We describe the calibration and performance of the Mars Odyssey spacecraft's Thermal Emission Imaging System visible imaging subsystem (THEMIS VIS) and present comparisons with other instruments in order to validate the results. The main challenge to the THEMIS VIS calibration process is the significant amount of stray light that accumulates during both integration and readout. The stray light is influenced by scene elements outside of the field of view of the THEMIS VIS detector, and so its magnitude can only be estimated. As a result, residual stray light artifacts are common in calibrated THEMIS VIS images and are especially prominent when the exposure time is short, or the scene contrast is high. Nevertheless, our absolute 2σ calibration uncertainty for the central region of the most frequently used THEMIS VIS channel, the 654 nm band, is better than 5% for all but the shortest exposures times, and our comparisons with Hubble Space Telescope and Mars Exploration Rover measurements show no evidence of systematic calibration inaccuracies.
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 The Mars Odyssey spacecraft's Thermal Emission Imaging System (THEMIS) consists of two independent multispectral imagers sharing a single telescope: a nine-band midinfrared microbolometer arrary (THEMIS IR) and a five-band visible/near-infrared interline transfer CCD (THEMIS visible imaging subsystem (VIS)). THEMIS has been acquiring visible and infrared images from Odyssey's 400 km circular polar orbit since February 2002 and has already provided important new insights about the geology and evolution of the Martian surface [e.g., Christensen et al., 2003; Titus et al., 2003; Pelkey et al., 2003, 2004; Milam et al., 2003]. Details of the THEMIS design, including THEMIS IR, and the optics shared by both subsystems, are given by Christensen et al. . The purpose of this paper is to describe and evaluate the calibration of THEMIS VIS and to describe aspects of the instrument design and on-orbit performance which are directly related to the calibration.
 THEMIS VIS is one of several visible light instruments currently in operation in Mars orbit. Of these, the Mars Orbiter Camera (MOC) on the Mars Global Surveyor (MGS) orbiter [Malin et al., 1992], and the High-Resolution Stereo Camera (HRSC) on the Mars Express orbiter [Neukum and Jaumann, 2004] are most similar to THEMIS VIS. The OMEGA imaging spectrometer on Mars Express [Bonello et al., 2004], the visible-band bolometer detectors of the Thermal Emission Spectrometer (TES) on MGS [Christensen et al., 1992], and the Wide-Field Planetary Camera 2 (WFPC2), Advanced Camera for Surveys (ACS), and Space Telescope Imaging Spectrometer (STIS) [e.g., Bell, 2003] on the Hubble Space Telescope (HST) are also currently providing (or have recently provided) visible light coverage of Mars, but at significantly lower resolutions than MOC and HRSC.
 MOC acquires monochrome images at scales as small as ∼1.5 m and two-color imagery at regional and global scales from a circular, 400 km altitude, 2:00 p.m. local solar time Sun-synchronous orbit [Malin et al., 1992]. HRSC provides stereo imagery and four-channel color from a highly elliptical orbit with a best resolution of ∼50 m per pixel [Neukum and Jaumann, 2004]. Both MOC and HRSC are line scan cameras, which use linear charge-coupled device (CCD) arrays to obtain information along one axis, and spacecraft ground track motion perpendicular to the CCD array direction to obtain the other spatial axis. THEMIS VIS acquires monochrome or color images in any combination of its five bands with resolution modes of 18, 36, and 72 m per pixel from its circular, 400 km, approximately Sun-synchronous orbit at typical local solar times of 4:00 p.m. to 5:00 p.m. The filter band passes have a full width at half maximum of roughly 50 nm and are centered at 423, 540, 654, 749, and 860 nm. Figure 1a shows the THEMIS VIS band-pass profiles. Unlike MOC and HRSC, THEMIS VIS is a two-dimensional (2-D) array framing camera; all 1024 × 1032 CCD pixels are exposed simultaneously. However, spacecraft ground track motion is also used to expand the spatial coverage of acquired imaging sequences. While THEMIS VIS is not designed to provide stereo capability, a limited amount of information on the altitude and cross-track velocity of high clouds can be derived by coregistering overlapping frames acquired at slightly changing viewing geometries over the course of a color imaging sequence [McConnochie et al., 2004].
 THEMIS VIS was found to have a serious problem from stray light contaminating the detector signal both during integration and during the CCD readout. Stray light accumulation during integration affects the entire array, but is most noticeable as broad and brighter stripes of additional signal near the edges of uncalibrated images. Stray light accumulation during the readout process creates an additive offset, and an effect similar to the “electronic shutter smear” seen in some shutterless CCD imaging systems, both of which are proportional to scene brightness but independent of the exposure time. Most of this paper is focused on the removal of both forms of stray light contamination from THEMIS VIS images.
 This paper describes and evaluates the process by which the radiance-calibrated data stored in the NASA Planetary Data System (PDS) THEMIS VIS reduced data records (RDRs; accessible at http://themis-data.asu.edu/) is generated from the raw PDS engineering data records (EDRs). (Note that THEMIS VIS RDRs are not geometrically calibrated. This paper does not address geometric calibration.) We begin by describing the relevant operational details of THEMIS VIS, as well as our preferred model for the stray light contamination mechanisms. We then discuss each step of the EDR-to-RDR calibration process. Finally, we evaluate the calibration by comparing the derived THEMIS VIS radiances to HST results, as well as to the surface-based multispectral measurements provided by the Mars Exploration Rovers' Panoramic Cameras (Pancams).
2. Operational Details and Labeling Conventions
 VIS uses a Kodak interline transfer CCD. For each column of photosites, there is a masked vertical register (v register) adjacent to it. At the start of a THEMIS VIS exposure, any charge accumulated on the detector is flushed. Charge accumulated in the photosites is transferred to the adjacent v registers twice during the commanded exposure period, once at the midpoint and once at the end. The v registers are masked to minimize accumulation of photoelectrons during readout. During the readout process, beginning immediately after the second photosite charge transfer, the charge is shifted “downstream,” i.e., down the v registers and transferred to the CCD's horizontal register (h register) one row at a time. The h register will be considered to be at the “bottom” of the detector array in this paper, so that “down” and “downstream” are in the same direction. That is, if one row of pixels is said to be below or downstream of another row of pixels, then the row that is “below” or downstream is closer to the h register. With this convention, the direction of spacecraft motion (south on the afternoon side of the orbit) is always down. We number rows starting from zero at the bottom. Note, however, that the opposite row ordering convention is used in PDS EDR and RDR products.
 Physically, the THEMIS VIS CCD detector has 1032 columns and 1024 rows. The 5 THEMIS VIS filters are strips that are 1032 columns wide and approximately 200 rows tall and are bonded directly to the detector. The front of the CCD housing is covered by a window, which therefore falls between the filter strips and the telescope optics, and which influences the overall spectral response in each THEMIS VIS band. Figure 1a gives the spectral response through each THEMIS VIS filter with the spectral throughput of the CCD window included. The spectral response of each filter by itself turns out to be important, as we will show, for predicting the magnitude of stray light contamination and is therefore plotted in Figure 1b.
 In this document, as in the EDR and RDR labels on the PDS files, the filters are labeled “filter 1,” “filter 2,” to “filter 5” in order of their position on the detector array, with filter 1 at the bottom, closest to the h register. In this naming scheme, filter 1 is 860 nm, filter 2 is 425 nm, “filter 3” is 654 nm, “filter 4” is 749 nm and filter 5 is 540 nm. An alternate filter designation exists, using the labels “band 1”, “band 2”, to “band 5” to refer to the filters in order of increasing center wavelength [425, 540, 654, 749, 860 nm]. Figure 2 shows the THEMIS VIS focal plane with “filters” f and “bands” k labeled.
 The ground track of the Odyssey spacecraft runs approximately parallel to the detector columns. This allows multispectral images to be acquired using a timed sequence of THEMIS VIS exposures. Any combination of one or more filters can be acquired by selectively reading out only those rows that correspond to the desired filters. Thus a single THEMIS VIS exposure can contain any or all of the five filters. We refer to the portion of an exposure that contains data for a single filter as a “framelet.” All exposures in a sequence have the same exposure duration, and all filters in an exposure must have the same exposure duration, and so all framelets in a given THEMIS VIS sequence have identical exposure durations.
 The different filter framelets in a full-frame THEMIS VIS exposure cover different regions of the Martian surface (nominally about 3.6 km apart for adjacent filters), hence the need for a sequence of multiple exposures to build up a multispectral image using spacecraft downtrack motion to image the same parts of the surface through different filters (and thus at slightly different times). The exposure sequences are designed so that all filters end up with the same number of framelets and so that when a filter moves past the targeted region of the surface, it is no longer read out, meaning that the first and last several exposures of an imaging sequence will have a smaller number of filters read out than exposures in the middle of a sequence. Figure 3 illustrates the way that the framelets of a series of exposures map onto the Martian surface and can be arranged to form a continuous image strip for each filter.
 The delay between exposures within a THEMIS VIS imaging sequence is set so as to maximize the areal coverage of an imaging sequence while allowing for adequate overlap between adjacent framelets of the same band. A 1 second delay leads to a typical overlap of between 10 and 20 detector pixels (depending on the surface elevation and the angle of the planet's rotation relative to the ground track) when projected onto the Martian surface. One side effect of this overlap is that framelets of different bands are not exactly spatially coincident when map projected, meaning that framelet boundaries (and calibration artifacts associated with those boundaries) for different bands are in different positions in mapped data, and that the total area covered by lower numbered filters within a sequence is offset relative to that covered by higher numbered filters by a number of pixels equal to the framelet-to-framelet overlap multiplied by the difference in filter number. The delay between exposures is given by the “INTERFRAME_DELAY” keyword in the THEMIS VIS PDS headers. A standard interframe delay of 1.00 seconds was for used image numbers up through V14885012. The standard interframe delay was changed to 0.90 seconds beginning with V14886001.
 Rows and columns near the edges of the filters are always cropped during readout. The result is that each framelet has dimensions of 1024 by 192 detector pixels. The rows that are cropped are referred to henceforth as interframelet rows. When THEMIS VIS EDR “cubes” are generated (three-dimensional data sets containing both spatial and spectral information, in PDS “QUB” file format), the THEMIS VIS exposures in a sequence are broken up into their constituent framelets and reassembled so that each framelet's position in the cube corresponds roughly to its spatial location on the surface of the planet. Thus, for a given filter, the corresponding plane of the EDR QUB is generated by concatenating the framelets from top to bottom in the order in which they were acquired. Assuming no spatial summing (spatial summing is discussed below), and using the PDS row ordering convention (opposite that used elsewhere in this paper), the first (top) framelet goes in EDR rows 0–191, the second in EDR rows 192–383, etc. Within a given EDR QUB plane, we assign the framelets a number, m, starting from the top, with the first framelet acquired in a given filter being m = 0.
 The procedure for removing stray light accumulated during readout requires that all of the framelets from a given exposure be grouped together. Consider a framelet at position mi taken through filter number f0. The position m1 of the framelet taken through filter number f1 that was acquired in the same exposure as framelet (m0, f0) can be found by
 If m1 < 0 or m1 ≥ n, where n is the number of framelets in each filter in the EDR cube, then there was no framelet of filter f1 acquired in the same exposure as framelet (m0, f0). Note that equation (1) is written in terms of filter numbers, which are, as mentioned previously, not the same as band numbers. Note also, however, that the EDR and RDR QUB planes are stored in band order, not in filter order, and the location of each filter within the EDR and RDR cubes is given by the BAND_BIN_FILTER keyword in the PDS labels. Figure 4 shows all five planes of the EDR from a five-band THEMIS VIS imaging sequence with each framelet labeled according to its band number k, filter number f, framelet number m, and exposure number a. Exposure number a is defined so that it is equal to m for the lowest numbered filter in a given EDR or RDR; thus:
 THEMIS VIS can operate in any of three spatial summing modes. The preceding discussion has assumed summing mode 1, i.e., no spatial summing, for clarity. In summing modes (i.e., for spatial summing factors of) 2 and 4, the THEMIS VIS pixels are binned 2 × 2 and 4 × 4, respectively. This creates framelets with 512 columns by 96 rows, or 256 columns by 48 rows, respectively. Summing mode 1 has a typical pixel scale of 18 m on the Martian surface for a nadir-pointed viewing geometry. Thus summing modes 2 and 4 have nadir-pointed pixel scales of 36 and 72 m per pixel. To date, the Martian surface has been targeted solely in the nadir-pointed configuration, so actual pixel scales will vary only slightly from the values quoted here.
 The amount of time required to read out a single THEMIS VIS exposure depends both on the spatial summing mode and on the number of filters being read out. Each framelet of an exposure requires 141, 76, or 39 ms to read out, for summing modes 1, 2, and 4, respectively. A small amount of time (∼ 4 ms) is also required to “dump” the charge from each filter that is not being read out. About 0.2 ms is required to dump each group of the interframelet rows, which are never read out. So, for example, it takes 703 ms to complete the readout process when summing mode 1 data from all five filters is being acquired. Since this readout time is long compared to the ∼5–10 ms typical exposure time, the opacity of the masks that protect the v registers from photoelectrons, as well as the amount of time that each row of data spends in the v registers during readout, are crucial factors for the THEMIS VIS calibration process. Also, since THEMIS VIS quickly dumps the charge in filters not being read out, the amount of time a given row spends in the v registers depends not only on its distance from the h register, but also on which of the filters that are below it on the array are being read out.
 Correcting for stray light and dark current accumulation during readout requires knowledge of the timing for each data pixel as it is clocked down the v registers to the h register during readout. For pixels in a given filter, this timing depends on which of the lower numbered (closer to h register) filters are being read out in a particular exposure, and which are being dumped and not read out. Pixels in filter number f have 2(f−1) possible timing “paths” during readout (e.g., for filter 3, these are the read out filter combinations [3,2,1], [3,2], [3,1], and (3)). The total number of possible paths is therefore
 These 31 possible paths will henceforth be referred to as “filter paths.” Each path is given a filter path code, F. For a given filter with filter number f0, F is determined by
where η(f) = 1 if filter where f is included in the exposure and η(f) = 0 if filter where f is not included in the exposure.
 A filter is included in an exposure if both of the following are true: (1) the filter is present in the EDR; and (2) the framelet position m of the filter (as calculated from equation (1)) satisfies m1 < n and m1 ≥ 0, where n is the number of framelets of each filter in the EDR cube.
 The calibration process is also influenced by the manner in which the THEMIS VIS electronics perform spatial summing and store the data. The number of pixels summed in the “horizontal” direction (along rows) is always equal to the number of pixels summed in the “vertical” direction (along columns), but the spatial summing in each direction is performed in a different manner. Vertical summing, if any, is performed in the h register. The charge from the appropriate number of v register pixels is simply added together into each h register site, and the h register sites are then passed to the analog-to-digital (A-to-D) converter and represented as digital (“data number” or DN) values with a gain measured at 25.4 electrons per DN. An additive bias equal to 104 DNs is removed at this stage, after which DN values can range from 0 to 2047 (i.e., 11 available bits) regardless of the spatial summing mode. At this point, summing has only been performed along the vertical axis, and the DN values represent charge from a number of pixels equal to the spatial summing factor. That is, for a given exposure time and scene brightness, 2 × 2 summing gives, neglecting charge accumulation during readout, twice as much h register charge, and thus a DN value which is larger by a factor of 2.
 Next, horizontal summing is performed on these DN values by THEMIS VIS onboard software, but now the appropriate number of pixels are averaged rather than summed, so that DN values in the 0–2047 range remain in the 0–2047 range. Thus, considering the total effect of the two disparate steps of the spatial summing process, the values returned by THEMIS VIS with spatial summing are, with a given scene brightness and exposure time, still only larger by a factor equal to the spatial summing factor (rather than to the spatial summing factor squared) once readout-accumulated charge has been removed, and the amount of exposure time required to reach saturation of the available 0–2047 DN representation range is correspondingly less by a factor equal to the spatial summing factor.
 Before the THEMIS VIS data are sent to the Odyssey spacecraft flight computer for storage and eventual downlink, the 0–2047 11-bit DN values are compressed into a 0–255 8-bit range using a square-root encoding algorithm. It is important to note that 0–255 square-root-encoded data are preserved in the EDR, that the encoding is nonlinear, and that therefore the encoding must be reversed before the THEMIS VIS EDR data are used in any quantitative way. (See Figure 15 in section 4.3.1.)
 The number of framelets in any THEMIS VIS imaging sequence is limited by the 3.8 megabyte capacity of the THEMIS VIS instrument's data storage buffer, since THEMIS VIS can transfer data to the spacecraft flight computer only after the completion of an imaging sequence. This maximum capacity corresponds to 19, 78, and 318 framelets for summing modes 1, 2, and 4, respectively, which means maxima of 3, 15, and 63 framelets per band for five-band imaging sequences, or, for example, 6, 26, and 106 framelets per band for three-band imaging sequences.
 In addition to the normal operating mode described above, THEMIS VIS has a “test” mode in which the entire 1024 × 1032 array is read out and stored, only a single exposure is acquired, and only spatial summing mode 1 is available. The in-flight test mode is otherwise identical to the normal operations. However, much of the calibration data used for this paper was acquired using a preflight test mode with several additional differences from in-flight operations: (1) only one photosite transfer, at the end of the commanded exposure time, is performed; (2) no bias subtraction is applied prior to data storage; and (3) the data are never compressed to an 8-bit representation.
 In order to compare preflight (or in-flight) test mode results with normal operating mode data, the test mode images are separated into framelets, with the test mode image yielding one framelet for each filter. These framelets are 1024 × 192, just like a normal framelet, and are obtained by extracting rows 2–193, 202–393, 403–594, 611–802, and 813–1004, for filters 1–5, respectively, so as to cover the same region of the detector as the operating-mode framelets. Rows are numbered starting with zero at the bottom of the detector.
3. Stray Light Model
 Our calibration method is based on the hypothesis that the source of the stray light signals is a light leak under the edges of the filters (Figure 5). Since the edges of the filters are not masked, and since light reaches the focal plane at fairly high incidence angles (the THEMIS optics are an f/1.7 system), a significant portion of the light that strikes the focal plane near the edges of the filters is scattered or reflected toward the gap between the filter and detector (this gap is filled with a transparent adhesive used to bond the filters to the array; see Figure 5). Since the interference filters strongly reflect broadband light, light that is scattered or reflected into the gap can be directed back down to the detector.
 Light directed back toward the detector will obviously contaminate the photosites of the detector array, and we will refer to this effect as photosite stray light. Photosite stray light is illustrated in Figure 6, which shows a preflight test mode image of a uniform integrating-sphere light source. Within the region covered by each filter the signal level is nearly uniform, and the brightness differences between the filters are the result of the calibration lamp spectrum and the wavelength dependence of the THEMIS VIS system's responsivity. Superimposed on this expected pattern are broad brighter stripes at the bottom and on either side of the array. These brighter stripes are the result of the photosite stray light. Notice that the signal in most of the area of the 860 nm filter, located at the “bottom” of the array, is dominated by the photosite stray light. It is not known why there is no similar bright stripe in the 540 nm filter at the “top” of the array. Also note that as shown by Figure 7, the amplitude of this stray light pattern is similar in each of the filters, despite the fact that the center field, mostly uncontaminated photosite signal varies widely from filter to filter.
 Light propagating in the filter-detector gap also contaminates the v registers and h register, since the narrow masks that protect the v registers and h register are less effective at high incidence angles. The v register and h register contamination is readily observed in the preflight calibration data by extrapolating DN vs. time to an exposure time of zero, thereby eliminating the photosite contribution to the signal. Figure 8 shows a column profile of this register contamination signal for several different calibration lamp brightnesses, with a 3-ms integration photosite signal shown for comparison. Figure 9 shows an image of the register contamination signal for a particular preflight calibration lamp brightness. The ramp-up of the register contamination signal toward high row numbers seen in Figures 8 and 9 is superficially similar to an electronic shutter smear effect, but it differs in that it depends on the brightness of the scene just outside of the detector field of view, rather than on the portion of the scene imaged by the detector, and so its magnitude and behavior are not directly related to the information contained in the image.
 Although details of the scattering/reflection processes and the properties of the register masks are not well known and so must be derived as part of the calibration procedure, the general features of our model yield several key assumptions for that procedure:
 1. The source for both types of stray light is just outside the field of view of the detector. Therefore the calibration procedure can never exactly remove stray light effects because the scene radiance in the region just outside the field of view can by definition only be estimated. For low-contrast scenes, this “guess” can be fairly accurate, but for high-contrast scenes, the error inherent in estimating out-of-field stray light can be significant.
 2. The register stray light is proportional to the scene radiance, and to the amount of time that each row spends in the v registers during the readout process, but it does not depend on the exposure time (since the readout time depends only on the spatial summing mode and filter combination).
 3. The photosite stray light is proportional to both the scene radiance and the exposure time.
 4. Both stray light signals are proportional to the “broadband” scene radiance, rather than the narrow-band radiance in any of the five filters, since the stray light bypasses the filters in our model of the effect. Thus, just as the “direct” (i.e., stray light free) signal in each filter is proportional to the scene radiance spectrum weighted by that particular filter's response function (shown in Figure 1a), the stray light signal in each filter is proportional to the scene radiance weighted by a different response function, one which includes all of the same factors as those for the filters, except for the transmissivity of the filters themselves. This response function is shown in Figure 10. The dominant factors in this broadband response function are the CCD window and the detector quantum efficiency.
4. Calibration Procedure
 The THEMIS VIS flight data calibration process (hereinafter referred to as the “calibration pipeline”) consists of seven steps: (1) 8-bit to 11-bit decoding; (2) identification of bad pixels; (3) bias subtraction; (4) register stray light subtraction; (5) correction for pixel response variations (i.e., “flat fielding”); (6) photosite stray light subtraction; and (7) conversion to radiance. Derivation of the calibration frames and calibration coefficients necessary for this pipeline proceeds in a somewhat different order, however. The first step of the calibration derivation is to use the preflight data set to determine the photosite stray light response and the direct response for the central region of each filter. The next step is to develop a model, which we will call the “broadband radiance model” that predicts the broadband scene radiance from the narrow-band scene radiances in each of the five THEMIS VIS band passes. The response coefficients and broadband radiance model are necessary for deriving the calibration frames of pipeline steps 4, 6, and 7. Once the response coefficients and broadband radiance model are in hand, the calibration frames are derived in the same order as the pipeline steps using only on-orbit data, and so we will conclude this section by describing each pipeline step together with the derivation of any required calibration frames.
 Many of the calibration coefficients, and in particular the response coefficients, apply to the entire area of a filter, and so they are calculated using mean values from a representative subregion of that filter's framelets. This subregion should also contain a minimal amount of stray light, so that the calibration coefficients based on it are influenced as little possible by any uncertainties in the stray light estimates. We have identified one such representative-yet-minimal-stray-light region, which will be referred to as the “calibration region of interest” (C-ROI), for each filter. The C-ROI boundaries are shown in Figure 6.
4.1. Derivation of Response Coefficients From Preflight Data
 Given the differences between the preflight and on-orbit operating modes, and considering the possibility of launch or postlaunch changes in instrument behavior, we have derived the THEMIS VIS calibration using on-orbit data wherever possible. However, in order to accurately determine the photosite radiometric responses, we require a source with known spectral radiance. In principle, this requirement could be met on-orbit using radiance measurements acquired by another instrument, but this other instrument would need to observe the same region of Mars at the same time as a THEMIS VIS imaging sequence, with the same viewing geometry, and have a well validated calibration, high spectral resolution, and subkilometer-scale spatial resolution. We have acquired concurrent THEMIS VIS–WFPC2 and THEMIS VIS–ACS measurements that come closest to satisfying these criteria, but given the spatial resolution discrepancy, and since the difference in phase angle between the HST measurements and the THEMIS VIS measurements is very large, and the high-phase photometric properties of the Martian surface-atmosphere system are potentially variable and not well known, we deem our ground based spectral radiance source to be the most reliable means for determining the radiometric response. The HST data sets will be used to validate the results.
 The preflight data set we have used was acquired with the THEMIS instrument in a temperature-controlled vacuum chamber viewing an integrating-sphere light source through a window in the chamber wall. The integrating-sphere exit port filled the THEMIS VIS field of view and extended well beyond it in all directions. We verified that changing the distance between the integrating sphere exit port and the THEMIS VIS window had no effect on the observed signal pattern or signal magnitude.
 The integrating sphere output spectral radiance is varied by changing the number of 8, 45, and 200 W lamps, and for many of the possible lamp combinations the integrating sphere output is traceable to National Institute of Standards and Technology (NIST) standards. For the THEMIS VIS calibration activities, we used from one to nine 8W lamps, and also settings with one 45W lamp and two 45W lamps. The lamp settings with the 8W lamps are all directly NIST traceable with ±2% precision. The one 45 W and two 45 W settings are not directly NIST traceable. We have estimated the spectral radiance for these settings by subtracting the NIST traceable ten 8 W setting spectral radiance from the NIST-traceable ten 8 W, one 45 W setting and ten 8 W, two 45 W setting spectral radiances. This subtraction is of course only valid under the assumption that the lamp settings are linear combinations of each other. We have tested the linearity assumption by differencing the NIST-traceable radiance measurements of sets of 8W lamp settings; for example, nine 8 W – six 8 W – three 8 W should give zero radiance with a precision degraded by a factor of roughly . We find that the linearity assumption is valid under such tests, and we therefore conclude that our differencing-based spectral radiance values for the 45 W lamps will have a precision of roughly × ±2%, i.e., better than ±3%, and will not be systematically biased. In any case, the 45 W lamps provide a different spectral shape from the 8 W lamps, and this different spectral shape is crucial for separating the direct response from the stray light response, so we have no choice but to use the 45 W lamps in the calibration process.
 The spectral radiance presented to THEMIS VIS is also of course affected by the transmissivity of the vacuum chamber window. Figure 11 shows the integrating sphere radiance for each lamp setting after correction for the window transmissivity. For each lamp setting, we calculate a weighted mean broadband radiance, as well as a weighted mean narrow-band radiance for each filter, by weighting the window-corrected sphere radiances with the response functions of Figures 10 and 1a, respectively. These weighted mean radiance values are listed in Tables 1a and 1b.
Table 1a. Radiance Calibration Data Set: Lamp
In-Band Radiance, W m−2 μm−1 sr−1
Broadband Radiance, W m−2 μm−1 sr−1
425 nm Band
540 nm Band
654 nm Band
749 nm Band
860 nm Band
Table 1b. Radiance Calibration Data Set: Detector
Signal, DN ms−1
425 nm Band
540 nm Band
654 nm Band
749 nm Band
860 nm Band
250 K Thermal Vac Test
268 K Thermal Vac Test
279 K Thermal Vac Test
295 K Thermal Vac Test
 A set of THEMIS VIS integrating-sphere tests was performed in a thermally controlled vacuum chamber temperature with the THEMIS VIS focal plane temperature maintained at each of six different settings. For each temperature setting, measurements were taken with the following twelve lamp settings: all lamps turned off, one through nine 8 W lamps, one 45 W lamp, and two 45 W lamps. Exposure times of 3, 6, 12, and 24 ms were used at each lamp setting, and ten images were acquired at each exposure time. All ten images are averaged together to produce a single low-noise image at each exposure time. We use this data set to estimate the photosite stray light response and the direct response as follows:
 1. Fit a line to the set of exposure times for each pixel of each lamp level in order to determine S*, the total photosite (stray light plus direct) signal, in DN per ms:
where D(t)ijk,l is the observed raw DN value for pixel i, j of band k at lamp level l, t is exposure time, and the residuals of the fit are represented by ɛ. Z gives the register stray light signal. Z is useful for understanding the behavior of the register stray light, but since the preflight test mode readout pattern and timing is different from the flight operating mode, Z is not directly applicable to the flight data calibration.
 Since the preflight test mode performs only one photosite-to-register transfer in the course of an exposure, detector full well was reached well before the A-to-D conversion saturation value of 2047. We observed nonlinear response associated with full well at DN levels greater than 1250. Therefore we inspected the D(t)ij images for each filter and lamp level and excluded any with substantial regions of >1250 DN. Additionally, in the linear fitting process, we exclude all individual pixels that are either greater than 1250 DN, or which have more than 5% of pixels >1250 in the surrounding 11 × 11 region.
 2. Determine the background-subtracted total photosite signal, S:
where S*ijk,0 refers to the lamp setting with all lamps turned off. The zero lamp signal is clearly not dark current, as the zero lamp images show a stray light pattern similar to the directly illuminated images, and their peak center filter values of 2.5 DN per ms at 262 K are too high for plausible dark current and show no temperature dependence. The zero lamp signal is also spectrally distinct from the integrating-sphere signal, which implies that its source may be background light in the testing room. In addition to subtracting this background light, we have also attempted to mitigate its impact by excluding the lower lamp settings from the model fits described below. Thus, in deriving the response coefficients, we use only those settings with the 45W lamps or at least six 8W lamps. With this restriction, the largest background signal contribution is to the 423 nm filter signal at six 8W lamps, where it makes up ∼15% of the total signal. Although the background subtraction should remove most of this contribution, without knowing the source of the background signal we have no way to assess its variability aside from the residuals of the model fits described below.
 3. For each band k, extract the C-ROI mean values of the background subtracted photosite signal, k,l (signal in Tables 1a and 1b). Then fit them to a linear model, with weighted-mean broadband radiance l (in Tables 1a and 1b), and weighted-mean band k narrow-band radiance k,l (also in Tables 1a and 1b) as the independent variables:
where xk and yk are the photosite stray light response coefficient and direct response coefficient, respectively. The straight bar in the symbols l and k,l indicates that these radiances should be interpreted as spatial means over the region sampled by a framelet.
 Since l and k,l are highly correlated, we have taken the precaution of solving for xk and yk using a grid search algorithm, which allows us to map the solution set and visualize the extent to which xk and yk can be simultaneously constrained. We perform the grid search with a grid spacing of 0.005 (DN ms−1) (W m−2 μm−1 sr−1)−1 in both xk and yk, calculating the χ2 probability at each point. The search range was 0–3 (DN ms−1) (W m−2 μm−1 sr−1)−1 in xk and 0–7 (DN ms−1) (W m−2 μm−1 sr−1)−1 in yk. In order to calculate the χ2 probability, each data point is assigned a standard error equal to the root-mean-square residual of the best fit. Figure 12 shows the resulting normalized probability density function (PDF) for four different vacuum chamber temperatures. Note that the model fits do a poor job of constraining the photosite stray light response for bands 2 and 4 and that they provide no constraint at all on the photosite stray light response of band 3. Also note the small upward trend of direct response with increasing temperature for bands 2 through 5. Since the 268 K and 279 K tests bracket the observed on-orbit focal plane temperatures of 268–278 K, we will use these tests to determine our response coefficients and associated confidence intervals.
 4. Identify the band 1 direct and stray light response coefficients directly from the 279 K PDFs. Since band 1 shows no temperature trend for either coefficient, we are free to choose the most precise estimate, which is the one calculated at 279 K. The normalized 1-D PDF for direct response is shown in Figure 13, and the normalized 1-D PDF for stray light response in Figure 14. The adopted values for the coefficients, and their confidence intervals, are given in Table 2. The adopted values are the mean of the population described by the 1-D PDF. The confidence intervals are ranges centered on the adopted value and encompassing 95% of the total probability in the 1-D PDF.
The given ranges represent 95% confidence intervals. Units are (DN ms−1) (W m−2 μm−1 sr−1)−1. For summing modes 1, 2, and 4, z = 5.50 ± 0.48, 6.70 ± 0.44, and 8.40 ± 0.99 DN (W m−2 μm−1 sr−1)−1, respectively.
4.180 ± 0.145
6.085 ± 0.075
5.605 ± 0.090
2.125 ± 0.060
0.6 ± 0.2
0.300 ± 0.025
0.300 ± 0.025
0.300 ± 0.025
0.300 ± 0.025
1.475 ± 0.225
 5. Identify the band 5 response coefficients by combining the 1-D PDFs for the 268 K and 279 K tests. This is necessary because the band 5 coefficients exhibit a temperature trend. The adopted values and confidence intervals are derived as previously described, but using the sum of the normalized 1-D PDFs for the two tests. However, as will be described later, the band 5 direct response coefficient derived in this manner was found to conflict with on-orbit validation measurements. We have therefore chosen to adopt the band 5 direct response coefficient implied by the validation measurements.
 6. Assume that band 2–4 have the same stray light response as band 1. Some kind of assumption about the band 2–4 stray light response is obviously required, since it is poorly constrained by the integrating sphere data set, and we have chosen the simplest possible assumption that is consistent with the 2-D PDFs. The row profiles of bands 1–4 shown in Figure 7 indicate that these four bands show similar stray light patterns with similar amplitudes, which suggests that similar amounts of light are reaching the central area of each filter, and that our assumption is therefore reasonable.
 7. Identify the direct response coefficients for bands 2–4 by combining the 1-D PDFs for the 268 K and 279 K tests. For these bands, the 1-D PDF is generated by summing over the region of the 2-D PDF that falls within the band 1 stray light response coefficient confidence interval.
4.2. Derivation of Broadband Radiance Model From HST Data
 Since the broadband radiance, , is not generally known during on-orbit operations, it must be estimated from the narrow band radiances, k. We do this by finding a set of coefficients αk such that
Alternatively, we can estimate directly from the photosite signal, S:
The ωk coefficients are related to the αk coefficients by
 The αk or ωk can be estimated from a data set of k and that is representative of the radiance of Mars during on-orbit operations. We have chosen a set of HST WFPC2 and ACS observations of Mars performed in 2003 during the Odyssey mission [Bell et al., 2004b], the details of which are given in Table 3. These HST observations use narrow-band filters to sample the Mars radiance spectrum, and so we must interpolate from these samples to form a complete spectrum suitable for convolution with the THEMIS VIS narrow-band and broadband spectral response functions. We do this by performing a cubic spline interpolation of the HST I/F samples and then multiplying by the solar radiance spectrum.
Table 3. The 1997 and 2003 HST WFPC2/PC and ACS/HRC Observations of Mars Used in This Study
 The HST measurements are calibrated using the methods described by Bell et al. . The images are projected onto a 2 pixel-per-degree simple cylindrical grid, and the Mars radiance spectrum at each grid point is used to generate k and for each grid point. All grid points from all of the HST observation sequences described in Table 3 form the q and k,q for the multiple linear regression model indicated by equation (8) or (9). We have used this regression model approach to find αk and ωk in two ways: (1) using equation (7) to calculate k,q from q and k,q, then equation (9) to solve for ωk, and then equation (10) to give αk from ωk; and (2) using equation (8) to solve for αk and then equation (10) to give ωk. Both methods produce essentially identical results.
 Our adopted ωk coefficients are shown in Table 4. In order to produce the best possible estimate of for a particular image sequence, the calibration pipeline needs to be able to use whatever k values are available, and so we need a different set of ωk for each of the 31 possible band combinations. The calibration pipeline will choose the set of coefficients for the band combination that includes only those bands that are being used to estimate , regardless of which bands happen to be present in an EDR or an exposure. Since the ωk are regression coefficients and not response coefficients, negative values are in principle allowed. However, all of the negative coefficients in Table 4 are very small relative to the others for their band combination, and so can be interpreted as zero, meaning that a particular band when part of a particular band combination does not contribute significantly to the estimate.
For an exposure in which all of the bands in an image are being read out, the filter path code (described in the text) of the band furthest from readout is equal to the band code.
1, 3, 5
1, 4, 5
3, 4, 5
1, 3, 4
1, 3, 4, 5
1, 2, 5
2, 3, 5
1, 2, 3
1, 2, 3, 5
2, 4, 5
1, 2, 4
1, 2, 4, 5
2, 3, 4
2, 3, 4, 5
1, 2, 3, 4
1, 2, 3, 4, 5
4.3. Flight Data Calibration Procedure
 In this section, we describe the calibration pipeline procedures in the order that they are performed.
4.3.1. The 8-Bit to 11-Bit Decoding
 The 8-bit-per-pixel square-root encoded EDR data is decoded to its full 11-bit-per-pixel linear range using the inverse look-up table shown in Figure 15. The decoded values can range from 0 to 2040.
4.3.2. “Bad” Pixel Identification
 Bad pixels, also known as null pixels, are assigned the value specified by the CORE_NULL keyword in the PDS RDR label, and are ignored in all subsequent THEMIS VIS processing. Bad pixels are identified in the decoded (11-bit-per-pixel) EDR data as follows:
188.8.131.52. Threshold Values
 All pixels with a DN level of 2040 (the highest possible value) or 0 are flagged as null pixels because they are likely saturated.
184.108.40.206. Bad Rows and Columns
 Some rows and columns near the edge of each framelet are always filled with unusable (noisy or saturated or zero) data. The pixels in these rows and columns (Table 5) are therefore flagged as null.
Table 5. Bad Rows and Columns
220.127.116.11. Saturated Regions
 In exposures where all or part of the detector is saturated, the DN value of some saturated pixels is “wrapped” by the instrument firmware to a value which is invariably small compared to the rest of the pixels in the array. These “wrapped” pixels are identified by considering each THEMIS VIS framelet separately. Any pixel whose value is less than the framelet median by at least 1200 DN has probably been wrapped and is therefore flagged as null. The median is calculated using all pixels not flagged as null in section 18.104.22.168 or 22.214.171.124.
126.96.36.199. Neighboring Pixels
 Pixels near to saturated regions are often observed to be anomalously high even if they themselves do not meet the criteria of section 188.8.131.52 or 184.108.40.206. Therefore, if too many of the pixels near a given pixel are flagged as null in section 220.127.116.11 or 18.104.22.168, that pixel is also flagged as null. A 5 by 5 pixel region centered on each pixel is considered. This region is truncated if it would extend pass the edge of the array. Pixels that lie in the “bad rows and columns” identified in section 22.214.171.124 are always treated as valid for this procedure. If more than 30% of the pixels in the 5 × 5 (truncated if necessary) region for a given pixel are null, then that pixel is also flagged as null.
4.3.3. Bias Subtraction
126.96.36.199. Nighttime VIS Images
 The bias subtraction procedure uses THEMIS VIS images acquired at night with nominal exposure times of zero. The bias is determined independently for each of the three available spatial summing modes. In practice there is no detectable difference between night images acquired with typical VIS exposure times (up to 30 ms) and night images with zero exposure time, so nonzero exposure time night images are also included in the averaging process that is used to determine the bias. This also means that the dark current for typical THEMIS VIS exposure times is effectively zero. (Typical on-orbit focal plane temperatures are ±5°C.) However, because the readout time is much longer than the exposure times, some of the signal for 0 ms nighttime exposures may be attributable to dark current accumulated during readout. The evidence for this is a slight ramp-up in the 0-ms signal toward the top of the detector. Fortunately, since the amount of dark current that accumulates during readout is not a function of exposure time, and since we have detected no temporal variability in the zero exposure nighttime images, there is no need to distinguish between bias and readout dark current for the purpose of calibration. We will therefore henceforth lump both effects together and refer to the combination of the two simply as “bias.”
188.8.131.52. Averaging to Create Bias Frames
 To properly account for the small ramp in the bias level, bias frames are created for each of the 31 filter paths and for each of the three spatial summing modes. Where possible, each bias frame is generated from a simple average of all available nighttime exposure framelets with the same summing mode and filter path.
184.108.40.206. Modeling to Handle Paths for Which No Empirical Data Are Available
 For many of the less common filter paths, nighttime exposure framelets have not yet been acquired. In these cases, we apply the following simple model of the detector readout process in order to approximate the expected bias signal:
 Consider what happens to the charge from all upstream filters while a downstream filter is being read out. As our first simplifying assumption, we ignore the interframelet rows (there are 7 ± 2 interframelet rows at each filter boundary) and pretend that the top row of one framelet is always adjacent to the bottom row of the next. As each row of the downstream filter is shifted into the h register, the upstream filter rows are shifted one row closer to the h register, so that by the time the entire downstream filter has been read out, the rows of each upstream filter f have shifted so that they now lie under the filter f − 1. The charge that these rows accumulate in that time is characteristic of filter f. When the next downstream filter is read out, these same rows are shifted from lying under filter f − 1 to lying under filter f − 2, with the charge accumulated during the shift being characteristic of the starting filter for that shift. When the rows of filter f are finally in the position where filter f rows are being transferred directly to the h register, the charge that they accumulate is characteristic of filter 1. Thus, if all of the filters downstream from f are read out in an exposure, then the rows in filter f accumulate charge from filters f, f − 1, f − 2, …1.
 If any of the filters downstream from f is not being read out, then the process of dumping that filter shifts f downstream very rapidly, so if the rows in filter f were under filter f − Δ at the start of the dump, very little of the characteristic bias charge of filter f − Δ is accumulated. Thus our second simplifying assumption is that the charge accumulated during such a dump is negligible.
 The bias charge built up during readout for a certain filter following a given filter path with filter path code f can therefore be calculated simply by summing the characteristic bias charges, Eij,f, of each downstream filter. The bias frame Bij,F is
The values of f0 and η(f) are properties of the filter path denoted by F and are defined by equation (4), i.e., f0 is the starting filter of the filter path and η(f) indicates whether the filter f is one of those being read out for that filter path. Eij,f and Bij,F are elements of arrays with the same dimensions as the framelets of the spatial summing mode for which the Bij,F are being derived. Note that η(f) is always equal to 1 when f = f0, so the characteristic bias of the first filter, Eij,f = 1, is always present in the bias frame.
Eij,f can be derived by using empirically known (via the averaging method described in section 220.127.116.11) Bij,F. We perform this derivation by using Bij,F values for those filter paths for which all downstream filters are read out. These are Bij,F=31, Bij,F=15, Bij,F=7, Bij,F=3, and Bij,F=1. Writing out equation (11) explicitly for these five Bij,F leads to the following trivial system of equations:
18.104.22.168. Bias Subtraction Algorithm
 Bias frames for each summing mode are stored in FITS files each with 31 planes corresponding to the 31 filter paths. The bias is removed by determining the filter path for each framelet of each filter and then subtracting from it the stored bias frame for that filter path and summing mode. The bias frames are stored in the FITS files in order of the F code, i.e., the bias frame for a given filter path is found at plane number F − 1 for a zero-based cube plane counting system.
4.3.4. Register Stray Light Subtraction
 The strategy for removing register stray light is to first identify the spatial pattern using zero duration exposures, and then determine the way that the intensity of that pattern scales with C-ROI mean broadband scene radiance .
22.214.171.124. Zero Duration Daytime THEMIS VIS Images
 On-orbit images acquired during daylight with a commanded exposure time of 0 ms show the same spatial pattern as the ground-calibration-derived zero exposure images (compare Figures 9 and 16). Of course, a commanded exposure time of zero does not really produce an exposure time of exactly zero. However, in the majority of zero exposure flight data, there is no evidence of surface features visible through the register stray light pattern. The spatial variation of the small amount of scene derived DN that is probably present in the zero exposure images will be washed out by averaging a sufficient number of images. Furthermore, the level of scene-derived contamination of the zero exposure images will be a negligible fraction of the signal of any given THEMIS VIS image as long as the effective exposure time for a commanded exposure time of zero is a negligible fraction of the effective exposure time for the THEMIS VIS image in question.
 The register stray light frames are generated using zero duration daytime THEMIS VIS images that have been calibrated through stage 3 (bias subtraction) of the pipeline. The zero exposure daytime images are first averaged according to filter path and spatial summing mode in the same manner as the nighttime images are averaged to create bias frames. Next, the register stray light frames are normalized by, first, setting the mean of a given spatial summing mode's F = 1 register stray light frame equal to unity. Then, the register stray light frame for every other filter path for that spatial summing mode is normalized so that its values, relative to the F = 1 frame, reflect the relative intensity of the register stray light for that filter path in the data set. Normalization to reflect relative intensities in the data set is crucial, because the only means that we have available for determining the scaling factor between the broadband radiance and register stray light is to measure the changes in mean DN from framelet to framelet within an image sequence as a result of changes in filter path, and then find a single scaling factor for all of the register stray light frames that eliminates these changes in mean DN.
 For each filter path, the necessary normalization factor is determined as follows: (1) Find all of the exposures that are in the set of images used to generate the register stray light frames, and contain both the given filter path and the F = 1 filter path. (2) For each of these exposures, calculate the ratio of the mean value of the framelet from the given filter path to the mean value of the F = 1 framelet. (3) The normalization factor is a single number by which the filter path's register stray light frame is scaled so that its mean equals the mean of the ratios from the previous step. Note that the means used to normalize the register stray light frames are calculated from the all nonnull pixels in the framelets, rather than from the C-ROI used elsewhere in the calibration procedures.
 A complication for this ratio-based normalization is that there are many filter paths that cannot occur in the same exposure as the F = 1 path. However, every filter path does occur in the same exposure as one of F = 16, F = 8, F = 4, F = 2, or F = 1 paths, which we will refer to as “clear” paths (these are the paths that apply to filters 5, 4, 3, 2, and 1, respectively, when there are no other filters downstream). The normalization factors of the clear filter paths are established using ratios with respect to the F = 1 path, according to the previously described procedure. Then the ratios of all other filter paths are measured with respect to whichever clear path they occur with, and then rescaled using the ratio of that clear path to the F = 1 path, which gives us the ratios of these paths with respect to the F = 1 path and hence the normalization factor.
 The clear-to-F = 1 ratios are established as follows: In THEMIS VIS images that use all five filters, an exposure containing F = 2 always occurs immediately following an F = 1 exposure, an exposure containing F = 4 always occurs immediately following an F = 2 exposure, etc. So the F = 2 ratio is calculated relative to the F = 1 from the previous exposure in the same image, the F = 4 is calculated relative to the previous exposure F = 2, F = 8 relative to the previous F = 4, and F = 16 relative to the previous F = 8. These ratios are then rescaled in sequence so that they are expressed relative to F = 1.
 This normalization scheme assumes that the register stray light signals from the various filter paths are related to each other by a constant proportionality factor. We have examined the register stray light signal ratios, and have found that they are consistent with the constant proportionality hypothesis. Obviously, the normalized register stray light frames are generated as described above only for the filter paths that are represented in our data set of zero exposure daylight images. Register stray light frames for filter paths that are not represented in the data set of zero exposure daylight images are generated by starting from the normalized frames and then applying the same method used for the bias frames in step 3c.
126.96.36.199. Deriving the Register Stray Light Response Coefficient
 In order to solve for the register stray light response coefficient, we postulate that on average, once the register stray light is removed, the framelet means will be independent of the filter path, provided that we consider framelets from the same filter. To take advantage of this postulate, we first determine, for each five-filter image of a given summing mode, the factor ψ that we need to multiply the normalized register stray light frames by so that when they are subtracted from the image framelets, the differences between adjacent framelet C-ROI means are minimized in a least squares sense.
 1. More explicitly, for each five-filter image, we identify the set of framelet pairs that satisfy (1) both members of the pair are from the same filter; (2) the members of the pair come from adjacent (in the time sequence) exposures; and (3) the members of the pair have different filter paths. Let the elements of this set be indexed by u.
 2. We calculate the difference in the mean (bias subtracted) DN for each pair, u, and the difference in the mean normalized register stray light frame for the filters paths of the members of each pair, u.
 3. We find the value of ψ that minimizes
 4. We record ψ, and each band's mean photosite signal k, for the image. If u′ is the C-ROI mean of the normalized register stray light frame for a framelet pair member's filter path, and u′ is the C-ROI mean uncorrected DN for that pair member, and t is the exposure time, then
The mean is over all elements of the u′ set of filter pair members for which the filter number is equal to the k in the k we are calculating.
 The register stray light response coefficient z is the factor that relates the mean broadband scene radiance to ψ:
Solving for ψ by minimizing equation (13) would of course be sufficient to remove register stray light in a THEMIS VIS sequence of a uniform scene, but in order to be able to calibrate any arbitrary sequence, we need to know z. To estimate z, we select a sample of five-band images with low scene variance and determine ψ and the k as described above. Let the elements of this sample be indexed by v, so we have a set of ψv and k,v.
 The k,v yield v by applying equation (9), and we use the resulting set of ψv and v with equation (15) to give a least squares solution for z. In finding the v, we are free to chose which of the bands k of the k,v, and thus which set of ωk to use. Naively, we might expect that using all five bands would give the best estimates and thus the smallest residuals on the z solution, because it provides the most information about the scene radiance spectrum. However, ψ is an imperfect solution for the register stray light, and so in general the k estimates will be contaminated by some register stray light, reducing the accuracy of the v. This contamination is least significant in the bands with the highest photosite signal, and thus we find that using only the k = 3 (650 nm) band gives the fit for z with the highest R2. The k =3 solution is our adopted z value, and is reported in Table 2. Excluding the k = 3, 4 band combination, which gives solutions and R2 identical to that of k = 3, the band combination with the next highest R2 is k = 2, 3. Since the difference in the z solutions of these two best fits is much higher than their formal errors, we use this difference to form our confidence interval, with the half width of the confidence interval being set equal to the magnitude of z(k = 1) − z(k = 2, 3).
188.8.131.52. Subtraction of Scaled Register Stray Light Frames
 Register stray light frames for each summing mode are stored in FITS files each with 31 planes corresponding to the 31 filter paths. These frames are stored in order of the F code, i.e., the frame for a given filter path is found at plane number F − 1 for a zero-based cube plane counting system.
 Let Dijfa be the bias-subtracted (i.e., calibrated through stage 3) DN for column i, row j, filter f, and exposure a of a THEMIS VIS imaging sequence with exposure duration t, and let Gij,F(f,a) be the i, j elements of the normalized register stray light frame with the appropriate filter path F. (F is determined from equation (4).) Then the photosite signal Sijfa is
 The means of determining broadband C-ROI mean radiance a is apparent if we take the C-ROI means of S, D, and G, in equation (16) and then substitute into equation (9), eliminating S:
 The sums are over whatever subset of filters we chose to use, and the ωf are those appropriate to that subset. The exposure number a′ used to calculate the broadband radiance estimate for exposure number a is not in general equal to a, because the source region for an exposure's register stray light may not be within its field of view. Since 50% or more of the register stray light is contributed by the h register, and since the h register is at the bottom of the array where it is most likely contaminated by scene radiance near the bottom of an exposure's field of view, we chose a′ to select a filter from a different exposure (a later exposure in the sequence) that covers the region just below the field of view of exposure a. We select that later exposure as follows:
 Although in principle we might wish to use as many of the THEMIS VIS filters as possible when calculating a from equation (17), in practice we have found that choosing a single filter produces better results, in the sense that when the results of single-filter-a register stray light removal and multifilter-a register stray light removal are visually inspected and compared, the multifilter-a image is more likely to be the one with more prominent register stray light artifacts. Therefore we calculate a using D and G from filter 3 (band 3) whenever it is present in the image. If filter 3 is not present, we use filter 4, and if not 4, then the next most preferable filter, with the full preference order being 3, 4, 5, 2, 1. Less preferable filters are those that have higher residuals when we fit for z in equation (15) using single-filter-derived values for .
 The series of a derived using equation (17) for an image will have missing elements on the edges due to the beginning and end of the exposure sequence, and sometimes in the middle due to invalid image data. Interior gaps are always filled by linear interpolation. Extrapolation is allowed for only one element past the edge of the valid a elements, and a is treated as constant from this edge point and outward.
4.3.5. Correction for Pixel Response Variation (Flat Fielding)
 The response of framelet pixels to scene radiance is variable, due both to variations in the sensitivity of detector pixels, and to variations in optical throughput. In the absence of stray light, an image of a uniform scene is a “flat field,” i.e., a map of these response variations.
 Unfortunately, the variability in THEMIS VIS signal in uniform scene images is dominated by stray light. The crucial distinction between response variations and stray light is that response variations are a multiplicative effect, while stray light is additive, so that regions of the detector that are brighter due to greater response will have proportionally enhanced contrast, while regions that are brighter due to more stray light will have not have enhanced contrast. This means that if we incorrectly attribute all of the nonuniform signal in a uniform scene image (after removal of register stray light) to photosite stray light, then regions of high response will have spuriously high contrast. We must therefore remove response variations prior to deriving and removing photosite stray light light. However, if we incorrectly attribute all of the nonuniform signal in a uniform scene image to response variations, our calibration process will have the converse effect of creating spuriously decreased contrast in regions of high stray light. We have therefore adopted a method for generating flat fields that relies on the scene contrast directly by using framelet to framelet signal differences rather than averages as in the usual flat fielding approach.
 A flat field is derived independently for each of the five filters using data calibrated through stage 4 (register stray light subtraction). Let S′ij,n and S″ij,n be the stage 4 signal for the early and later members, respectively, of framelet pair n. Let σ′ij,n and σ″ij,n be the standard errors for the DN values of the members of the pair. These standard errors are based on Poisson counting noise statistics, i.e., they are simply the square root of the numbers of electrons detected at each pixel, so:
The “raw DN level” refers to the DN level measured prior to bias subtraction, i.e., calibrated through stage 2. The gain is 25.4 electrons per DN for spatial summing mode 1, 25.4/2 for spatial summing mode 2, and 25.4/4 for spatial summing mode 4. A framelet pair consists of any two adjacent framelets from the same filter and same THEMIS VIS data cube. Let the total number of such framelet pairs in the data set for a given filter be labeled N. We assume that the photosite stray light signal is slowly varying from framelet to framelet, so that the difference between framelets depends only on the changes in the scene and on counting noise. Since the magnitude of the differences caused by scene changes (as a fraction of the mean signal) is proportional to responsivity, we estimate the responsivity R by starting with the variance of the framelet-to-framelet differences:
and then subtracting the contribution of counting noise to those differences,
The responsivity R is the square-root of the counting-noise corrected variances:
 Unfortunately, the assumption of slowly varying photosite stray light is not always valid, leading to some contamination of our responsivity estimate by stray light effects. This contamination is most severe where the stray light is most severe, on the right and left edges of all filters, and across the entire filter 1 (band 5, 860 nm). Given the strong column dependence of the stray light, the weak variation of derived responsivity with column in less contaminated regions, and the tendency of the variation with column to mimic that of the stray light, we are not confident that any of the column variation in derived responsivity is real. However, the row dependence of responsivity is much stronger, and not correlated with the stray light patterns. We therefore treat the responsivity as being a function only of detector row, allowing us to derive it by averaging across the central, least contaminated columns. Figure 17 (dashed lines) shows a family of profiles along columns of Rij, to illustrate the dominance of variation with row. The strong drop in responsivity below row 15 is due to vignetting by the edge of the filter.
 We derived the column-averaged flat field using summing mode 2 Rij values, because the summing mode 2 data set is much larger than that in the other modes, and so produces a higher quality flat field. The resulting profile is normalized so that it has unit mean within the C-ROI bounds. This normalization ensures that flat fielding does not on average alter . Thus the flat field as a function of row, Rj, is
Each column of each framelet is corrected by dividing it by Rj. Since Rj is originally generated using only summing mode 2 data and is thus a 96 element vector, it is resampled to 192 elements via linear interpolation for use with summing mode 1 data, and resampled to 48 elements via pixel averaging for use with summing mode 4 data. The solid line in Figure 17 shows Rj for filter 3, summing mode 2. Rj for filters 2, 4, and 5 is similar. For filter 1, the stray light contamination is so severe that we do not attempt to derive a flat field, and so Rj for filter 1 is set equal to one for all rows.
4.3.6. Photosite Stray Light Subtraction
 We model the photosite stray light signal in each framelet as the sum of a spatially variable component Xijk and a spatially uniform component xk. So Sijk,p, the total flat-fielded signal (i.e., the signal calibrated through stage 5) for column i, row j, of the pth band k framelet in the THEMIS VIS data set, is modeled as:
The parameters xk and yk are derived from preflight data as previously described and as used in equation (7). k,p is the mean scene radiance for band k framelet p, also as used in equation (7), and I′ijk,p represents deviations from that mean scene radiance. All spatial means, denoted by overbars, are taken over the C-ROI region of each framelet. By combining equation (22) with equation (7), we get:
Assuming a sufficiently large sample of framelets, the pattern of radiance in those scenes, I′ijk,p, will be essentially random, and so we can combine the I′ijk,p term with the k,p error term. Using S′ijk,p to denote deviations from the framelet mean signal, this yields:
Thus we solve for Xijk using a least-squares fit of equation (24). The framelet groups indexed by p are sets of framelets from different bands that come from the same imaging sequence and share the same framelet number m. Each framelet group has a set of k from which is calculated using equation (9). Xijk is derived independently for each summing mode using all framelet groups in the THEMIS VIS data set that contain all five bands. For the fitting process, 5, the mean band 5 (860 nm) signal is not used in the calculation of .
 Using the derived Xijk, determining the corrected photosite DN signal, Qijk,p,
for any arbitrary framelet group is simply a matter of determining for that group. We do so using equation (9) with the k from that group. All valid k values from a group are used to calculate , except that band 5 is excluded, unless band 5 has the only valid value in that framelet group. We exclude band 5 because, by visual inspection of the calibration results, we have found that excluding band 5 leads to a decrease in the prominence of residual photosite stray light artifacts. An k is valid if at least 50% of the pixels in its C-ROI are nonnull. If all k for a framelet group are invalid, then all pixels in that framelet group will be set to null.
4.3.7. Conversion to Radiance
 Given Qijk from stage 6, and since yk is derived from preflight measurements, Iijk, the calibrated radiance in a THEMIS VIS framelet, is simply:
 In order to verify the accuracy of the THEMIS VIS calibration, we use concurrent observations with HST from five different dates in August 2003 and one date in March 2003. (We have also made comparisons, which we will described in the next section, with historical HST observations from March 1997.) For each set of HST observations in 2003 (see Table 3 for details), we targeted five-band THEMIS VIS images within 2 hours of the first or last HST image. These THEMIS VIS images (Table 6) were acquired with all three of the available spatial summing modes, and they sample incidence angles of 60° and higher (all THEMIS VIS emission angles were near zero), and terrains with the full range of Martian albedo values.
Table 6. THEMIS VIS Images Used for Figure 18 Comparison With Concurrent HST Images
Image Center Coordinates
Exposure Duration, ms
Number of Framelets
East Longitude, deg
Incidence Angle, deg
Figure 18 compares radiances derived from the HST image sets with C-ROI mean radiances from the concurrently targeted THEMIS VIS framelets. This comparison uses only framelets from THEMIS VIS images with an effective exposure time greater than 3 ms. (The effective exposure time is the actual exposure time multiplied by the spatial summing mode.) In order to perform the comparison, the HST images are first projected onto a simple cylindrical grid with a 5 pixel-per-degree sampling interval, so that each HST grid box covers 0.2° × 0.2° of latitude and longitude, starting from 90° South, 180° West at the bottom left corner of the bottom left grid box. (We use a 3 pixel-per-degree grid for the lower-resolution March 2003 HST images.) We define usable HST grid boxes to be those with both incidence and emission angles of less than 80°. Similarly, we define THEMIS VIS pixels to be usable if they contain valid data and have incidence and emission angles of less than 80°. If less than 75% of the C-ROI pixels are usable in any band of a framelet, we do not use that framelet.
 To generate HST radiances for the comparison, we do the following separately for each band of each framelet:
 1. Find the set of usable C-ROI pixels.
 2. Identify the THEMIS VIS radiance as the mean radiance of the usable C-ROI pixels.
 3. Identify the THEMIS VIS incidence, emission, and phase angles as the mean values of these quantities over the C-ROI pixels.
 4. For each narrow-band HST image, identify the HST I/F, incidence angle, and emission angle as the weighted mean of the quantities over the HST grid boxes into which the usable THEMIS VIS C-ROI pixels fall. The weights are the fraction of usable THEMIS VIS C-ROI pixels in each grid box. (The HST phase angle is, of course, a constant within each HST image.)
 5. Apply a photometric correction to the HST I/F values, using a Minnaert photometric model to adjust them to values appropriate for the THEMIS VIS observations. The correction factor is
where μ0 is the incidence angle, μ is the emission angle, h is the phase angle. The Minnaert coefficients, β and κ (both are functions of the phase angle h) are given in Table 7, and were derived using the HST data set and methods described by J. M. Soderblom et al. (Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data, submitted to Icarus, 2005). The coefficients were derived using narrow-band filters at the wavelengths listed in the table. To correct HST radiances at other wavelengths, we interpolate the coefficients linearly in wavelength, unless the target wavelength is outside of the wavelength range of tabulated coefficients, in which case we use the coefficients from the nearest-neighbor wavelength. Although separate Minnaert coefficients are used for each phase angle bin, correcting for phase effects is still problematic, because the HST data used to derive the correction do not exceed 40° phase and all of the THEMIS VIS data used for the radiance comparison have phase angles greater than 60°. No Earth-based instruments can observe at phase angles as large as those of THEMIS VIS in its current nadir-viewing configuration and 5pm orbit.
Table 7. Minnaert Photometric Model Coefficients
Phase Angle h Range
 6. Estimate the spectral radiance by fitting a cubic spline in wavelength space to the HST narrow-band I/F values and then multiplying by the solar spectral radiance.
 7. Identify the HST radiance by weighting the HST-estimated spectral radiance by the spectral response function (Figure 1) of the given THEMIS VIS band.
 For bands 1–4, the comparison in Figure 18 provides satisfactory validation of our calibration results. There are clearly systematic differences between the HST and THEMIS VIS radiances, as well as substantial scatter, but because we are extrapolating the photometric behavior of Mars out to phase angles at which it is not adequately measured by our concurrent HST study, and because of the large difference in resolution between HST and THEMIS VIS, the typical mismatches of 15% cannot be construed as evidence of a calibration inaccuracy. In fact, it may be possible in the future to constrain the photometric properties of Mars using this kind of comparison.
 For band 5, the preflight-calibration-based y coefficient produced radiance values that were much too high relative to the HST radiance estimates. We have no explanation for this, but given the very high level of stray light in band 5, it is not entirely surprising that our preflight calibration strategy did not produce a useful result. Table 2 lists our arbitrarily adopted band 5 y coefficient of 0.6 ± 0.2 (DN ms−1)(W m−2 μm−1 sr−1)−1, which was selected because it produces the reasonable agreement with HST shown in Figure 18.
6. Error Analysis
Figures 19 and 20 illustrate the significance of the two stray light contributions to the C-ROI signal in a THEMIS VIS EDR, while Figures 21 and 22 illustrate the way that these contributions are distributed over the area of an EDR framelet. Together with the confidence intervals on the x, y, and z calibration coefficients given in Table 2, Figures 19–22 allow estimation of the uncertainty in any given RDR framelet's radiance values. The x and y confidence intervals enclose 95% of the one-dimensional probability density and so can be regarded as 2σ error estimates. The z confidence interval is more problematic, since it is simply based on the difference between the two most plausible models, but in the following discussion we will treat it too as a 2σ error estimate.
 For Figures 19 and 20 we have extracted the register stray light and photosite stray light C-ROI signals, respectively, from the calibration pipeline, and compared them to the direct C-ROI signal derived by the calibration pipeline. The contribution of each type of stray light to the fractional uncertainty for a particular C-ROI radiance can be estimated by multiplying the expected fractional contribution of that type of stray light to the total signal by the fractional uncertainty in the relevant calibration coefficient. Consider, for example, a spatial summing 2, band 1 (423 nm) image with a typical perihelion period exposure time of 2.5 ms. This is an effective exposure time of 5 ms (“effective” exposure time equals spatial summing mode times actual exposure time), and so according to the histogram in Figure 19, the register stray light signal is typically equal to the direct signal. The z calibration coefficient for summing mode 2 has an uncertainty of 6.4%, and so 6.4% × 1.0 gives a register stray light contribution to the uncertainty of 6.4%. For the same band and summing mode, the photosite stray light signal is typically 19%, and the x coefficient uncertainty is 8.3%, so 8.3% × 0.19 give a photosite stray light contribution to the uncertainty of 1.6%. For comparison, the uncertainty in y contributes 3.5%, and the overall 2σ uncertainty can be regarded as = 7.5% , meaning that register stray light is the dominant source of error for band 1 at this particular exposure time. Tables 8a and 8b show similar calculations for all bands and summing modes at various exposure times.
Table 8a. The 2σ Uncertainty as a Function of Band
y Coefficient Contribution
Photosite Stray Light Contribution
Table 8b. The 2σ Uncertainty as a Function Summing Mode and Effective Exposure Time
Summing Mode 1
Summing Mode 2
Summing Mode 4
Register Stray Light Contribution
Register Stray Light Contribution
Register Stray Light Contribution
2 ms Effective Exposure Time
5 ms Effective Exposure Time
10 ms Effective Exposure Time
50 ms Effective Exposure Time
 It is important to note that since the stray light signals are additive, the residual (i. e., uncorrected portion) stray light signal included in our error estimates is also additive. Thus uncertainty introduced by residual stray light will affect the signal ratio between two regions of an image, but, in instances where we are prepared to regard residual stray light as uniform over a framelet, its uncertainty does not affect the signal difference between two regions of a framelet. Of course, uncertainty in the direct response coefficient y will affect differences, but not ratios.
 Strictly speaking, all three of the sources of uncertainty (photosite stray light, register stray light, direct response coefficients) discussed thus far represent the possibility of global (i.e., over the whole data set) biases. However, the success of the stray light removal for any given THEMIS VIS image is inherently stochastic because the stray light signal depends on the brightness of the scene in regions outside of the detector field of view. The uncertainties in the global stray light calibration coefficients reflect the extent to which the stray light is unpredictable. Thus the fractional uncertainties derived from the uncertainties in the calibration coefficients can be construed as applicable to individual images as well as to the data set as a whole. Any future quantitative work with the THEMIS VIS data set must consider stray light errors that may be both correlated and uncorrelated across the data set.
 When considering regions of a framelet other than the C-ROI, the contour plots of Figures 21 and 22 provide a scaling factor for the amount of stray light uncertainty in a calibrated RDR. These contour plots are derived from the register stray light and photosite stray light (respectively) calibration frames and coefficients. For each filter path, or each band, they are constructed by subtracting the C-ROI mean stray light signal from the calibration frames, and then dividing by the C-ROI mean stray light signal. Thus the contour plots show the amount of stray light contamination in a region over and above that in the C-ROI, as a fraction of the C-ROI contamination. For example, when the contour plot shows a value of 6, the amount of stray light is 6 + 1 = 7 times as great as in the C-ROI, and the uncertainty introduced in that region by that type of stray light is 7 times as large as it is for the C-ROI. Clearly, the amount of photosite stray light increases dramatically at the edges of the framelets, whereas the increase in register stray light toward the edges of the framelets is still significant, but much less.
 For brevity, we have shown these contour plots for summing mode 2 only, and for only a representative selection of filter paths. At the level of precision necessary for error estimation, these plots can nevertheless be used for any summing mode, with appropriate scaling of the axes. Similarly, they can be used for any filter path, in which case the panel of Figure 21 with the same number of filters in the path as the framelet under consideration should be used. For example, the register stray light in a framelet with filters 2, 4, and 5 in its path is most similar to the values in the panel for filter 1, 2, and 3.
 When calibrated (RDR) framelets can be inspected individually, and residual stray light artifacts are recognizable, Figures 21 and 22 can also be used to provide additional information about stray light residuals. If, for example, we observe a stray light artifact in a band 1 framelet with a shape similar to the stray light lobes on the edges of the band 1 contour plot in Figure 22, and we estimate the amplitude of the artifact to be 100 in some arbitrary radiance units, then since the peak of the these lobes is about 7 in Figure 22, the residual photosite stray light in the C-ROI is likely no more than 100/7 = 14 of the same radiance units. Since estimating the “amplitude” of the artifact is inherently imprecise, and since stray light artifacts frequently vary dramatically in magnitude and sign across a framelet and between neighboring framelets, this type of calculation should normally be treated as a constraint on the error rather than as a correction to the RDR radiance. It may ultimately be possible to derive corrections to RDR radiances by iteratively adjusting, scaling, and subtracting calibration frames until stray light artifacts are no longer visible, but that work is beyond the scope of this paper.
 Most calibrated THEMIS VIS images show stray light artifacts to some degree, although, as Tables 8a and 8b make clear, these artifacts are more significant in band 1, and least significant in band 3. In band 5, the residual stray light artifacts normally dominate the direct signal, so that band 5 only very rarely provides usable data. In Figures 23 and 24 we present two examples of THEMIS VIS RDRs with very prominent stray light artifacts. Figure 23 is dominated by register stray light residuals, whereas Figure 24 shows a combination of both effects. The most diagnostic characteristic of register stray light is the pattern of narrow stripes that it produces at the edges of each framelet. As is often the case, because it is far from the readout register, and its direct signal is relatively low, band 4 shows the most prominent pattern of narrow stripes. However, as is apparent from Figures 9 and 21, the narrow stripes only become a significant part of the register stray light pattern for rows far from the h register, and so given that band 4 also shows a uniformly positive register stray light residual, it is probable that band 1 also has a significant register stray light residual even though band 1 shows no obvious pattern. Another symptom of register stray light is a “roll-off” (or “roll-up”) near the top or bottom of each framelet caused by the interaction of the flat field with the register stray light subtraction. This roll-off often appears as a bright or dark stripe. When such a roll-off is present, it is most likely caused by register stray light, simply because it is the most important source of error. Photosite stray light residuals may also lead to a roll-off, however.
 Unlike Figure 23, where the residuals are clearly one particular kind of stray light, and are quite uniform throughout the image, Figure 24 shows highly variable residuals and a combination of register and photosite stray light. The abrupt changes in brightness between framelets could be caused by changes in either photosite stray light or register stray light residuals, but register stray light is always the most likely culprit for ambiguous artifacts. However, some features in the Figure 24 RDR are readily distinguishable as photosite stray light, because residual photosite stray light produces a broad swath or lobe at the framelet edges for bands 1–4 rather than a narrow stripe. Of course, these lobes can be either bright or dark depending on the sign of the residual. The bottom six framelets of band k = 1 show the clearest example of these lobes, with a broad swath of high positive residual on the right hand side accompanied by negative residuals on the left hand side. Presumably this asymmetry is caused by bright and dark scene elements just outside the detector's field of view. In band 5 the broad curving photosite stray light stripe is almost always clearly visible as a residual, which appears to be modulated by the brightness of the scene just below the framelet's field of view.
7. Comparison With Other Data Sets
 We conclude by evaluating the extent to which I/F values based on our calibrated radiances are comparable with historical HST measurements of Mars, and with Mars Exploration Rover Pancam measurements from the Martian surface. The solar irradiance used to calculate I/F for a given THEMIS VIS band is obtained by taking a weighted average of the solar irradiance spectrum, where the weights are that band's relative response function (Figure 1a). Since the historical HST data and the Pancam data were acquired at different times and with different viewing geometries from any THEMIS VIS images we might compare them to, we do not expect them to match in terms of absolute radiances. Instead, we will compare normalized I/F spectra in order to demonstrate that the relative spectral calibration of THEMIS VIS is reasonable, and to demonstrate some of THEMIS VIS's scientific potential.
 To minimize the impact of atmospheric variability on the HST comparison, we have selected a set of HST observations from northern summer (10 March 1997, Ls 89°, see Table 3 for details), and chosen our THEMIS VIS images from middle to northern latitudes in the 90°–120° Ls period. Consulting Smith , we note that this is Mars's most consistently aerosol-free region and season. Figure 25 shows the March 1997 HST data and the locations of the HST data points which we compare to the THEMIS VIS data. Each location is labeled with the number of its corresponding THEMIS VIS image.
Figures 26–31 present the HST-to-THEMIS- VIS comparisons. For the first comparison, Figure 26 and 27, we have selected two HST data points in the vicinity of Elysium Mons, one inside of (at 25° N) and one outside of (at 29° N) the cloud feature on the volcano's flank. The THEMIS VIS image (Figure 26) also shows a clear region to the north and water-ice cloud in the south, and so we plot spectra (Figure 27) from both areas. Since the THEMIS VIS and HST spectra are from different times of day (∼5 pm local solar time for THEMIS and near the subsolar point for HST), slightly different locations relative to a variable cloud feature, and two Martian years apart in time, it is not surprising that the spectral match is not exact. However, the difference in normalized I/F caused by the cloud is very similar in the HST and the THEMIS VIS data, suggesting THEMIS VIS is correctly measuring cloud-induced spectral contrasts. This validative evidence is important, because THEMIS VIS spectral data is frequently notable for its lack of spectral contrast; see, for example, Figures 30 and 31 or Figures 32 and 33. By showing the correct degree of spectral contrast in Figure 27, we demonstrate that the spectral uniformity on THEMIS VIS scales is in many locations a real characteristic of the Martian surface.
Figures 28 and 29 show that without the confounding influence of aerosols, the THEMIS VIS data very closely match the historical HST results. For the brighter, redder terrain unit, which predominates in this region of northern Arabia Terra and thus controls the much lower resolution HST spectrum, the THEMIS VIS spectrum is a very close match. In Figures 30 and 31, however, we do not have a good match between the HST and THEMIS VIS spectra. We do not have an explanation for this mismatch, but the combination of variable atmospheric hazes with high atmospheric path length in the HST data (emission angle of 49°), and changes in surface dust cover, are both possibilities.
Figures 32–35 compare THEMIS VIS spectra with published Pancam data [Bell et al., 2004a, 2004c]. In Figures 32–33 we select examples of the two main surface albedo units in the vicinity of the Spirit Rover landing site, avoiding the region of excess residual stray light at the bottom of the THEMIS VIS image (Figure 32), and find that these surface units have a distinct but very small color contrast when viewed from orbit. The THEMIS VIS spectra are slightly bluer than the Pancam soil measurement, which could be explained by the extra atmospheric path length in the THEMIS VIS observations, or by areal mixing with the sparse population of exposed rock.
 THEMIS VIS V03671001 in Figure 34 does not quite cover the Opportunity Rover landing site, but it is the only multiband image of the landing site vicinity without severe atmospheric aerosol contamination. As in Figures 32–33, our spectra sample two distinct albedo/color units, and avoid the residual stray light. Pancam clearly shows much greater spectral diversity than THEMIS VIS, but the THEMIS VIS spectra fall well within the range of the Pancam samples. The dark terrain THEMIS VIS spectrum (“A”) is a good match to the dark Eagle Crater soil (“5”), and the lighter THEMIS VIS terrain (“B”) is redder just as the bright Pancam outcrop (“1”) is, but to a much lesser degree.
 Despite significant stray light contamination, we have obtained an absolute calibration for THEMIS VIS 654 nm band-pass images with an uncertainty that we estimate to be better than 5% (2σ) in most situations. The calibration uncertainty is greater for the 425 nm, 540 nm, and 749 nm bands, but even for these bands the uncertainty (Tables 8a and 8b) can be better than 5% (2σ) in low-light conditions where long exposure times are possible. For the 860 nm band, however, the uncertainties in the radiances are so high that they are effectively unusable. For all of the THEMIS VIS bands, relative radiance comparisons are no more precise than absolute measurements, because the additive and variable residual stray light is responsible for most of the uncertainty.
 Generally speaking, the exposure duration for an imaging sequence is limited by saturation in the most sensitive bands (654 nm and 540 nm, in that order). Thus, since all bands in a sequence have the same exposure duration, leaving the most sensitive bands out of a sequence can reduce the contribution of register stray light in the remaining bands, significantly improving their uncertainty over bright surfaces. Of course, sacrificing the more sensitive bands for this purpose is only rarely desirable.
 THEMIS VIS spectra are consistent with Pancam measurements, and are mostly consistent with historical HST measurements. Furthermore, our concurrent-imaging calibration validation strategy indicates that THEMIS VIS absolute radiances and HST absolute radiances are broadly consistent. However, the concurrent imaging does not in and of itself provide a strong constraint on the accuracy of THEMIS VIS absolute radiances because of the limitations in the ability of the photometric models to “correct” the observations to the same standard viewing geometry. The comparison does suggest, however, that THEMIS VIS could in the future play a role in constraining Mars's photometric properties.
 Residual stray light artifacts can frequently be quite prominent in RDR images that have be stretched to maximize contrast, but because photosite stray light is so much more intense near the framelet edges, the presence of visible photosite stray light artifacts does not normally indicate a significant problem with center field radiances. Obvious register stray light artifacts are more rare, but they are a more serious concern for the accuracy of center field radiances. When quantitative radiance values are important, THEMIS VIS data users should check their accuracy by comparing any prominent artifacts with Figures 21 and 22.
exposure number (0 – …).
bias frame, i.e., bias charge for a framelet (DN).
region of interest for which calibration coefficients are defined.
difference in mean raw DN values for a pair of framelets.
raw (bias subtracted, for cases where it matters) 11-bit DN value.
C-ROI mean raw 11-bit DN value.
characteristic bias charge from the portion of the chip under a certain filter (DN).
filter number (1–5).
filter path code (1–31).
difference in the mean normalized register stray light frames of a pair of framelets.
normalized register stray light frame, i.e., spatial pattern of register stray light for a framelet (unitless).
C-ROI mean of a normalized register stray light frame.
column number (0 – …).
radiance in a particular band.
C-ROI mean radiance in a particular band.
deviations from the mean radiance.
C-ROI mean broadband radiance.
row number (0 – …).
band number (1–5).
ground calibration lamp level index.
framelet number (0 – …).
number of framelets in an EDR or RDR.
number of elements in the set of framelets used to estimate R.
an index for the set of all framelets in the data set.
an index for a set of framelets or data points, used in various contexts.
calibrated THEMIS VIS signal (DN ms−1).
responsivity (i.e., the flat field) (unitless).
C-ROI mean photosite signal (DN ms−1).
photosite signal (DN ms−1).
ground calibration photosite signal prior to background subtraction (DN ms−1).
index for the set of framelet pairs used to generate the register stray light response coefficient.
index for the set of framelet pair members.
elements of the sample of framelets used to estimate ψ.
photosite stray light response coefficient.
photosite stray light calibration frame, i.e., the spatial pattern of photosite stray light for a framelet.
direct response coefficient.
register stray light response coefficient.
register stray light (DN) frame in ground calibration.
coefficients for predicting from k.
Minnaert photometric model coefficient.
used in various contexts to represent the residuals of a curve fit.
equal to 1 or 0 depending on whether filter f is present in a certain filter path (see equation (4)).
Minnaert photometric model coefficient.
framelet standard errors.
register stray light scaling factor.
coefficients for predicting from .
 We acknowledge Steve Silverman and Hansford Cutlip of Raytheon Santa Barbara Remote Sensing for their help in gathering necessary records from the preflight calibration activities. We acknowledge Jason Soderblom of Cornell University for providing us with Minnaert photometric coefficients for the HST data, and we thank Alex Hayes and Min Hubbard for additional help with the HST data set. We thank Kim Murray and Noel Gorelick of Arizona State University for their feedback on the results of the calibration pipeline. We also thank Ken Herkenhoff and Paul Geissler for thoughtful and influential reviews of the manuscript. This work was supported in part by a National Science Foundation Graduate Research Fellowship and by a grant to J.F.B. from the NASA Mars Odyssey Orbiter Participating Scientist program (JPL contract 1241434).