1. Introduction

Ocean color instruments operating on satellites and low-altitude platforms measure the intensity of spectral water-leaving radiances, ocean surface reflected radiances, and atmospherically scattered radiances at the top-of-atmosphere (TOA). Atmospheric correction is the process of removing atmospheric and sea surface effects in satellite ocean color imagery to retrieve the desired water-leaving radiance or remote sensing reflectance (when the former is normalized by the incident solar irradiance). Achieving the goal with the uncertainty of the water-leaving radiance retrieval within ±5% (see IOCCG, 2013 [1]) is fundamentally difficult due to the dominant contribution of atmospheric scattering radiance to the TOA radiance.

Sufficiently accurate atmospheric correction algorithm should be based on the solutions of radiative transfer equation, which include generating look-up tables (LUTs) for Rayleigh and aerosol properties and / or simulating the TOA radiances and related atmospheric parameters [2]. Presently, standard atmospheric correction algorithms used for global ocean color data processing are based on the LUTs generated for the vertically inhomogeneous / horizontally infinite homogeneous plane parallel (PP) atmospheric layers, which limit ocean color work applicability to low-moderate solar zenith angles (SZAs) (e.g., < 70°) and sensor viewing zenith angles (VZAs) (e.g., < 60°). In the past studies, the effects of Earth curvature / sphericity of the atmosphere were not properly accounted for the radiative transfer simulations, which were used to generate LUTs for standard atmospheric correction algorithms. For retrieving the water constituents at high solar and sensor geometries, it is most appropriate to use the vector radiative transfer model (RTM) calculations with multiple-scattering and polarization for a spherical shell (SS) atmosphere geometry in the incoming and outgoing solar radiation or for a SS atmosphere in the incoming solar radiation and a PP atmosphere in the outgoing radiation after scattered by the ocean-atmospheric system in another simplified method (known as pseudo-spherical correction, [3–6]), despite these radiative transfer calculations being computationally more expensive than the radiative transfer simulations for a plane parallel atmosphere [7,8].

Several studies have performed radiative transfer simulations with the inclusion of relevant physical effects that influence the accuracy of modelled radiance profiles. In earlier studies, the effects of Earth curvature were incorporated into a vector RTM and corrected based on the SS geometry. Lenoble and Sekera [9] have developed a model for radiative transfer equation in SS geometry and used the perturbation method to approximately provide the radiative transfer equation solutions. Herman et al. [10] used the Guass-Seidel iteration method [11] to define SS atmospheric layers and calculated single-scattering and multiple-scattering radiances based on the matrix equations with the splitting schemes. The calculations of the Rayleigh and Mie scattering radiances by their model ignored polarization effects and the sea surface contributions. Xu et al. [12] built an RTM based on the Markov chain method with the assumption of pseudo-sphericity for titan’s atmosphere. Based on Monte Carlo method, Adams and Kattawar [13] and Kattawar and Adams [14] built an RTM for PP and SS atmospheres for calculating TOA radiances using Rayleigh and Henyey-Greenstein (HG) phase functions. Similarly, Adams and Kattawar [13] calculated single-scattering and multiple-scattering radiances for PP and SS atmospheres through the Rayleigh-scattering phase function. For multiple-scattering scenarios, their results demonstrated that for SZA of 0°, VZA of 60° and relative azimuth angle of 0°, the Earth curvature effect is about 3.89%. The relative errors for PP and SS geometries increased to 26.44% with the increasing VZAs. For VZA of 89°, the effects of Earth curvature and atmospheric refraction caused the radiance decay and gave much higher radiances (at least by a factor of two) for PP geometry. Kattawar and Adams [14] used the HG phase function to calculate radiances of the anisotropic scattering properties of the atmospheric dust particles. These results revealed that the error was 7.04% for the same conditions and even increased to 30.92% for some cases. Korkin et al. [15] improved the Adams and Kattawar [13] results by using a Monte Carlo code from the libRadtran package (MYSTIC) [16,17] for SS geometry and the discrete ordinates code (VLIDORT) [18] for PP geometry. However, Adams and Kattawar [13] and Korkin et al. [15] used scalar models without the polarization effects on Rayleigh-scattering radiances [19]. He et al. [20] established a coupled ocean-atmosphere vector RTM based on the matrix-operator method that incorporates the Earth curvature effects for a pseudo-spherical shell (PSS) atmosphere (PCOART-SA) [21]. The comparison was generally good among the PCOART-SA, benchmark model [13,14], CDISORT code [22], and AccuRT code [23] for high SZAs (> 70°). However, the Earth curvature effects on Rayleigh-scattering radiances were significantly high (i.e., 12%), but the PSS method did not use the full multiple-scattering for a pseudo-spherical atmosphere and the PCOART-SA model did not incorporate the Earth curvature effects at very high VZAs. Zhai and Hu [24] built an improved pseudo-spherical shell (IPSS) approximation method to correct the error of pseudo-spherical algorithm on multiple-scattering radiances at high VZAs. It was reported that the ratio of the single-scattering radiances to the multiple-scattering radiances for PP geometry was consistent with that for SS geometry. To verify the accuracy of their method, the simulated Rayleigh-scattering radiances were compared with results from the Adams and Kattawar [13] model. The IPSS method showed an error of less than 2% for SZA = 84.26° and all VZAs. Later, Korkin et al. [25] incorporated the polarization information based on Korkin et al. [15] using the MYSTIC and VLIDORT codes. Earlier studies performed Monte Carlo simulations for a spherical shell atmosphere [24] but without a detailed analysis on the Earth curvature effects on Mie-scattering radiances at high VZAs.

Recently, we implemented a refined spherical vector radiative transfer model using Monte Carlo method (hereafter our MC-SRTM) [26] for simulating TOA radiances at high SZAs and VZAs. However, these results concerned with the Earth curvature effects on Rayleigh-scattering radiances of the ideal molecule atmosphere and revealed that for SZAs < 70° and VZAs > 60°, the Earth curvature effects increased to 3% [25,26]. Other studies have performed radiative transfer calculations for a spherical shell atmosphere and showed the Earth curvature effects on TOA radiances at high SZAs (> 70°). To date, there is a lack of work done on the effects of Earth curvature on both the Rayleigh-scattering and Mie-scattering components at high VZAs (> 60°). Future atmospheric correction algorithms will depend on such radiative transfer simulations for a spherical or pseudo-spherical atmosphere and the required atmospheric LUTs.

In this study, we present an enhanced vector radiative transfer model for a spherical shell atmosphere geometry using Monte Carlo method (MC-SRTM) with considering the effects of Earth curvature, atmospheric conditions, polarization, high solar and sensor geometries, altitudes and wavelengths. The vector radiative transfer simulations are utilized to analyze the effects of Earth curvature on Rayleigh-scattering radiances (due to ideal molecule atmosphere) and Mie-scattering radiances (due to pure aerosol atmosphere) radiances at high SZAs (> 70°) or VZAs (> 60°). The simulated results from MC-SRTM are compared with results from the Kattawar and Adams and PCOART-SA models. Based on MC-SRTM, this study is focused on the Earth curvature effects on the Rayleigh-scattering and Mie-scattering radiances in the presence of ideal molecule atmosphere, pure aerosol atmosphere and aerosol-molecule mixed atmosphere at VZAs > 60°.

2. Model and method

This section presents a refined spherical vector radiative transfer model using Monte Carlo method to simulate TOA radiances at high SZAs and VZAs. The MC-SRTM is an efficient tool to deal with the complex photon interaction processes and perform the detailed multiple-scattering calculations for atmospheric in both PP and SS geometries (Fig. 1). The MC-SRTM simulates the movement of every photon in PP and SS geometries with a typically flat sea surface boundary [26]. The atmosphere consists of vertically inhomogeneous layers but horizontally homogeneous and infinite in a PP geometry. As shown in Fig. 1, for high VZAs, the photons reached at point A1 and A2 in PP and SS geometries. The atmosphere is divided into 32 layers with different heights and optical thicknesses. For high VZAs, the photons have a longer distance and may be scattered many times depending on the size distribution of scattering particles and complex refractive index in a given wavelength.

 

Fig. 1. Schematic of the radiative transfer processes based on the backward Monte Carlo method. Dotted and solid lines represent the radiative transfer processes in PP and SS geometries, respectively. The radiative transfer processes at low-moderate and high VZAs are depicted in red color and black color respectively. Since the path length of the photon increases at high VZAs, many multiple-scattering events could occur.

Download Full Size | PDF

The atmosphere is composed of many components including gaseous molecules, aerosols, ozone, and other constituents. In our earlier studies [26], we analyzed the effects of Earth curvature and ideal molecule atmosphere. In the present study, an aerosol-molecule mixed atmosphere composed of molecule and aerosol particles is considered to study the Mie-scattering influences on TOA radiances as described in the following sections. The input parameters of the MC-SRTM including: 1) the number of the photons, 2) the optical thickness with atmosphere, 3) the solar zenith angles, viewing zenith angles and relative azimuth angels, 4) the radius of the Earth, 5) the depolarization factor with Rayleigh-scattering and 6) the coefficients of the Legendre polynomials with each aerosol model.

2.1 Aerosol-molecule mixed atmosphere scattering method

For an aerosol-molecule mixed atmosphere, we assumed that there are gaseous molecules and aerosol particles in the atmosphere, where Rayleigh-scattering is caused by gaseous molecules and Mie-scattering by aerosol particles. Wang and Gordon [27] presented that total upward radiance at the TOA (${I_t}$) is the sum of five components: 1) Rayleigh-scattering radiance ${I_r}$ due to multiple collisions between solar photons and gaseous molecules, 2) Mie-scattering radiance ${I_a}$ due to multiple collisions between solar photons and aerosol particles, 3) photons’ multiple-interaction scattering radiance ${I_{ra}}$ in an aerosol-molecule mixed atmosphere, 4) the radiance reflecting from the sea surface ${I_s}$, and 5) the water-leaving radiance ${I_w}$ as given by

In the present study, the total radiance ${I_t}$ at the top of the aerosol-molecule mixed atmosphere consists of four components (${I_r}$, ${I_a}$, ${I_{ra}}$, and ${I_s}$). The total radiances in PP and SS geometries are denoted as ${I_{PP}}$ and ${I_{SS}}$, respectively. And the Earth curvature effect is analyzed by a factor P.

Consequently, we calculate three factors for the aerosol-molecule mixed atmosphere ${P_t}$, ideal molecule atmosphere ${P_r}$, and pure aerosol atmosphere ${P_a}$ as

Since the term ${I_a} + {I_{ra}}$ is regarded as the total aerosol radiance, the Earth curvature effect on Mie-scattering radiances due to the total aerosol particles is given by ${P_a} + {P_{ra}}$

To analyze the Earth curvature effect at high VZAs, the values of these parameters are presented in section 4.

Having described the Rayleigh-scattering calculations in our previous paper [26], we introduced the Mie-scattering calculations based on phase matrix using the MC-SRTM model. Using the vector radiative transfer theory [26], the scattering radiance ${{\boldsymbol I}_f}$ can be calculated from the incident radiance ${{\boldsymbol I}_i}$ [28]

Based on the Rayleigh-scattering method [31,32], the ideal Rayleigh-scattering phase matrix is obtained from

Because anisotropy is a fundamental property of atmospheric molecule, the Rayleigh-scattering phase matrix is rewritten as

For anisotropic scattering, the scaled HG scattering phase function is given by

According to the Monte Carlo method, $\phi (\psi )$ is assumed as a uniform density function given by [14].

The distribution function $Q(\psi )$ is given by

Based on the Monte Carlo method, R is defined as a random number between 0 and 1. Hence, $\psi $ is obtained by

According to the previous studies [34,35], a strongly forward-peaked Mie-scattering phase matrix (Eq. (6)) that includes six elements (${a_1}$, ${a_2}$, ${a_3}$, ${a_4}$, ${b_1}$, and ${b_2}$) can be decomposed by Legendre polynomial as

2.2 Aerosol model

Shettle and Fenn [36] and others built a number of aerosol models representative of the different types of aerosols based on their microphysical and optical properties, which are usually described by particle size distribution, scattering phase function, and single scattering albedo. Based on the basic aerosol models from Shettle and Fenn [36], He et al. [37] had established twenty aerosol models for the HY-1B/COCTS sensor with the different mixing ratios and relative humidities, which include five aerosol types such as Oceanic aerosol (O), Coastal aerosol (C), Maritime aerosol (M), Urban aerosol (U) and Tropospheric aerosol (T). In other studies, for the SeaWiFS and MODIS sensors, eighty aerosol models have been established for regional and global applications [38]. In this study, we used the backward Monte Carlo method to simulate the radiative transfer process to improve computational efficiency. Thus, it was assumed that the photons are emitted from the sensor through the radiative transfer process (scattering, sea surface reflection, etc.) and turn to the sunlight incidence direction. Thus, all photons can contribute to the received signal.

In the present study, the effects of Earth curvature on Mie-scattering radiances were analyzed at the visible and near-infrared (NIR) wavelengths (only considered 412 and 865 nm for brevity) using the two typical aerosol models (represented by coastal and tropospheric models) (Table 1).

Table 1. The aerosol models we used in this study

View Table | View all tables in this article

According to Wu et al. [39] and Song et al. [40], the Gaussian-like vertical distribution of the aerosol optical thickness is calculated as

Based on the US Standard Atmosphere Model and previous study by Song et al. [40], the vertical column of the ideal molecule and pure aerosol atmospheres is divided into 32 layers with respect to the different mean optical thicknesses of the gaseous molecules and aerosols (Table 2). For these calculations, we considered the following parameters: optical thickness of total gaseous molecules = 0.3099 and 0.0154 at 412 and 865 nm respectively, optical thickness of aerosols = 0.5, and total atmospheric height = 100 km. For aerosol-molecule mixed atmosphere, we assumed that the atmosphere is divided into 32 layers in vertical direction, and the optical thickness of each layer are the added of the gaseous molecule and aerosol. The probability of Rayleigh-scattering and Mie-scattering of photons in the atmosphere is calculated by the percentage of the optical thickness of gaseous molecules and aerosols in each layer.

Table 2. The optical thicknesses of gaseous molecules and aerosols for each atmospheric layer

View Table | View all tables in this article

3. Results and discussion

Based on the MC-SRTM, the scalar and vector radiances were simulated for PP and SS geometries and validated with results from the Kattawar and Adams [14] and PCOART-SA models. In Kattawar and Adams [14], backward Monte Carlo simulations with the HG phase function provided scalar results at TOA for PP and SS geometries. Similarly, the PCOART-SA model (with the assumption of pseudo-spherical shell approximation) was used to simulate the vector TOA radiances for pure aerosol atmosphere. Validations revealed that our MC-SRTM simulations were in good agreement with PCOART-SA results and literature benchmarks at high solar and sensor geometries.

3.1 Comparison with the Kattawar and Adams results

According to our previous work [26], the MC-SRTM performed Rayleigh-scattering simulations for the ideal molecule atmosphere with great accuracy at high SZAs and VZAs. In the present study, the effects of anisotropic scattering of the atmospheric aerosols are incorporated into our MC-SRTM based on the HG phase function. Kattawar and Adams [14] gave the detailed values of scalar TOA radiance using the HG phase function. For the HG phase function, the asymmetry factor ($g$) is an important aerosol radiative forcing assessment parameter and set as 0.5 and 0.7 in this study. The SZAs, VZAs and relative azimuth angles are set as same as our previous study [26]. The origin of the local coordinates is the center of the observation point, and the z-axis is the line connecting the center of the Earth and the observation point. The X-axis was defined by the solar radiation projected on the XOY plane. The relative azimuth angle of 0° means that the projection of the direction of the detector receiving photons was X-axis. And the relative azimuth angle of 180° was that the projection was the opposite direction of X-axis. Here we showed the schematic of the relative azimuth angles (ϕ) of 0° and 180°in Fig. 2. The results from our MC-SRTM and Kattawar and Adams (hereafter K & A) model are shown in Figs. 3 and 4 (for $g$ = 0.5). The relative difference among these two models is generally less than 5%, which is slightly higher than that among the previous Rayleigh-scattering simulations.

 

Fig. 2. Schematic of the definition of relative azimuth angles (0° and 180°). Red line represents the solar radiation. Blue and green lines represent the detector receiving directions with relative azimuth angles of 0° and 180°, respectively.

Download Full Size | PDF

 

Fig. 3. Comparison of the TOA radiances simulated by our MC-SRTM and K & A model. The asymmetry factor g was set as 0.5. The optical thicknesses of (a) to (c) and (d) to (f) are 0.25 and 1.0, respectively. The blue hollow circle curve and the black hollow square curve represent the results of K & A model for PP and SS geometries, respectively. The green star curve and red star curve represent the results of our MC-SRTM for PP and SS geometries, respectively.

Download Full Size | PDF

 

Fig. 4. The relative differences (%) calculated from the MC-SRTM and K & A model results for PP and SS geometries with $g$ = 0.5. The optical thicknesses of (a) to (c) and (d) to (f) are 0.25 and 1.0, respectively. The relative difference is defined as $100\textrm{\%} \times ({{I_{reference\textrm{}model}} - {I_{our\textrm{}model}}} )/{I_{reference\textrm{}model}}$.

Download Full Size | PDF

Validation results (as shown in Figs. 3 and 4) showed that when the total optical thickness ($\tau $) is 0.25, the mean relative differences between MC-SRTM and K & A results were 2.24% and 2.53% for PP and SS geometries, respectively. For SZA of 0$^\circ $ and low-moderate VZAs (< 60$^\circ $), the mean relative differences were 0.97% and 1.26% for PP and SS geometries, respectively. The maximum relative differences were 3.04% and 4.22%. For same SZA, the mean relative differences were 0.18% and 2.22% at high VZAs (> 60$^\circ $) for PP and SS geometries, respectively. And the maximum relative differences were 0.23% and 5.15%. When $\tau $ reached a higher value of 1.0, the mean relative differences between MC-SRTM and K & A results increased to 3.05% and 3.19% at all SZAs and VZAs for PP and SS geometries, respectively. For SZA of 0$^\circ $ and low-moderate VZAs (< 60$^\circ $), the mean relative differences were 1.42% and 2.5% for PP and SS geometries, respectively. And the mean relative differences were 0.66% and 2.12% for SZA of 0$^\circ $ and high VZAs (> 60$^\circ $).

In another scenario with $g = 0.7$ and $\tau $ = 0.25, the mean relative differences between MC-SRTM and K & A results were 4.69% and 4.20% for PP and SS geometries, respectively (as shown in Figs. 5 and 6). These differences increased rapidly to 33.91% and 13.7% as the asymmetry factor increased and hence the scattering of atmospheric dust aerosols became significant and predominantly peaked on the forward angles. When $g = 0.7$ and $\tau $ = 1.0, the mean relative differences were 3.59% and 4.84% for PP and SS geometries, respectively. Note that for SZA = 0$^\circ $, the relative differences with the two asymmetry factors and two optical thicknesses became small as compared to the results for SZA > 70$^\circ $. At high SZAs, photons travel a long distance and experience multiple collisions; at SZA = 0$^\circ $, photons travel a short distance and experience fewer collisions. Because photons scattered multiple times caused the higher level of uncertainty in the radiative transfer process, the relative differences increased for the higher SZAs.

 

Fig. 5. Same as Fig. 3 but for $g$ = 0.7.

Download Full Size | PDF

 

Fig. 6. Same as Fig. 4 but for $g$ = 0.7.

Download Full Size | PDF

Kattawar and Adams [14] reported that photons of different scattering orders would arrive at different places and a different delay time on the inhomogeneous atmospheric layer. Their scattering calculations showed that multiple-scattering of the HG phase function will increase the randomness of the scattering signal as compared to single-scattering, thereby increasing the uncertainty of radiation. We calculated the single-scattering radiances of HG scattering and compared with K & A results to decrease the randomness by multiple-scattering. And the comparisons were presented with some detail in the Appendix A. Besides, limited by the computation power in early years, the results from K & A contained high uncertainty. Since the number of the photons also has significant impact on the radiances. We analyzed the effects of the photons’ number in the Appendix A. These results proved that the relative differences of multiple-scattering are caused by the randomness of the Monte Carlo simulation.

A comparison of the PCOART-SA and K & A results showed a mean relative difference of larger than 4% at low-moderate VZAs (< 60°) [14]. This difference is mainly caused by the uncertainty associated with Monte Carlo simulations as confirmed in our previous study [26].

3.2 Comparison with the PCOART-SA model results

The PCOART-SA model was previously used to study the effects of Earth curvature based on the matrix-operator method with the assumption of pseudo-spherical shell approximation. According to He et al. [20,21], the PCOART-SA model can calculate the TOA vector radiances generated by multiple-scattering atmospheric aerosol effects, which is different from the RTM with the assumption of PP geometry that only considers the vertical variation of atmospheric optical properties. In the present study, we used two aerosol models for the PCOART-SA model to produce the results at two wavelengths (as shown in Table 1). These results were generated with the inputs of $\tau $ = 0.5, SZAs = 20°, 50°, 70°, and 80°, VZAs = 0° to 83.93°, and relative azimuth angles = 0° to 180°. The underlying boundary interface between the atmosphere and ocean surface was assumed as a flat sea surface with the water refractive index of 1.34.

The relative differences in the simulated radiance between PCOART-SA and MC-SRTM are shown in Fig. 7 (with the C50-412 aerosol model). Since there is strong sun glint impact when SZAs are close to VZAs, the relative differences were not calculated for VZAs equal to SZAs ± 5°. For example, the relative differences were ignored at VZAs = 15° to 25° for SZA = 20° due to the influence of the sun glint. As shown in Fig. 7, the relative differences between the PCOART-SA and MC-SRTM are generally lower than 1.5% for SZAs of 20°, 50°, 70° and 80°. The maximum and mean relative differences are 2.16% and 0.60% for SZAs > 70° and VZAs < 60°, respectively. For high SZAs (> 70°) and VZAs (> 60°), the maximum and mean relative differences are 2.83% and 1.32% respectively. These results suggest that for relatively low-moderate VZAs (< 60°) and high SZAs (> 70°), MC-SRTM results with the Mie-scattering component are consistent with PCOART-SA results.

 

Fig. 7. The relative differences (%) of the Mie-scattering radiances between PCOART-SA model and our MC-SRTM with the C50-412 aerosol model. The SZA represents the solar zenith angle. The radius of each plot represents the viewing zenith angle, and the circumferential direction represents the relative azimuth angle.

Download Full Size | PDF

Figures 8–10 show the relative differences between MC-SRTM and PCOART-SA results for C50-865, T90-412, and T90-865 aerosol models, respectively. Note that the relative differences are closely consistent with the results of C50-412 aerosol model. For SZAs > 70° and VZAs < 60°, the relative differences are mostly less than 1.5% for all aerosol models. For the same conditions, the maximum differences are 2.57%, 1.81%, and 2.67% and the mean relative differences are 0.57%, 0.58%, and 0.81% for the C50-865, T90-412, and T90-865 aerosol models. However, for high SZAs (> 70°) and VZAs (> 60°), the relative differences between these two models reached up to 3.89%, 3.14%, and 3.75% for the three aerosol models. The corresponding mean values are 0.75%, 0.75%, and 0.81% for all SZAs and VZAs. These results verified the accuracy of our MC-SRTM for the aerosol-loaded atmosphere. The increased relative differences at higher SZAs and VZAs outputs are caused due to the fact that the PCOART-SA model is based on the pseudo-spherical hypothesis, in which the multiple-scattering radiances is still calculated in a PP geometry.

 

Fig. 8. Same as Fig. 7 but for the C50-865 aerosol model.

Download Full Size | PDF

 

Fig. 9. Same as Fig. 7 but for the T90-412 aerosol model.

Download Full Size | PDF

 

Fig. 10. Same as Fig. 7 but for the T90-865 aerosol model.

Download Full Size | PDF

4. Earth curvature effects on the Rayleigh-scattering and Mie-scattering computation

Although Korkin et al. [25] and Zhai and Hu [24] have studied the effects of Earth curvature on Rayleigh-scattering radiances with the inclusion of polarization properties for high VZAs, their study lacked the analysis of Mie-scattering effects. For an aerosol-molecule mixed atmosphere, there is the interaction of Rayleigh and Mie scattering processes and an uncertainty in the radiance calculations caused by the Earth curvature effects at high SZAs and VZAs. Using our MC-SRTM, the Earth curvature effects on TOA radiances were calculated for ideal molecule, pure aerosol, and aerosol-molecule mixed atmospheres. For these calculations, SZAs = 20°, 50°, and 80°, VZAs = 0° to 84.55°, and relative azimuth angles = 0° to 180°. Further, it was considered that the Earth curvature effects on Mie-scattering radiances are the result of the aerosol-molecule mixed atmosphere minus the ideal molecule atmosphere. This resulted the effects of Earth curvature on Mie-scattering radiances due to the total aerosol particles.

4.1 Earth curvature effects on the aerosol-molecule mixed atmosphere

To analyze the Earth curvature effects on an aerosol-molecule mixed atmosphere, the MC-SRTM was implemented on both PP and SS geometries to calculate TOA radiances of the aerosol-molecule mixed atmosphere (including Rayleigh-scattering radiances due to gaseous molecules and Mie-scattering radiances due to two aerosol models) at two wavelengths (412 nm and 865 nm). The relative errors of the MC-SRTM results for PP and SS geometries are shown in Fig. 11. The Earth curvature effects on TOA radiances of the aerosol-molecule mixed atmosphere with C50-412, C50-865, T90-412, and T90-865 aerosol models are notably similar without many variations. The Earth curvature effects with these aerosols are less than 2.24% when SZAs < 80°. However, it increased to 12.45% at high SZA (80°). The Earth curvature effects on TOA radiances of the aerosol-molecule mixed atmosphere are around 2.23% for SZAs < 70° and VZAs > 60°. For high SZA (80°) and low-moderate VZAs (< 60°), the maximum relative error is 4.63%. These results suggest that the Earth curvature effects on TOA radiances of the aerosol-molecule mixed atmosphere cannot be ignored for high SZAs or VZAs. Our results showed that the Earth curvature effects with all aerosol models reached up to 12.45% at SZA = 80° and VZAs > 60°.

 

Fig. 11. (a) to (c) The relative errors (%) of the MC-SRTM simulations for both PP and SS geometries at three solar zenith angles (20°, 50°, and 80°). (1) to (4) Four aerosol-molecule mixed atmosphere with two aerosol models at two wavelengths (C50-412, C50-865, T90-412, and T90-865). The radius of each plot represents the viewing zenith angle, and the circumferential direction represents the relative azimuth angle. The relative error is defined as $100\textrm{\%} \times ({{I_{PP}} - {I_{SS}}} )/{I_{PP}}$.

Download Full Size | PDF

Here, we selected only five relative azimuth angles (0°, 45°, 90°, 135° and 180°) to depict the fluctuation trends of the Earth curvature effects on an aerosol-molecule mixed atmosphere (Fig. 12). For SZAs < 80° and VZAs < 60°, the Earth curvature effects on an aerosol-molecule mixed atmosphere increased from 0% to 1% with increasing VZAs. The relative errors became negative at VZA = 84.55°, which means that the TOA radiances for a PP geometry are small compared to the TOA radiances for a SS geometry. For SZA = 80°, the relative errors are negative at VZAs < 60° which are similar to the results at SZAs < 70° and VZAs > 60°. And the Earth curvature effects are greater than 5% at SZAs > 70° and VZAs > 60° due to the photons’ long path length in SS geometry.

 

Fig. 12. Same as Fig. 11 but for only five relative azimuth angles (0°, 45°, 90°, 135°, and 180°).

Download Full Size | PDF

4.2 Earth curvature effects on pure aerosol atmosphere

Based on our MC-SRTM with the Mie-scattering component for PP and SS geometries, the TOA radiances of pure aerosol atmosphere were analyzed due to the Earth curvature effect. As shown in Figs. 13 and 14, the relative errors of the MC-SRTM simulations for PP and SS geometries are similar to the results of the aerosol-molecule mixed atmosphere. For low-moderate SZAs (20° and 50°) and VZAs (< 60°), the relative errors are less than 1.86% (as shown in Fig. 13). Under the same SZAs, the relative errors reached up to 2.60% at high VZAs. For SZA = 80°, the relative errors reached a maximum of 5.36% and 10.06% for low-moderate VZAs and high VZAs, respectively. These results demonstrate that the Earth curvature has a significantly greater effect on the TOA radiances of pure aerosol atmosphere at SZAs than at VZAs.

 

Fig. 13. Same as Fig. 11 but for pure aerosol atmosphere.

Download Full Size | PDF

 

Fig. 14. Same as Fig. 12 but for pure aerosol atmosphere.

Download Full Size | PDF

As shown in Fig. 14, the Earth curvature has a similar effect on both pure aerosol and aerosol-molecule mixed atmospheres. At SZAs < 70° and VZAs < 60°, its effect is less than 1%; however, the relative errors reached up to 5.36% at SZAs < 70° and VZAs > 60°. The maximum relative error at five relative azimuth angles is 10.06% for SZA = 80° and VZAs > 60°.

4.3 Earth curvature effects on the ideal molecule atmosphere

The Earth curvature effects on the TOA radiance (at 412 nm wavelength) of the ideal molecule atmosphere were calculated according to Xu et al. [26]. Based on our previous study, we added the relative errors of the MC-SRTM simulations between PP and SS geometries at 865 nm as shown in Fig. 15. For a visible wavelength at 412 nm, the Earth curvature effects were less than 2.56% for all VZAs and SZAs < 70°. For a NIR wavelength (865 nm), the Earth curvature effects reached up to 9.93% at the same SZAs and VZAs. These effects were caused by the differences of the optical thickness. The total optical thickness was 0.31 at 412 nm, which is much higher than that at 865 nm and caused by a number of multiple-scattering events that reduced the effects of Earth curvature.

As shown in Fig. 16, the relative errors are larger than 1% and even reached to 3.76% at 865 nm for SZAs < 70° and VZAs < 60°. For the same solar and viewing geometry, the Earth curvature effects are generally less than 1% at 412 nm. For SZA = 80° and VZAs < 60°, the relative errors are about 2% at 412 nm. And the relative errors at 865 nm even reached up to 4.05%. For SZA = 80° and VZAs > 60°, the maximum relative errors are 7.47% and 10.81% at 412 nm and 865 nm, respectively.

 

Fig. 15. Same as Fig. 11 but for the ideal molecule atmosphere at 412 and 865 nm.

Download Full Size | PDF

 

Fig. 16. Same as Fig. 12 but for the ideal molecule atmosphere at 412 and 865 nm.

Download Full Size | PDF

4.4 Earth curvature effects on Mie-scattering radiances

Based on our MC-SRTM, the Earth curvature effects on Mie-scattering radiances are caused by the effects of total aerosol particles and hence calculated using Eq. (4). Considering the different optical thicknesses of the atmospheric gaseous molecules and aerosols, one can rearrange the influencing terms as

Figures 17 and 18 show the effects of Earth curvature on Mie-scattering radiances. At 412 nm, the optical thickness of the gaseous molecules is close to that of the aerosols. The Earth curvature effects were slightly small relative to that of the aerosol-molecule mixed atmosphere at low-moderate SZAs (20° and 50°) and VZAs. The increasing trend of this effect with respect to VZAs also became weak. In contrast, when VZA reached up to 84.55°, the Earth curvature effects on Mie-scattering radiance became larger than that of the aerosol-molecule mixed atmosphere.

 

Fig. 17. Same as Fig. 11 but for the Earth curvature effects on Mie-scattering radiances. The relative differences (%) of single-scattering radiances by MC-SRTM and K & A model with asymmetry factor $g = 0.5$. The optical thicknesses of (a) to (c) and (d) to (f) are 0.25 and 1.0, respectively.

Download Full Size | PDF

 

Fig. 18. Same as Fig. 12 but for the Earth curvature effects on Mie-scattering radiances.

Download Full Size | PDF

Note that at 865 nm, the optical thickness of the gaseous molecules is very low relative to that of aerosols and the Earth curvature effects on Mie-scattering radiances follow the same trend as that of the aerosol-molecule mixed atmosphere.

5. Conclusion

The vector radiative transfer model, as previously used in our study, was refined (MC-SRTM) with the inclusion of Mie-scattering component and other parameters in the presents study. The MC-SRTM efficiently generated a vast number of simulations for analyzing the effects of Earth curvature on Mie-scattering radiances for an aerosol atmosphere at high solar or sensor geometries. The simulated Mie-scattering radiances from our MC-SRTM were compared with the results of Kattawar and Adams [14] and the PCOART-SA models [21]. These results showed that the relative differences between MC-SRTM and K & A model results with the HG phase function is generally small (< 5%). Further, model simulations were performed with polarization effects and two aerosol models – representative of the aerosol-loaded atmosphere in the absence of gaseous molecules and these results were analyzed at two visible and NIR wavelengths. The relative difference between MC-SRTM and PCOART-SA results was less than 2% and reached up to 2.67% for SZAs > 70° and VZAs < 60°. The comparison of our MC-SRTM and K & A and PCOART-SA results verified the accuracy of the former approach for high solar and sensor geometries.

Our MC-SRTM simulations on the effects of Earth curvature on the three atmospheric conditions (determined by ideal molecule, pure aerosol, and aerosol-molecule mixed atmospheres), revealed that the Earth curvature is weak on these atmospheres at low-moderate SZAs (< 70°) and VZAs (< 60°). However, the Earth curvature effects on TOA radiances for three atmospheric conditions became slightly high (4.09%, 5.36%, and 4.63%) at high SZAs and low-moderate VZAs. For SZAs < 70° and VZAs > 60°, the Earth curvature effects reached up to 9.93%, 2.60%, and 2.23%. The maximum relative errors for PP and SS geometries were 10.81%, 10.06% and 12.43% at all SZAs and VZAs for three atmospheric conditions respectively. Finally, we analyzed the Earth curvature effects on the Mie-scattering radiances of the aerosol-molecule mixed atmosphere and found similar effects as pure aerosol atmosphere.

Overall, the refined MC-SRTM incorporated all important optical processes and produced simulations useful for analyzing the Earth curvature effects on the TOA radiances for different atmospheric conditions. The present analysis neglected the effects of sea surface roughness and whitecaps on the TOA radiances. It is also highly desired to incorporate more aerosols into our MC-SRTM to analyze the effects on the TOA radiances, which will help in establishing look up tables for each aerosol models and that can be applied for atmospheric correction of satellite ocean color remote sensing data.

Appendix A. Comparison between MC-SRTM and Kattawar & Adams.

The relative differences of the HG scattering radiances between MC-SRTM and K & A have a significant fluctuation. To avoid the randomness of the scattering signal caused by multiple-scattering, we calculated the single-scattering radiances and compared with K & A as shown in Fig. 19 and 20. As shown in Fig. 19, the relative differences between two models with asymmetry factors $g$ = 0.5 are mostly less than 0.2%. The relative differences larger than 1% such as SZA of 70.47°, VZA of 70° and relative azimuth angle of 0° were most probably because of the uncertainty of the Monte Carlo simulation. Kattawar & Adams established the model and calculated the results in 1978 when the computation power was limited. So their results contained high uncertainty. Besides, the relative differences with $g$ = 0.7 in Fig. 20 are larger than them with $g$ = 0.5 because of the stronger forward-peaked scattering. And the range of fluctuations of single-scattering are smaller than multiple-scattering significantly. The comparison results above showed that the relative differences of multiple-scattering are caused by the randomness of the Monte Carlo simulation.

 

Fig. 19. The relative differences (%) of single-scattering radiances by MC-SRTM and K & A model with asymmetry factor $g = 0.5$. The optical thicknesses of (a) to (c) and (d) to (f) are 0.25 and 1.0, respectively.

Download Full Size | PDF

 

Fig. 20. Same as Fig. A but for $g = 0.7$

Download Full Size | PDF

To minimize the uncertainty, these simulations needed more photons for multiple-scattering calculations. For the K & A model, the number of photons was unknown. To validate the errors caused by the photons number, we calculated the radiances using HG phase function with 100000, 1 million and 2 million photons. The relative errors with the calculations for 100000 and 1 million and 2 million photons with the asymmetry factors $g$ = 0.7 and $\tau $ = 0.25 or 1.0 are shown in Figs. 21 and 22. The relative error was generally less than 2% for SZA = 0°. The errors reached up to 6.36% for $\tau $ = 0.25 and SZA = 84.26°. When $\tau $ = 1.0, the errors increased even more rapidly to 17.21% at SZA = 84.26°. These results indicate the higher influence of the photons number with the increasing optical thickness and SZAs.

 

Fig. 21. The relative errors (%) associated with our MC-SRTM results for the different photon number and $\tau $ = 0.25 and $g$ = 0.7. (a) to (c) are the relative errors between 100000 and 1 million, and (d) to (f) are the relative errors between 100000 and 2 million.

Download Full Size | PDF

 

Fig. 22. Same as Fig. 21 but for $\tau $ = 1.0.

Download Full Size | PDF

Funding

National Natural Science Foundation of China (#41825014, #42176177, #U22B2012); Natural Science Foundation of Zhejiang Province (#LDT23D06021D06); "Pioneer" R&D Program of Zhejiang (2023C03011); PI Project of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2021GD0809).

Acknowledgment

We thank the staffs of the satellite ground station, satellite data processing & sharing center, and marine satellite data online analysis platform of the State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Ministry of Natural Resources (SOED/SIO/MNR) for their help with simulations.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. R. Frouin. (ed.), “In-flight Calibration of satellite ocean-colour sensors,” Reports of the International Ocean-Colour Coordinating Group, No. 14. Dartmouth, Canada: IOCCG, (2013).

2. R. J. Frouin, B. A. Franz, A. Ibrahim, et al., “Atmospheric correction of satellite ocean-color imagery during the PACE Era,” Frontiers in Earth Science 7(2019).

3. J. J. Deluisi and C. L. Mateer, “On the application of the optimum statistical inversion technique to the evaluation of Umkehr observations,” J. Appl. Meteor. 10(2), 328–334 (1971). [CrossRef]  

4. J. V. Dave, Development of programs for computing characteristics of ultraviolet radiationMd. Gaithersburg, (IBM Corporation, Federal Systems Division, 1972).

5. A. Dahlback and K. Stamnes, “A new spherical model for computing the radiation field available for photolysis and heating at twilight,” Planet. Space. Sci. 39(5), 671–683 (1991). [CrossRef]  

6. I. Petropavlovskikh, R. Loughman, J. Deluisi, et al., “A comparison of UV intensities calculated by spherical-atmosphere radiation transfer codes: Application to the aerosol corrections,” J. Geophys. Res. 105(D11), 14737–14746 (2000). [CrossRef]  

7. D. G. Collins, W. G. Blättner, M. B. Wells, et al., “Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres,” Appl. Opt. 11(11), 2684–2696 (1972). [CrossRef]  

8. B. L. Granovskii and S. M. Ermakov, “The Monte Carlo method,” J. Sov. Math. 7(2), 161–192 (1977). [CrossRef]  

9. J. Lenoble and Z. Sekera, “Equation of radiative transfer in a planetary spherical atmosphere,” Proc. Natl. Acad. Sci. U.S.A. 47(3), 372–378 (1961). [CrossRef]  

10. B. M. Herman, A. Ben-David, and K. J. Thome, “Numerical technique for solving the radiative transfer equation for a spherical shell atmosphere,” Appl. Opt. 33(9), 1760–1770 (1994). [CrossRef]  

11. B. Herman and S. Browning, “A Numerical Solution to the Equation of Radiative Transfer,” J. Atmos. Sci. 22(5), 559–566 (1965). [CrossRef]  

12. F. Xu, R. A. West, and A. B. Davis, “A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere,” J. Q. Spectrosc. Radiat. Transf. 117, 59–70 (2013). [CrossRef]  

13. C. N. Adams and G. W. Kattawar, “Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering,” Icarus 35(1), 139–151 (1978). [CrossRef]  

14. G. W. Kattawar and C. N. Adams, “Radiative transfer in spherical shell atmospheres: II. Asymmetric phase functions,” Icarus 35(3), 436–449 (1978). [CrossRef]  

15. S. Korkin, E.-S. Yang, R. Spurr, et al., “Revised and extended benchmark results for Rayleigh scattering of sunlight in spherical atmospheres,” J. Q. Spectrosc. Radiat. Transf. 254, 107181 (2020). [CrossRef]  

16. C. Emde, R. Buras, B. Mayer, et al., “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10(2), 383–396 (2010). [CrossRef]  

17. C. Emde, R. Buras-Schnell, A. Kylling, et al., “The libRadtran software package for radiative transfer calculations (version 2.0.1),” Geosci. Model Dev. 9(5), 1647–1672 (2016). [CrossRef]  

18. R. J. D. Spurr, “VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media,” J. Q. Spectrosc. Radiat. Transf. 102(2), 316–342 (2006). [CrossRef]  

19. M. I. Mishchenko, A. A. Lacis, and L. D. Travis, “Errors induced by the neglect of polarization in radiance calculations for rayleigh-scattering atmospheres,” J. Q. Spectrosc. Radiat. Transf. 51(3), 491–510 (1994). [CrossRef]  

20. X. He, Y. Bai, Q. Zhu, et al., “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Q. Spectrosc. Radiat. Transf. 111(10), 1426–1448 (2010). [CrossRef]  

21. X. He, K. Stamnes, Y. Bai, et al., “Effects of Earth curvature on atmospheric correction for ocean color remote sensing,” Remote Sens. Environ. 209, 118–133 (2018). [CrossRef]  

22. R. Buras, T. Dowling, and C. Emde, “New secondary-scattering correction in DISORT with increased efficiency for forward scattering,” J. Q. Spectrosc. Radiat. Transf. 112(12), 2028–2034 (2011). [CrossRef]  

23. B. Hamre, S. Stamnes, K. Stamnes, et al., “AccuRT: A versatile tool for radiative transfer simulations in the coupled atmosphere-ocean system,” AIP Conf. Proc. 1810, 120002 (2017). [CrossRef]  

24. P.-W. Zhai and Y. Hu, “An improved pseudo spherical shell algorithm for vector radiative transfer,” J. Q. Spectrosc. Radiat. Transf. 282, 108132 (2022). [CrossRef]  

25. S. Korkin, E. S. Yang, R. Spurr, et al., “Numerical results for polarized light scattering in a spherical atmosphere,” J. Q. Spectrosc. Radiat. Transf. 287, 108194 (2022). [CrossRef]  

26. F. Xu, X. He, X. Jin, et al., “Spherical vector radiative transfer model for satellite ocean color remote sensing,” Opt. Express 31(7), 11192–11212 (2023). [CrossRef]  

27. H. R. Gordon and M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33(3), 443–452 (1994). [CrossRef]  

28. G. W. Kattawar and C. N. Adams, “Stokes Vector Calculations of the Submarine Light Field in an Atmosphere- Ocean with Scattering According to a Rayleigh Phase Matrix: Effect of Interface Refractive Index on Radiance and Polarization,” Limnol. Oceanogr. 34(8), 1453–1472 (1989). [CrossRef]  

29. W. Rooij and C. Stap, “Expansion of Mie scattering matrices in generalized spherical function,” Astron. Astrophys. 131, 237–248 (1984).

30. M. Frost and E. Huber, “Light scattering by small particles,” Aqua 47, 87–94 (1998). [CrossRef]  

31. S. Chandrasekhar and S. Chandrasekhar, Radiative Transfer, (Clarendon, Oxford, 1950), Chap. 1.

32. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16(4), 527–610 (1974). [CrossRef]  

33. A. T. Young, “Revised depolarization corrections for atmospheric extinction,” Appl. Opt. 19(20), 3427–3428 (1980). [CrossRef]  

34. W. Wiscombe, “The Delta– M Method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34(9), 1408–1422 (1977). [CrossRef]  

35. J. Joseph, W. Wiscombe, and J. Weinman, “The Delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 8 (1977). [CrossRef]  

36. E. P. Shettle and R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” Air Force Geophysics Laboratory, Air Force Systems Command, United States Air Force79(214), (1979).

37. X. He, Y. Bai, D. Pan, et al., “Primary analysis of the ocean color remote sensing data of the HY-lB/COCTS,” Proc. SPIE 7150, 715003 (2008). [CrossRef]  

38. Z. Ahmad, B. A. Franz, C. R. McClain, et al., “New aerosol models for the retrieval of aerosol optical thickness and normalized water-leaving radiances from the SeaWiFS and MODIS sensors over coastal regions and open oceans,” Appl. Opt. 49(29), 5545–5560 (2010). [CrossRef]  

39. Y. Wu, M. de Graaf, and M. Menenti, “The impact of aerosol vertical distribution on aerosol optical depth retrieval using CALIPSO and MODIS data: Case study over dust and smoke regions,” J. Geophys. Res-Atmos. 122(16), 8801–8815 (2017). [CrossRef]  

40. Z. Song, X. He, Y. Bai, et al., “Effect of the vertical distribution of absorbing aerosols on the atmospheric correction for satellite ocean color remote sensing,” IEEE T. Geosci. Remote. 60, 1–12 (2022). [CrossRef]