We built an ocean-atmosphere microwave radiation brightness temperature model to simulate the brightness temperature received by the 1D-MSAMR ( T B , p , θ ) as follows: T B ,   p , θ = T B U , θ + τ θ ∙ E p , θ ∙ T S + τ θ ∙ T B Ω , θ T B Ω , θ = R p , θ ∙ [ T B D , θ + τ ∙ T c o l d ] + T B , s c a t , p , θ T B , s c a t , p , θ = = Ω p , θ ( τ , W ) ∙ [ T B D , θ + τ θ ∙ T c o l d ] ∙ R p , θ where p denotes the polarization mode, p = υ , h . τ θ is the atmospheric transmittance of the entire atmosphere, R p , θ = 1 − E p , θ is sea surface reflectance, and E p , θ is sea surface emissivity. T B U , θ is upward atmospheric radiation brightness temperature and T B D , θ is downward atmospheric radiation brightness temperature. Further, T c o l d is the effective cold space temperature, which is assumed to be a fixed value of 2.7 K, T s is the SST, T B Ω , θ is the upward sky radiation brightness temperature scattered from the ocean surface, and τ ⋅ T B , s c a t , p , θ is used to correct atmospheric path in the downwelling scattered sky radiation. Ω p , θ is the empirical correction parameter.

The ocean-atmosphere microwave radiation brightness temperature model is divided into two parts, the atmospheric absorption emission model, and the sea surface emissivity model. The atmospheric absorption emission model used in this study was developed by Wentz (2000) for the AMSR. The sea surface emissivity model used in this study was developed by Meissner and Wentz (2012). The model assumes that the sea surface roughness is only related to the sea surface wind vector, and the specific formula for calculation of sea surface emissivity was not provided.

The 1D-MSAMR simulator is used to simulate the entire process of 1D-MSAMR detection of bright temperature. The schematic diagram of the detection process is shown in Figure 1. The 1D-MSAMR uses the principle of binary interference imaging wherein P i and P k are two small antennas, d σ is a pixel in the radiation source A, and source radiation is received by the small antennas at the same time. The output signals of the two small antennas are processed using complex correlation to obtain the visibility function. The visibility function is then inversely calculated to obtain the brightness temperature image of the observation scene (Schanda, 1979; Ruf et al., 1988). The output voltages are denoted by U i ( t ) and U k ( t ) , where t is a limited period of time, and H i ( f ) and H k ( f ) are the frequency responses of the receiving channel. Furthermore, D is the distance of the small antenna, D i and D k are the distances from each small antenna to the origin o , R i and R k are the distances from the pixel to each small antenna, and R σ is the distance from the pixel to o . The zenith and azimuth angles are denoted by θ and φ, respectively. After correlation calculations, it was concluded that V is only related to the distance between the two antennas (Corbella et al., 2004).

The formula for calculation of the visibility function and image reconstruction of the microwave interferometric radiometer with the 1D-MSAMR simulator is as follows (Wentz and Meissner, 2007; Lise et al., 2018): V m = 1 Ω m ∫ ξ 2 ≤ 1 T B ( ξ ) 1 − ξ 2 ∙ F m ( ξ ) ∙ r ∼ m ( − u m ξ f c ) ∙ exp ( − j 2 π u m ξ ) d ξ T ^ ( ξ ) = ∑ m = − N N V m exp ( j 2 π u m ξ ) where Ω m is the stereo angle of the antenna, V m ( u m ) is the visibility function, u m = m ⋅ Δ u is the baseline composed of two antennas at different positions, F m ( ξ ) is the normalized voltage pattern, T B ( ξ ) is the brightness temperature of the observation scene, r ∼ m is the stripe elimination function, ξ = sin ⁡ θ , and f c is the operating frequency.

We used the brightness temperature received by the 1D-MSAMR to calculate the model brightness temperature T B , mod with the 1D-MSAMR simulator. The antennas and channel of the 1D-MSAMR simulator used in this study were all in an ideal state, so the measured brightness temperature T θ , p m e a s was simulated by adding E M b to the model brightness temperature T B , mod . E M b were assumed to follow a Gaussian distribution; in the absence of satellite observation data, it is a reasonable pre-research method used by many researchers (Bobylev et al., 2010).

Figure 2 shows the schematic diagram of the SST retrieval experiment. First, we established a complete data set describing the background field of the atmosphere and ocean environment. The data were then divided into training and test sets. The second step was to establish an ocean-atmosphere microwave radiation brightness temperature model to calculate the brightness temperature received by the 1D-MSAMR. The simulator with different Gaussian noises was then used to simulate the measured brightness temperature ( T θ , p m e a s ) of the 1D-MSAMR. The last step was to establish a regression relationship between the T θ , p m e a s and SST. The obtained regression coefficient was then verified by using the test data set.

The 1D-MSAMR has the characteristics of multiple incidence angles that influence the brightness temperature considerably. Therefore, SST must be retrieved separately from different incidence angles. We used two different SST retrieval algorithms: MLR and TSSR (based on MLR). Calculation of MLR (Wentz and Meissner, 2007) was as follows: T s , θ = A θ , 0 + ∑ i = 1 10   o r   8 a θ , p , i ∙ t θ , p , i t θ , p , i = T θ , p , i m e a s   f o r   6.9 ,   10.65 G H z t θ , p , i = − ln ( 290 − T θ , p , i m e a s )   f o r   18.7 ,   23.8 ,   36.5   G H z where A θ , 0 is a constant term, a θ , p , i is the regression coefficient, and the subscript i corresponds to different frequencies.

For TSSR, T θ , p m e a s and MLR were used first to retrieve the preliminary SST retrieval value ( T S , f i r s t ), which was then used as “a priori” in a second step to divide T θ , p m e a s data. T θ , p m e a s data were divided into different data sets in accordance with T S , f i r s t for 2 K intervals within 273.15 K–313.15 K. Then we trained the divided data sets to obtain the regression coefficients of each set using MLR. The last step was to use the regression coefficients to retrieve SST ( T S r e t r i e v a l ). In the retrieval process, the training set was used to calculate the regression coefficients, and the test set was used to verify the regression coefficients. The specific process is shown in Figure 3.

This study used the data products of WindSat in Remote Sensing Systems (RSS) to establish the background field data. The data were 7-day average data product worldwide from 2016 to 2018. Specifically, the data included SST ( T s ), total atmospheric column water vapor content (V), total atmospheric column cloud liquid water content (L), rainfall rate (R), wind direction at 10 m height (φ), and wind speed at 10 m height (W).

This research focused on the retrieval of SST in the case of non-precipitation; therefore, we excluded data with R > 0 m m / h . We further excluded data corresponding to V > 50 m m and L > 0.2 m m due to the limitations of the ocean-atmosphere microwave radiation brightness temperature model. The remaining data contain about 2.1 million sets of data. Our data is randomly divided into training and test sets. Figure 4 shows data histograms. Since seawater salinity is more sensitive to the L-band (1.4 GHz) and has little influence on the frequencies used in this study, we also set the seawater salinity to the value of 35 psu (Feng et al., 2021.).

We used the root mean square error (RMSE) to represent the retrieval error: σ = 1 N ∑ k = 1 N ( T S , k t r u e − T S , k r e t r i e v a l ) 2 where σ represents the RMSE, T S , k t r u e is the true value of the SST, T S , k r e t r i e v a l denotes the retrieval value of the SST, and N represents the number of T S , k t r u e .

Figure 5 shows the relationship between the SST retrieval errors and the incidence angle when all channels of the 1D-MSAMR have the same E M b (a massif of random Gaussian distribution with a zero mean and σ = 0∼1 K). The figure demonstrates that the SST retrieval errors of the training set and the test set were almost the same for the two SST retrieval algorithms. This shows that the two SST retrieval algorithms are generalizable. In addition, the SST retrieval errors of TSSR were less than MLR, especially at low incidence angle and large E M b . When E M b of each channel of the 1D-MSAMR was within the range of 0∼1 K, the SST retrieval error ranges of the test set and training set for MLR were 0.21 K–2.14 K and 0.21 K–2.15 K, respectively, and the retrieval error ranges of TSSR were 0.04 K–1.67 K and 0.04 K–1.66 K, respectively. The Gaussian error added by each channel was generated separately.

Figure 6 further illustrates the SST retrieval errors versus the incidence angle in the case of different E M b . The SST retrieval errors of TSSR and MLR decreased with increasing incidence angle. This suggests that larger incidence angles should be set for the 1D-MSAMR to improve SST retrieval accuracy. Feng et al. (2022). reached a similar conclusion when analyzing the influence of frequency combination on the retrieval accuracy of SST and pointed out that an incidence angle set between 30°–60° is most conducive to the retrieval of SST. For different E M b , the SST retrieval errors of test sets are provided in Table 1. These data indicate that the SST retrieval accuracy decreased with increasing E M b and the average SST retrieval accuracy of TSSR was about 0.3 K (25%) higher than that of MLR. These results suggest that the 1D-MSAMR system error should be reduced as much as possible with redesign of its hardware system.

C-band is the band that is most sensitive to SST. Sensitivity to the 6.9 GHz frequency ( C s e n s ) is proposed to quantitatively describe the relationship between the E M b of 6.9 GHz and SST retrieval accuracy. C s e n s is calculated as follows: C s e n s = ∂ σ ∂ E M b where σ is the RMSE of retrieval error and E M b is brightness temperature measurement error.

The relationship between C s e n s of the two SST retrieval algorithms and the E M b is given in Figure 7. The C s e n s decreased rapidly with increasing E M b . This shows that with the increasing E M b of 6.9 GHz, the C s e n s curve decreases rapidly and finally trends to zero, which indicates that the retrieval accuracy of SST decreases rapidly when there is a measurement error in the 6.9 GHz channel. However, when the measurement error is greater than a certain value, the retrieval accuracy of SST remains stable. The C s e n s curve (black line) corresponding to TSSR is always below the MLR curve (red line), which shows that the TSSR is less affected by the E M b of 6.9 GHz. That is, TSSR can reduce the negative impact of 6.9 GHz channel measurement error on the retrieval accuracy of SST, and with an increase in 6.9 GHz channel measurement error, the retrieval accuracy of SST will regain stability faster.

Among future on-board sensors for remote sensing of SST, 1D-MSAMR has excellent potential. This study used the MLR algorithm, which has fast computation speed and small computational memory requirements, to develop an on-board SST retrieval algorithm for 1D-MSAMR. Although the systematic error terms of each channel of 1D-MSAMR cannot be accurately obtained, the maximum error was set at the upper limit of 1 K according to the systematic error published by the on-orbit real-aperture microwave radiometer. Surprisingly, TSSR improved the retrieval accuracy by about 25% on the basis of only one more set of regression coefficients compared to MLR, and TSSR was less affected by the EMb of 6.9 GHz. Therefore, the requirement of 6.9 GHz channel design can be appropriately reduced when TSSR is adopted. One reason for the improved performance of TSSR versus MLR is that, with TSSR, the initial retrieval value of SST is divided into intervals and then each interval is retrieved separately. The SST has obvious zonal distribution characteristics (high SST at low latitude), and the sensitivity of each channel to SST is nonlinear (Feng et al., 2021). Therefore, it is more reasonable to adopt different coefficients for different SSTs. TSSR has the capacity for automatic operation without manual intervention. After running for a period of time, the algorithm can automatically update the coefficients of the two-step method and improve the retrieval accuracy by using the temperature data of the sea buoy. However, this study was based on simulation data. Although the Gaussian error we added to each channel was generated separately, the error of each channel has the same distribution (mean value is 0, variance is the same), which is not common in the actual 1D-MSAMR system. Although our simulation model is scientific, it still needs future verification and adjustment based on actual 1D-MSAMR measurement data.

Due to the different baselines of small antennas at different frequencies, it is difficult for the spatiotemporal synchronous observation areas to coincide completely, which leads to the problem of high spatial resolution at high frequencies and low spatial resolution at low frequencies. However, the premise of TSSR algorithm application is that the measured brightness temperature of each channel has spatiotemporal matching. Therefore, before applying TSSR, it is necessary to preprocess the measured brightness temperature so that the measured brightness temperature of each channel corresponds to the same detection area.

There is an advantage to using a microwave radiometer to monitor SST under conditions of weak precipitation. Precipitation will increase the roughness and change the emissivity of the sea surface. Due to limitations of the simulation model, the sea surface emissivity in the presence of precipitation cannot be calculated, so we did not study the retrieval of SST under those conditions; future studies will address measurement of SST via the 1D-MSAMR in the presence of precipitation. The 1D-MSAMR used in our study has five frequencies. The detected brightness temperature contains more information about the sea surface and atmospheric environment, and the five frequencies include those that are sensitive to water vapor content and weak precipitation (23.8 GHz and 36.5 GHz, respectively). Therefore, 1D-MSAMR retrieval of SST under conditions of weak precipitation is theoretically feasible, and a direction to be explored in our future research.