Chapter 2 - Data and Reduction

The Data Set

The data were taken using the NSFCAM at the NASA Infrared Telescope Facility (IRTF). The IRTF is a 3 meter infrared telescope located on the summit of Mauna Kea, Hawaii at an altitude of 13,796 feet. The low humidity of the site makes it an ideal location for infrared work in general and its altitude positions it above 40% of the Earth's atmosphere. The instrument has a 256×256 InSb array detector camera sensitive from 1 to 5.5µm with a read noise of about 50 electrons (e^-) or about 4 data numbers (DN) and a dark current of <1e^-/s. The camera has a reimaging lens that gives it three possible plate scale settings (0.06, 0.15, and 0.30"/pixel), of which the 0.06"/pixel scale was used for most of this work. There are also several standard broadband filters and two circular variable filters (CVF) used in this study. The CVF properties are summarized in Table 2.1 (Leggett and Denault 1996)

These CVF's were stepped through several distinct wavelengths diagnostic of various mineral and volatile absorption features as well as some scans at a full Nyquist sampling, where the CVF is stepped in increments of half the spectral resolution.

The data were gathered by J. F. Bell III and C. D. Kaminski on several nights around the 1995 opposition and were spaced to maximize global coverage as well as longitudinal repetition. Table 2.2 summarizes the Mars data sets used in this study. A full disk, cylindrical projection albedo map from the Royal Astronomical Society of Canada (Bishop 1995) is presented at the end of the chapter in Figure 2.19 to allow matching albedo features and place names used throughout this dissertation.

Presented in figures 2.1 through 2.14 are images derived from the Viking broadband IRTM albedo and topography data for Mars (Viking Mars Consortium Data) transformed to orthographic views to match the view of Mars at the time of the observations. The albedo images are provided for comparison to data images to help in identifying Martian albedo regions. The topography maps will help in identifying whether certain band depth maps correlate with atmosphere pathlength indicating whether or not the absorption could be due to atmospheric species. In the topography images brighter regions have higher elevation. The sub-Earth point and some of the major features are listed below each image.

In addition to the images of Mars and the solar-type standard star BS4030, also known as 35 Leo, dome flat and linearity sets were also taken. Dome flats are images of a particular out of focus spot on the closed dome with all lights on, thus giving a uniformly lit image on the detector. These images are used to correct pixel-to-pixel gain variations. The linearity sets are flat-field images taken in pairs at ever-increasing integration times in order to find the light levels where the pixel response is no longer linear and the light level where the pixels saturate. They are also used to calculate the overall electron to data number (e^-/DN) gain value.

Image Reduction Technique

Before images gathered at the telescope can be used for scientific study they must first be reduced to a form such that the value of any pixel within the object is a linear function of the amount of light coming from that part of the object. Raw data frames contain this information, but it is buried within many other effects which much be corrected.

These effects include non-linear pixel effects above certain light levels, dark current (charge that will accumulate over time with or without light exposure), sky flux (due to thermal emission from the air), as well as pixel-to-pixel gain variations due to differences in the array pixels, non-flat thermal emissions, dust on detector filters and windows, and many other effects that are not uniform across the scene. Fortunately these effects can be modeled and corrected.

Gain Calculation

The first step in the data reduction was to calculate the proper conversion factor, or gain between DN and e^- so that shot noise errors of single frame aperture photometry can be calculated. Since the photon arrival rate from an object to the detector is a random process it follows from Poisson statistics that the error in the measurement of the number of photons is the square root of the number of photons detected. The semi-conductor nature of the detector creates one electron-hole pair for each photon that arrives. However, in the detector electronics these electron counts are converted to a digital data number which is usually not in a one-to-one correspondence with the number of electrons. Thus the gain needs to be calculated.

A set of images of an identical flat-field scene was taken at increasing integration times, with two images at each particular integration time. These images were averaged to get a better signal-to-noise ratio of each pixel at each integration time. The images were also differenced because the difference between two images of an identical scene at the same integration time will be proportional to the error in the pixel as follows:

              if

              where x, y, and z are image arrays

              then

              and if x = y then

Thus a frame created from the difference of two flat field frames of the same integration time will contain values which are a factor of greater than the error of the original frame.

To get the gain (k = e^-/DN) we look at the following:

then:

and finally

From equation (2.3) we know:

therefore

So, plotting verses the average frame will give a line whose slope is one over the gain. Creating this plot, as seen in Figure 2.15, for the linearity set of 19 FEB 95 gives an average value of 12.5 e^-/DN (as compared to the 10 e^-/DN in Leggett and Denault 1996). Each of the two areas were chosen so that the varations within them were a minimum but that they would contain enough pixels for reliable statistics (1681 pixels in section 1 and 2601 pixels in section 2).

Linearity Calculation

The pixels in an array detector are linear, i.e. there is a linear relationship between the number of photons hitting a pixel and the number DN recorded, as calculated in the gain factor above, but this linearity is only valid up to a limited value. Beyond this upper linear value limit, and before the pixel becomes saturated, there is a regime in which the behavior may be non-linear but still well behaved. If that regime can be modeled then the information can be linearized and thus made useful. Therefore, the second task in the data reduction is linearization (McCaughrean 1989).

To linearize the data we take the linearity set of flat field images and plot the DN as a function of integration time. Using plots like those shown in Figure 2.16 we can determine the maximum linear DN value and the saturation DN value. In the linear regime we solve for the linear coefficients for each pixel, creating a slope image and an intercept image. The images between the linear regime and the saturation level are then modeled using a cubic function and those coefficient images are also found using least square error techniques. A cubic is used because it is the lowest order polynomial that can be fit to the data with a high degree of correlation.

The next step to actually linearize the data images requires that each pixel in which the DN level is greater than the saturation level be flagged so that it can be replaced later. Then for each pixel with a DN greater than the maximum linear value we solve for an "equivalent integration time", t_i, using the previously calculated cubic coefficients where t_i is the time required under flat field conditions to reach the given DN value. The solution to this cubic is found using an iterative technique which converges to t_i to within the nearest tenth of a second. Finally the linearized DN is calculated by using t_i and the linear coefficients.

Dark Current & Sky Flux Removal and Flat Fielding

After linearization the next steps in the data reduction are to remove any dark current, and sky flux from the object images. The inclusion of sky flux due to a non-black sky in the infrared actually makes the removal of the dark current easier than it would be for optical imaging. For each object image exposed at the telescope, a corresponding adjacent blank sky frame is taken at the same integration time. This frame then contains the same bias level, dark current, and sky photon flux per pixel as the object image and thus all we need do is perform a simple image subtraction to correct the object image.

If all pixels had the same electronic gain, equivalent to the average gain in e^-/DN calculated above, and all effects within the scene were uniform, then the basic data reduction would be complete. However each pixel has its own electronic gain and so the pixel to pixel variations, and many other effects such as dust on, or aberrations in the detector window and filters which are constant but non-uniform, must be corrected. By exposing images of a uniformly lit surface (e.g. a flat field target inside the dome) we have an image which should be uniform and have the same DN value in each pixel. Thus, the true DN values are representative of the different pixel gains. Normalizing this image to a mean of one gives an image where the value of the pixel is just the electronic gain variation. Dividing our sky-subtracted images by this flat field image then gives us images where each pixel has the same gain, and pixel to pixel variations are removed. The images are also normalized to an integration time of one second so images can be compared in consistent units of DN/sec.

Bad Pixel Fixing

The final step in the basic image reduction is to repair any bad pixels. These are those pixels previously flagged as saturated or any other pixels that may have been affected by fluctuations that cannot be modeled (e.g. cosmic ray hits). The technique used on this data set is a three step process, each of which replaces determined bad pixels with the median of the surrounding eight pixels. This can be done because the plate scale of the images substantially oversamples the seeing conditions. The first step is to automatically replace those pixels known to be bad and marked in a bad pixel map, as defined below. The second step is to replace any pixel whose absolute value is greater than 10⁵, a pixel value that can cause overflow/underflow errors in future calculations, and a value that will never be real under the observing conditions used. Finally, an iterative search is made through the image looking for any pixel that differs by more than two standard deviations from the median of the surrounding eight pixels.

The second and third parts of the bad pixel fixing routine can be easily automated without any other information. The first part, however, requires the creation of a bad pixel map. This is done for each night of data using the linearity flat field frames. The set is searched for pairs of frames where the second frame has twice the integration time of the first. These frames are then divided so that, within the noise, the value of the resulting frame should be two in every pixel. The mean and standard deviation of these frames is then calculated and any pixel that is more than two standard deviations from the mean value of the frame is marked as bad in the bad pixel map. The end result is an image where the pixel value is zero for all but the bad pixels which are given a value of one.

Solar Spectrum Removal

Along with the Mars images there were sets of star images taken with the same spectral resolution and plate scale as the Mars images. The standard star chosen was BS4030, also know as 35 Leo, a G2IV star. Since this spectral class is similar to the Sun's class of G2V and the infrared colors differ by only a few hundredths of a magnitude (Strecker et al. 1979, Zombeck 1990, Carrasco et al. 1991), the Mars images can be calibrated to a scaled reflectance unit of Mars/Star. This unit then only differs from a true reflectance by a multiplicative factor of Star/Sun.

To create the calibrated images the integrated flux of the stars must be calculated. The star images are first processed just as the Mars images using all the above steps. The reduced star image flux can be calculated by setting an aperture on the image and totaling the DN values of all the pixels within the aperture. Residual sky flux DN values within an annulus surrounding the aperture is totaled and this is subtracted from the aperture sum. The aperture is slowly increased until the total DN flux remains constant, assuring that all the star light is included in the flux count (Figure 2.17). This total DN value is then the star brightness and is divided into each pixel of each image to calibrate Mars in units of scaled reflectance and the images are then fully reduced. Figure 2.18 presents the stellar spectra used to calibrate the Mars images.

Image Registration

In order to automate the analysis techniques it is necessary to align all the Mars images since, during the data acquisition, the image of the disk can shift by a few 10's of pixels due to telescope tracking and pointing difficulties.

The process of registration was performed in several steps. First, a reference image from within the spectral scan was chosen. The reference image is chosen to come from somewhere near the middle of a spectral scan to minimize the effects of Martian rotation from one end of the scan to the other. Second, an automated procedure was run on the spectral scan that would attempt to register images based on the cross-correlation between the images in the spectral scan and the reference image. This technique worked quite well for most of the images in the scan, except those in the strong 2.00µm absorption band where the limbs and north polar region of Mars are faint. Third, the registered set is checked by eye using a combination of a difference image and a ratio image between the images and the reference image as well as a blink comparison. Overall, registration could be performed to within about 3 pixels with most of the error in the registration due to Martian rotation effects not allowing a simultaneous registration of the planet limbs and the planet albedo features.

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