This is an open access article under the terms of the

The intercalibration between a broadband and a hyperspectral satellite Earth observation system requires the convolution of the hyperspectral data with the spectral response functions (SRFs) of the corresponding broadband channels. There are two potential issues associated with the convolution procedure. First, the finite resolution of a hyperspectral spectrum, that is, the deviation from the highly accurate *line‐by‐line* monochromatic radiances, will contribute to convolution errors. The magnitude of the errors depends on the spectral resolution and the SRF shape of the hyperspectral instrument. This type of the convolution error has not been well recognized, and there is a lack of corresponding discussion in most published papers. Although it is small as compared with the instrument accuracy of existing hyperspectral sounders, the error is deemed to be significant when it is compared with the stringent calibration requirement imposed by future climate missions like the Climate Absolute Radiance and Refractivity Observatory. Second, some broadband channels are insufficiently covered by the hyperspectral data, causing spectral gaps that lead to convolution errors. Although several methods have been developed to fill the spectral gaps and hence compensate for the second type of convolution error, the correction accuracy may still need improvement especially when a large spectral gap needs to be filled. This paper presents a methodology to accurately quantify and compensate for both types of convolution errors. This methodology utilizes the available hyperspectral information to correct the scene‐dependent convolution errors due to either the limited spectral resolution or spectral gaps. We use simulations to characterize the intercalibration errors between the Moderate resolution Imaging Spectroradiometer (MODIS) and current operational infrared sounders. We demonstrate that convolution errors can be effectively removed to meet the highly accurate intersatellite calibration requirement proposed by the Climate Absolute Radiance and Refractivity Observatory. Our methodology is also validated using real satellite data for the intercalibration between Aqua MODIS and Aqua Atmospheric Infrared Sounders (AIRS). Our study demonstrates that the accurate characterization and correction for the convolution errors greatly reduces the scene‐dependent and spectrally dependent errors, being critical to the consistency check between Infrared Atmospheric Sounding Interferometer (IASI) and AIRS using the double‐difference method. The convolution correction also facilitates the evaluation for other intercalibration errors (e.g., the drift of MODIS SRFs). Our derived SRF shift values from MODIS‐AIRS (after convolution error corrections) and from MODIS‐IASI intercalibration are consistent with each other. We further extend the methodology to study the calibration of a broadband channel which is either completely or largely uncovered by a hyperspectral measurement. The large spectral gap filling methodology is validated by demonstrating the accurate prediction of the MODIS radiance of band 29 using the Cross‐track Infrared Sounder spectra, with the real IASI spectral data being used as the reference.

Intercalibration of different satellite sensors using a high accuracy hyperspectral data is critical to ensure the accuracy and reliability in using satellite observations for weather, climate, and environmental applications. Hyperspectral sensors have much smaller spectral response uncertainties than broadband sensors. Using hyperspectral sounders to calibrate broadband infrared (IR) systems is an important part of the international intercalibration efforts that include the intercalibration between the geostationary satellites and low Earth‐orbiting and the intercalibration between low Earth‐orbiting sensors. In order to maximize the advantage provided by the hyperspectral instruments, intercalibration errors in principle should be minimized to be, ideally, smaller than the radiometric calibration accuracy of the hyperspectral sensors. There are currently several hyperspectral IR sounders in orbit: the Infrared Atmospheric Sounding Interferometer (IASI‐A and IASI‐B) in a Sun‐synchronous midmorning orbit, Atmospheric Infrared Sounders (AIRS), and Cross‐track Infrared Sounder (CrIS) in an afternoon orbit. The IASI‐A, IASI‐B, and AIRS have been used as the hyperspectral intercalibration references in the Global Space‐based Inter‐Calibration System (Chander et al., ; Goldberg et al., ; Hewison et al., ). CrIS has also been accepted as one of the Global Space‐based Inter‐Calibration System references recently. Most recently, Hyperspectral Infrared Atmospheric Sounder that is carried by the FY‐3D satellite has joined IASI and CrIS as the third Fourier transform spectrometer that can provide global scale atmospheric sounding. A lot of studies have been carried out in the past to characterize and/or correct the intercalibration errors between those hyperspectral sensors and other broadband sensors (Cao et al., ; Gunshor et al., ; Tobin et al., ; Wang & Cao, ; L. Wang et al., , , , ; X. Wu & Yu, ; Yu & Wu, ). Although methodologies used in those studies have provided means to characterize and/or reduce the intercalibration errors, those studies are not aiming at achieving an intercalibration with *climate* accuracy. In the future, hyperspectral sensors like the Climate Absolute Radiance and Refractivity Observatory (CLARREO) with radiometric traceability to the International System of Units will provide on‐orbit calibration for the existing Earth observation missions so that the measurement accuracy suitable for climate studies can be achieved (Wielicki et al., ). The spectral calibration accuracy of CLARREO is designed to calibrate other Earth observation missions with the desired *climate observation accuracy* (Liu et al., ). The ultrahigh calibration accuracy of CLARREO will impose much more stringent intercalibration accuracy requirement that may not be satisfied by the existing intercalibration schemes.

Intercalibration between hyperspectral and broadband sensors requires the use of synchronized, collocated, and viewing‐geometry matched measurements. Any mismatch in those observation factors will introduce errors. In addition to spatial‐temporal matching errors and calibration errors of both broadband and hyperspectral instruments, there is another potential error due to finite instrument spectral resolution and spectral gaps in the hyperspectral instrument. We call this error source as *spectral convolution error*. The intercalibration requires the transformation of hyperspectral measurements to *hyperspectral convolved* measurements with the same spectral response feature of the target broadband sensor. The transformation is typically done by convolving the spectral response function (SRFs) of the broadband sensor with the hyperspectral measurements. In the spectral domain,*k*th broadband channel centered at wavenumber *υ*_{k}; *R*_{h}(*υ*) represents the hyperspectral channel radiance at wavenumber *υ*; and *S*_{b} is the SRF of the *k*th broadband channel with the band edges being defined by *k*th broadband channel is*R*_{m} is the monochromatic radiance. The difference between the hyperspectral convolved broadband radiance,

The causes of the convolution error, *R*_{h}, which has a finite spectral resolution, and the *line‐by‐line* monochromatic radiance, *R*_{m}; the missing information due to the existence of spectral gaps introduces errors in

The convolution error caused by spectral gaps in hyperspectral measurements has been widely recognized and studied. The spectral gaps are spectral regions within a broadband channel that are not covered by the hyperspectral measurement. For example, Figure demonstrates the spectral gaps in AIRS observations that will affect the Moderate resolution Imaging Spectroradiometer (MODIS)‐AIRS intercalibration. AIRS is a hyperspectral grating spectrometer with 2,378 spectral channels covering the 650–1,136, 1,217–1,614, and 2,181–2,665 cm^{−1} spectral regions (Aumann et al., ). MODIS is a high–spatial resolution, multispectral scanning radiometer with 36 spectral bands. Sixteen of the MODIS bands have their central wavelengths in the IR spectral region (695–2,732 cm^{−1} Barnes et al., ). The hyperspectral channels of AIRS are not evenly distributed within the measurement bands. The spectral gaps illustrated in Figure cover all regions where the interval between two adjacent hyperspectral channels is greater than 5 cm^{−1}. Figure also demonstrates the spectral gaps of CrIS, a Fourier transform spectrometer that provides atmospheric sounding with 1,305 spectral channels, over three wavelength regions: 650–1,095, 1,210–1,750, and 2,155–2,550 cm^{−1} (Han et al., ; Lee et al., ).

To address the convolution errors introduced by spectral gaps, one can either fill the missing spectral region before the convolution transformation or apply corrections on the convolved radiances. Tobin et al. () simulated the clear‐sky radiances measured by AIRS and MODIS using six standard atmospheres as inputs. They calculated the convolution errors based on those simulation results, and the simulated bias values were used to correct the intercalibration of real measurements. This method successfully reduced some systematic errors but did not fully account for the scene‐dependent uncertainties due to inadequate representation of the real atmospheric variations using the six standard atmospheres. Gunshor et al. () used the U.S. Standard Atmosphere to simulate the missing spectra of AIRS and tried to compensate for the scene‐dependent uncertainties according to the end points values of the spectral gaps; however, they pointed out that their method may not accurately address the variations of real atmosphere from the U.S. Standard Atmosphere, did not include cloudy spectra, and could fail when the end points of a spectral gap fall on an atmospheric absorption line. Tahara and Kato, () derived a linear regression relationship between the observed channel radiances of the IASI or AIRS and the simulated radiances for eight atmospheric model profiles. Those eight atmospheric profiles included representative atmospheric profiles in various altitude regions under both clear and cloudy sky conditions. Their method improved the spectral gap filling accuracy by effectively addressing the scene‐dependent feature of the convolution errors. It can fill in missing IASI and AIRS radiance channels for narrow spectral gaps. However, the usage of only eight sample spectra do not provide adequate account for the large scene variability and the gap‐filling error gets large when the spectral gap becomes large.

In comparison to errors due to the spectral gaps, the convolution error caused by the finite spectral resolution of the hyperspectral measurements is less recognized. Theoretically, the radiance measured by a hyperspectral channel is*S*_{h} is the SRF of the *l*th hyperspectral channel that centers at wavenumber *υ*_{l} and spans from

If the spectral resolution of a hyperspectral measurement is high compared to the Doppler and Lorentz half widths, then the hyperspectral channel radiance becomes equivalent to the monochromatic radiance so that there is no convolution error due to the finite spectral resolution. In reality, the fundamental difference between *R*_{h} and *R*_{m} can be understood from two perspectives:

There has been a lack of systematic characterization and corresponding correction for such convolution errors in the intercalibration studies. However, we demonstrate in section 3 that such errors need to be corrected for a high‐accuracy intercalibration, like that proposed for the CLARREO mission. CLARREO is designed to serve as a high‐accuracy, International System of Units‐traceable calibration reference on orbit for other EO systems. CLARREO's radiometric calibration accuracy for the IR instrument will be in the range of 0.04–0.06 K (95% confidence interval) (Liu et al., ; Wielicki et al., ). This requirement is based on the climate trend detection uncertainty analysis. Due to the natural variabilities of climate variables such as atmospheric temperature and water vapor, it will take a certain length of time to detect an accurate trend even if there is no error in the observation system. The CLARREO instrument is designed so that the instrument calibration error will only add a minimum delay in the trend detection relative to a perfect observation system (Liu et al., ).

A correction algorithm that can be used to correct scene‐dependent convolution errors is introduced in this paper. The algorithm uses a pretrained regression relationship and the available hyperspectral information to predict the convolution errors or the missing spectral information. Section 3 shows that convolution errors can be estimated and corrected with a very high accuracy using this algorithm. The convolution errors for the intercalibration between MODIS and three operational sounders, that is, IASI, AIRS, and CrIS, are characterized using simulated results. We demonstrate that a high‐accuracy correction for convolution errors due to either the small spectral gaps or the finite spectral resolution of the hyperspectral measurements can be achieved using the correction algorithm. To highlight the accuracy of our algorithm for the convolution error correction, we compare the residual errors after the correction with the calibration requirement defined by the CLARREO mission (Liu et al., ). We evaluate our algorithm by checking if the convolution errors larger than the CLARREO calibration requirement can be effectively corrected. We also illustrate the successful correction for convolution errors in the MODIS‐AIRS intercalibration using the real data. We show in section 4 that the MODIS‐AIRS (both on Aqua) intercalibration errors after the correction for convolution errors become consistent with the corresponding MODIS (Aqua)‐IASI (Metop‐A) intercalibration errors. The accurate characterization and correction for convolution errors avoid the overestimation or underestimation for the real instrument calibration errors and facilitate the investigation about potential contributors to MODIS calibration bias (e.g., potential spectral shifts in MODIS SRFs).

When a large portion of a hyperspectrum needed for the intercalibration of a broadband channel are missing, for example, the intercalibration between MODIS band 29 radiances and AIRS (or CrIS) spectra, the hyperspectral convolved broadband radiance,^{−1} region within the MODIS band 29 using the CrIS measurements that do not provide the spectral coverage.

Section 2 presents details about the regression‐correction methodology. Section 3 systematically characterizes the convolution errors between MODIS and the hyperspectral IR sounders and demonstrates the correction for those large errors according to the CLARREO standard. Section 4 presents the application of the algorithm for the intercalibration between real satellite measurements of AIRS and MODIS. The spectral gap filling study for the intercalibration of MODIS band 29 using CrIS spectra is discussed in section 5.

As is shown by equations – in section 1, the convolution errors can be viewed as a function of the true monochromatic radiance, *R*_{m}, and the hyperspectral radiance, *R*_{h}. There is a strong correlation relationship between the monochromatic radiance and the hyperspectral radiance. Therefore, a linear regression relationship between the convolution errors and the hyperspectral channel radiance can be derived. The regression relationship is established by using simulated broadband and hyperspectral radiances. Monochromatic spectra, *R*_{m}, are calculated using the line‐by‐line code LBLRTM (Clough & Iacono, ). A total of 14,600 spectra are simulated using realistic geophysical properties under both clear sky and cloudy sky conditions. Half of the total spectra are daytime spectra which include the contribution from solar scattering, and the remaining half are nighttime spectra. The global distributed atmospheric temperature, water vapor, and other trace gas profiles with wide dynamic range are selected from databases that cover some extreme conditions. Both ice and water clouds are included in the simulation. The simulation for multiple scattering properties of ice and water clouds is done off‐line using 32‐stream DISORT (Stamnes et al., ) and incorporated into the forward radiative transfer calculation via an effective parameterization scheme (Liu et al., ; W. Wu et al., ). The cloud top heights are randomly distributed in a range from close to surface to 50 hPa. The spectra include simulated observations of both over land and over water. The simulation covers the observations under various viewing geometry conditions, with satellite zenith angle ranges from −60° to 60°.

Following equations –, we calculate the MODIS band radiances, *R*_{b}, the convolution errors, Δ*R*^{k}, and the hyperspectral radiances, *R*_{h}, for IASI, AIRS, and CrIS, respectively. Figure shows the flow diagram of the convolution error simulation process. The regression relationship between the convolution errors of the *k*th broadband and the corresponding hyperspectral radiances (within the *k*th MODIS band) is mathematically defined as

where ** nch** is the number of hyperspectral channel radiances within the

In addition to the direct correction for convolution errors, a similar method can be applied to fill large spectral gaps. The direct convolution correction application using equation may not work when most or all of the hyperspectral radiances within a certain band are not covered by a measurement (e.g., the intercalibration between AIRS or CrIS and MODIS band 29). For those cases, the available spectral radiances outside of a large spectral gap can be used to predict the missing channel radiances in the spectral gap,*A* can be determined by

Here *A* is a nch2 by nch1 matrix. The number of available hyperspectral channels is nch1, and the number of channels in the spectral gap region is nch2. This method works when there are correlations between the available spectral radiances outside of the gap and the missing spectral gap radiances. As we are going to discuss in section 5, the hyperspectral radiances within the MODIS band 29 can be well predicted using the available CrIS spectral outside of the MODIS band 29. As compared with Tahara's method described in section 1, this method uses a large number of spectra samples to better account for the scene variability. The prediction for spectral gap radiances is channel dependent. Unlike Tahara's method where a different regression coefficient has to be obtained for each observation, the spectral correlation relationship is defined using a fixed regression coefficient matrix *A*, which was trained in an offline process. Therefore, the implementation of this method for the gap radiance prediction is faster and easier.

After using LBLRTM to simulate 14,600 monochromatic spectra, the corresponding hyperspectral radiances and MODIS band measurements are obtained by convolving the monochromatic spectra with the hyperspectral and broadband SRFs, respectively. equation is used to calculate the convolution errors discussed in section 2. Table lists the mean and the standard deviation (std) values of the convolution errors based on these 14,600 simulated cases. These cases cover a wide range of globally distributed observation scenes. The mean value corresponds to the bias that would be introduced into the intercalibration of each MODIS band even after averaging over a large number of collocated events. The std defines the variation range of the convolution errors.

IASI has a 0.25 cm^{−1} spectral sampling interval with a 0.5 cm^{−1} apodized spectral resolution. The magnitude of the convolution errors for all MODIS bands is in the order of millikelvin (mK) or less. As compared with the CLARREO's spectral calibration requirement of 0.02–0.03 K (*k* = 1, 68.3% confidence level), the convolution errors for the intercalibration between MODIS and IASI are small enough and will not pose as significant calibration errors.

The convolution errors for the MODIS‐CrIS intercalibration are much larger than that for MODIS‐IASI, due to the lower spectral resolution and spectral gaps in CrIS measurement. The spectral resolution for three measurement bands of CrIS in nominal spectral resolution (NSR) mode are 0.625 for longwave IR (LWIR), for midwave IR 1.25 for shortwave IR (MWIR), and 2.5 cm^{−1} (SWIR), respectively. In December 2014, the National Oceanic and Atmospheric Administration started to operate the Suomi National Polar‐orbiting Partnership S‐NPP CrIS in the full spectral resolution (FSR) mode (Chen & Han, ), in which the MWIR and SWIR bands have the same spectral resolution (0.625 cm^{−1}) as the LWIR band. MODIS bands 21–28 are calibrated using the MWIR and the SWIR measurement of CrIS, and bands 30–36 are calibrated using the LWIR measurement of CrIS. When a FSR CrIS spectrum is used to calibrate MODIS, the convolution errors of MODIS bands 23–28 become smaller due to the improved CrIS spectral resolution. When the same sampling resolution is used for all spectral bands, that is, the case for FSR CrIS, the convolution errors associated with the MODIS bands 33–36 (in the CO_{2} absorption region) are larger than that for MODIS bands 23–32 (in atmospheric window spectral regions with relatively small spectral features). It reflects the fact that the variation in the monochromatic radiance in the CO_{2} absorption region deviates more from that which can be captured by the hyperspectral measurement (see Figure ). The same characteristics can be seen in MODIS‐IASI convolution errors. Except for the convolution errors of bands 21 and 22, the errors for other bands are simply due to the limited spectral resolution and the spectral overlapping among adjacent channels of the CrIS measurement. The convolution errors for apodized CrIS measurement of bands 24, 25, 33, 34, 35, and 36 can still be viewed as significant as compared with the CLARREO calibration requirement (0.02–0.03 K). The corresponding errors for unapodized CrIS measurements are much smaller. This is because the spectral overlapping between adjacent SRFs of Blackman apodization (see Figure ) is larger than that of the unapodized SRFs (see Figure ). For both apodized and unapodized CrIS measurements, the band 21 and band 22 errors are in similar scale, indicating that the spectral gap (see Figure ), rather than the apodization difference, is the dominant factor responsible for the convolution errors.

AIRS is a grating spectrometer with an unevenly distributed spectral grid spacing in wavenumber unit. Besides the MODIS band 29 which is largely uncovered by the AIRS measurement, MODIS bands 23, 25, 27, 30, and 31 are not sufficiently covered due to the spectral gaps with a spectral interval larger than 5 cm^{−1}. These gaps are due to nonoverlapping spectral converge between different detector arrays and due to the removal of channels with non‐Gaussian noise (popping noise) in many of the AIRS detector array elements. Figure shows sample radiance spectra from AIRS measurements on 17 September 2007. In order to strictly ensure the quality of the spectral information, measurements from more than 400 bad channels may not be used. Therefore, the AIRS spectral coverage for MODIS bands 28, 32, 33, 34, and 35 can also be distorted due to the missing information of bad channels. The convolution errors for the MODIS bands listed above are all significantly large according to the CLARREO standard.

Although the convolution errors are quantified by mean and std values listed in Table , it is noted that they are not randomly distributed errors. Figure uses the intercalibration between MODIS bands 21, 22, 24, 25, 33, 34, 35, and 36 and CrIS (NSR) as examples to demonstrate the dependence of convolution errors on observation scenes. The distribution of convolution errors varies from instrument to instrument, and from band to band, being illustrated by the difference between Blackman apodized CrIS and the unapodized CrIS and the difference between different MODIS bands. The histograms demonstrate that convolution errors can be better characterized by a multimodal distribution than a random distribution. Such kind of errors cannot be effectively corrected using simple bias correction or large‐sample averaging but can be well corrected using the algorithm described in section 2. Figure demonstrates examples of using our method to correct the convolution errors for the intercalibration between MODIS bands 21, 22, 24, and 34 and the apodized CrIS in FSR mode. Figure demonstrates the correction for the intercalibration between MODIS bands 28, 32, 33, 34, and 35 and AIRS. Our simulation study demonstrates that the convolution correction scheme effectively reduces the errors for all these cases to become orders of magnitude smaller than that required by the CLARREO mission.

Collocated MODIS‐AIRS and MODIS‐IASI nadir data sets are used to test the convolution correction algorithm. Both MODIS and AIRS are carried by Aqua satellite, and their observations are always simultaneous. In principle, the results shown in this section for the MODIS‐AIRS intercalibration should also work for the off‐nadir observations since we trained our algorithm for ±60° viewing zenith angles. Considering that the MODIS‐IASI convolution errors are negligible (see Table ), the collocated MODIS‐IASI intercalibration results can be used to validate the correction for the MODIS‐AIRS convolution errors. The MODIS Level‐1B Thermal Emissive Band product has 1 km resolution at nadir, while AIRS and IASI provide measurements with a spatial resolution of 13 and 12 km at nadir, respectively. The MODIS‐AIRS data sets consist of AIRS spectra taken from six footprints located around the center of a scan (within a range of ±3.3° from nadir) and MODIS pixel radiances within a range of ±10.0° from nadir. For each AIRS footprint, MODIS pixels that lie within a 6.75 km radius around the center of the AIRS nadir footprint are selected. The aggregated value of those MODIS pixel radiances is used for the intercalibration. Typically, one AIRS footprint includes 140 MODIS pixels. We select AIRS footprints that are covered by more than 120 MODIS pixels to ensure the spatial coverage of the MODIS measurements for each AIRS footprint. When AIRS spectra are used to calculate the *convolved* MODIS band radiances, low quality AIRS channels are removed based on the channel quality assurance property given in AIRS Version 5 Documentation (). Specifically, only the channels with a quality assurance value less than 3 were used. Similar approaches have been used for various other studies and therefore been confirmed as an efficient and reliable method suitable for the MODIS‐AIRS intercalibration study here (Efremova et al., ; Li et al., ; Li et al., ; A. Wu et al., ). The Simultaneous Nadir Overpass (SNO) data sets of IASI and MODIS are collected from two pairs of IASI footprints from each side of nadir and MODIS pixels in a range of ±10° near the nadir frame. The time difference for each MODIS‐IASI SNO event is limited to be within 30 s. The radiances of all MODIS pixels located within a 6‐km radius of an IASI footprint center are averaged to match the IASI observation. The Metop‐A satellite's *morning* orbit only crosses with the Aqua orbit at approximately ±73.9° latitude line. There are limited number of MODIS‐IASI SNO samples within 1 day. Instead of collecting SNO data with a rigorous temporal and spatial collocation relationship between IASI and AIRS, we collect the MODIS‐AIRS samples within a ± 2.4° latitude zone near the ±73.9° latitude line for the study.

The hyperspectral measurement convolved MODIS radiances are calculated from IASI and AIRS observations following equation (). The convolution of IASI and AIRS spectra with MODIS SRFs follows the trapezoidal integration rule (sample SRFs defined in wavenumbers are plotted in Figure ). Those convolved radiances are then compared with the collocated MODIS radiances to obtain MODIS‐IASI and MODIS‐AIRS intercalibration errors. Considering that those are nadir observations, the convolved radiances are compared with the mean value of the radiances from the aggregated MODIS pixels that fall within the AIRS or IASI single field of view. The comparison between MODIS‐IASI and MODIS‐AIRS intercalibration of 1 July 2008 is demonstrated in Figure . We focus on the calibration for MODIS bands (25, 27, 28, 31, 33, and 34) which have the largest convolution errors according to our simulation study (see Table ). Figure confirms the accurate correction for the MODIS‐AIRS convolution error that is demonstrated as the reduction in the systematic difference between the MODIS‐IASI and MODIS‐AIRS. A great measurement consistency between AIRS and IASI can be achieved after applying the convolution correction for the MODIS‐AIRS intercalibration using the regression coefficients from the simulation study. Before the convolution error correction, the systematic difference between IASI convolved radiance and AIRS convolved radiance is larger than 0.1 K for MODIS bands 25, 27, 28, 33, and 34 (see red and blue histograms in Figure ). Band 27 has the largest MODIS‐AIRS intercalibration error of 1.1 K, which is consistent with the simulation results. Despite the data sampling difference, the systematic difference between the IASI and the AIRS calibrated MODIS observations in those MODIS bands can all be reduced to be smaller than 0.1 K by applying the correction for the MODIS‐AIRS convolution errors. The correction is more effective when the convolution error is the dominant error source of the MODIS‐AIRS intercalibration errors (e.g., MODIS bands 25 and 27). It is interesting to see that the small intercalibration errors (0.01 K) between AIRS and MODIS band 34 is actually the combined effect of the convolution error and the real MODIS measurement error with a 0.21 K bias. This value is smaller than the collocated MODIS‐IASI mean bias error of 0.29 K, indicating that IASI radiances are 0.08 K smaller than the collocated AIRS radiances near 730.8 cm^{−1} spectral region. This is consistent with the 0.088 K value derived from double‐difference comparisons between GOES12 band 6 and AIRS and IASI radiances (L. Wang et al., ). This study provides a good example why an accurate convolution correction is needed in order to reveal some hidden intercalibration issues.

Since the Aqua and Metop‐A satellites have limited SNO opportunities, we choose to further demonstrate our convolution error correction algorithm by using aggregated MODIS‐AIRS global data taken from the Aqua satellite on 17 September 2007. Figure illustrates the accurate convolution correction for the MODIS‐AIRS intercalibration using these data. The multimodal distribution feature of the convolution errors for bands 25 and 27, which is scene dependent (see Figure ), becomes very obvious in these global‐distributed observations. The convolution correction clearly reduces the systematic biases for the calibration of five bands (25, 27, 28, 31, and 33). The distributions of intercalibration errors of bands 25, 27, 28, and 33 after the correction become more randomly distributed and less scene dependent, demonstrating the effective removal of convolution errors. It should be noted that even after the convolution error corrections, there are still residual errors such as random instrument noise, temporal‐spatial matching error, and systematic calibration errors of the MODIS and the AIRS instrument themselves.

The spectral shift of MODIS SRFs has been suspected as a cause for the discrepancy between MODIS and IASI measurements (Sohn et al., ). The magnitude of the spectral shift for each MODIS band can be determined based on the MODIS‐IASI intercalibration errors shown in Figure , that is, the SRF shifts are optimized to eliminate the bias of MODIS‐IASI difference. For the MODIS‐AIRS intercalibration, the same SRF shifts of MODIS are applied before the convolution correction. Figure illustrates that the combined consideration for both the SRF shifts and the convolution errors can provide a very accurate compensation for the difference between AIRS and MODIS observations in both CO_{2} and water vapor bands (MODIS bands 25, 27, 28, and 34). It is noted here that the −78.0 nm (~4.2 cm^{−1}) SRF shift demonstrated in the lower right subplot of Figure seems to be large, although it effectively offsets the convolution errors between MODIS band 34 and AIRS. It is possible that there are other uncharacterized MODIS measurement error sources besides the SRF shifting. The validation for the spectral shift hypothesis requires additional information about other instrument related issues such as the radiometric nonlinearity (Xiong et al., ), electronic crosstalk (Sun et al., ), and out‐of‐band filter leaks (M. Wang et al., ).

Using only the SRF shifting to account for the scene‐dependent intercalibration errors has been evaluated (Tobin et al., ). However, our study here reveals that the convolution error is a critical component of the scene‐dependent intercalibration errors so that the SRF shifting effect can be overestimated without an accurate correction for convolution errors. By comparing Figure with Figure , we can separate the convolution error from the error due to the potential SRF shifts. The top middle subplot in Figure shows that the convolution correction without the SRF shifting of MODIS band 27 can already reduce the calibration bias by ~1.5 K (out of a total bias close to 2.0 K). Therefore, the convolution errors make up the major component of the total scene‐dependent intercalibration errors in some cases. Figure further compares the compensation for the MODIS‐AIRS (band 27) difference via two means: (1) by applying convolution correction along with shifting the MODIS SRF by −9.0 nm (~2.0 cm^{−1}, the green histograms) and (2) by simply shifting the MODIS SRF by −35.0 nm (~7.7 cm^{−1}, the red histograms). Although shifting MODIS band 27 SRF without correcting for the convolution error can produce a zero‐biased match between MODIS and AIRS radiances, the std of the resulting error distribution (red histogram) is much wider than that which can be achieved using method 1 (green histogram in Figure ). Most importantly, the −9.0 nm shift value is more consistent with a shift value derived from the MODIS‐IASI intercalibration study shown in Figure , where we have to shift the MODIS band 27 SRF by −10.5 nm in order to match the observed MODIS and IASI radiances.

The spectral region of MODIS band 29 spans from 1,114 to 1,231 cm^{−1}. There are only 34 channels out of the 2,211 FSR CrIS channels within MODIS band 29, covering less than 20% of the band region. AIRS has 70 channels within MODIS band 29 that cover less than 32% of the band region. The MODIS‐AIRS or MODIS‐CrIS intercalibration of band 29 requires the compensation for missing spectral information. The missing spectral information can be predicted using the available CrIS or AIRS spectral radiance with the regression relationship described in section 2. This section demonstrates the application of the spectral filling methodology for the calibration of MODIS band 29 using FSR CrIS and evaluates the filling accuracy and uncertainty. The application discussed here can be easily extended for the calibration of other broadband sensor channels that provide measurement in the overlapping spectral region (e.g., the ABI channel 11, AHI channel 11, SEVIRI channel 7, and VIIRS channel M14).

The MODIS band 29 is covered by IASI measurements, and the MODIS‐IASI convolution error for band 29 is negligible as we have demonstrated via the simulation study in section 3. Therefore, the IASI data can be used as a solid reference to test the spectral gap filling methodology. The spectral gap filling errors can be characterized explicitly by directly comparing the IASI spectra within the band 29 region and the values predicted using the collocated CrIS spectra. The regression matrix *A* (see equations and ) used for the spectral gap filling is trained using the sample IASI channel radiances that lie within the MODIS band 29 but not covered by CrIS and the corresponding CrIS spectra. Both IASI and CrIS spectra are generated by convolving the 14,600 monochromatic LBLRTM spectra (see section 2) with corresponding hyperspectral channel SRFs. In principle, the relationship between the radiances from 386 IASI channel and the radiances from 2,211 CrIS channels can then be explicitly established as*i* is the index of 386 IASI channels and *k* is the index of CrIS channels. The regression coefficient *a*_{i, k}, as the matrix element of regression matrix *A* can be given by the ordinary least squares solution*N*_{s} matrix that represents the ensemble of CrIS spectra, and *N*_{s} matrix that represents the ensemble of IASI gap spectra, *N*_{s} is the number of sample spectra used for the training. However, The matrix ** U** is the PC matrix generated by applying a singular value decomposition on

The regression coefficient matrix *A* trained for spectral gap filling is validated using independent data sets from real satellite observations. In order to provide a clean evaluation for the spectral gap filling errors that can be separated from other error sources, we convert IASI spectra to CrIS proxy spectra using a double Fourier transform method to ensure a *perfect collocation*, instead of using real collocated CrIS measurements. The CrIS spectra are generated by a standard procedure that includes several steps: (1) Fourier transformation of IASI spectra to interferograms of 2.0 cm optical path difference; (2) deapodization of the IASI interferograms; (3) truncation of deapodized interferograms according to the FSR CrIS optical path difference of 0.8 cm; (4) applying the Blackman apodization to CrIS interferograms; and (5) inverse Fourier transformation of the interferograms to construct the CrIS spectra. A similar procedure has been used for postlaunch calibration assessment between IASI and CrIS, and the resampling error has been confirmed to be less than 0.02 K (L. Wang et al., ).

The IASI spectra used for validation are selected from observations on 1 September 2007. The subplot (a) of Figure illustrates the distribution of the spectral radiances. The subplot (b) shows the FSR CrIS spectra generated from the IASI spectra. The subplot (d) plots the IASI spectral gap radiances predicted from the CrIS spectra. Those predicted spectra gap radiances well recovers the true spectral information that is demonstrated in the subplot (c). Subplot (e) of Figure shows that the bias of the difference between the real in‐band IASI spectra and the predicted ones is close to 0 in all spectral gap channels. The uncertainty of the prediction error is consistent with the IASI instrument noise level. The true IASI spectra are then convolved with the SRF of MODIS band 29 to obtain the collocated MODIS radiances. The convolution is also applied to the synthetic spectra that combines the CrIS spectra and the predicted IASI spectral gap radiances, and the results are compared with MODIS radiances obtained from the IASI spectra. The difference is shown as a histogram plot in subplot (f). The mean and standard value of the difference shown in subplot (f) confirms that the remaining systematic bias after the spectral gap filling is negligible and the uncertainty is about 0.05 K. The randomly distributed errors are mainly due to the instrument random noise in IASI data, and will not show up as intercalibration errors between space‐time averaged observations.

We have studied the impact of convolution errors on the intercalibration between hyperspectral and broadband sensors. In order to fully benefit from the future calibration reference standard provide by CLARREO and to generate accurate long‐term time series record for climate applications, large convolution errors need to be reduced to be smaller than or at least on the same order of magnitude of the CLARREO accuracy. Intercalibration convolution errors are fundamentally caused by the missing information due to spectral gaps, and the *coarseness* of the hyperspectral measurement characterized by the limited spectral resolution and the spectral overlapping between SRFs. IASI's high spectral resolution and continuous spectral coverage between 645 and 2,760 cm^{−1} makes the convolution error of MODIS‐IASI intercalibration a negligible concern. The nominal spectral resolution of CrIS is coarser than IASI; therefore, the convolution errors between CrIS and MODIS are much bigger. Although the convolution errors for apodized CrIS measurements of bands 24, 25, 33, 34, 35, and 36 can still be viewed as significant as compared with the CLARREO calibration requirement, the corresponding errors for unapodized CrIS measurements are much smaller. The unapodized CrIS measurement causes much smaller errors in those bands because of its less spectral overlapping between adjacent SRFs. CrIS does not cover the complete spectral regions of the MODIS bands 21–22. The convolution errors introduced by both apodized and unapodized CrIS measurements for the intercalibration of those two MODIS bands are significant and similar in scale. AIRS provides a better spectral resolution than CrIS of the nominal resolution (but lower spectral resolution than the full‐resolution CrIS for MWIR and SWIR bands) but can introduce larger convolution errors for the intercalibration of several MODIS bands due to the instrument spectral gaps or the missing information of bad channels.

We have also developed a correction algorithm that can reduce these errors. One of the significant advancements of our algorithm compared with previous studies is that we can accurately account for both the scene and the spectrally dependent convolution errors by taking advantage of the hyperspectral measurements. This is possible because of the spectral correlations in the measured hyperspectral data. Large scene‐dependent, spectrally dependent convolution errors associated with MODIS‐CrIS or MODIS‐AIRS intercalibration can be well addressed by the regression‐correction scheme using the coefficients obtained via the simulation study. The convolution errors after correction become negligible even according to the most stringent calibration standard that is required for the climate studies. The relationship between the convolution errors and the corresponding hyperspectral information derived from the radiative transfer simulation results is further validated using the real MODIS‐AIRS data. Our application study on real data demonstrates that intercalibration errors can be effectively reduced after correcting the MODIS‐AIRS convolution errors. A better than 0.1 K consistency is demonstrated between the SNO observations of AIRS and IASI. The accurate characterization for the scene and spectrally dependent feature of the convolution errors also helps to reduce calibration uncertainties and therefore better evaluate other calibration errors. We demonstrate that the potential drifts in the SRFs of MODIS bands 25, 27, and 28 can be overestimated. For example, a shift of MODIS band 27 SRF by as much as −35 nm is needed in order to match the observed MODIS and AIRS observations. However, a shift of only −9.0 nm is needed after performing convolution error correction. This SRF shift value is more consistent with the shift obtained from the MODIS‐IASI study. The spectral correlation relationship can also be used for the large spectral gap filling or spectral range extension. Our study using the real IASI data as the reference confirms that the spectral information that is missing but needed for the calibration of MODIS band 29 can be very accurately filled using the available CrIS measurements. The same spectral filling methodology can be applied for the intercalibration between MODIS band 29 and AIRS measurements. Methods developed in this paper can also be applied to intersatellite calibration in solar spectral regions.

This research was supported by the NASA CLARREO project and by the NOAA JPSS program. We thank AER Inc. for providing the LBRTM model used in this study. The NASA SMD High‐End Computing (HEC) resources were used to support the LBLRTM radiative transfer calculations. The AIRS data (version 5) used for this study are from