A hybrid stream PCRTM-SOLAR model has been proposed for fast and accurate radiative transfer simulation. It calculates the reflected solar (RS) radiances with a fast coarse way and then, with the help of a pre-saved matrix, transforms the results to obtain the desired high accurate RS spectrum. The methodology has been demonstrated with the hybrid stream discrete ordinate (HSDO) radiative transfer (RT) model. The HSDO method calculates the monochromatic radiances using a 4-stream discrete ordinate method, where only a small number of monochromatic radiances are simulated with both 4-stream and a larger N-stream (N ≥ 16) discrete ordinate RT algorithm. The accuracy of the obtained channel radiance is comparable to the result from N-stream moderate resolution atmospheric transmission version 5 (MODTRAN5). The root-mean-square errors are usually less than 5x 10-4 mW/cm2/sr/cm−1. The computational speed is three to four-orders of magnitude faster than the medium speed correlated-k option MODTRAN5. This method is very efficient to simulate thousands of RS spectra under multi-layer clouds/aerosols and solar radiation conditions for climate change study and numerical weather prediction applications.
© 2016 Optical Society of America
To understand the Earth’s climate and climate change, one has to comprehensively understand the radiation emitted from the Sun, solar energy absorbed by the atmospheric constituents and the surface, and the emission of thermal infrared energy from the Earth-Atmosphere system. The Climate Absolute Radiance and Refractivity Observatory (CLARREO) system , which is proposed and developed by NASA, will directly measure the Earth's thermal infrared spectrum (IR), the spectrum of solar radiation reflected by the Earth and its atmosphere (RS), and radio occultation (RO). IR, RS, and RO measurements provide information on the most critical but least understood climate forcings, responses, and feedbacks associated with the vertical distribution of atmospheric temperature and water vapor, broadband reflected and emitted radiative fluxes, cloud properties, surface albedo, and surface skin temperature. Since long-term and global observational RS spectral data with climate quality are not available for now , one has to use observational system simulation experiments (OSSEs) to simulate the RS spectral data for long-term climate investigations [2,3]. The OSSE uses the observed geophysical variables, such as the aerosol and cloud properties, to simulate global RS spectra. The OSSE is a very powerful tool for climate study but it needs at least 10 years of RS spectra to study the climate change [2,3]. The computational burden is so heavy that any current available radiative transfer (RT) algorithm cannot complete the job for all the measured geophysical variables in 10 years. The climate change spectral fingerprinting requires spectra averaged over large-scale space and time, thus the development of a fast and accurate RT model (RTM) is extremely important for both OSSE and climate change fingerprinting study.
We reported a principal component-based radiative transfer model, PCRTM-SOLAR, which is part of the effort devoted to the development of fast radiative transfer models for simulating reflected solar spectrum . In , we also compared the different approaches developed by other research groups. To obtain the top of atmosphere (TOA) channel radiance, the regular PCRTM-SOLAR algorithm solves RT equations at a few hundreds preselected frequencies rather than at the hundreds of thousands of monochromatic frequencies. Therefore, PCRTM-SOLAR is up to one thousand times faster than the widely used MODTRAN. In addition, PCRTM-SOLAR may be integrated with any RT equation solver, such as the successive-orders-of-scattering (SOS) approach [5,6], Monte Carlo (MC) method [6–8], adding-doubling (AD) method [6,9,10], or discrete ordinate (DO) method [6,11,12]. The discrete ordinate method is widely used since it has been found to be efficient and accurate for calculations of scattered intensities and fluxes . To get high accuracy using discrete ordinate method to solve RT equation, one has to use a high enough stream number in the calculation. However, the computation time goes roughly as the cubic of the number of streams . Large number of streams, like 16 and higher, is unaffordable for both OSSEs and climate change fingerprinting. Feldman et al used the number of streams of 8 in their CLARREO shortwave observing system simulation experiments of the twenty-first century .
In this work, we propose a new methodology, converting coarse RT results to highly accurate results by the help of a pre-saved matrix. We demonstrate the idea in this paper by developing a fast and accurate hybrid stream discrete ordinate (HSDO) method for radiative transfer simulation. The regular RT algorithm, such as MODTRAN5, calculates a very high-resolution monochromatic radiance spectrum with a high enough stream number. The results are then convolved with the instrument lineshape function to get the desired highly accurate TOA channel radiance spectrum. To achieve the same accuracy, the HSDO method calculates the monochromatic radiance with a very low stream number and only few of them need to be calculated with a high stream number. Therefore, the HSDO method will greatly reduce the computational burden. When HSDO method is integrated with the principal component RT model, a hybrid stream PCRTM is obtained. The hybrid stream PCRTM-SOLAR (HS-PCRTM-SOLAR) model, as in the regular PCRTM-SOLAR algorithm, simulates the RS channel radiance spectrum by calculating monochromatic radiances at preselected monochromatic frequencies. The difference is that the regular PCRTM-SOLAR has to fulfill the monochromatic radiance calculations at each frequency with very high accuracy to obtain high accuracy channel radiances. The HS-PCRTM-SOLAR method just needs a coarse low stream number calculation at each of the preselected frequencies and a finer high stream number calculation at a small part of these frequencies. The computational speed is accelerated greatly since most of the calculations are completed with a very low stream number in the radiative transfer equation solver.
This hybrid stream strategy can be easily expanded to include other RT equation solvers.
2. Accuracy and speed dilemma
The monochromatic radiance at vertical optical depth (measured from upper boundary to bottom) in discrete ordinate is given by 
To obtain the above results, we assumed that the radiance expansion was the same length as the phase function expansion. In discrete ordinate, the integral in Eq. (3) is approximated by a Gaussian quadrature sum
The integral is approximated to be the summation of 2N streams, thus we may call Eq. (7) 2N-stream approximation. Usually, higher accuracy will be obtained with a larger stream number. However, the computation time in popular discrete ordinate algorithms, such as the discrete-ordinate method radiative transfer (DISORT), increases roughly as the cubic power of the stream number . There is a dilemma between accuracy and speed. One has to compromise accuracy by using a smaller stream number to simulate thousands of hundreds of spectra for climate change study. The number of stream used in Feldman’s study was 8 to simulate the needed spectra. For strongly forward-peaked scattering, a much larger stream number, which is over 16 or even 128, was used . Using 16-stream in MODTRAN5 with a medium speed correlated-k option, it would take over 30 years to simulate one hundred of thousands of TOA channel radiance spectra on a LINUX machine with a CPU speed of 2.3 GHz. The regular PCRTM-SOLAR we developed recently may reduce the time to a few days . Using the HS-PCRTM-SOLAR we proposed in this work, the computation time may be further reduced to a few hours.
3. Hybrid stream discrete ordinate (HSDO) method
We noticed that the simulated radiances obtained from N1-stream approximation and N2-stream approximation are actually have the same inputs: same atmosphere profile, aerosol, cloud, surface, and sun-satellite geometry. When the stream numbers are big enough, the stream number dependence of the simulated radiance should be negligible and both calculations would give results with desired accuracy. However, when one stream number is small, like 2 or 4, and another stream number is big, like over 16, the difference in the obtained radiances may be too big to be ignored. This is due to the fact, as shown in Eq. (7), that the stream number determines the accuracy of the integral calculation.
Figure 1 shows a typical stream number dependence of the TOA radiance of a system with three-layer clouds and a land surface. The simulation was carried out in the solar spectral region with solar radiation. Both ice and water clouds were included in the calculation. The relative error of 4-stream results to 16-stream results was about 3%. In the case of a stronger forward ice cloud case, this error will become much bigger. In general, the error becomes greater with the asymmetric factor and optical depth . Therefore, much higher stream numbers are required to get desired accuracy for climate study using currently available RT algorithms.
However, since the radiances with different stream numbers were obtained from the same inputs, they should be highly related and the coarse low-stream result may be used to predict the finer high-stream result. To identify this idea, we simulated the TOA monochromatic radiances for about 18,000 different cases by using 4-stream, 16-stream and 32-stream discrete ordinate methods. Each of the cases has different atmospheric profile, solar-satellite geometry, surface altitude, surface reflection, and cloud conditions. The single scattering properties were obtained using the parameterization method and lookup tables for water and ice cloud, respectively .
The correlation coefficient between 4-stream and 16-stream results for the same case is very high, as shown in Fig. 2, where most of them are larger than 0.99. Similar results were obtained between 4-stream and 32-stream results.
To investigate the relationship between 4-stream and N-stream results, we assume that the monochromatic radiances need to be simulated at M1 frequencies using the discrete ordinate method, and N is a stream number not less than 16 for high accuracy RT simulation. The relationship between the N-stream monochromatic radiances () and the 4-stream monochromatic radiances () is
One may expect that contains most of the information of , such as the spectral features and spectral magnitude, since they are from the same physical processes. The differences between them are simply from the summation approximation of the integral in Eq. (7). Therefore, these differences must be small compared to the radiances themselves.
Figure 3 shows typical TOA monochromatic radiances (rMono) at 241 preselected frequencies for the same input parameters. They were obtained by using the 4- and 16-stream MODTRAN5, respectively. Obviously, the 4-stream results comprise most of the information of the 16-stream results.
In , Natraj et al used a principal component analysis based spectral binning method to speed up the computational speed while maintaining an overall accuracy of 0.01%. In a given wavelength range, the TOA radiances for the mean optical properties and few perturbed optical property profiles were calculated using both two-stream and N-stream RT solvers. In the shortwave region of the spectrum from 290 to 3,000 nm, the runtimes of their fast RT model are shorter by factors between 10 and 100 compared to the 32-stream LIDORT . In this work, we calculate the radiances in the frequency domain directly at pre-selected frequencies using both low-stream and high-stream RT solvers. It is 3 to 4 orders faster than the 16- and/or 32-stream MODTRAN5 for the 8 nm RS spectrum simulations.
It is well known that the monochromatic radiances are highly correlated; a very small part of them can be used to reproduce the whole spectrum with very high accuracy [4,19]. One may expect that spectrum also has a number of redundant information since it is caused by the same mathematic approximation. Thus one may expect that at M1 monochromatic frequencies may be regressed to at M2 preselected monochromatic frequencies with M2 << M1. That is
Equations (8) and (9) indicate that we just need to calculate the monochromatic radiances with 4-stream approximation () at all M1 monochromatic frequencies and with N-stream approximation at the few preselected M2 monochromatic frequencies. The N-stream accuracy monochromatic radiances are then obtained by submitting Eq. (9) into Eq. (8). When N is large, most of the computation time is from the N-stream calculations. The new method will thus increase the speed greatly and reduce the computation time to about M2/M1 of the already very fast regular PCRTM-SOLAR.
The TOA channel radiances may be obtained using
Here matrix and matrix were obtained using the same training procedure described in the regular PCRTM-SOLAR paper (4). The number of channel is represented by while the number of principal components is given by . The monochromatic radiances, , at the pre-selected frequencies with the desired N-stream accuracy, may be obtained using Eqs. (8) and (9).
Though only very few monochromatic radiances are calculated using N-stream discrete ordinate method, the channel radiances obtained using the three pre-saved matrixes (, , and ) and the 4-stream monochromatic radiances will have N-stream accuracy. The N-stream accuracy channel radiances obtained with the 4-stream calculations greatly reduces the computational burden.
4. Results and discussions
4.1 Training Results for Regular PCRTM-SOLAR
The CLARREO RS spectrum will be 4 nm spectral sampling with an 8-nm spectral resolution. We used both Boxcar and Gaussian apodization functions to simulate the CLARREO RS channel radiances at TOA. The channel radiance at wavelength for the Boxcar and the Gaussian apodizations are given by Eqs. (11) and (12), respectively.
The 1-wavenumber resolution TOA monochromatic radiances from 300 nm to 2500 nm were simulated using MODTRAN5 for thousands of cases with various atmosphere profiles, clouds/aerosols, trace gases, and surfaces . The CLARREO 8 nm resolution channel radiances were then obtained by using Eq. (11) or (12). It was found that the Boxcar apodized channel radiances could be converted from the Gaussian apodized channel radiances easily and vice versa. Therefore, we only discuss the results for the Boxcar apodized channel radiances in this paper.
The monochromatic radiances obtained using MODTRAN5 and the channel radiances obtained using Eqs. (11) and (12) were used in a training process to get the matrix and matrix for the regular PCRTM-SOLAR application. Among the simulated TOA channel radiances and the corresponding monochromatic radiances, 16,250 samples were chosen as the training data while 16,249 samples were chosen for validation for the land surface case. The training results indicate that the monochromatic radiances need to be calculated at only 263 selected frequencies to simulate the RS spectrum using the regular PCRTM-SOLAR. On the other hand, MODTRAN5 needs to calculate monochromatic radiances at 259,029 monochromatic frequencies, followed by a time-consuming convolution calculation. This makes the regular PCRTM-SOLAR around 1,000 times faster than MODTRAN5.
For the ocean surface, 15,178 RS spectra and their corresponding monochromatic radiance spectra were calculated using MODTRAN5 and a convolution algorithm. Half of the data were trained to get the matrix, matrix, and the positions of the selected monochromatic frequencies. The others were used to validate the training results. Only 241 monochromatic frequencies were selected for the regular PCRTM-SOLAR algorithm for the ocean surface case.
The regular PCRTM-SOLAR simulated TOA channel radiances using the training results described above have very high accuracy, as shown by the validation data in Fig. 4. The maximum root-mean-square (RMS) error is less than 10−3 for both land and ocean surfaces. More details on the regular PCRTM-SOLAR results may be found in .
Based on these results, one could further reduce the computational burden but keep the computation accuracy using the HSDO method described in Part 3.
4.2 Training Results for HSDO Method
Though the regular PCRTM-SOLAR is already about 1,000 times faster than MODTRAN5 for the CLARREO RS spectrum simulation, it would be greatly appreciated by the OSSE community if we could further accelerate the calculation. To achieve this goal, we propose to reproduce the time-consuming high-stream monochromatic radiance spectrum using a coarse but fast low-stream monochromatic radiance spectrum. The low-stream spectrum contains most of the information of the high-stream spectrum. The missing information is obtained from the high-stream monochromatic radiances at only a few preselected frequencies. Since only a few high-stream number calculations are needed, this method will further reduce the calculation burden of the regular PCRTM-SOLAR algorithm.
To fulfill this idea, we calculated the monochromatic radiances at the preselected M1 frequencies using a 4-stream, 16-stream, and 32-stream discrete ordinate method, respectively. Their differences were obtained and trained according to Eq. (9). The matrix B, which will be used to simulate the TOA channel radiances using the HSDO method later, was obtained from this training. To obtain the 16-stream accuracy monochromatic radiances at the 263 preselected frequencies, one needs to simulate the 16-stream accuracy monochromatic radiances at only 49 selected monochromatic frequencies for the land surface case. The training also shows that the 16-stream simulation needs to be done at only 23 frequencies to get the 16-stream accuracy monochromatic radiance spectrum at the 241 monochromatic frequencies for ocean surface case. These results are shown in Table 1.
For the high stream number accuracy case, the time needed for the 4-stream calculation is very small compared to that for the N-stream calculation. Therefore, the speedup of the HS-PCRTM-SOLAR to the regular PCRTM-SOLAR may be approximated as for this case. Thus the HS-PCRTM-SOLAR is 5 to 10 times faster than the regular PCRTM-SOLAR. The speedup of the HS-PCRTM-SOLAR to MODTRAN5 are over five thousands times for land surface and over ten thousands times for ocean surface, as shown in Table 1.
To obtain the desired accuracy, we made sure that the maximum RMS error of the HSDO method is smaller than 10−3 compared to the 16-stream monochromatic radiances during training for both land and ocean surfaces. Since the results are similar for both cases, we will show the results for ocean surface only in the following.
As shown in Fig. 5, the RMS errors of the 4-stream MODTRAN5 results are much larger than those of the HSDO results. The corresponding biases of the 4-stream MODTRAN5 results are bigger too. A systematic error is shown in the 4-stream results, which may be caused by the mathematical approximation. The biases of the HSDO results are negligible and no systematic error is found.
The HSDO method is very versatile. The obtained B matrix not only works for 16-stream accuracy but also works for higher-stream accuracy. Figure 6 shows the RMS errors and biases for the 32-stream accuracy case. Very similar results were obtained compared to the 16-stream accuracy case.
4.3 Accuracy of HSDO Method
The accuracy of the HSDO method was checked using an independent validation data set. Figures 7 and 8 show the RMS errors and biases in the monochromatic radiances at the preselected 241 frequencies. The blue lines represent results from the 4-stream MODTRAN5 while the green lines show results obtained using the HSDO method. The 4-stream RMS errors of the validation data are comparable to the RMS errors of the training data. They both are too big to be used for climate applications. The systematic errors also exist in the validation data.
The RMS errors of the results from the HSDO calculations are smaller than 10−3 for both the 16- and the 32-stream accuracy cases. The corresponding biases are very small and no systematic error occurs.
The really used data in OSSEs are the channel radiances rather than the monochromatic radiances. Usually, one has to calculate hundreds of thousands of monochromatic radiances and then using a convolution algorithm to get the channel radiances. This is a very much time-consuming process. However, the HSDO method may obtain the channel radiances easily by using Eq. (10) as soon as the N-stream monochromatic radiances at the 241 frequencies are known.
The HSDO method simulates TOA channel radiance spectrum in a fast and accurate way. Figures 9 and 10 show the RMS errors and the biases in the reproduced channel radiances. In Fig. 9, the results, which are from the 16-stream MODTRAN5 simulations for the monochromatic radiances and from the corresponding convolution calculations for the channel radiances, are considered as the true values. Obviously the 4-stream results are different from the 16-stream results; the RMS errors are about 10−2 . This is consistent with their corresponding monochromatic radiances cases. However, the biases are much smaller than the monochromatic cases. In contrast, the RMS errors for the results from the HSDO method are much smaller than those of the 4-stream results. It is usually smaller than .
In Fig. 10, the 32-stream MODTRAN5 results are considered to be the true values. For this much higher stream number, the RMS errors of the HSDO results are still smaller than for most of the wavelengths. The biases of the HSDO results in both Figs. 9 and 10 are negligible. These results clearly show that HSDO is a highly accurate method for cloudy sky RS spectrum simulation.
We developed a fast and accurate radiative transfer model using a hybrid stream discrete ordinate method for reflected solar spectrum simulation. This model is based on our previous regular PCRTM-SOLAR model and could be integrated with PCRTM-SOLAR easily. The regular PCRTM-SOLAR may use the discrete ordinate algorithm as well as other algorithms to solve the RT equations. When the discrete ordinate method is chosen, the regular PCRTM-SOLAR needs to calculate the monochromatic radiances at the few hundred preselected frequencies with a large enough stream number, N, to get the desired accuracy. In the HSDO method we discussed in this paper, the monochromatic radiances are calculated with the 4-stream discrete ordinate algorithm and only few of them are calculated with both the 4-stream and the N-stream discrete ordinate method. This methodology further reduces the computational burden while it conserves the high accuracy.
The HSDO method is three to four orders of magnitude faster than the medium speed correlated-k option MODTRAN5 and the RMS errors in the reproduced TOA channel radiances are usually smaller than . The HSDO RT model is versatile and can handle multi-layer clouds/aerosols systems with any desired stream number accuracy. It significantly eliminates the accuracy-speed dilemma in the radiative transfer solver using the discrete ordinate method.
The developed HS-PCRTM-SOLAR works in very wide atmospheric and geometrical ranges since the matrix was obtained by training radiance data under various atmospheric profiles, with all kinds of surface conditions, and up to 3-layer of clouds. This model works from 300 nm to 2500 nm. The valid solar zenith angles are from 0 to 80 degrees, the instrument view zenith angles are from 0 to 70 degrees, and the relative azimuthal angles are from 0 to 360 degrees.
The fast and accurate natures of the HSDO method make it a powerful tool for OSSEs and climate fingerprinting studies in the CLARREO project. It also provides a valuable approach for retrieval of cloud properties, aerosols, trace gases and other atmospheric parameters in the solar spectral region. This may greatly improve the efficiency in climate change study and numerical weather prediction.
The authors thank for the NASA CLARREO project, the NASA NPP program, and the NASA SMD high-End Computing (HEC) resources for supporting this work.
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