Abstract

This paper discusses the new radiometric calibration methods and results that we have developed to apply to day/night whole sky imager data and extinction imager data. Day/night whole sky imagers detect the presence and location of clouds and categorize them by opacity. The new day and night cloud algorithms for these instruments are documented elsewhere, and these algorithms depend on the use of the new calibrations we have developed. In this paper, we document our calibration methods for determining the absolute radiance of the whole sky imager scene at each pixel in the images as well as the results of the calibrations. The extinction imagers are new systems we developed that determine beam transmittance and path extinction for horizontal paths through the atmosphere. These instruments and their algorithms are documented elsewhere, and they also depend on the new calibration results. The extinction imager algorithms only require relative radiances, as opposed to absolute radiances. The calibration steps for the two instruments are quite similar, and they include dark correction; linearity calibration, which corrects for nonlinearities in the signal versus flux relationship; absolute calibration (for the whole sky imager); and uniformity calibration, which corrects for pixel-to-pixel nonuniformities. These developments include methods for handling uniformity calibrations in a system using a fisheye lens and the calibration of both visible and short-wave IR systems. This paper presents both the new methods and results in support of the papers documenting the algorithms.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

This paper discusses new radiometric calibration methods and results that we developed to apply to our day/night whole sky imagers (D/N WSI) and extinction imagers (EI). We developed D/N WSI systems to detect the presence and location of clouds in the day and night sky and categorize them by opacity. A general overview of the WSI systems and their hardware is provided in an earlier paper [1], and the details of our cloud algorithms are provided in a second reference [2]. We developed the EI systems and their algorithms recently, and they and their algorithms are documented elsewhere [3,4]. They determine beam transmittance and path extinction (effective optical extinction coefficient) for horizontal paths of sight through the atmosphere using a single-ended system. Both the WSI and the EI algorithms require the results of the new radiometric calibrations.

By applying calibration results to the data in the algorithms, we remove artifacts that are due simply to the characteristics of the measuring system and enable the development of accurate algorithms that depend on the physics of atmospheric optics. The WSI systems require absolute radiance calibrations, although the daytime cloud algorithm only requires relative calibrations. The EI systems only require relative calibrations.

The calibration methods for the WSI and EI instruments are similar, so we have presented them in order by calibration types, which include dark calibration, linearity calibrations, absolute calibrations, and uniformity calibrations. Although the results are specific to these instruments, the methods are generally applicable to digital imagers operating in the visible, the near infrared (NIR) near 800 nm, and the short-wave IR (SWIR) near 1.6 μm.

The D/N WSIs are part of a family of fully automated digital WSIs developed over many years by the Atmospheric Optics Group (AOG) at the Marine Physical Lab (MPL), Scripps Institution of Oceanography (SIO), at the University of California San Diego (UCSD) [1]. This development is documented in numerous reports, which may be traced from the references listed in that paper. The WSI imagers are ground-based sensors that acquire digital imagery of the full sky down to the horizon in several spectral bands in order to detect the presence and distribution of clouds and categorize them by opacity. The D/N WSIs automatically acquire high-quality digital imagery of the sky under all conditions, including full sunlight through moonlight and including starlight conditions. We developed and fielded cloud algorithms for detecting the clouds on a pixel-by-pixel basis for both day and night [2]. Our extinction imagers [3,4] are narrow-angle systems viewing dark targets such as a black box or the ocean surface near the horizon. They include algorithms we developed to determine the beam transmittance and path extinction of the path of sight. EI systems were developed for both the visible and the SWIR, and both systems will be included in this discussion. The visible EI system is designated the multispectral extinction imager (MSI), and the SWIR EI system is designated the short-wave IR imager (SRI). We have also calibrated imaging systems for a daylight visible/NIR WSI (VN WSI) [5], an airborne imager, and other systems that will not be discussed in detail.

2. OVERVIEW AND BACKGROUND

Radiometric calibration theory and general methods are well documented [6], and numerous papers document the different waysthese methods have been applied to a variety of instruments [710]. The AOG imager calibration methods initially evolved from our methods for calibrating photometers [11] for an airborne program involving well over 100 flights with several airborne instruments (e.g., [11]). We developed methods to adapt and enhance the photometer calibration techniques so that they could be applied to CCD systems and shared them with a colleague who adapted them for his instrument [8]. Then we further developed the techniques to yield both the calibration methods used with the VN WSI [7] and the calibration methods documented here for the D/N WSI and EI systems.

Our current calibrations include the following steps, which will be discussed in this paper. Dark calibration characterizes the output signal as a function of gain and exposure settings at each pixel when there is no input flux. Linearity calibration characterizes the relative response to varying input flux in a region of interest (ROI) near the center of the image. These linearity results are typically dependent on exposure and gain settings. Absolute calibration determines the absolute radiance in the ROI for a nominal dark- and linearity-corrected signal such as 1000 for each combination of filter, lens setting, exposure, and gain. Uniformity calibration characterizes the relative response as a function of pixel position relative to the ROI near the center of the image. Our calibration procedures also included angular or geometric calibrations to enable one to assign an angle in object space to each pixel in image space. The geometric calibration for the EI systems was a straightforward assessment of pixel position versus angle position from test imagery. The geometric calibration for the WSI system was more accurate and more complicated, and it is discussed in the paper on the WSI cloud algorithms [2].

A. Imaging System Considerations

Before we discuss the calibration steps in more detail, it is worth noting some of the features of the WSI and EI systems that made calibration more accurate and indeed possible. Both types of systems used digital imagers and were set up to avoid the use of any features such as auto gain that would have affected the signal-to-radiance relationships. In particular, in the SRI, we had to disable the camera’s auto-uniformity feature, because that feature altered the linearity relationships on an image-by-image basis. The D/N WSI and MSI used a very accurate 16-bit CCD, and we will also present some results for the 12-bit CCD used in the VN WSI for comparison. The SRI used a hybrid sensor with InGaAs detectors hybridized onto a CMOS-integrated readout chip with 12-bit resolution.

All of the systems used chips that were cooled and temperature-stabilized and thus enabled consistent calibration characteristics. They all had a live zero, meaning that there was a nonzero signal when no flux was applied, enabling characterization of the dark images. The WSI systems were sealed and purged with dry nitrogen for most sponsors in order to help avoid the degradation of interference filters that would have influenced the stability of the calibrations. The EI systems were not sealed, but they were not fielded for long-term deployments where this would have been important. Stray light can be another consideration, and the WSI system included a solar/lunar occulter (sun shade) to avoid direct solar or lunar flux on the lens. The impact of the residual stray light was also carefully evaluated and handled.

The absolute calibrations also depend on a knowledge of the effective spectral response of the instruments, and for this purpose the spectral responses of the filters and other components were carefully measured in the WSI systems. Also, to minimize the impact of any long-term changes in calibration characteristics, we frequently acquired dark images in the field, and we designed our algorithms to use relative radiance to the extent possible. In processing extensive databases, we re-evaluated the algorithm results for roughly each year of data in order to detect and handle any long-term drifts such as might be caused by degradation of interference filters or other components. Also, it was important that field data be kept on scale, i.e., not too bright or dark, so that the data were within the calibrated range. For the WSI systems, we developed automated flux control algorithms to keep the data on scale [1]. These algorithms were based on solar and lunar position as well as on lunar phase and distance. A flux control algorithm was not required in the EI systems, but it would be if future systems that operate both day and night are developed.

It is not always practical to design instrumentation for optimal calibration accuracy, given budget constraints. As a result, part of the purpose of calibrations is to enable one to understand the instrument characteristics and limitations, not just to determine the steps needed to convert raw signals to calibrated radiances.

B. Calibration Facility

The terms “radiance” and “irradiance” are well defined in numerous texts (e.g., [6]). Following these references, we use the term “radiance” to mean the radiant energy in a given direction, per unit area and per steradian, in the limit as the unit area and angle approach zero. Our instruments were calibrated to provide the spectral radiance, i.e., radiance per wavelength increment, in units of watt per square meter per steradian per micrometer (wattm2sr1μm1). Similarly, we use the term “spectral irradiance” to mean the spectral radiant energy per unit area of a surface. Most of our calibrations were accomplished with a calibration facility including a 3 m calibration bar, although we also developed an automated field calibration device based on Labsphere integrating spheres. The field calibration device is beyond the scope of this paper; however, the principles remain the same.

The primary light source for the calibration bar was a FEL 1000 watt quartz-halogen tungsten coiled-filament lamp standard of spectral irradiance traceable to the National Institute of Standards and Technology (NIST). When the front surface of the lamp’s mounting pin is at a distance of 50 cm, the lamp’s filament is at a distance we shall define as d0. According to the manufacturer, d0 is approximately 50.32 cm. We used an alignment jig (available from the lamp manufacturer) to measure this value and found that the values ranged from 50.2 to 50.35 for different lamps. The manufacturers provide the values for the spectral irradiance Eλ(d0) on a plaque.

The lamp was mounted on a rolling precision mount and placed on a rail or “bar” that was 3 m long. The lamp illuminated a 12×12 Spectralon standard plaque from Labsphere placed at the 0 position on the 3 m bar. For each wavelength (or for a monochromatic system), when the lamp was placed on the calibration bar with its filament at a distance d from the reflectance plaque, it created a spectral irradiance Eλ(d) given by Eq. (1) as follows:

Eλ(d)=Eλ(d0)*d02/d2.
The reflectance of the calibration plaque Rλ was provided by the manufacturer as a function of wavelength. The plaque we used had a reflectance of 0.99 in our wavebands. When a known irradiance Eλ was present on the reflectance plaque, the resulting radiance Lλ of the plaque was given by
Lλ(d)=RλEλ(d)/π,
where Lλ is the spectral radiance and Rλ is the plaque reflectance. For radiance measurements, the imaging system was placed where it could image the plaque. Since the plaque was a diffuse reflector and essentially Lambertian, this radiance was relatively invariant with distance from the plaque and angle with respect to the plaque, so placement of the system with respect to the plaque was not critical. The derivation of the effective spectral radiance of the lamp in the instruments’ passbands will be discussed in the absolute calibration section, since neither the dark calibration nor the linearity calibration depend on the absolute radiance in each filter setting.

The lamp was controlled by an Optronics Laboratory Precision Current Source, which provided a fixed current across the filament. Alignment devices enabled precise alignment of both the plaque and the lamp filament with respect to the bar. The imaging system viewed the plaque from a roughly 45° angle. While the details of the use of the calibration facility are beyond the scope of this paper, we will further note that we used the WSI in night mode to detect and enable us to minimize sources of stray light in the room. The next sections discuss each of the primary calibrations that we acquired. After these sections, our method to apply the results to field data is given.

3. DARK CALIBRATION

The first step in the radiometric calibration was the dark calibration. A dark calibration is a measure of the signal generated by the imager when there is no input light flux. It is normally a combination of an electronic bias set by the manufacturer and a thermally generated dark noise generated by the chip. The dark image typically may vary as a function of exposure time, gain setting (if any), and temperature of the chip. The thermally generated dark noise also typically has some pixel-to-pixel variance as a result of the crystal structure of the sensor chip. In addition, with some sensor chips, such as the CMOS used in the SRI, the readout method impacts the dark image, resulting in features such as vertical lines in the dark image. The dark signal averaged over a ROI near the center of each image has been plotted as a function of exposure for two generations of cameras used in the D/N WSI camera in Fig. 1 and a SWIR CMOS system in Fig. 2.

 

Fig. 1. Dark signal versus exposure, for two Photometrics CCD imagers, a Series 200 and a Series 300.

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Fig. 2. Dark signal versus exposure from an Alpha NIR SWIR camera, in low gain and high gain.

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The early D/N WSIs used Series 200 Photometrics cameras, and the later WSI systems used the Series 300 cameras. These cameras include a mechanical shutter, and thus a dark image could be acquired simply by setting the desired integration period on the chip, but with the shutter closed. As seen in Fig. 1, the Series 300 cameras had a lower electronic bias and a lower thermally generated signal. In the Series 200, the signal increased by about 100 counts as the exposure was changed from near 0 to 1 min, but in the Series 300 the increase was only about 5 counts. Both cameras were thermally stabilized near 40°C. These cameras had no variable gain setting. The Series 300 camera was also used in the MSI.

The SRI used the Alpha NIR camera shown in Fig. 2. Prior to dark calibration, we had to adjust the readout settings to produce a “live zero” and also disable the auto-uniformity feature. This camera had two gain settings, and the dark signal versus exposure shows several discontinuities in the plot, which the manufacturer called “glitch points.” They indicated that these occur as a result of the readout process. The manufacturers were not initially aware that there were so many glitch points, but they were able to duplicate our results. They recommended not using these exposure values for field data acquisition. Thus one of the modifications we had to do in order to use a CMOS sensor was to use only preselected exposure values, rather than allowing flux control algorithms to choose from any exposure setting.

Although measurements of the dark calibrations in the calibration room were important for understanding system performance, in practice we used dark images acquired in real time in the field in our algorithms. The WSI systems had mechanical shutters, and a dark image was acquired in the field whenever the exposure changed by closing the shutter and letting the chip integrate for the normal time. The SRI had an electronic filter so we used field images acquired at night, when flux levels were well below the response range, to represent the dark image in the algorithm processing.

4. LINEARITY CALIBRATION

The linearity calibrations measure the relationship between the input flux and the dark-corrected image signal. (To avoid confusion, we should note that we have become aware that some groups use the term “linearity” for what we call “uniformity.”) Even in cameras that are rated to be very linear, we often find significant nonlinearities that can create significant errors in the results if not corrected for. Typically, the linearity is affected by chip characteristics both at the low end of the sensitivity range and the high end of the response range, and this nonlinearity may also be a function of exposure and gain. In this section, we will present the results for the D/N WSI and MSI camera, which was extremely linear, the camera used in the VN WSI, which required significant linearity corrections, and the camera used in the SRI, which had very large nonlinearities.

A. Linearity for the 16-Bit Camera Used in the D/N WSIs and MSI

The D/N WSI systems and the MSI used a very “well-behaved” 16-bit sensor, i.e., the Photometrics 16-bit Series 200 camera or series 300 camera. These cameras had only one gain setting, with low noise and excellent linearity, but these results are presented to explain the new methods that we used. All of the Series 200 and 300 cameras had results very similar to those illustrated in this subsection.

We performed two basic types of linearity calibrations: a calibration in which we changed exposures at a fixed lamp position, called “linearity versus exposure”, and a calibration in which we changed lamp positions at a fixed exposure, called “linearity versus radiance.” This dual method provides redundancy and the ability to evaluate stray light. The linearity versus exposure results are shown in Figs. 3 and 4, and the linearity versus radiance results are shown in Figs. 5 and 6. These results are for the average in a ROI near the center of each dark-corrected calibration image.

 

Fig. 3. Dark-corrected signal as a function of exposure from the exposure calibration, for photometrics series 200 camera, unit 8.

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Fig. 4. Percent nonlinearity as a function of dark-corrected signal from the exposure calibration data in Fig. 3.

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Fig. 5. Dark-corrected signal as a function of relative flux from the Lin versus radiance measurements.

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Fig. 6. Percent nonlinearity as a function of dark-corrected signal from the Lin versus radiance data in Fig. 5.

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For Fig. 3, we set the camera up in the calibration room at a fixed lamp position and varied the camera exposure setting. The left edge of the plot shows the signal measured at the minimum exposure used. The relative value of these two curves is arbitrary, as it depends on how we set up the measurement (i.e., which lamp position was used for each curve). What is relevant is that the response appears to be highly linear.

The fractional nonlinearity shown in Fig. 4 was derived as follows.

  • (a) We dark correct the data to yield a corrected signal S (which is shown in Fig. 3).
  • (b) We determine by interpolation the exposure E(10,000) that corresponds with a dark-corrected signal of 10,000 for each of the above curves. For the blue filter curve, this point was near an exposure of 55,000 ms, and for the red filter curve, this point was near 2000 ms.
  • (c) For each point, we determine nonlinearity from the equation
    NonLin(Sig)=Sig*E(10,000)10,000*E(Sig).
    This results in fractional values near 1, which are then converted to percent nonlinearity for the plot in Fig. 4. One can think of the values in Fig. 4 as the error in the dark-corrected signal with respect to a fully linear system, if the linearity is not corrected for. From this curve, we see that there appear to be nonlinearities of less than 1% over most of the dynamic range, with nonlinearities up to about 2% at the high end. We should also note that a 15 ms exposure correction has been made in these data because the camera’s input exposure time has about a 15 ms offset with respect to the effective opening time of the mechanical shutter. (The exposure correction is automatically made prior to grabbing an image in the field data acquisition programs, but it is not automatically made during calibration, so that we can more accurately assess the actual offset.)

For the calibration shown in Fig. 5, we set the camera up in the calibration room at a fixed exposure and varied calibration lamp position to provide known changes in relative light on the calibration plaque. This measurement also shows that the system is quite linear. The fractional nonlinearity shown in Fig. 6 was derived from Fig. 5 using a similar procedure to that used for Fig. 4. These calculations also included the 15 ms exposure correction mentioned above.

Ideally, Figs. 4 and 6 would be precisely consistent; however, Fig. 4 depends in effect on the true linearity of the system response being independent of exposure, whereas Fig. 6 depends in effect of having no stray light in the calibration room. The fact that these two plots are consistent with each other to within about a percent over most conditions, and two percent at high exposures, indicates that both of these conditions are nearly correct for this system. From evaluation of the raw data, however, we found that there was increasing stray light for lamp positions closer to the plaque than 1 log, so we avoided these lamp positions when possible and did not include them in Fig. 6.

In further analysis, we found that the linearity results for the ROI could be applied to the whole image and that pixel-dependent variances were well handled by the dark calibration and uniformity calibration. We did not apply any linearity corrections to the D/N WSI because the nonlinearities were insignificant. For the MSI, we decided to apply the linearity results by characterizing the nonlinearity curve in a lookup table that listed the corrected signal for each dark-corrected signal from 0 to 65,535.

B. Linearity for the 12-Bit CCD Camera Used in the VN WSI

Even though the primary purpose of this paper is to document the calibrations used with the D/N WSI systems and the EI systems, we would like to document the linearity results for the VN WSI, which used a 12-bit CCD camera. These results are much more representative of the linearity results that we have obtained with a variety of 12-bit CCD cameras such as might be used in future sky imagers. We only document the linearities for this system because the other calibrations for this system were similar to those for the D/N WSI systems.

The VN WSI used a Photometrics Sensys camera. We found that this camera response was not as linear as the more costly 16-bit cameras as might be expected. The manufacturer indicated that, in addition to the exposure offset, there was a signal offset called the “toe effect” because the chip must acquire a certain number of electrons before a signal is created. By acquiring a number of linearities at low exposure and high signal values, we were able to determine the exposure offset to be 10 ms. The toe effect was less easy to characterize in the form of a signal offset. Instead, we chose to simply include the impact of the toe effect as part of the measured linearity curve. The results are shown in Figs. 7 and 8, which include an exposure offset Eo of 10 ms. (Note change of plot scale with respect to the earlier figures.)

 

Fig. 7. Dark-corrected signal as a function of exposure from the exposure calibration for the Photometrics Sensys camera.

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Fig. 8. Percent nonlinearity as a function of dark-corrected signal from the exposure calibration data in Fig. 7.

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In Fig. 7, the results at first glance appear to be reasonably linear. However as shown in Fig. 8, when the fractional nonlinearities are computed from these data using the methods discussed earlier, one can see that the results are actually quite nonlinear as well as gain-dependent, especially below dark-corrected signals of about 300. [These values were normalized to a value of 1000, rather than the 10,000 value used in Eq. (3).] These results were new to us, and they were unexpected. We feel that these plots illustrate clearly that it’s very easy to assume from preliminary analysis that a system is linear, and yet the errors if this assumption are made can be very significant. Even in Gain 2, the gain for which the camera was optimized, the nonlinearity is greater than 10% for signals below 100. These linearity results were applied to the data in the cloud algorithms through use of a separate lookup table for each gain setting.

This camera was rated for a 0.5% nonlinearity; however, the manufacturer was able to duplicate our results. As a general rule, we have found that digital cameras that are rated to be linear can result in significant error if linearity is assumed. We have discussed this with various manufacturers and found that there are several reasons for this. First, the manufacturers we have discussed this with did not take into account the effect of exposure complications such as the exposure offsets. Second, we were told that they did not measure the bottom or top of the range. And third, we were told that the nonlinearity was calculated as a fraction of the full scale value. For our applications, in which digital accuracy was important, it was important to look at the nonlinearity as a fraction of the signal, not as a fraction of the full-scale signal. We would also like to note that with some visible 12-bit cameras, we were not able to characterize the impact of exposure, so we had to choose in advance which exposures would be used in the field and then generate separate linearity curves for each exposure and gain combination that we used.

C. Linearity for the 12-Bit Short-Wave Infrared CMOS Sensor Used in the SRI

As noted in the section on dark current, the hybridized InGaAs CMOS sensor used in our SWIR systems had several “glitch points” when the dark levels were measured as a function of exposure. The glitch points also affected the linearity, whether dark-corrected or not, especially in high-gain mode. As a result of these points, the linearity was not well behaved as a function of exposure, as shown in Fig. 9. However, at a fixed exposure and gain, the linearity was somewhat better behaved as a function of input flux. Thus we measured the linearity as a function of flux level for each exposure used in the field. The SRI only required one exposure, and the results of the calibration for this exposure are shown in Fig. 10.

 

Fig. 9. Dark-corrected signal as a function of exposure for the alpha NIR camera used in the SRI.

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Fig. 10. Dark-corrected signal as a function of flux for the alpha NIR camera used in the SRI, low gain.

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Like Figs. 3, 5, and 7, Fig. 10 looks fairly linear. However, for our purposes it was necessary to derive the nonlinearity corrections as a fraction of the signal in order to provide accurate quantitative data results, as done in Figs. 4, 6, and 8. When the actual nonlinearity was derived from the data in Fig. 10 (normalizing to a signal of 1000), the results showed that the system response is actually very nonlinear, as shown in Fig. 11 (note change of scale with respect to earlier figures).

 

Fig. 11. Percent nonlinearity as a function of dark-corrected signal from the linearity versus radiance calibration for the alpha NIR.

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In Fig. 11, we can see that the nonlinearities as a fraction of the signal are quite significant. The nonlinearities are as high as about 10% for some signals near the bright end of the sensitivity range and up to 80% for signals near the darker end of the sensitivity range. Even at a dark-corrected signal of 100, the nonlinearity relative to a signal of 1000 was greater than 50%. Our understanding is that in low-gain mode, the Alpha NIR sensor had nonlinear behavior at low well fill because the integration capacitor in each unit cell was a solid state device grown on silicon that had non-ideal properties. As the capacitor filled with charge, the “spacing” between the capacitor “plates” changed since the depletion zone would change its thickness. The capacitors were pre-filled with the skim voltage control to help get it out of that nonlinear region at low light levels [12].

Repeated measurements with slightly varied settings and different cameras indicated that the nonlinearities were stable and repeatable, so they could be corrected for. As a result, we felt that these nonlinearities could be handled in the algorithm and were an acceptable trade for the ability to measure data in the SWIR wavelengths, which the CCD systems could not do at that time. These calibration results were applied to the SRI data in the EI algorithm with a lookup table.

These new linearity calibration results are very important, as we feel that without these calibration corrections, we would not have been able to use EI systems in the SWIR wavelengths. Similarly, with 12-bit CCD systems, we feel that linearity calibrations are very important and contribute very significantly to the accuracy of measurements, including our VN WSI results. Also, the linearity calibrations gave us a much better understanding of instrument performance and have alerted us to the fact that with the SRI we could not simply change the exposure and expect the linearity of the response to be consistent.

5. ABSOLUTE CALIBRATION

Absolute calibration is designed to determine the spectral radiance in wattm2sr1μm1 corresponding to a value of 10,000 (16-bit systems) or 1000 (12-bit systems) in a dark- and linearity-corrected image near the center of the image, for each combination of filter, lens settings, etc. used by the instrument. The EI systems only required relative radiance, because the EI algorithms are based on relative radiance between a dark target and the horizon sky. Thus they did not need an absolute calibration, although they did need the uniformity calibration discussed in the next section. The WSI systems were calibrated to provide absolute radiance. This was partly because the sponsors desired absolute radiance distributions for the sky as a secondary product. The night cloud algorithm also required absolute radiance values. The daytime cloud algorithm for the WSI systems was based on relative radiances in the various spectral filters; however, the relative spectral filter corrections were determined using absolute calibrations.

A. Effective Plaque Radiance in the Sensor’s Passband

For a monochromatic system, the spectral radiance of the plaque as a function of lamp position is given in Eqs. (1) and (2). The passbands used in the D/N WSI and other imagers were typically wide enough (e.g., 70 nm bandwidth) that the lamp irradiance varied somewhat within each passband. As a result, we replaced the value of Eλ in Eqs. (1) and (2) with the effective spectral irradiance in each combination of spectral (SP) and neutral density (ND) filter, for each individual WSI system as follows:

E(SP,ND)¯=EλSλTλ(SP)Tλ(ND)dλSλTλ(SP)Tλ(ND)dλ,
In Eq. (4), Eλ is the spectral irradiance of the lamp, Sλ is the spectral sensitivity of the chip, and Tλ is the spectral transmittance for each SP and ND filter. The spectral sensitivity of the chip is proportional to the wavelength times the quantum efficiency, which is provided by the camera manufacturer. Since it appears in both numerator and denominator, only relative values of Sλ as a function of wavelength are required.

The D/N WSI included two filter wheels. The SP wheel held a NIR, open, red, and blue filter in spectral positions 1–4. The ND wheel held an open, 2 log, 3 log, and open filter in positions 1–4. (A 2-log ND filter attenuates the signal by roughly a factor of 100, but the attenuation is somewhat spectrally dependent.) We typically used the 3-log ND filter with the red, blue, and NIR in the daytime, and used an open hole at night.

The spectral transmittance curve for each filter was measured with a spectrophotometer prior to assembly. The WSI system also included an IR-blocking filter to block the light from the photodiodes used in the filter changer and a nearly neutral filter that was part of the fisheye lens. The transmittance curves for each of these were also included in the derivation, but are not specifically listed in Eq. (4) for simplicity.

B. Acquiring and Processing Absolute Calibration Data

To acquire absolute calibration constants, theoretically a single measurement for each combination of SP and ND filter used in the D/N WSI would be required. (For other instruments such as the VN WSI, a single measurement for each combination of filters, gain, aperture, etc. would be required.) In practice, we acquired about five measurements in each filter combination to provide redundancy and an estimate of uncertainty resulting from stray light and other factors.

For each measurement, we compute the effective radiance of the plaque, using Eqs. (1), (2), and (4). For each measurement, an estimate of the absolute constant corresponding to a dark- and linearity- corrected signal of 10,000 and an exposure of 100 for the D/N WSI was derived from Eq. (5) as follows:

A(SP,ND,d)=L(SP,ND,d)*10,000Sig*Exp100.
In this equation, A(SP,ND,d) is the derived estimate of the absolute radiance corresponding to a dark- and linearity-corrected signal of 10,000 and a camera effective exposure of 100 ms for the SP and ND filter selection and the lamp position. L(SP,ND,d) is the plaque radiance for the specific lamp position. Sig is the signal corrected by the dark current and the linearity and averaged over a 20×20 ROI near the center of the image. Exp is the effective exposure used in the measurement. (For the D/N WSI, the linearity correction was not needed, but for other systems it was used in this step.)

A sample calculation from D/N WSI Unit 13 calibration for SP3 (red) ND3 (3 log) is shown in Table 1. This sample was chosen because Unit 13 was intermediate in accuracy with respect to other cameras, and the SP3 ND3 combination was often used for the daytime cloud algorithm.

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Table 1. Sample Absolute Calibration Constant Measurement and Derivation

In Table 1, Columns 1 and 2 are the lamp position in logs and centimeters (cm). That is, the 2-log position is at 300 cm and results in light a factor of 100 or 2 logs darker than the light at 30 cm, which is the bar reference point. Column 3 was derived from Eqs. (1), (2), and (4). Column 4 is the measured signal with dark and linearity corrections. Column 5 is the effective camera exposure. Column 6 is the calibration constant derived using Eq. (5). And column 6 shows the offset with respect to the average constant of 110.9.

In the above example, measurements were taken at five lamp positions ranging from 2 to 1 log darker than the 30 cm position, and the averaged corrected ROI signals ranged from about 1851 to 18,765 counts. In the absence of stray light or other measurement issues, the value of the derived calibration constant should be the same for all of the lamp positions. In this example, the derived calibration constants differed from the average calibration constant by a maximum of ±0.8% and had a STD of 0.5% with respect to the average. The average value is normally used, unless there is an outlier for some reason, but there were no outliers in this case.

For this instrument, the filter combinations that were used in the cloud algorithms and the variation (STD) in the measured absolute calibration constants are listed in Table 2.

Tables Icon

Table 2. Absolute Calibration Measurement Uncertainties by Filter

In Table 2, the visible spectrum STDs (blue and red filters) are less than a percent, and the filter selections that include the NIR spectrum near 800 nm (NIR filter and open hole selection) are higher, with STDs of about 1%–2%. The STDs in Table 2 include the cumulative effects of stray light in the calibration room or the optical system and variations in the linearity and/or dark image. The results in Table 2 were typical results, and they reflect a high degree of accuracy in the absolute calibration.

The final uncertainty in the calibrated radiances is impacted by the uncertainties indicated in Table 2 as well as the cumulative effects of uncertainty in the lamp irradiance (1% according to the manufacturer), the plaque reflectance, the linearities, and so on. Although it is difficult to determine the overall uncertainty of the final calibrated radiances without more detailed tests and cross-calibrations, our best estimate is that there was an overall uncertainty of approximately 3% for the blue, red, and open hole filters, and approximately 4% for the NIR filter or slightly more depending on field conditions (such as dust on the optics).

The calibration and procedures, when applied to field data, yield the effective spectral radiance as viewed with a given instrument’s passbands. When several instruments are being used that have slightly different spectral passbands (due to slight variations in camera responsivity or filter transmittance), the measured radiances will be slightly different for each system, even in the absence of measurement or calibration uncertainties. As a final step, we defined a standard instrument response and corrected the results for each instrument to the results that would have been obtained with the standard response. Although these details are beyond the scope of this paper, we note that these corrections were less than 2% in the blue, red, and NIR, and 3% in the open hole for this instrument, and similar values were derived for the other D/N WSI systems.

The output of the absolute calibrations consisted of the calibration constant corresponding with each combination of filters, gain, etc. to be used in the field. These calibration constants were the absolute radiance corresponding with a dark- and linearity-corrected signal of 10,000 (16-bit systems) or 1000 (12-bit systems). The application of these constants to the field data is discussed in Section 8.

6. UNIFORMITY CALIBRATION

The uniformity calibration measures the spatially dependent variance in the response of the system and enables correction for any nonuniformities in the response. The basic concept is to measure a uniform source such as a white diffusely reflecting plaque, apply the dark and linearity corrections, and then evaluate the nonuniformities in the resulting image. Typically, these nonuniformities may be due to losses such as Fresnel losses in the optics, losses due to vignetting, pixel-to-pixel variations in the chip response, and nonuniformities that are an artefact of the readout process. We have found with all the uniformity calibrations we have taken that the net impact acts as a percent change in throughput that differs from pixel to pixel. That is, because all of these effects introduce a fractional change in the signal, measurements taken at a given input flux level can be effectively applied to imagery taken at other flux levels.

The uniformity calibration is reasonably straightforward for reasonably narrow-angle lenses such as used in the MSI and SRI, which had a field of view of about 4 deg. For these EI systems, we took redundant measurements with different targets and used the image from a diffusely reflecting plaque acquired outside on an overcast day. The MSI nonuniformities were in large part due to the lens and some vignetting within the filter changer optics. The MSI had a different uniformity image for each spectral filter. In the SRI, much of the nonuniformity is due to the camera readout, and it takes the form of vertical lines. We used the SRI at only one exposure in the field, with only one lens aperture setting, and with no spectral filter, so only one uniformity image was required for processing. A sample raw SRI image is shown in Fig. 12, and an image with dark, linearity, and uniformity correction is shown in Fig. 13.

 

Fig. 12. Raw SRI image acquired near 1.6 μm, looking South to Los Coronados Islands from Pt. Loma, CA.

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Fig. 13. Processed image, with dark, linearity, and uniformity corrections.

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Determination of the uniformity calibration for the D/N WSI was more complicated, due to its wide field of view. The D/N WSI uses a Nikon Fisheye-Nikkor 8 mm f/2.8 lens, with approximately a 181° field of view. To measure the uniformity for this wide field of view, we lowered the system into a 1-m integrating sphere so that the lens was at the center of the sphere. This presented a nearly uniform flux level at all angles. However, the presence of the camera inside the sphere slightly distorted the light in the sphere, and there were also slight nonuniformities due to minor flaws in the sphere interior surface. To address these issues, we took multiple images, rotating the camera system 15° about the axis of the lens between each image. We averaged these images together to eliminate any azimuthal dependence of the light in the sphere, creating a uniformity image. This uniformity image included nonuniformities caused by the lens, filters, and camera. However, it also potentially included the impact of a zenith angle dependence of the light in the sphere, which we did not want to include.

To test whether there was a zenith angle dependency that was the result of the sphere and not the WSI hardware, we first extracted the zenith angle dependence from the uniformity image measured in each filter. Next we placed the camera system on a rotary table looking at the calibration bar plaque and measured the decrease in the signal as the look angle was changed from 0° zenith angle to near 90° zenith angle. The rotary table measurements were at low resolution, but comparison with the hemisphere results allowed us to verify that the integrating sphere, as used in these tests, had no zenith angle dependence, and thus the zenith angle dependence in the uniformity image was a true measure of the system response. This test of the zenith dependency of the integrating sphere only had to be completed once. Thus the average of all the images taken at 15° increments could be used as a measure of the zenith angle dependency in the throughput of the WSI system. The uniformity image characterized both the pixel-to-pixel differences within the camera system and the optical effects caused by the lens and filter characteristics.

The output of the uniformity calibration consisted of a dark- and linearity-corrected image in each spectral filter and the average value of this uniformity image within the ROI used for linearity and absolute calibrations. Section 8 illustrates how to apply these calibrations to the field data.

7. SPECIALTY CALIBRATIONS

Often a variety of specialty calibrations were acquired. For example, by measuring linearity versus exposure at very short exposures, we could more accurately determine the exposure offset in the WSI systems. Noise characteristics were determined by taking sets of repeat images at a single exposure. From these data we could independently determine the magnitude of the spatial variance and the temporal variance. Knowing the readout noise and electrons per digital count provided by the manufacturer, in combination with our noise measurements, we were also able to derive shot noise and total noise. Also, any time we did not understand the behavior of a camera system, we took additional measurements as needed. In the case of the VN WSI, this resulted in detecting nonlinearities that the manufacturer was not aware of but was able to duplicate, and in the case of the SRI, these additional measurements resulted in detecting glitch points that the manufacturers had not known about but were able to duplicate.

8. APPLICATION OF CALIBRATION RESULTS TO FIELD DATA

The new procedures for applying the calibration results to images acquired by the instrument were the same for the WSI and EI systems as well as all the other systems we calibrated for ourselves and for others. These are:

  • (a) Subtract the dark image on a pixel-by-pixel basis at the correct exposure and gain setting as follows:
    Sig(x,y)=Sig(x,y)Dark(x,y).
  • (b) Apply the linearity correction, which was done by applying a lookup table of corrected signal versus dark-corrected raw signal (although it could have been done by means of a best-fit equation). Note that the linearity data were also dark-corrected prior to extracting the lookup table or best fit curve;
    Sig(x,y)=f(Sig(x,y)).
    In this equation the function is represented by the lookup table. It is based on the calibration measurements within an ROI near the center of the image, but is applied to all pixels.
  • (c) If absolute calibration is to be applied to field data, as it was with the WSI systems, the result typically now becomes floating point and depends on instrument settings. For the D/N WSI, the result is dependent on the spectral filter, neutral density filter, and exposure, and it is derived with the following equation:
    l(x,y,SP,ND,Exp)=A(SP,ND)*Sig(x,y)10,000*100Exp.
    In this equation, we have used a lower case “l” because the derivation of actual radiance “L” is not yet complete. In this equation, we did not have to apply the exposure offset correction because the field data acquisition program automatically applied the offset correction. Like the linearity calibration, the absolute calibration is based on the calibration results within an ROI near the center of the image but is applied to all pixels.
  • (d) The final step in the derivation of the radiance at each pixel is application of the uniformity result. In this step, we divide each pixel by the uniformity image value at that pixel and multiply by the average of the uniformity image within the ROI. This is the step, in combination with the pixel-by-pixel dark correction, that extends the results to the full image. Thus, the final equation result is
    L(x,y,SP,ND,Exp)=l(x,y,SP,ND,Exp)*U(SP,ROI)U(SP,x,y).
    If absolute calibration is not required, the uniformity correction is applied directly using the following equation:
    Sig(x,y,SP,ND)=Sig(x,y)*U(SP,ROI)U(SP,x,y).
    Here we have applied the uniformity result that is appropriate for the given filter selection.

In the specific case of the D/N WSI systems, we used the general process as listed above, except that this system was so linear that we measured—but did not need to apply—the linearity calibration results. For the VN WSI, our dark images and our linearities were a function of gain as well as exposure, and the absolute calibration constants were functions of gain and filter. We would like to note that the D/N WSI had a tremendous dynamic range and was able to measure radiances at night, in the darkest part of the sky between stars, at our darkest sites, with a signal to noise of 40:1. During the daytime, a 3-log ND filter and narrower spectral filters were used, and exposures of 100 ms versus 1 min at night were used. With the additional dynamic range inherent in the 16-bit camera (with grey levels from 0 to 65,535 and a readout noise of 1 count or less), we were able to measure calibrated radiances over a range of more than 1010 from daylight to starlight.

For the visible EI system (MSI), we used the same camera as for the D/N WSI but with a narrow-angle lens, but the algorithms did not require the use of absolute radiance. With this system we went ahead and made the linearity correction, even though it was a small correction. Thus, we applied the dark, linearity, and uniformity corrections. The SWIR EI system (SRI) data were similarly processed using the dark, linearity, and uniformity corrections.

For the WSI and EI systems, the calibrated data were primarily used as an intermediate product that was input into algorithms to derive cloud distributions, beam transmittance, and other data products, as discussed in the AOG references (e.g., [14]). We regret that we never had the opportunity to evaluate the resulting radiance distributions due to sponsor priorities. However, there are some radiance results shown by Feister [13], and they appear to be reasonable. As discussed in the references, the resulting algorithm products were very good.

9. SIGNIFICANCE OF THE CALIBRATIONS

The decision to calibrate the sensors is not taken lightly because it is indeed a time-consuming task. However, we have found it to contribute vitally especially in three regards. First, calibrations helped us to understand the operating constraints of the instruments and how they could be used in the field, as noted earlier. Second, calibration enabled us to better assess the accuracy of the data. For example, measurement of pre- and post-deployment linearities enabled an assessment of how stable the system was. And measurements of noise levels enabled further assessment of instrument performance.

Third, the calibrations were extremely important in providing accurate data for use in algorithms. It would have been very difficult to develop the cloud algorithms without measurements of the relative response in the blue and red filters, for example. And development of the EI algorithms in the SWIR wavelengths would have been perhaps impossible without the linearity corrections. By calibrating and correcting for the impact of the instrument characteristics, we were able to develop accurate algorithms based on the physics of the atmosphere.

10. SUMMARY

In this paper, we have described the new calibration methods and results that we developed for use with our D/N WSI systems and EI systems. The steps in the calibration include dark calibration, linearity calibration, absolute calibration (if needed), and uniformity correction. The dark corrections were particularly interesting in detecting exposures that should not be used in the SWIR systems. The linearity calibrations for the VN WSI presented the issue of decoupling exposure effects from electronic chip effects. Also, the linearity calibrations for the SRI, which used a CMOS chip, illustrated that these SWIR cameras are sufficiently stable to use in our application requiring quantitative result, but that the application of the calibrations was vital. The uniformity calibrations were especially interesting, because special techniques had to be developed to handle the fisheye lenses used in the D/N WSI systems.

We found that we were able to successfully calibrate not only the CCD systems, which thus enabled higher-quality cloud algorithms and visible EI extinction results. We were also able to calibrate and characterize the SRI, which used a CMOS chip, which thus enabled extension of the EI capabilities into the SWIR wavelengths. These methods can also be applied to other systems and applications. We have discussed how we processed the calibration data and how we applied them to field data. Finally, we believe we have addressed why calibration can be very important.

Funding

Office of Naval Research (ONR) (N00014-16-1-2317, N00014-07-1-1060); Lincoln Laboratory, Massachusetts Institute of Technology (3032802); Pacific Northwest National Laboratory (PNNL) (PO 227480-ARW); Air Force Research Laboratory (AFRL); Kirtland’s Starfire Optical Range (FA9451-008-C-0226).

Acknowledgment

As noted earlier, the calibration techniques were developed over many decades, and thus it is not practical to list all of the contracts and grants, nor all the individuals, that have aided in this effort. We acknowledge the technical and fiscal support of our recent sponsors. For the EI work, ONR; and MIT Lincoln Laboratory; for the VN WSI work, PNNL, AFRL, and Kirtland’s Starfire Optical Range, and several earlier AFRL contracts. We appreciate the support of our primary sponsor contacts Sarwat Chappell, Quenten Saulter, Sadegh Siahatgar, Paul Berger, John T. Schriempf, Uwe Feister, Tim Tooman, Ann Slavin, and Earl Spillar. We also acknowledge and thank the AOG individuals who contributed to this work: Richard Johnson, Justin Baker, Art Burden, and Jack Varah.

REFERENCES

1. J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013). [CrossRef]  

2. J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”

3. J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” Final Report for JTO/ONR Grant N00014-07-1-1060, DTIC No. ADA 613006, 2014, https://apps.dtic.mil/docs/citations/ADA613006.

4. J. E. Shields and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Extinction imagers for measurements of atmospheric beam transmittance.”

5. U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005). [CrossRef]  

6. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

7. J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

8. K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989). [CrossRef]  

9. R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012). [CrossRef]  

10. E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014). [CrossRef]  

11. S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” AFGL- TR-0188, DTIC No. ADA 035571, 1976, https://apps.dtic.mil/docs/citations/ADA035571.

12. A. Richards, Senior Research Scientist, FLIR Systems, Inc., 6769 Hollister Ave., Goleta, CA 93117 (personal communication, 2019).

13. U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010). [CrossRef]  

References

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  1. J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013).
    [Crossref]
  2. J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”
  3. J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .
  4. J. E. Shields and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Extinction imagers for measurements of atmospheric beam transmittance.”
  5. U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005).
    [Crossref]
  6. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).
  7. J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).
  8. K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989).
    [Crossref]
  9. R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
    [Crossref]
  10. E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
    [Crossref]
  11. S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” , 1976, https://apps.dtic.mil/docs/citations/ADA035571 .
  12. A. Richards, Senior Research Scientist, FLIR Systems, Inc., 6769 Hollister Ave., Goleta, CA 93117 (personal communication, 2019).
  13. U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
    [Crossref]

2014 (1)

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

2013 (1)

2012 (1)

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

2010 (1)

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

2005 (1)

U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005).
[Crossref]

2003 (1)

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

1989 (1)

K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989).
[Crossref]

Alados-Arboledas, L.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Anton, M.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Baddeley, L.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Baker, J. G.

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

Berger, P. J.

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

Bilbao, J.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Biles, D.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Bjørkland, H.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Brändström, U.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Burden, A. R.

J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013).
[Crossref]

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”

Cazorla, A.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Chen, X.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Chernouss, S.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

de Miguel, A.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Deehr, C. S.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Duntley, S. Q.

S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” , 1976, https://apps.dtic.mil/docs/citations/ADA035571 .

Dyrland, M.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Feister, U.

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005).
[Crossref]

Frederickson, P. A.

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

Gordon, J. I.

S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” , 1976, https://apps.dtic.mil/docs/citations/ADA035571 .

Görsdorf, U.

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

Güldner, J.

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

Hodgkiss, W. S.

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

Holmen, S. E.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Johnson, R. W.

J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013).
[Crossref]

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” , 1976, https://apps.dtic.mil/docs/citations/ADA035571 .

Karr, M. E.

J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013).
[Crossref]

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

J. E. Shields and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Extinction imagers for measurements of atmospheric beam transmittance.”

Lind, R. J.

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

Lorentzen, D. A.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Lybekk, B.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

McCluney, R.

R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

Mikuls, V. W.

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

Moen, J.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Möller, H.

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

Olmo, F. J.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Richards, A.

A. Richards, Senior Research Scientist, FLIR Systems, Inc., 6769 Hollister Ave., Goleta, CA 93117 (personal communication, 2019).

Roman, R.

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Sattler, T.

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

Shields, J.

U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005).
[Crossref]

Shields, J. E.

J. E. Shields, M. E. Karr, R. W. Johnson, and A. R. Burden, “Day/night whole sky imagers for 24-h cloud and sky assessment: history and overview,” Appl. Opt. 52, 1605–1616 (2013).
[Crossref]

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

J. E. Shields and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Extinction imagers for measurements of atmospheric beam transmittance.”

Sigernes, E.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Trondsen, E.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Trondsen, T.

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

Voss, K. J.

K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989).
[Crossref]

Zibordi, G.

K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989).
[Crossref]

Appl. Opt. (1)

Atmos. Meas. Tech. (1)

R. Roman, M. Anton, A. Cazorla, A. de Miguel, F. J. Olmo, J. Bilbao, and L. Alados-Arboledas, “Calibration of an all-sky camera for obtaining sky radiance at three wavelengths,” Atmos. Meas. Tech. 5, 2013–2024 (2012).
[Crossref]

Atmos. Res. (1)

U. Feister, H. Möller, T. Sattler, J. E. Shields, U. Görsdorf, and J. Güldner, “Comparison of macroscopic cloud data from ground-based measurements using VIS/NIR and IR instruments at Lindenberg, Germany,” Atmos. Res. 96, 395–407 (2010).
[Crossref]

Geosci. Instrum. Methods Data Syst. (1)

E. Sigernes, S. E. Holmen, D. Biles, H. Bjørkland, X. Chen, M. Dyrland, D. A. Lorentzen, L. Baddeley, T. Trondsen, U. Brändström, E. Trondsen, B. Lybekk, J. Moen, S. Chernouss, and C. S. Deehr, “Auroral all-sky camera calibraton,” Geosci. Instrum. Methods Data Syst. 3, 241–245 (2014).
[Crossref]

J. Atmos. Ocean. Technol. (1)

K. J. Voss and G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fisheye’ camera radiance distribution system,” J. Atmos. Ocean. Technol. 6, 652–662 (1989).
[Crossref]

Meteorol. Z. (1)

U. Feister and J. Shields, “Cloud and radiance measurements with the VIS/NIR daylight whole sky imager at Lindenberg (Germany),” Meteorol. Z. 14, 627–639 (2005).
[Crossref]

Proc. SPIE (1)

J. E. Shields, R. W. Johnson, M. E. Karr, A. R. Burden, and J. G. Baker, “Daylight visible/NIR whole sky imagers for cloud and radiance monitoring in support of UV research programs,” Proc. SPIE 5156, 155–167 (2003).

Other (6)

R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

J. E. Shields, A. R. Burden, and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Atmospheric cloud algorithms for day/night whole sky imagers.”

J. E. Shields, M. E. Karr, V. W. Mikuls, P. J. Berger, P. A. Frederickson, R. J. Lind, and W. S. Hodgkiss, “Development of extinction imagers for the determination of atmospheric optical extinction,” , 2014, https://apps.dtic.mil/docs/citations/ADA613006 .

J. E. Shields and M. E. Karr, University of California San Diego, 9500 Gilman Dr. 0701, La Jolla, CA 92093-0701, are preparing a manuscript to be called “Extinction imagers for measurements of atmospheric beam transmittance.”

S. Q. Duntley, R. W. Johnson, and J. I. Gordon, “Airborne measurements of optical atmospheric properties in Northern Germany,” , 1976, https://apps.dtic.mil/docs/citations/ADA035571 .

A. Richards, Senior Research Scientist, FLIR Systems, Inc., 6769 Hollister Ave., Goleta, CA 93117 (personal communication, 2019).

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Figures (13)

Fig. 1.
Fig. 1. Dark signal versus exposure, for two Photometrics CCD imagers, a Series 200 and a Series 300.
Fig. 2.
Fig. 2. Dark signal versus exposure from an Alpha NIR SWIR camera, in low gain and high gain.
Fig. 3.
Fig. 3. Dark-corrected signal as a function of exposure from the exposure calibration, for photometrics series 200 camera, unit 8.
Fig. 4.
Fig. 4. Percent nonlinearity as a function of dark-corrected signal from the exposure calibration data in Fig. 3.
Fig. 5.
Fig. 5. Dark-corrected signal as a function of relative flux from the Lin versus radiance measurements.
Fig. 6.
Fig. 6. Percent nonlinearity as a function of dark-corrected signal from the Lin versus radiance data in Fig. 5.
Fig. 7.
Fig. 7. Dark-corrected signal as a function of exposure from the exposure calibration for the Photometrics Sensys camera.
Fig. 8.
Fig. 8. Percent nonlinearity as a function of dark-corrected signal from the exposure calibration data in Fig. 7.
Fig. 9.
Fig. 9. Dark-corrected signal as a function of exposure for the alpha NIR camera used in the SRI.
Fig. 10.
Fig. 10. Dark-corrected signal as a function of flux for the alpha NIR camera used in the SRI, low gain.
Fig. 11.
Fig. 11. Percent nonlinearity as a function of dark-corrected signal from the linearity versus radiance calibration for the alpha NIR.
Fig. 12.
Fig. 12. Raw SRI image acquired near 1.6 μm, looking South to Los Coronados Islands from Pt. Loma, CA.
Fig. 13.
Fig. 13. Processed image, with dark, linearity, and uniformity corrections.

Tables (2)

Tables Icon

Table 1. Sample Absolute Calibration Constant Measurement and Derivation

Tables Icon

Table 2. Absolute Calibration Measurement Uncertainties by Filter

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

E λ ( d ) = E λ ( d 0 ) * d 0 2 / d 2 .
L λ ( d ) = R λ E λ ( d ) / π ,
NonLin ( Sig ) = Sig * E ( 10 , 000 ) 10 , 000 * E ( Sig ) .
E ( SP , ND ) ¯ = E λ S λ T λ ( SP ) T λ ( ND ) d λ S λ T λ ( SP ) T λ ( ND ) d λ ,
A ( SP , ND , d ) = L ( SP , ND , d ) * 10 , 000 Sig * Exp 100 .
Sig ( x , y ) = Sig ( x , y ) Dark ( x , y ) .
Sig ( x , y ) = f ( Sig ( x , y ) ) .
l ( x , y , SP , ND , Exp ) = A ( SP , ND ) * Sig ( x , y ) 10 , 000 * 100 Exp .
L ( x , y , SP , ND , Exp ) = l ( x , y , SP , ND , Exp ) * U ( SP , ROI ) U ( SP , x , y ) .
Sig ( x , y , SP , ND ) = Sig ( x , y ) * U ( SP , ROI ) U ( SP , x , y ) .

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