## Abstract

A key characteristic of a spectroradiometer is the instrumental spectral response function (ISRF) that is determined during spectral characterization and calibration. The response shape of the ISRF is commonly assumed to be Gaussian, though this is known to not always be the best description. We show that in the context of laboratory calibration, the largest source of uncertainty lies in the ISRF assumption. We perform the spectral calibration of laboratory measurements obtained with four analytical spectral device field spectroradiometers using several different ISRF “modes” to investigate their respective fitting performance, and examine the impact of choosing an ISRF that differs from a Gaussian when calibrating a MODTRAN6 spectrum. Finally, we conduct the uncertainty analysis of our calibration by propagating uncertainty via a Monte Carlo method.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

A spectroradiometer is a light measurement tool that uses a spectral disperser to measure the amplitude at the wavelengths of the light emitted, reflected, and/or transmitted by a target or remote scene [1]. A full spectrum is obtained with a single acquisition, by letting in light during a fixed interval of time known as the integration time, decomposing the incoming signal via a prism or grating and spreading the result across a detector array that discriminates each wavelength and then scales it based on the detector-elements’ sensitivity and the incoming signals’ amplitude. A (single point) spectroradiometer (a type of hyperspectral device, i.e., capable of capturing the spectral content of a target in hundreds or even thousands of narrow, adjacent spectral bands) typically provides a base measurement of counts (the digital numbers, or DNs, in arbitrary units mainly defined by the integration time, incoming light intensity, and quantum efficiency of the detector elements), which must be calibrated in order to retrieve information on spectral radiance and/or spectral flux. The calibration process itself is the comparison of measurement values delivered by the device under test with those of a calibrated standard of known uncertainty, with metrological traceability leading back to the primary measurement standards maintained by a national metrology institute (NMI) [2].

Field spectroradiometers are simply portable spectroradiometers that are commonly used not only for obtaining ground-based data for fundamental research, but also as a means for calibrating and validating airborne- and satellite-based optical sensor systems [3]. In particular, Earth observation (EO) communities regularly make use of such instruments during field campaigns in order to evaluate by independent means the accuracy of satellite-derived EO data and determine their uncertainties by analytical comparison with reference data [4]. Thus, field spectroradiometers are expected to deliver data with high accuracy, hence making their characterization (profiling) and calibration (defining the system response quantitatively) of paramount importance. Ideally, both the EO and reference data can be traced back to a metrological reference through an unbroken chain of calibrations or comparisons, each contributing to the stated measurement uncertainty [5,6]. In the case of spectroradiometers, we speak of traceable spectroradiometric calibration (spectral and radiometric calibration).

Arguably the most important component of reliable spectroscopy is the
instrumental spectral performance [7,8]. Spectral
characterization and calibration are aimed at determining the
wavelength-dependent response of the instrument, which is known as the
instrumental spectral response function (ISRF). In this paper, we focus on
the laboratory spectral calibration and related uncertainty analysis of
four field spectroradiometers by following metrological best-practice per
the Guide to the Expression of Uncertainty in Measurement (GUM) [9] guidelines. In particular, we aim to
assess the performance of different types of spectral response function
parameterizations in terms of fitting the measured DNs obtained from
calibration lamp measurements. Regarding our choice of instruments, we
selected the analytical spectral devices (ASDs) field spectroradiometers,
which feature three spectral detectors to capture spectra in the
350–2500 nm spectral range: these instruments have become a *de facto* standard within the field spectroscopy
community [10], and hence the
results of our study should prove useful to several research
groups—particularly as we characterize and calibrate four of these
instruments, allowing for cross-examination and some assessment of the
consistency and interoperability of ASD sensors. This is all the more
relevant when considering that ASD spectroradiometers have sometimes been
used as a reference to calibrate other instruments: for example, JB
Hyperspectral Devices previously made use of ASD devices for calibrating
its automated fluorescence box (FloX) instruments [11].

While the flagship instruments of major science missions may benefit from more careful analysis of their ISRFs, the common field spectroradiometer user typically relies on the basic Gaussian assumption for the ISRF during spectral calibration (when not simply relying on the manufacturer’s nominal specifications of the instrument and foregoing calibration altogether). We aim to investigate the consequences of the knowledge gaps resulting from this assumption: if there is an ISRF parameterization with a better fitting performance than the standard Gaussian, how much does using the former change the final instrumental spectral performance compared to using the latter? In other words: what is the impact on the retrieved spectra? Generally, this will depend on the type of spectroradiometer; hence, in stressing the impact of choosing an ISRF that differs from a Gaussian, we chose to work with the ASD series of field spectroradiometers that are in widespread use. We further highlight the impact of our finds by calibrating a MODTRAN6 [12] spectrum using different ASD ISRF parameterizations.

We compare the Gaussian ISRF parameterization fitting performance with that of the lognormal (and its reverse), symmetric super-Gaussian (SSG), and asymmetric super-Gaussian (ASG) ISRF parameterizations. The lognormal distribution was chosen for its ability to describe skewed shapes using just two parameters, and for its potential to be very similar to a Gaussian when its variance is small compared with its mean. The super-Gaussian (SG) distributions (both symmetric and asymmetric variants) are direct extensions of the classical Gaussian [13], and have previously been used for parameterizing the ISRFs of differential optical absorption spectroscopy (DOAS) instruments [14]. The SG parameterizations are advantageous in that they can describe a wide variety of shapes while remaining comparatively simple (two parameters for the SSG, three to four parameters for the ASG). It should be said that while the ideal ISRF simulation would describe the range of spectral response shapes with the smallest number of parameters, more tailored parameterizations with many parameters may sometimes be necessary to meet a stringent accuracy requirement, for example, on a mission-specific instrument’s ISRF (such as the TROPOMI-SWIR ISRF which was modeled by the weighted sum of functions for the peak and the tails, using up to seven parameters [15]).

While our focus is on the ISRF parameterization, we also detail our spectral calibration protocol, since there are currently no comprehensive and universally accepted protocols for the calibration of field spectrometers [16]. Commercial confidentiality hampers the release of the calibration procedures used by spectrometer manufacturers, leading to individual research groups developing their own protocols.

Finally, consistent with the aforementioned GUM guidelines, we perform the uncertainty analysis of the spectrally calibrated data, taking into account both the noise associated with the measured DNs around selected emission lines and the uncertainty on the center wavelength (CW) positions of said emission lines, and propagating the uncertainties to the interpolated bands using a Monte Carlo method. Our hypothesis is that within the context of laboratory calibration, where environmental conditions can be controlled to a fine degree, the largest source of uncertainty lies in the ISRF assumption.

## 2. METHODS

#### A. Spectroradiometer Calibration Overview

For airborne or spaceborne instruments, there is laboratory calibration prior to launch, in-flight/in-orbit calibration, and vicarious calibration [17]. Since we focus on portable spectroradiometers for field measurements, we only describe laboratory calibration.

Laboratory spectroradiometric calibration typically consists of two separate parts: spectral calibration and radiometric calibration. Radiometric calibration determines the radiance response of each band by resolving the conversion coefficients between the DN output and the (nominally) uniform-radiance field at the instrument’s entrance pupil, which is commonly assumed to be a linear sensor system [18]. However, the CW and spectral response shape of each band need to be ascertained first, and hence spectral calibration precedes radiometric calibration [19]. Any change in the instrumental spectral performance directly affects the accuracy of the radiometric calibration [8]. The actual calibration measurements can be done in any order, provided that they are carried out in close succession and under the same environmental conditions, and that the instruments are not subjected to rough transportation in the intervening time, in order to reasonably assume that the devices remain in the same “state.”

Spectral calibration defines the spectral sensitivity of each pixel of the instrument. For nonimaging spectroscopy, the ISRF (also known as the instrument line shape, or slit function) is defined by a CW position (peak response wavelength) and a response shape [20]. For both nonimaging and imaging spectrometers, ISRFs are commonly approximated by Gaussian shapes [18,20–22], with the associated full width at half-maximum (FWHM) being used to express the spectral resolution. In addition, the spectral sampling interval (SSI) refers to the spectral distance between the CWs of adjacent spectral bands or pixels, respectively [23].

#### B. Laboratory Measurements

The calibration measurements were carried out at the German Aerospace Centre (Deutsches Zentrum für Luft- und Raumfahrt; DLR) Calibration Home Base (CHB [24]) laboratory in Oberpfaffenhofen, Germany, which is routinely used to calibrate the APEX airborne imaging spectroradiometer [25].

The spectral calibration data are obtained by making measurements of light from a gas discharge lamp producing emission spectral lines [see Fig. 1(a)]. Since the positions of those lines are well known, it becomes possible to link band numbers to wavelengths (in the first approximation, the relation is a linear one). Hence, such a lamp producing narrow, intense lines from the excitation of a gas or metal vapor is known as a spectral calibration lamp, or emission line lamp. The spectrum of the lamp is convolved with the instrument response, and that product should ideally match the spectrum that is actually being measured. We rely on a spectra database to identify the position of prominent emission lines captured by the spectroradiometer (and verify that they do, in fact, constitute single spectral lines instead of an unresolved combination of emission peaks). The most comprehensive database freely available is the National Institute of Standards and Technology (NIST) online Atomic Spectra Database [26]: it provides SI-traceable CW values with associated uncertainties for the spectral lines of most of the known chemical elements. The NIST database includes information on the relative intensities of the spectral lines, which helps to isolate the most important single peaks (relative intensities only help when comparing strengths of spectral lines that are not separated widely, as there is no common scale for relative intensities [27]).

The three calibration lamps we used are mercury (Hg), neon (Ne), and xenon (Xe) sources. By using three different line lamps, we increase the number of valid candidate (i.e., particularly prominent, or “clean”) spectral lines to select from [see Fig. 1(b)] in order to examine the instrument line shape over a greater number of calibration bands, hence increasing the reliability of ISRF interpolation across all the remaining bands. Table 1 details the CWs and associated uncertainties of the emission peaks hand-picked for spectral calibration.

We calibrated two *ASD FieldSpec 3*
spectroradiometers [28] and two
*ASD FieldSpec 4 Standard-Res*
spectroradiometers [29]:

- • all four instruments feature three channels (detectors), which henceforth shall be referred to as the
*VNIR*(350–1000 nm), the*SWIR1*(1000–1800 nm) and the*SWIR2*(1800–2500 nm) channels; - • per product specification, the instruments have the same (channel-dependent) spectral resolution (the
*FieldSpec 4*series of instruments also comes with higher resolution variants, which we did not use): nominally, the FWHM is 3 nm in the*VNIR*channel, and 10 nm in both*SWIR*channels; - • the
*ASD FieldSpec 3*instruments are referred to by their serial number:*16006*and*16007*; - • likewise, the
*ASD FieldSpec 4 Standard-Res*instruments are:*18130*and*18140*.

We note here that while the ASD devices ostensibly record spectra based
on the information of 2151 bands, the latter actually correspond to
the number of wavelengths after being interpolated to 1 nm spacing.
The detector bands refer to either physical detectors in the case of
the *VNIR* channel (512 detectors), or to
the sampling points in the case of the *SWIR* channels (515 bands for each). The bands are converted
to wavelengths by the calibration polynomials and then interpolated
back down to 1 nm spacing by the ASD interpolation algorithm. Since
the calibration first-order polynomial is larger than 1 by a
sufficiently large amount, there are fewer real bands than what is
displayed as interpolated wavelengths, even when the truncation of the
splice positions is taken into account.

Once we have the measurement data, we can proceed with data processing—the actual calibration.

#### C. ISRF Characterization

The optical properties of the instrument (entry and exit slits, the spectral disperser, the detector characteristics, etc.) define the ISRF, since the latter is the convolution of the slit image, the detector spatial response in the spectral direction, and the optical point spread function (PSF). In particular, diffraction and optical aberrations are the two primary sources of disturbances for the ISRF [21].

While this function is usually too complex to be accurately reproduced by a physical model [14], it can be measured accurately in the laboratory using emission line lamps or light that has passed a monochromator. The measured spectral shapes can be retrieved as vectors in order to skip parameterization (and storing vector representations could be of interest in an age of spectral databases like SPECCHIO [30]); however, unless there is an opportunity to use a scanning monochromator (capable of covering the entire wavelength range of the instrument and hence allowing the retrieval of a measured spectral shape vector for every band), the measured shapes at specific bands will have to be interpolated across all the remaining bands in order to obtain the complete wavelength-dependent response of the instrument, and the lack of ISRF parameterization would make this task difficult. One must also state here that many users do not have access to (or the time to use) a monochromator to measure ISRFs. Hence, the most common, “poor man’s” approach is to use emission line lamps.

Thus, for this study, we focus on this common practice and endeavor to highlight the effect of the most basic of assumptions: indeed, the most routine parameterization of the slit function is a standard Gaussian [14,20]. This simple parameterization, which describes the measured line shapes by only one free parameter, $\sigma$, often works well enough: for a traditional spectrometer, where the ISRF focuses solely on the dispersion direction, the slit image can be determined by using the tangential line spread function (${{\rm{LSF}}_{\rm{T}}}$) [21]; when ${{\rm{LSF}}_{\rm{T}}}$ is of comparable width to that of the spectrometer’s exit slit, then the ISRF resembles a Gaussian with a bandwidth larger than said width. In that case, the CW and FWHM are enough to describe the spectral response characteristics accurately. But due to the aforementioned optical effects and variations in instrument design, this Gaussian assumption does not always work (e.g., the ozone monitoring instrument (OMI) ISRF was parameterized by the sum of a Gaussian and a “broadened Gaussian” with an exponent of 4 instead of 2 [31], which allows it to reflect different shapes, including skewed patterns). Another instrument, the fluorescence imaging spectrometer (FLORIS) for the upcoming ESA European Space Agency (ESA) FLuorescence EXplorer (FLEX) mission, has an ISRF that was found to be best parameterized by a convolution between two rectangular functions [32].

By spectrally calibrating four ASD spectroradiometers across the full 350–2500 nm spectral range, we can verify not only what ISRF parameterization best describes the measured spectral shapes on a per-channel basis, but also how consistent this parameterization is between different ASD units.

The first step involves the use of different ISRF parameterizations for fitting the measured data (through a least-squares minimization process). The root-mean-square error (RMSE, calculated as the square root of the sum of the squared difference between the measured instrument response and the simulated instrument response for each fitted emission peak) results then tell us which parameterization best fits the measured shapes. Thereafter, we interpolate the parameterizations to the remaining (noncalibration) bands.

In order to simulate the instrumental spectral response to a narrowband optical signal, we generated high-resolution spectral emission lines and convolved them with simulated ASD sensors using MATLAB [33]. The spectral convolution itself can be described as a weighted average, where the weight is defined by the ISRF of the sensor that is simulated [34].

The base sensor model’s spectral range and SSI are determined by the nominal ASD specifications (2151 bands as interpolated wavelengths with 1 nm spacing), and the choice of ISRF parameterization defines the sensor type (since the ISRF provides the coefficients used for the convolution operation). For the synthesized spectral lines, we created a high-resolution linearly spaced vector (with a step size of 0.01 nm, i.e., of much higher resolution than the ASD sensor) for the full 350–2500 nm spectral range, and inserted very narrow boxcar signals of arbitrary amplitude with the same CWs as the chosen calibration emission lines. In order to match the ISRF coefficients to the data points of the high-resolution vector, the ISRFs of the ASD sensors are resampled to the wavebands of said vector. The synthesized spectral lines are then convolved with the simulated sensors (hence downsampling the high-resolution wavelength bands to the spectral bands of the ASD sensors), with the result being normalized convolved peaks for each ISRF parameterization (or ISRF mode). Each band is convolved separately, and the band convolution can be expressed as follows:

Subsequently, the mode-specific normalized convolved peaks are used to fit the corresponding selected peaks of the mean spectra measured by each ASD spectroradiometer (following a simple amplitude-scaling operation).

Figure 2 displays the workflow for the ISRF characterisation phase of the spectral calibration and contains some visual examples of the sensor ISRF types used in this study.

#### D. Simulated ISRF Types

The sensor ISRF types that we simulated include the baseline Gaussian, the lognormal (and its reverse), the SSG and the ASG functions.

The normalized Gaussian distribution of a random variable $X$ can be described by the following equation [36]:

The (normalized) lognormal distribution of a random variable $X$ can be expressed using the standard
deviation of the transformed data parameter (*ln
standard deviation*, also the location parameter) $\sigma ^\prime
$ [37],

*fliplr*, which flips arrays about a vertical axis).

As for the SSG, we use the following formulation [14]:

where $A(w,s)$ is the amplitude used for normalizing the SSG to an integral of 1 for application as an ISRF (with the finite integrals needed for normalization calculated numerically), with the two independent parameters $w$ and $s$ determining the width and shape of the SSG, respectively. Similar to the Gaussian case, the normalizing constant is proportional to the inverse width and depends on the shape parameter with a maximum for $s = 2$, when the SSG is equal to the Gaussian and ${w_{s = 2}} = \sqrt 2 \sigma$. The SSG becomes*leptokurtic*[38], i.e., more peaked than the Gaussian (with long tails on both sides), when $s \lt 2$. For $s \gt 2$, the SSG acquires a “flattopped” shape, converging to a boxcar shape of width $2w$ for $s \to \infty$. In the generalized super-Gaussian (SG) case, we note that the FWHM is dependent on both the width and shape parameters, which is consistent with the definition of the FWHM in the Gaussian case.

The SG can be extended to an asymmetric function [14],

#### E. Unit Test and MODTRAN Spectrum Calibration

In order to verify that our spectral calibration algorithm works as intended, a simple “unit test” was performed: having modeled ASD sensors with known response shapes, we checked that the code can retrieve the same shapes from the sensors’ synthesized DNs. The test follows a process similar to the one described for the ISRF characterization: a set of virtual sensors is created using the ASD nominal spectral range and SSI, with each sensor using one of the ISRF modes; DNs are generated from the convolution of the ISRFs with simulated emission lines (the same as those used for the calibration of measured data); finally, the virtual DNs are calibrated just like the measured DNs.

We then calibrated a MODTRAN-simulated spectrum by convolving it with
the following sensor models: ASD nominal (1 nm SSI, FWHM values of
3 nm in the *VNIR* channel, and 10 nm in
both *SWIR* channels); best-fitting
Gaussian and SSG ISRF parameterizations (as determined by the RMSE
minimization process during calibration of ASD instruments *16006*, *16007*,
*18130*, and *18140*) interpolated across the noncalibration bands. We
ignored the lognormal and ASG ISRF models after the calibration
results (see Fig. 3) showed
them to be underperforming and overfitting, respectively (Fig. 4 shows the random distribution of
asymmetric parameter values).

The uncertainty of the various ISRF parameters (explained further) is also propagated to the MODTRAN spectrum through the convolution, leading to a series of convolved MODTRAN spectra for each ISRF mode, from which we calculated a mean spectrum per mode and derived an associated uncertainty “envelope” for plotting purposes (see Fig. 8).

#### F. Uncertainty Propagation

We take into account two sources of uncertainty associated with the measured emission line peaks selected for calibration:

- • the system noise in the DN domain, which per the central limit theorem (CLT) is assumed to be normally distributed;
- • the uncertainty on the CW values as provided by NIST’s Atomic Spectra Database (see Table 1).

In practice, this is achieved by using a Monte Carlo simulation that
generates draws from a probability distribution in order to simulate
DNs and CW fluctuations. For each run of the Monte Carlo simulation,
we generate DNs using the instrument noise and emission line CW
fluctuation values using the known uncertainty of the emission line
positions in order to calculate new CWs. We get the system noise from
the estimated standard deviation of the DNs obtained from 30 spectra
(with each spectrum being the average of 30 readings), ${\sigma
_{{\rm{DN}}}}$, on a per-calibration band basis. The
estimated standard deviation (the “true” value for the standard
deviation can only be found from an infinite set of readings) is also
known as the *standard uncertainty*. When
propagating the uncertainty due to noise, the simulation uses ${\sigma
_{{\rm{DN}}}}$ for its Gaussian distribution used to
generate (pseudo-)random DNs. The preceding ISRF characterization
phase informs the fitting of the Monte Carlo realizations using the
ISRF mode-specific best-fitting parameters, such that we fit the Monte
Carlo realizations with parameter values chosen in a small interval
around the mean of the best-fitting parameter values, keeping the
minimization process fast.

We produced 30 Monte Carlo realizations per calibration band and per best-fitting ISRF parameterization, which once fitted result in 30 different best-fitting RMSE values per calibration band and per ISRF mode. Thus, the uncertainties of the best-fitting parameters for each ISRF mode (as determined during the ISRF characterization phase) are the standard deviation of the associated parameter values derived from all Monte Carlo realizations. These ISRF uncertainties are then propagated through the interpolation process across the remaining ASD bands to estimate ISRF uncertainties for all spectral bands (thus achieving full instrument characterization).

The standard uncertainties may be thought of as equivalent to “1 standard deviation,” for a confidence level of 68.27%, where the coverage factor is $k = 1$ in the case of a Gaussian distribution. Due to our standard uncertainties being very small (see Figs. 3, 5–7), for plotting purposes, we rescaled the combined standard uncertainties with a coverage factor of $k = 2$ for a confidence level of 95.45%.

## 3. RESULTS

The best-fitting results from the calibration of measured peaks per ISRF
mode and for all four ASD instruments are provided in Fig. 3. The smallest RMSE values are almost
always achieved with the ASG by a significant margin regardless of the
instrument or channel, with the SSG being a general second-best. The
Gaussian RMSE fitting results follow a pattern similar to that of their
SSG counterpart and have similar RMSE values. The lognormal ISRFs appear
to be the worst performing in the *SWIR*
channels and can otherwise compete intermittently with the other ISRFs in
the *VNIR* channel.

Since the ASG parameterization of the ASD ISRF appears to best describe the measured spectral shapes, it is worth looking at the distribution of values of the associated asymmetric parameters, ${a_w}$ and ${a_s}$, in order to highlight any pattern in the ISRF profiles. Figure 4 shows the asymmetric parameters from each detector region plotted versus each other and versus wavelength, but no clear pattern emerges, a result that will be discussed further.

Figures 5 and 6 show the results of the width parameter fitting to the
measured spectral lines for the Gaussian and SSG ISRF parameterizations,
respectively, along with the interpolation of the width across
noncalibration bands for all three detector channels. For both the
Gaussian and SSG ISRF cases, we note the expected behavior of the width
parameter in all wavelength regions: for the *VNIR* channel, the concave formed by the width when plotted
against wavelength is due to a geometrical effect arising from the
dispersion of the incoming radiance onto a linear array (the distance to
the imaging plane varies from band to band). The *SWIR* channels make use of a swiveling device for scanning, hence
correcting the aforementioned effect. However, whereas the bandwidth
remains broadly stable for the *FieldSpec 4*
series of ASD instruments in the *SWIR1*
channel, we find that the width for the *FieldSpec
3* series of devices is sloped. Given the small number of data
points in the *SWIR2* channel, we choose to
retain the average width only, since the first-order fit does not reveal
any consistent trend.

The best-fitting SSG ISRF shows a clear leptokurtic tendency ($s \lt 2$) in the *SWIR* channels (see Fig. 7). For the *VNIR* channel, with the
exception of instrument *16006*, the shape
parameter starts off at $s \gt 2$ in the shorter wavelengths (top-hat
shape), and decreases to $s \lt 2$ in the longer wavelengths (Fig. 7). The shape parameter results for the
ASD *16006* may have been skewed by the fact
that its *VNIR* channel was performing
suboptimally (detected radiance levels were much lower than with the other
instruments, which normally would only impact radiometric calibration, but
the measured spectral shapes may also be more influenced by the instrument
noise).

Figure 8 shows the result of the calibration of a MODTRAN spectrum: when the simulated radiance is convolved with the ISRF of the ASD nominal sensor model, the finer spectral absorption features can mostly be retrieved; however, convolving the spectrum using the best-fitting Gaussian and SSG ISRF parameterizations (determined through spectral calibration of the ASD-measured line spectra) results in significantly smoothened spectral features. The resolution is at its worst with the SSG ISRF model, which best describes the ASD-measured spectral shapes (once the overfitting ASG ISRF model is discarded).

Figure 9 shows some selected examples of mean ISRF types derived from the interpolated ISRFs for simulated instruments, along with the associated uncertainty buffers.

## 4. DISCUSSION

With the results of the spectral calibration of four ASD spectroradiometers using different ISRF modes, we may not only find a hypothetical overall “best” ISRF parameterization for these instruments, but also by considering several possible types of “best” parameterizations, we can make our conclusions more robust to epistemic error, since the results of only one “best” parameterization may be overconstrained by the choice of this single parameterization, even if the ISRF parameters include estimates of uncertainty. This is due to the choice of one “best” parameterization depending on a particular realization of the epistemic error in the calibration data set.

While the ASG ISRF mode achieves the best-fitting performance (see
Fig. 3), the absence of clear
asymmetric pattern resulting from the distribution of asymmetric
parameters ${a_w}$ and ${a_s}$ (see Fig. 4) leads us to conclude that the ASG parameterization is
overfitting: thanks to a greater degree of freedom than the other
functions, the ASG can fit the “wobbliness” of the peaks, highlighting
fluctuations due to a stochastic effect, like noise, rather than a
systemic property of the ASD sensors. The unit test results seem
reasonable in that they indicate that our fitting algorithm can, in fact,
retrieve the shape of an ISRF, and hence this method may indicate that the
small instabilities leading to slightly asymmetric peaks are due to the
internal resampling of the ASD instruments to a 1 nm SSI. Comparing the
values of the shape parameter $s$ obtained by both the ASG and SSG in the
*VNIR* and *SWIR*
channels in order to check whether the ASG replicates, to a first
approximation, the broader symmetric shapes obtained with the SSG,
confirmed that the spectral response shape in the *SWIR* channels is leptokurtic, whereas in the *VNIR* channel, the shape varies between flattopped in
the shorter wavelengths and leptokurtic in the longer wavelengths (except
for instrument *16006*, which has a *VNIR* detector with an anomalously low SNR).

For all four ASD instruments, the estimated bandwidth values obtained with
both the Gaussian and SSG ISRF modes are much greater (usually by 20%–40%)
than the nominal ASD FWHMs of 3 nm in the *VNIR* channel (except around the middle region of the detector)
and 10 nm in both *SWIR* channels. This places
stronger limitations on the retrieval of fine spectral absorption features
with an ASD device than would be expected based on the product
specifications, as highlighted by the calibration of a MODTRAN6 simulated
at 3 km altitude spectrum (see Fig. 8). Compared with laboratory calibration, where bandwidth effects
are minimized due to the continuous emission spectra that approximate an
ideal Planck source, we can say that in-field radiometric calibration
would be far less accurate, as the incoming spectrum would have a wide
range of spectral absorption lines (including the Fraunhofer lines
resulting from gas in the Sun’s photosphere).

We did not set out to compare the differences between our (NIST-traceable) CWs and the CWs provided by the ASD spectroradiometers, as the instruments have not been recently calibrated in-factory. Hence, an assessment of our calibrated CWs versus the ASD-provided CWs would not be relevant beyond showing some potential degradation of the instruments over time.

We cannot yet quantify how much better our CW retrieval is compared to the standard (i.e., common) spectral calibration approach, which does not rely on an iterative method through the minimization process (hence foregoing the band convolution simulation). In this sense, the instrument models we developed constitute feasible hypotheses for fitting measured hyperspectral data. As a cross-check of our fitting algorithm, a further step would be to use a monochromator or tunable laser to retrieve directly the response shapes and compare with our results. This would also be the only way to know an ISRF with a quantified accuracy across the full 350–2500 nm spectral range.

Finally, having determined that the uncertainties related to system noise and CWs lead to minimal fluctuations in normal laboratory conditions, it would be of interest to carry out a similar analysis with spectroradiometers under environmental stress.

## 5. CONCLUSION

We performed an SI-traceable laboratory spectral calibration of four ASD
spectroradiometers using emission line lamps and different assumptions for
the ISRF besides the standard Gaussian, and the results of the fitting of
the measured spectral shapes through a least-squares minimization process
clearly show that a leptokurtic SSG is consistently better at describing
the response shape of the instruments’ ISRF than the standard Gaussian in
the *SWIR* channels. In the *VNIR* channel, the SSG remains the better-fitting
ISRF parameterization, but its shape varies between flattopped in the
shorter wavelengths and leptokurtic in the longer wavelengths. The ASG
ISRF model technically leads to the smallest RMSE fitting residuals;
however, that is a result of overfitting and not a reliable description of
the ASD ISRFs. The calibrated bandwidth values obtained with both the
Gaussian and SSG ISRF parameterizations are much greater than the nominal
ASD bandwidth values, usually by 20%–40%. Moreover, the propagation of the
uncertainties related to system noise and emission line CWs lead to
fluctuations so small as to be insignificant, whereas the choice of ISRF
significantly impacts the instrument model. The calibration of a MODTRAN6
spectrum illustrates small but noticeable differences between
ASD-convolved radiances using nominal, Gaussian, and SSG ISRF models:
notably, the best-fitting SSG ISRF model leads to the lowest resolution
and worst retrieval of spectral absorption features.

The results suggest that the use of ASD field spectroradiometers to radiometrically calibrate instruments with different ISRFs under field conditions can lead to significant errors.

The next step would be to quantify how accurate our knowledge of the ASD ISRFs is by comparing our results with spectral response shape measurements made across the full 350–2500 nm spectral range with the help of a scanning monochromator or tunable laser. Furthermore, processing the data to level 2 products in order to examine how sensitive reflectance values are to changes in ISRFs would further highlight the importance of careful ISRF characterization.

As a further outlook, one could compare a measured bottom-of-atmosphere solar irradiance spectrum with MODTRAN-simulated spectra convolved with the ISRF types introduced in this work. This could further prove that the best ISRF model determined during laboratory calibration also applies in field conditions.

Finally, we provide access to the source code of the spectral calibration and uncertainty propagation processes for ASD data via GitHub [39].

## Funding

European Association of National Metrology Institutes (EURAMET) (16ENV03 MetEOC-3, 19ENV07 MetEOC-4).

## Acknowledgment

This work was supported by MetEOC-3 and MetEOC-4. The projects 16ENV03 MetEOC-3 and 19ENV07 MetEOC-4 have received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. We thank Andreas Baumgartner for assistance in performing the spectroradiometric calibration measurements at DLR’s CHB laboratory in Oberpfaffenhofen. We thank Prof. Dr. Michael E. Schaepman for providing advice throughout the duration of this work. We also thank Prof. Dr. Meredith C. Schuman for useful discussions and comments on the manuscript.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

The spectral calibration measurement data underlying the results presented in this paper are available in Ref. [40].

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