1. INTRODUCTION

The top of atmosphere (TOA) radiance provides a wealth of information on the atmosphere as well as on the Earth’s surface. The absorption, emission, and scattering of radiation by gases, aerosols, clouds, and precipitation in the atmosphere and media on the Earth’s surface make the radiative transfer modeling extremely complicated.

The radiative transfer equation, which describes the physical radiation process, may be numerically solved using the successive-orders-of-scattering (SOS) approach [1–13], the Monte Carlo (MC) method [5,14–21], the adding-doubling (AD) method [5,22–30], the discrete ordinate method (DO) [5,31–35], and the matrix operator (MO) method [36–40]. In the SOS method, the total intensity is obtained by summing light scatterings from different orders, which can be calculated individually. It has been used to solve radiative transfer equations for specific geometries. However, it may be time-consuming to solve for radiative transfer equation in plane-parallel atmospheres with a large optical thickness [41]. The MC method may be used to simulate the transfer of radiation in any geometric configuration [18]. However, it requires enormous amounts of computing time to achieve reliable accuracy, since the MC method is subject to statistical fluctuations. The AD method works naturally with layered geometries and yields reflection and transmission readily. The AD method is incapable of providing radiance at arbitrary positions within the atmosphere and needs lengthy computing time when optical thickness is big. The DO method divides all coordinates, including scattering, zenith, and azimuthal angles, into discrete grid points. The DO method has been found to be efficient and accurate for calculations of scattered intensities and fluxes [41]. To reduce computational time, analytic two- and four-stream approximations have been developed for flux calculations [41]. However, the limited resolution of the computational angles introduces errors in the simulation. The MO is another commonly used method to solve radiative transfer equation; the solution for the vertically inhomogeneous atmosphere is obtained recurrently from the analytical solutions for the divided sublayers [39,40].

There are advantages and disadvantages of the methods mentioned above. We choose to use the DO method in this work, since this is what medium speed correlated-k option MODTRAN5 uses and we are using MODTRAN5 as our reference model. In the current work of developing a fast and accurate PCRTM solar radiative transfer model, we are not focused on the radiative transfer solver itself; instead, we are trying to reduce the number of radiative transfer calculations needed when dealing with TOA radiance or reflectance spectra that span a large spectral range.

The results obtained from the solution of a radiative transfer equation are the monochromatic radiances (monoradiances). The radiances obtained from satellite detector (channel radiances) are the convolution of monoradiances with the instrument spectral response function (SRF). Due to the narrow Doppler and Lorentz half-widths of the atmospheric molecules, usually millions of monoradiances have to be simulated in order to get one TOA radiance spectrum. Thus a full line-by-line radiative transfer simulation for the monoradiances and the convolution calculation for the channel radiances are too time-consuming to be used in an operational satellite data assimilation or retrieval system.

A few fast radiative transfer algorithms have been developed to simulate satellite measured channel radiances. The community radiative transfer model (CRTM) and the radiative transfer for TOV (RTTOV) are the most popularly used fast models in satellite data assimilation [42,43].

Version 1 of CRTM uses the advanced doubling-adding method while version 2 employs the advanced matrix operator method [40,44]. Version 2 is faster than version 1 and is capable for visible spectral region calculation.

The so-called polychromatic transmittance approximation uses preselected transmittance predictors and the corresponding regression coefficients to calculate the convolved layer-to-space transmittances [45–55]. The thermal contribution to the channel radiance is then calculated using these transmittances. The layer-to-space transmittances may be calculated in two different ways: the pressure level optical depth method (PLOD) or the optical path transmittance method (OPTRAN) [47,53]. To obtain the channel radiance measured by the instrument, one has to calculate the contributions from scattering and surface reflection separately.

Another type of channel-based fast radiative transfer models (RTMs), such as correlated-k distribution (CKD), exponential sum fitting transmittance (ESFT), radiance sampling method (RSM), and optimal spectral sampling (OSS), uses a few representative monoradiances to predict one-channel-radiance [56–72]. For example, the MODTRAN5 uses an improved CKD method to calculate the spectral features due to atmospheric emission, transmission, and reflection. More than three hundred thousand monochromatic calculations are needed to cover a spectral range from 300 to 2500 nm with a spectral resolution of one wavenumber.

Liu et al. developed a fast and accurate principal component-based radiative transfer model (PCRTM), which calculates much fewer monoradiances [73–80]. In the PCRTM model, the few thousand channel radiances observed by hyperspectral instruments may be simulated from just a few hundred rather than thousands or even millions of monoradiances [80]. This greatly increased the computational efficiency. When we develop the PCRTM model, the principal components (PCs) of the channel radiances of the instrument were selected during a training process. Typically, the number of selected PCs is much smaller than the number of channels of that instrument. The corresponding PC scores could be calculated from monoradiances at few hundreds of preselected frequencies. The few thousands of channel radiances are then easily obtained by linearly combining the preselected PCs with PC scores as weights [76]. The PRCTM is much faster than channel-based radiative transfer models. It has been successfully developed for many satellite instruments as well as aircraft instruments, such as Atmospheric Infrared Sounder (AIRS), Infrared Atmospheric Sounding Interferometer (IASI), Cross-Track Infrared Sounder (CrIS), the NPOESS airborne sounder tested interferometer (NAST-I), and Scanning High-resolution Interferometer Sounder (S-HIS) [80]. The PCRTM approach has great potential for superfast retrieval and for numerical weather prediction (NWP) [81–105].

In this work, we expand the PCRTM to the solar spectral region (PCRTM-SOLAR). The solar source, which is a quasi-collimated direct beam, makes the channel radiance highly azimuthal angle-dependent. The complicated multiple scattering of solar light by cloud particles and/or aerosol as well as the nonLambertian nature of the Earth’s surface makes the radiative transfer process very complicated. It requires significant computational effort to obtain the TOA radiance using the well-known radiative transfer equation solver [15,16,25,26,32]. Thus it is very important to develop a fast and accurate RTM to simulate reflected solar spectrum, which is critical for NWP data assimilation and other climate related projects, such as the climate absolute radiance and refractivity observatory (CLARREO) mission. Both the accuracy and the speed of current available fast RTMs in solar spectral region are still not good enough for near real-time applications [106]. Our new PCRTM-SOLAR model provides high accuracy radiance as well as up to 3 orders faster speed than 16-stream MODTRAN5. The PCRTM-SOLAR is capable of simulating TOA radiance and reflectance with arbitrary layers of cloud/aerosol in the solar spectral region. It is valid for solar zenith angles (SZAs) up to 80 deg, satellite view zenith angles (VZAs) up to 70 deg, and arbitrary relative azimuthal angles.

The PCRTM-SOLAR was developed by training and validating over 60 thousands of RS spectra obtained using MODTRAN5.

2. TOA RADIANCE SIMULATION USING MODTRAN5

The radiances we used in this work were obtained by running MODTRAN5 with medium-speed correlated-k option under various atmospheric and Sun–satellite geometrical conditions.

We simulated the RS spectrum for 32,499 land surface cases and 30,356 ocean surface cases using MODTRAN5. The stream number used was 16 in the simulation. There were clear sky and cloudy sky cases. The cloudy sky case might have one to three layer(s) of cloud. In the three-layer cloud case, the lowest was always water cloud, while the highest was ice cloud. The middle layer was either ice or water cloud.

A. Atmospheric Profile Set

The six MODTRAN5 provided geographical-seasonal model atmospheres (tropical atmosphere, midlatitude summer, midlatitude winter, subarctic summer, subarctic winter, and 1976 U.S. standard atmosphere) were randomly selected in each of the simulations. The atmospheric parameters, such as pressure and temperature, were given by these atmospheric profiles. The ${\mathrm{CO}}_2$ mixing ratio of 370 ppmv was used, the vertical water vapor column was in the range of 0 to $5.0\mathrm{g}/{\mathrm{cm}}^2$, the vertical ozone column was in the range of 0 to 0.5 ATM-cm, and the aerosol optical depth at 550 nm was in the range of 0 to 1.

B. Aerosol Model

The aerosol profile was selected dependent on seasons. The aerosol types used for land surface were rural, urban, and tropospheric. Both 5 km and 23 km visibility models were included in the rural aerosols. The maritime aerosol was used for ocean surfaces. The vertical aerosol optical depth at 550 nm was randomly changed from 0 to 1 for both land and ocean surface cases.

C. Cloud Model

The MODTRAN5 has been modified so that we do not need to input the single-scattering properties of the clouds directly. In fact, the single-scattering properties of the water cloud were calculated using Mie theory. It was found that the optical properties of water cloud were determined mainly by its effective radius and could be parameterized successfully [107]. The parameterization is almost as accurate as the exact Mie calculation. Therefore, the parameterization method for water cloud was used in this work. To increase the computing speed, the single-scattering properties of ice cloud were interpolated from presaved lookup tables.

The modified MODTRAN5 and PCRTM-SOLAR need only the cloud type (water or ice) and cloud particle effective radius for the cloud properties’ calculation.

1. Water Cloud

The optical properties of water cloud were calculated for each band from the Hu and Stammes parameterization [107]. The extinction coefficient ${\beta }_{\text{ext}}$, the single-scattering albedo $\omega$, and the asymmetry parameter $g$ are given by

All of the coefficients $a_i,b_i,c_i$ are functions of wavelength and particle size. The water clouds were grouped according to their radii: (1) small size, 2.5 to 12 μm; (2) medium size, 12 to 30 μm; and (3) large size, 30 to 60 μm. In each group, the coefficients $a$, $b$, and $c$ were given at 84-wavelength bands in the solar spectral range from 290 to 2525 nm.

Figure 1 shows the typical results of the obtained extinction coefficients, absorption coefficients, and asymmetry factors for the cloud water droplet from Eqs. (1)–(3).

 

Fig. 1. Typical results of the optical properties of the cloud water droplets. (a) Water droplet extinction coefficient (WDEC, ${\mathrm{km}}^{-1}{\mathrm{m}}^3/\mathrm{g}$). (b) Coalbedo $(1-\omega )$. (c) Water droplet asymmetry factor (WDAF). The legends represent the water droplet effective diameter (μm).

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2. Ice Cloud

The single-scattering properties of the ice cloud were calculated as a function of cloud particle size and wavelength using the method developed by Yang et al. [108] and Baum et al. [109].

The scattering, absorption, and polarization properties of ice particles were computed based on a combination of the Amsterdam discrete dipole approximation, the T-matrix method, and the improved geometric optics method. The full set of single scattering properties considered the three-dimensional random orientations for 11 ice crystal habits. The surface of each ice crystal habit includes smooth, moderately roughened, and severely roughened surfaces. The results with 23 different particle sizes, from 10 μm to 120 μm, at 221 different wavelengths, from 300 to 2500 nm, were saved and used as a lookup table. The single-scattering properties of the ice cloud in MODTRAN5 and PCRTM-SOLAR simulation were obtained through interpolation by using these saved data.

The typical ice particle extinction coefficient (IPEC), ice particle absorption coefficient (IPAC), and ice particle asymmetry factor (IPAF) randomly selected from the over 60 thousands of cases are shown in Fig. 2.

 

Fig. 2. Typical results of the optical properties of the cloud ice particles. (a) Ice particle extinction coefficient (IPEC, ${\mathrm{km}}^{-1}{\mathrm{m}}^3/\mathrm{g}$). (b) Coalbedo $(1-\omega )$. (c) Ice particle asymmetry factor (IPAF). The legends represent the ice effective diameter (μm).

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D. Surface Reflection Model

The nonLambertian bidirectional reflectance distribution function (BRDF) is used for both land and ocean surfaces. The semiempirical RossThick–LiSparse (Ross–Li) model was used for the land surface BRDF [110–113]. The BRDF is expressed as a sum of few kernel functions and given by

The RossThick kernel is given by

The LiSparse-Reciprocal kernel is

When ${\theta }_v={\theta }_s=0$, one may have $K_{\mathrm{RT}}=K_{\mathrm{LSR}}=0$. Therefore, the BRDF is equal to $P_1$, the Lambertian scattering component of BRDF. Parameter $P_2$ is the coefficient of the LiSparse-Reciprocal geometric-scattering kernel, and parameter $P_3$ is the coefficient for the RossThick volume-scattering kernel. The recommended value for parameter $P_5$ is 1 [113].

The ocean surface BRDF is greatly influenced by wind speed. It is calculated using the method given in [114]. Cox and Munk [115] described the wind-blown ocean surface as a collection of individual mirror facets. The slope distribution of the facets is usually expressed as a Gaussian function with a wind-speed-determined distribution width:

The reflectance and transmittance at the rough ocean surface is closely related to the slope distribution of the surface facets. For example, the single-scattering reflectance at the air–water interface is given by

The total reflectance is given by the summation of the single- and multiple-scattering reflectance of the solar radiation [114].

E. Typical Channel Radiances

The obtained channel radiances are different from each other, since each case has different atmospheric profile, ozone, water vapor, cloud, aerosol, surface BRDF, SZAs, and satellite angles. Figure 3 shows the typical TOA radiances for both land and ocean surfaces. The spectra were calculated in the range from 300 to 2500 nm with a resolution of $1{\mathrm{cm}}^{-1}$.

 

Fig. 3. Typical TOA radiances obtained using MODTRAN5 for both land and ocean surfaces, respectively. There are many fine features in the spectrum. The high-resolution channel spectrum makes the channel-based RTMs inefficient. Each color represents a randomly selected case.

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There are many fine spectral features in the high-resolution channel radiance spectrum. The rich spectral features as well as the various spectral shapes make it very difficult to develop a channel-based accurate fast RTM. However, the PCRTM-SOLAR model developed in this work has been shown to be a powerful tool for fast and accurate radiative transfer simulation in the solar region.

The 29,311 channel radiances from 300 to 2500 nm for one whole spectrum make the channel-based fast RTMs inefficient. On the other hand, our PCRTM-SOLAR only needs to simulate 1359 monoradiances for land surfaces and 958 monoradiances for ocean surfaces to obtain the whole 29,311 channel radiances. Though the selected monofrequencies are coarse in the frequency axis, the fine spectral details are reproduced excellently with very high accuracy using PCRTM-SOLAR, the PCRTM.

3. PCRTM

A. PC Analysis for TOA Radiances

It is well known that the major atmospheric gases absorb photons at many frequencies. However, the spectral features at these frequencies are highly correlated [76]. In principle, if we know the transmittance for a given amount of gas at a limited number of selected frequencies, we should be able to represent the transmittance spectrum as a linear combination of the selected transmittances provided that the numbers are large enough. This is an effective way to remove redundant information. For those window channels where there are no or little atmospheric molecular absorptions, we can represent these channels with only just a few monochromatic frequencies, since the absorption and scattering of the clouds and aerosols have broad spectral features.

Not only absorption and scattering, but also emission spectrum, has a lot of redundant information. Consequently, the TOA monoradiance spectrum as well as the channel radiance spectrum possess a great deal of redundant information. The PCRTM method was developed to remove this redundancy and significantly reduce the computation burden.

Principal component analysis (PCA) is a powerful tool [117,118] that transforms a number of correlated variables into a much smaller number of uncorrelated variables. These uncorrelated variables are called the PCs. The first PC represents as much of the variability in the data as possible; the second PC is orthogonal to the first one and accounts for the next highest variance. Each successive PC is responsible for the maximum proportion of the remaining variance in the data. The advantage of PCA is that one needs only the significant PCs rather than all of the PCs to reproduce the data with high accuracy.

For the satellite TOA channel radiance spectra with ns samples and nch channels, the data set may be written as the following, using singular value decomposition (SVD):

To quantitatively understand how many significant PCs are needed in the PC analysis, we define a normalized cumulative SV (NCSV) as

The upper plot in Fig. 4 shows the SV for each PC of the TOA channel radiance in the range of 625.8606 (15978) to 980.4883 nm ($10199{\mathrm{cm}}^{-1}$) for ocean surface cases. There are 5780 channels/wavenumber/variables in this range; thus we have 5780 PCs. Among the 5780 PCs, the first PC contains 74.46% of the total SV; the second PC represents 11.49%, and the third PC has 4.51%. The proportion decreases very fast with the PC number.

 

Fig. 4. Singular value for each principal component (upper) and normalized cumulative singular value as function of the selected number of PCs (lower) in the band range of 625 nm to 980 nm for ocean surface case.

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As shown in the lower plot in Fig. 4, the first three PCs include more than 90% of the summation of the SV, while the first 10 PCs enclose over 96% of it. Obviously, the major advantage of PCA is that most of the information of the data set can be reproduced by the first few PCs, which greatly speeds up the computation process. Namely, the channel radiance may be given with very high accuracy by

To demonstrate this idea, Fig. 5 shows the original TOA channel radiance and the reconstructed radiance using equation (15) with $\mathrm{npc}=299$. Obviously, the reconstructed radiance accurately reproduced the original radiance with root mean square (RMS) error smaller than $5\times {10}^{-4}$, as shown in the lower plot in Fig. 5.

 

Fig. 5. Typical TOA radiance and the reconstructed radiance using 299 PCs for ocean surface case in wavelength range from 625 to 980 nm (upper). The RMS errors of the reconstructed TOA radiances compared to the original radiances (lower).

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We reconstructed TOA channel radiances using Eq. (15) for both ocean and land surface cases with different spectral resolutions, including $1{\mathrm{cm}}^{-1}$, 8 nm, and other customized resolutions. The results show that similar results were obtained in the whole spectral range from 300 to 2500 nm. Our previous work indicates that Eq. (15) works not only in the solar region but also in the thermal region.

B. Regression of Score Matrix on Selected Monoradiances

From Eq. (15), the PC score matrix may be obtained by [76,77]

The channel radiance is the convolution of the monoradiance with the instrument line shape (ILS) or SRF, ${\mathrm{\Phi }}_k(k=1,2,\cdots N)$. That is,

Since both $U_{\mathrm{npc}\times \mathrm{nch}}$ and ${\mathrm{\Phi }}_k$ do not change with different atmospheric states (ns number), the PC score is a linear function of the monoradiance, which may be obtained by combining Eqs. (16) and (17):

It is worth noting that the number of selected monofrequencies, nsmo, is much smaller than the total number of monofrequencies, nmo. That is, $\mathrm{nsmo}\ll \mathrm{nmo}$. The nsmo preselected monofrequencies were obtained by calculating the variance and correlation coefficients of the monoradiances. The frequency with the largest variance in the monoradiance data set was picked and saved as one of the selected frequencies. The correlation coefficients of all other frequencies to the selected frequency were then calculated. All frequencies with correlation coefficients larger than a preset criterion were removed. The same procedure was repeated with the remaining frequencies until each of the frequencies either was picked or removed. After this procedure, the picked frequencies were used to train the channel radiances so that we might further decrease the number of selected monofrequencies. The training was carried out so that it satisfied a preset RMS error criterion in the reconstructed channel radiances.

C. Training Results

We found that the monoradiances at only a few hundreds of frequencies were needed to successfully represent the whole monoradiance spectrum at millions of frequencies.

For the $1{\mathrm{cm}}^{-1}$ spectral resolution PCRTM-SOLOR training, we have divided the 300–2500 nm range into five spectral regions. Figure 5 shows a typical spectrum in the second spectral region. In this region, there are 82,789 monofrequencies. You can clearly see the ${\mathrm{O}}_2\text{-}A$ band spectral features at 676 nm and a water vapor spectral feature elsewhere. After the selection procedure, monoradiances at only 232 selected frequencies are needed to accurately generate the PC score matrix. Therefore, only 232 rather than 82,789 radiative transfer (RT) calculations are needed.

Table 1 shows the training results for the selected number of PCs, and the number of preselected monofrequencies for both land and ocean surfaces with both $1{\mathrm{cm}}^{-1}$ and 8 nm spectral resolutions, respectively. These results cover spectral range from 300 to 2500 nm. For the 8 nm spectral resolution cases, the spectral sampling size is 4 nm.

Table 1. Number of Monofrequencies (nmo), Number of Channel Frequencies (nch), Number of Significant PCs (npc), and Number of Preselected Monofrequencies (nsmo), for Both Land and Ocean Surfaces with Both $\mathsf{1}{\mathsf{cm}}^{-\mathsf{1}}$ and 8 nm Resolutions

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4. RESULTS AND DISCUSSION

A. Speedup of PCRTM-SOLAR

The training process for both the channel and monoradiances discussed above provides us both a $U_{\mathrm{npc}\times \mathrm{nch}}$ matrix and a $A_{\mathrm{npc}\times \mathrm{nsmo}}$ matrix. To calculate the TOA channel radiances using Eqs. (15) and (18), the only unknown parameter is $R_{\mathrm{nsmo}\times \mathrm{ns}}^{\mathrm{mono}}$. Therefore, one simply performs radiative transfer calculation at the few hundreds selected monofrequencies to obtain the channel radiance spectrum. This makes the simulation very fast.

In MODTRAN5, we simulated the monoradiances at 259,029 frequencies and then performed a convolution operation with appropriate ILS/SRF to get the TOA channel radiances at 29,311 frequencies for the $1{\mathrm{cm}}^{-1}$ resolution cases. In PCRTM-SOLAR, we just calculated the monoradiances at the preselected 958 frequencies and multiplied the results by the two presaved matrices to get the channel radiances at the 29,311 frequencies for ocean surfaces. The limiting factor in computational speed is the number of monoradiative transfer calculations needed. For the case of $1{\mathrm{cm}}^{-1}$ resolution ocean PCRTM-SOLAR, the speed acceleration is about 270 times relative to original MODTRAN5.

Table 2 gives the summary of the acceleration of PCRTM-SOLAR relative to MODTRAN5. Again for MODTRAN5, the radiative transfer calculations are done at 259,029 preselected CKD monofrequencies, while the PCRTM-SOLAR only performs RT calculations at a few hundred to about one thousand monofrequencies.

Table 2. Increase in Speed of PCRTM-SOLAR Compared to MODTRAN5

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The actual computational time of one whole channel radiance spectrum for the 8-nm resolution ocean surface case is less than 10 s on a LINUX machine with a 2.3 GHz CPU speed. The MODTRAN5 needs about 3 h for the same work on the same machine.

The speed of PCRTM-SOLAR is also much faster than the channel-based MODTRAN5, since the latter has to perform radiative transfer calculations at 17 monochromatic frequencies for each of the 29,311 channels in our $1{\mathrm{cm}}^{-1}$ resolution cases. On the other hand, we just need to calculate at 1359 monofrequencies for land surface cases or at 958 monofrequencies for ocean surface cases. In addition, some of the channel-based fast RTMs usually need 1–30 predictors to predict the radiance of each channel [76].

B. Accuracy of PCRTM-SOLAR

A solid fast RTM must have very high accuracy compared to the reference line-by-line or high spectral resolution CKD calculations. To check the accuracy of our PCRTM-SOLAR, we divided our data sets for both the channel radiances and the monoradiances into two groups. One group was used for training and the other group, which is independent of the first one, was used for the validation.

Figure 6 shows a randomly picked TOA channel radiance for an ocean surface case. The blue color represents the original data obtained using MODTRAN5, while the green color is the regenerated data using our fast PCRTM-SOLAR. It is very hard to see the blue line, since it overlaps with the green line so well. To clearly see the difference, we plot it in the lower panel in Fig. 6. The majority of the difference between the results obtained from MODTRAN5 and PCRTM-SOLAR is less than ${10}^{-3}\mathrm{mW}/{\mathrm{cm}}^2·\mathrm{sr}·{\mathrm{cm}}^{-1}$. When the overall radiances become smaller, the difference will also become smaller. This phenomenon was found for both training data and validation data. The validation data shown in Fig. 7 happened to be a case with smaller radiances compared to the data shown in Fig. 6.

 

Fig. 6. TOA radiances obtained using MODTRAN5 (blue) and fast PCRTM-SOLAR (green) for an ocean surface case with $1{\mathrm{cm}}^{-1}$ resolution (upper). The difference between the TOA channel radiances obtained using MODTRAN5 and those obtained using PCRTM-SOLAR (lower). This case was included in training.

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Fig. 7. Typical TOA radiances obtained using MODTRAN5 (blue) and fast PCRTM-SOLAR (green) for an ocean surface case with $1{\mathrm{cm}}^{-1}$ resolution (upper). The difference between the TOA channel radiances obtained using MODTRAN5 and those obtained using PCRTM-SOLAR (lower). This case was not included in training.

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Figure 7 shows the TOA channel radiances for a validation case, which was not used during training to obtain the $U_{\mathrm{npc}\times \mathrm{nch}}$ matrix and $A_{\mathrm{npc}\times \mathrm{nsmo}}$ matrix. This case was randomly selected from the thousands of validation cases. The blue line stands for data obtained by using MODTRAN5, while the green line is for data obtained using the fast PCRTM-SOLAR program. The difference between the two lines is smaller than ${10}^{-3}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$ at all of the 29,311 wavelengths, most of them actually smaller than ${10}^{-4}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$. Therefore, it looks like a straight line in the figure.

To better understand the accuracy of the fast PCRTM-SOLAR, we calculated both the mean absolute (MA) error and the RMS error for both training and validation cases. Both of them are regularly employed in model evaluation studies [119]. They are defined by

MA error gives the same weight to all errors, while RMS error gives more weight to larger errors. Therefore, RMS error is always larger or equal to MA error.

The MA errors of TOA channel radiances for ocean surfaces with $1{\mathrm{cm}}^{-1}$ resolution in the wavelength range of 300 to 400 nm are around ${10}^{-5}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$, as shown in Fig. 8. The maximum MA error is around ${10}^{-3}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$, and the major part of the MA errors is in the range of ${10}^{-5}$ to ${10}^{-3}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$, which is about 3 orders of magnitude smaller than the original radiances.

 

Fig. 8. Mean absolute errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue line is for the validation data set while the green one is for the training data set.

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The RMS errors, shown in Fig. 9, provide very similar information as the MA errors. However, RMS error is a little bit larger than the corresponding MA error, since it uses higher weighting for the larger error. The maximum RMS error is also around ${10}^{-3}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$. The MA errors are very close for the training data set and the validation data set. The RMS errors for the validation data are little bit larger than those for the training data due to the higher weight of the larger error in the RMS error calculation. However, they both are very small and comparable, indicating good training results were obtained.

 

Fig. 9. RMS errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue line is for the validation data set while the green one is for the training data set.

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Very similar results were obtained for the accuracy study for a land surface with $1{\mathrm{cm}}^{-1}$ resolution case as well as for cases with 8 nm resolution for both land and ocean surfaces. Figure 10 shows the RMS error for these three cases.

 

Fig. 10. RMS errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue lines are for the validation data set while the green ones are for the training data set. (a) Land surface with $1{\mathrm{cm}}^{-1}$ resolution; (b) land surface with 8 nm resolution; (c) ocean surface with 8 nm resolution.

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The systematic model biases of PCRTM-SOLAR are very small for all the cases. Figure 11 shows the biases for the land surface 8 nm resolution case. The biases are smaller than $2\times {10}^2\mathrm{mw}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$.

 

Fig. 11. Typical systematic model biases of PCRTM-SOLAR model. The data shown are for land surface with 8 nm resolution. The blue line is for the validation data set while the green one is for the training data set.

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The CLARREO RS instrument, which covers from 320 to 2300 nm, has a spectral sampling of 4 nm and a resolution of 8 nm [120]. It is required to have an accuracy of 0.3% in broadband reflectance and signal-to-noise ratio larger than 33 at SZA of 75° for the mean 0.3 reflectance [120,121]. We developed the 8 nm resolution version of PCRTM-SOLAR especially for the CLARREO RS mission. The results shown in Figs. 10 and 11 indicate the simulation error of PCRTM-SOLAR is very small. The relative error is usually less than 0.2%, which is smaller than the instrument sensitivity requirements. Therefore, PCRTM-SOLAR is extremely useful for the CLARREO mission.

PCRTM-SOLAR is a very useful fast model for hyperspectral CLARREO sensors. It can be easily expanded to include other sensors, like the moderate resolution imaging spectroradiometer (MODIS) and the visible infrared imaging radiometer suite (VIIRS). The MODIS sensor has only 20 spectral bands in the wavelength range from 405 to 2155 nm and VIIRS has only 9 visible/near IR bands; thus one may expect many fewer monofrequencies are needed for these sensors.

5. CONCLUSIONS

In conclusion, an accurate, fast PCRTM-SOLAR model has been developed for reflected solar spectrum simulation. The main advantages of this model are the very fast speed and very high accuracy.

PCRTM-SOLAR does not need to solve the RT equation at hundreds of thousands of monofrequencies or at every channel frequency. Instead, it calculates the RT equation at only 1359 preselected monofrequencies for the $1{\mathrm{cm}}^{-1}$ resolution RS radiance spectrum for land surface, at 958 frequencies for ocean surface, at 262 frequencies for the 8 nm resolution land surface, and at 240 frequencies for the 8 nm resolution ocean surface case. The much less radiative transfer calculations greatly reduce the computation burden and will greatly benefit the observing system simulation experiments (OSSEs) for climate change study.

The accuracy of the PCRTM-SOLAR is guaranteed by the prestored $U_{\mathrm{npc}\times \mathrm{nch}}$ matrix and $A_{\mathrm{npc}\times \mathrm{nsmo}}$ matrix. The $U_{\mathrm{npc}\times \mathrm{nch}}$ matrix contains information for each of the satellite sensor channel, while the $A_{\mathrm{npc}\times \mathrm{nsmo}}$ matrix links the PC scores to the preselected monoradiances. The typical MA errors and RMS errors are around ${10}^{-4}\mathrm{mW}/{\mathrm{cm}}^2/\mathrm{sr}/{\mathrm{cm}}^{-1}$, which are about 3 to 4 orders of magnitude smaller than the typical RS radiances.

The PCRTM-SOLAR model has been validated by comparing thousands of independent TOA channel radiance spectra obtained from both PCRTM-SOLAR and MODTRAN5. The validation errors were comparable to the training errors, indicating excellent performance of PCRTM-SOLAR to simulate RS spectrum under various atmospheric, cloud, aerosol, and surface conditions with various SZAs and instrument view angles.

PCRTM-SOLAR is a physical-based model that achieves accuracy and speed under a wide range of conditions. It works even for situations that are not included in the training data set, such as a system with a water cloud on top of an ice cloud. To make the model more diverse and solid, we may include more aerosol models, cloud models, and surface conditions in the next version of PCRTM-SOLAR.

The validated fast and accurate PCRTM-SOLAR provides a valuable approach for retrieval of cloud properties, aerosols, atmospheric trace gases, and surface conditions in the solar spectral region. It will be the key part in OSSEs and climate fingerprinting. The fast nature of PCRTM-SOLAR makes it especially useful for retrieval of long-term recorded RS spectrum data as well as real-time simulation of the real-time RS data. This may significantly help climate change study and numerical weather prediction.