of OCO is afeature of the HyViSI Hawaii-1RG FPA when used in a rollingread-out mode at 180 K and is not simply an indication of afaulty flight detector. The lifetime of the residual image is afunction of the number of frames rather than the time after anabrupt change in the input radiance; for OCO with a frame rateof 3 Hz, the residual image could last many seconds and wouldhave so compromised the spectra that meeting the missionobjectives would have been impossible.

A model was developed to relate the measured spectrumto the current true spectrum and a history of all preceding


changes in the spectra. This model led naturally to a time-marching correction algorithm, in which the true spectrum wasreconstructed from the history of previous measured spectraand their derivatives in the spectral dimension. Parameters ofthe model, and hence, also of the deconvolution algorithm,may be derived from spectra measured during “matador” tests,in which the input beam is blocked at regular intervals byinsertion of an opaque plate. The algorithm appears to berobust, because the exponential nature of the decay of residualimage means that error early in the time series is damped out.The algorithm was tested with data acquired by OCO while itwas in the vacuum chamber at the Jet Propulsion Laboratory(JPL); sunlight was directed into the vacuum chamber by aheliostat on the roof of the laboratory. Generally, the algorithmreduced the contribution of residual image to a level only a littlelarger than the noise level expected for the detector. Numericalsimulations suggest that, once the spectra have been corrected,the impact of residual image upon retrieved XCO2

will be wellwithin the error budget for the mission.

ACKNOWLEDGMENT

The authors would like to thank B. Hancock at the JetPropulsion Laboratory (JPL) for his guidance and invaluablesuggestions and C. O’Dell at Colorado State University andJ. McDuffie at JPL for applying the OCO “full-physics” re-trieval algorithm to the heliostat spectra and simulated spectra,as described in Section VIII. A portion of the research describedin this paper was carried out at JPL, California Institute ofTechnology, under a contract with the National Aeronautics andSpace Administration.

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Denis M. O’Brien received the M.Sc. degree intheoretical physics from the Australian NationalUniversity, Canberra, Australia, and the Ph.D. de-gree in mathematical physics from the University ofAdelaide, Adelaide, Australia.

After holding postdoctoral positions in Scotlandand Australia, he was with the Commonwealth Sci-entific and Industrial Research Organization of Aus-tralia, where his focus was radiation, remote sensing,and instrumentation. In 2004, he moved to ColoradoState University, Fort Collins, as a Senior Research

Scientist. He is a member of the Orbiting Carbon Observatory and the Atmo-spheric Carbon Observations from Space science teams.

Randy Pollock received the B.S. degree in engineer-ing and applied science from California Institute ofTechnology, Pasadena, in 1991.

He is an Instrument Architect for the OrbitingCarbon Observatory at the NASA Jet Propul-sion Laboratory, California Institute of Technology,Pasadena, CA.

Igor Polonsky received the M.S. degree in physicsfrom the Belarusian State University, Minsk,Belarus, in 1983 and the Ph.D. degree in physicsfrom the B. I. Stepanov Institute of Physics, NationalAcademy of Sciences of Belarus, Minsk, in 1998.

In 1983, he was with the B. I. Stepanov Instituteof Physics and worked there for 17 years. That time,his major research interests were in the approximatesolution of radiative transport problems and activeand passive remote sensing of the atmosphere–oceansystem. In 2004, he received a Los Alamos National

Laboratory director fellowship and worked there until 2007. He joined Col-orado State University, Fort Collins, in 2007, and currently he is involved inCloudSat and Orbiting Carbon Observatory projects as an Algorithm Devel-oper. He has published over 35 refereed papers, books, and chapters in editedvolumes where the most recent ones are focused on 3-D radiative transfer inclouds, remote sensing of clouds and aerosols, and interaction between cloudsand aerosols.

Matt Rogers received the Ph.D. degree in at-mospheric science from Colorado State University(CSU), Fort Collins, in 2008.

He is currently a Research Scientist I with CSU.His primary areas of research include cloud–climateinteraction and remote sensing of atmospheric prop-erties from space. Apart from his work on the Orbit-ing Carbon Observatory (OCO), he is involved withthe CloudSat mission, which was launched in April2006. His recent research includes using CloudSatdata in concert with observations from the MODIS

and AMSR-E instruments aboard the Aqua spacecraft to classify tropical cloudregimes and to compute the regime-averaged properties of each. His work withthe OCO mission included sensor analysis and testing of the O2 A-band sensoraboard the OCO spacecraft.

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