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Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer

Mengyan Feng Weihua Ai Wen Lu Chengju Shan Shuo Ma Guanyu Chen

Mengyan Feng, Weihua Ai, Wen Lu, Chengju Shan, Shuo Ma, Guanyu Chen. Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-021-1712-7
Citation: Mengyan Feng, Weihua Ai, Wen Lu, Chengju Shan, Shuo Ma, Guanyu Chen. Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-021-1712-7

doi: 10.1007/s13131-021-1712-7

Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer

Funds: The National Natural Science Foundation of China under contract Nos 41475019 and 41705007.
More Information
  • Figure  1.  The concept of one-dimensional synthetic aperture microwave radiometer.

    Figure  2.  The schematic diagram of the incidence angle of a one-dimensional synthetic aperture radiometer.

    Figure  3.  Distribution histogram of environment parameters. a. Sea surface temperature, b. wind direction, c. wind speed, d. total column cloud water vapor content, and e. total column cloud liquid water content.

    Figure  4.  Diagram of the process to retrieve sea surface temperature.

    Figure  5.  Schematic diagram of the BP neural network.

    Figure  6.  The relationship between sensitivity of brightness temperature to SST and incidence angle.

    Figure  7.  The relationship between sensitivity and SST at $\theta ={30^ \circ }$.

    Figure  9.  Relationships between the retrieval errors and sea surface temperature when the added noise is 0.50 K and $\theta =30^\circ$.

    Figure  10.  Scatterplots of retrieval sea surface temperature and true sea surface temperature for Gaussian error of 0.5 K and $\theta =30^\circ$.

    Table  1.   Parameters of the one-dimensional synthetic aperture radiometer

    ParameterValues
    Frequency/GHz6.9
    Bandwidth/MHz200
    Polarization modesvertical and horizontal polarization
    Integral time/s0.5
    Number of antenna elements55
    Minimum spacing between antenna elements0.73λ
    Angular range of the field of view–43° to 43°
    Angular resolution0.43°
    Antenna element positions[0 1 2 3 7 11 12 14 21 24 26 28 38 43 58 59 60 65 68 70 73 75 76 78 83 86 88 93 101 103 105 112 118
    121 122 123 132 133 134 137 147 148 153 156 164 166 172 173 174 175 178 180 181 182 183]
    Parabolic size12 m×10 m
    Spatial resolution at nadir/km5
    Range of incidence angle θ with vertical observation–51° to 51°
    下载: 导出CSV

    Table  2.   Geophysical parameter values for the background field used in the sensitivity analysis

    SceneSTS/KW/(m·s)–1 φ/(°)V/mmL/mm
    Scene235273.15–313.151530300.1
    下载: 导出CSV

    Table  3.   Mean RMS errors, where the top values are the training set and the bottom values are the test set

    Noise/KMean RMS error/K
    BPRFRE1RE2
    0.250.300.76 0.60 0.61
    0.30 0.78 0.60 0.61
    0.50 0.42 0.900.68 0.69
    0.42 0.92 0.68 0.68
    0.75 0.50 1.04 0.74 0.75
    0.51 1.07 0.74 0.75
    下载: 导出CSV

    Table  4.   The average mean biases, where the top values are the training set and the bottom values are the test set

    Noise/KMean bias/K
    BPRFRE1RE2
    0.25−1.0×10−4−6.0×10−42.1×10−4−1.5×10−4
    −1.1×10–4−0.001−1.2×10−3−1.3×10−3
    0.50−3.7×10−4−6.0×10−4−3.2×10−57.1×10−5
    −5.7×10−4−4.7×10−4−7.7×10−4−8.9×10−4
    0.75−0.005−5.4×10−4−1.2×10−5−2.6×10−5
    −0.005−6.4×10−46.6×10−4−6.6×10−4
    下载: 导出CSV

    Table  5.   SST interval and sample size for each interval in the training set and the corresponding RMS error and mean bias

    SST/KNRMS error and mean bias/K
    BPRFRE1RE2
    271.15–274.8458 4590.29/−0.030.67/−0.320.77/0.470.72/0.50
    274.85–278.5329 0870.28/0.020.74/−0.090.49/−0.050.48/−0.08
    278.54–282.2128 5180.28/0.010.81/−0.050.70/−0.500.72/−0.53
    282.22–285.9024 1940.29/−0.040.89/−0.060.84/−0.710.86/−0.71
    285.91–289.5925 8500.32/−0.090.99/−0.130.83/−0.670.87/−0.69
    289.60–293.2731 8020.32/−0.031.06/−0.150.65/−0.400.70/−0.43
    293.28–296.9644 6250.33/0.051.08/−0.090.49/−0.050.55/−0.06
    296.97–300.6570 2490.36/0.100.97/−0.010.59/0.320.64/0.33
    300.66–304.3352 6580.37/0.0011.05/0.740.66/0.290.71/0.32
    304.34–308.02154 0.36/0.204.09/3.861.01/0.791.13/0.91
    All365 6240.38/−0.010.93/0.0040.67/0.000.69/0.00
    Note: The RMS error and the mean bias are separated by /.
    下载: 导出CSV

    Table  6.   SST interval and sample size for each interval in the test set and the corresponding RMS error and mean bias

    SST/KNRMS error and mean bias/K
    BP RF RE1 RE2
    271.15–274.8458 0010.29/−0.040.69/−0.340.78/0.470.72/0.50
    274.85–278.5329 0910.28/0.030.76/−0.090.49/−0.060.49/−0.08
    278.54–282.2128 6620.28/0.010.85/−0.070.70/−0.500.72/−0.53
    282.22–285.9024 1390.29/−0.040.92/−0.060.84/−0.700.85/−0.71
    285.91–289.5925 9930.32/−0.091.03/−0.150.82/−0.670.86/−0.69
    289.60–293.2731 7390.33/−0.031.11/−0.160.66/−0.410.71/−0.44
    293.28–296.9644 5340.34/0.051.11/−0.090.50/−0.060.56/−0.07
    296.97–300.6570 5950.36/0.131.01/0.000.59/0.330.65/0.34
    300.66–304.3352 7030.37/0.001.07/0.760.70/0.300.72/0.33
    304.34–308.02167 0.43/0.164.03/3.760.98/0.691.07/0.78
    All365 6240.39/−0.010.97/0.0040.67/0.000.69/−0.001
    Note: The RMS error and the mean bias are separated by “/”.
    下载: 导出CSV
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  • 收稿日期:  2020-04-09
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Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer

doi: 10.1007/s13131-021-1712-7
    基金项目:  The National Natural Science Foundation of China under contract Nos 41475019 and 41705007.
    通讯作者: E-mail: a1044175130@163.com

English Abstract

Mengyan Feng, Weihua Ai, Wen Lu, Chengju Shan, Shuo Ma, Guanyu Chen. Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-021-1712-7
Citation: Mengyan Feng, Weihua Ai, Wen Lu, Chengju Shan, Shuo Ma, Guanyu Chen. Sea surface temperature retrieval based on simulated microwave polarimetric measurements of a one-dimensionalsynthetic aperture microwave radiometer[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-021-1712-7
    • Sea surface temperature (SST) is an important parameter to understand air-sea interactions and climate change (Alsweiss et al., 2017). It plays a significant role in the hydrological cycle and energy cycle, and it is closely related to the formation of El Niño, LaNiña and typhoons (Emanuel, 1999; Martin-Neira et al., 1994; McClain et al., 1985; McPhaden, 1999). Therefore, it is important to obtain high-quality sea surface temperature data.

      Many conventional methods such as measurements from ship and marine buoys cannot obtain large-scale and long-term sea surface temperature data. Therefore, satellite remote sensing data are used to derive global sea surface temperature. Compared with infrared remote sensing, microwave remote sensing can pass through the clouds (Ulaby et al., 1983; Chelton and Wentz, 2005; Mätzler, 2006). It can observe sea surface temperature during the entire day and in all weather conditions.

      At present, in-orbit microwave radiometers such as the WindSat, AMSR-2 and TMI, are conically scanning real aperture microwave radiometer imagers. They have only one incidence angle. The WindSat and AMSR-2 radiometers include C-band, which has the highest sensitivity to sea surface temperature. However, real aperture microwave radiometers suffer from inherent limitations concerning their wavelength and antenna, leading to a low spatial resolution. For example, the spatial resolution of the C-band is approximately 50 km. Although increasing the size of the antenna can improve the spatial resolution, it also increases the volume and quality of the radiometer (Kilic et al., 2018).

      Compared with traditional real aperture microwave radiometers, synthetic aperture microwave radiometers have higher spatial resolution and multiple incidence angles. The principle of synthetic aperture microwave radiometers is to use a small aperture antenna array instead of a large real aperture antenna. The cross-correlating of the signals from pairs of antennas is measured at many different baselines. Each baseline yields a sample point in the Fourier transform of the brightness temperature map of the scene, and the brightness temperature images are reconstructed after all measurements are completed by inverting the transform. The spatial resolution is determined by spectrum sampling, not by the size of antennas (Le Vine, 1990; Camps et al., 2016). It is similar in principle to earth rotation synthesis employed in radio astronomy (Ruf et al., 1988; Schanda, 1979).

      Recently, the spaceborne synthetic aperture microwave radiometer has been successfully applied only in the remote sensing of salinity. The Microwave Imaging Radiometer using Aperture Synthesis (MIRAS) instrument mounted on board the Soil Moisture and Ocean Salinity (SMOS) satellite is a two-dimensional synthetic aperture microwave radiometer. It operates at the L-band, which is sensitive to salinity (Martin-Neira et al., 1994, 2002). However, two-dimensional synthetic aperture microwave radiometers have the disadvantages of complex imaging methods and difficult calibration. In contrast, the systematic complexity of one-dimensional synthetic aperture microwave radiometers is greatly reduced (Le Vine et al., 1990, 1994); thus, it is easy to achieve high spatial resolution detection (Le Vine et al., 1990).

      Currently, the one-dimensional synthetic aperture microwave radiometer has only been used for airborne and ground-based observation experiments (Le Vine et al., 1994, 2001). To retrieve sea surface temperature by one-dimensional synthetic aperture microwave radiometers from space, we propose a one-dimensional synthetic aperture microwave radiometer operate at frequencies of 6.9 GHz, 10.65 GHz, 18.7 GHz, and 23.8 GHz. Each frequency adopts a dual polarization observation mode. Since there is no one-dimensional synthetic aperture microwave radiometer in orbit, in this paper, a radiation transmission forward model is used to simulate the brightness temperature measured by the spaceborne one-dimensional synthetic aperture microwave radiometer. The concept of one-dimensional synthetic aperture microwave radiometer is shown in Fig. 1.

      Figure 1.  The concept of one-dimensional synthetic aperture microwave radiometer.

      At present, the sea surface temperature retrieval algorithms for microwave remote sensing mainly include physical algorithms and empirical algorithms. The physical basis for the physical algorithms are the radiative transfer model. It retrieve sea surface temperature by establishing a cost function of observations and model simulations (Meissner and Wentz, 2005; Koner et al., 2015). For example, both AMSR (Wentz and Meissner, 2000) and WindSat (Bettenhausen et al., 2006; Brown et al., 2006) use physical algorithms. Empirical algorithms include the D-Matrix algorithm (Goodberlet et al., 1990), linear regression algorithm (Wentz and Meissner, 2000) and neural network method (Obligis et al., 2005; Krasnopolsky et al., 2000), which establish an empirical relationship between brightness temperature and sea surface temperature. However, these algorithms cannot be used directly for the one-dimensional synthetic aperture microwave radiometer because it includes multiple incidence angles. In this paper, we mainly study empirical algorithms and propose two machine learning algorithms (backpropagation neural network and random forest) and two multiple linear regression algorithms to retrieve sea surface temperature.

      The rest of this paper is organized as follows. Section 2 introduces the parameters of the one-dimensional synthetic aperture microwave radiometer, and Section 3 presents the methods and data. The methods include the radiation transmission forward model and retrieval method. The results and analyses are discussed in Section 4, and the conclusions are drawn in Section 5.

    • The synthetic aperture radiometer imaging is an indirect method to measure the brightness temperature of a scene. It measures the spatial frequency component of the scene by measuring all of the coherent products of the output voltage from pairs of antennas (Le Vine, 1990), which are called visibility functions (Corbella et al., 2004), the brightness temperature image is obtained by an inverse Fourier transform.

      In the one-dimensional synthetic aperture microwave radiometer, the frequencies are 6.9 GHz, 10.65 GHz, 18.7 GHz and 23.8 GHz, and every frequency operates at vertical and horizontal polarization. Moreover, 6.9 GHz is the most significant frequency because the sensitivity of 6.9 GHz to sea surface temperature is high. Table 1 lists some parameters of 6.9 GHz.

      Table 1.  Parameters of the one-dimensional synthetic aperture radiometer

      ParameterValues
      Frequency/GHz6.9
      Bandwidth/MHz200
      Polarization modesvertical and horizontal polarization
      Integral time/s0.5
      Number of antenna elements55
      Minimum spacing between antenna elements0.73λ
      Angular range of the field of view–43° to 43°
      Angular resolution0.43°
      Antenna element positions[0 1 2 3 7 11 12 14 21 24 26 28 38 43 58 59 60 65 68 70 73 75 76 78 83 86 88 93 101 103 105 112 118
      121 122 123 132 133 134 137 147 148 153 156 164 166 172 173 174 175 178 180 181 182 183]
      Parabolic size12 m×10 m
      Spatial resolution at nadir/km5
      Range of incidence angle θ with vertical observation–51° to 51°

      According to the positions of the antenna elements, we define the minimum wavelength spacing between antenna elements to be ${\rm{0}}{\rm{.73}}\lambda $, and the maximum distance is ${\rm{183}}\times{\rm{0}}{\rm{.73}}\lambda$, where $\lambda $ is the wavelength. According to Lim’s study (Lim, 2009), the angular resolution is $\Delta \theta \approx{\rm{1/(183}}\times{\rm{0}}{\rm{.73}}\lambda)$, which provides an angular resolution of approximately 0.43°. The parabolic cylindrical antenna has a size of 12 m×10 m, and all small antenna elements share a parabolic cylindrical antenna. We assume that the satellite is operating along a track at an altitude of 900 km. At nadir, the spatial resolution can be calculated at approximately 5 km by

      $$\Delta x \approx 1.2\frac{\lambda }{D}H,$$ (1)

      where $\Delta x$ is spatial resolution, $D$ is the size of the parabolic antenna and $H$ is the height of the satellite orbit.

      Figure 2 shows the schematic diagram of the incidence angle of a spaceborne one-dimensional synthetic aperture radiometer.

      Figure 2.  The schematic diagram of the incidence angle of a one-dimensional synthetic aperture radiometer.

      According to the curvature of the earth, satellite height and angular range of the field of view, we can calculate the incidence angles as $- 51^\circ \leqslant\theta \leqslant51^\circ$. The equations are as follows:

      $$L = ({R_{\rm{e}}} + H)\cos \beta - \sqrt {R_{\rm{e}}^2 - {{({R_{\rm{e}}} + H)}^2}(1 - {{\cos }^2}\beta)},$$ (2)
      $$\theta = \pi - \arccos \left[{\frac{{{L^2} + R_{\rm{e}}^2 - {{({R_{\rm{e}}} + H)}^2}}}{{2L{R_{\rm{e}}}}}} \right], $$ (3)

      where $\beta $ is the field of view of the one-dimensional synthetic aperture microwave radiometer, $H$ is the orbital height, ${R_{\rm{e}}}$ is the radius of the earth, and $L$ is the distance from the spaceborne one-dimensional synthetic aperture microwave radiometer to the observation scene.

      When the spaceborne one-dimensional synthetic aperture microwave radiometer observes the earth, the vertical polarization and horizontal polarization observation modes work alternately. The antenna elements of all frequencies have the same spatial resolution.

    • In this paper, the newest version of the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data, ERA5, is used to calculate the model brightness temperature with the radiation transmission forward model. The data acquisition times are 0:00 and 12:00 on January 15, April 15, July 15, and October 15, 2018. The data contain the 10-m wind speed components $u$ and $v$, total column cloud liquid water content $L$, total column water vapor content $V$, average rainfall rate $R$ and sea surface temperature ${T_S}$. The spatial resolution of the data is 0.25°. We calculate the wind speed W and wind direction $\varphi $ using the 10-m wind speed components $u$ and $v$. The data under precipitation conditions $R > 0$, ${T_S} < 271.15$ conditions and unreasonable data $L < 0$ are eliminated. Since the seawater salinity $S$ has little effect on the frequencies of the one-dimensional synthetic aperture microwave radiometer, we set the seawater salinity to 35.

      We obtain approximately 730 000 sets of data after processing. Statistical histograms of the sea surface temperature, wind direction, wind speed, total column cloud liquid water content and total column water vapor content are shown in Fig. 3. Finally, we divide the data into a training set and a test set on average. The training set is used to train the retrieval algorithm, and the test set is used to test the performance of the retrieval algorithm.

      Figure 3.  Distribution histogram of environment parameters. a. Sea surface temperature, b. wind direction, c. wind speed, d. total column cloud water vapor content, and e. total column cloud liquid water content.

    • The brightness temperature ${T_{{\rm{B,p}}}}$ received by the spaceborne one-dimensional synthetic aperture microwave radiometer can be expressed as follows (Meissner and Wentz, 2009, 2012):

      $$\left\{ \begin{aligned} & {T_{{\rm{B,p}}}} = {T_{{\rm{BU}}}} + \tau {E_{\rm{p}}} {T_{\rm{S}}} + \tau {T_{{\rm{B}}\Omega }} \\ & {T_{{\rm{B}}\Omega }} = {R_{\rm{p}}} \left( {{T_{{\rm{BD}}}} + \tau {T_{{\rm{cold}}}}} \right) + {T_{{\rm{B,scat,p}}}} \\ & {T_{{\rm{B,scat,p}}}} = {\Omega _{\rm{p}}}\left({\tau,W} \right) ({{T_{{\rm{BD}}}} + \tau {T_{{\rm{cold}}}} - {T_{{\rm{cold}}}}}) {R_{\rm{p}}} \end{aligned} \right\},$$ (4)

      where the subscript ${\rm{p}}$ denotes the polarization mode, p means v or h, represents vertical or horizantal polorization ${T_{{\rm{BU}}}}$ and ${T_{{\rm{BD}}}}$ are the upwelling and downwelling atmospheric brightness temperatures, respectively, $\tau $ is the entire layer of atmospheric transmittance, ${E_{\rm{p}}}$ is the sea surface emissivity, ${T_{{\rm{cold}}}}$ is the effective cold space temperature, which is usually assumed to be a fixed value of 2.7 K, ${T_{\rm{S}}}$ is sea surface temperature, ${T_{{\rm{B}}\Omega }}$ is the downwelling sky radiation that is scattered from the ocean surface, ${R_{\rm{p}}} = 1 - {E_{\rm{p}}}$ represents the sea surface reflectance, $\tau {T_{{\rm{B,scat,p}}}}$ accounts for the atmosphere path length correction in the downwelling scattered sky radiation, and ${\Omega _{\rm{p}}}$ represents an empirical correction factor for downwelling radiation that is reflected from nonspecular directions (Meissner and Wentz, 2009). ${\Omega _{\rm{p}}}$ depends on the frequency $f$, incidence angle $\theta $, transmittance $\tau $ and wind speed $W$.

    • The sea surface emissivity model is the premise of this research. It is difficult to formulate a consistent theoretical model for the wind dependency on sea surface temperature (Stogryn, 1967; Hollinger, 1971). In this paper, the sea surface emissivity with nonspecular reflection is calculated from a sea surface emissivity model established by Meissner and Wentz (Meissner and Wentz, 2012). The model can calculate the sea surface emissivity ${E_{\rm{p}}}$ for incidence angles within the range of 0°–65° and sea surface temperature within the range of approximately –2°C to 40°C. This model divides the sea surface emissivity into three parts: the specular sea surface emissivity (${E_{0,{\rm{p}}}}$), the emissivity increase caused by the wind speed W ($\Delta {E_{W{\rm{,p}}}}$), and the emissivity increase caused by the wind direction $\varphi $ ($\Delta {E_{{\rm{\varphi}},{\rm{p}}}}$). The equation is as follows (Meissner and Wentz, 2012):

      $${E_{\rm{p}}} = {E_{0,{\rm{p}}}} + \Delta {E_{W,{\rm{p}}}} + \Delta {E_{{\rm{\varphi}},{\rm{p}}}},$$ (5)

      where ${E_{0,{\rm{p}}}}$ is by far the largest part, and it is a function of the frequency $f$, incidence angle $\theta $, sea surface temperature ${T_S}$ and salinity $S$. In addition, it is related to the complex permittivity of seawater ε. We calculate the complex permittivity of seawater using the algorithm established by Meissner and Wentz in 2004 (Meissner and Wentz, 2004), which is extensively validated and valid for fresh water and sea water in the temperature range of –2°C to 40°C and –2°C to 29°C, respectively. It was subsequently corrected in 2012 (Meissner and Wentz, 2004, 2012). The emissivity increase caused by the wind speed $\Delta {E_{W,{\rm{p}}}}$ is related to the wind speed $W$, and the increase in emissivity caused by the wind direction $\Delta {E_{{\rm{\varphi}},{\rm{p}}}}$ is related to the wind direction $\varphi $ (Meissner and Wentz, 2012). The specific equation is not described in detail here.

    • For atmosphere without rain where scattering can be neglected, there are three atmospheric absorption components in the microwave range of frequencies below 100 GHz: water vapor, liquid water and oxygen. The atmospheric parts ${T_{{\rm{BU}}}}$, ${T_{{\rm{BD}}}}$ and $\tau $ in Eq. (5) can be calculated by an empirical model (Wentz and Meissner, 2000; Wang et al., 2005, 2014).

      In this paper, we use an empirical model established by Wentz and Meissner (Wentz and Meissner, 2000). This model needs to input frequency $f$, incidence angle $\theta $, sea surface temperature ${T_{\rm{S}}}$, total column cloud liquid water content $L$, and total column water vapor content $V$. The equations are not described in detail here.

      However, the cloud liquid water absorption algorithm we used was developed by Wang et al. (2014) because the Wentz model (Wentz and Meissner, 2000) needs to input an average cloud temperature ${T_L}$, which is difficult to obtain. The equation is as follows:

      $${A_L} = {a_{L0}}(1 + {a_{L1}}V)L,$$ (6)

      where ${a_{L0}}$ and ${a_{L1}}$ are empirical coefficients. In this paper, all the empirical coefficients are not listed here.

    • We use the filtered data to calculate the model brightness temperature ${T_{{\rm{B,mod}} }}$ with the radiation transmission forward model. The measured brightness temperature ${T_{{\rm{B}},{\rm{meas}}}}$ is simulated by adding Gaussian noises to the model brightness temperature ${T_{{\rm{B,mod}} }}$. The standard deviations of the Gaussian noises are 0.25 K, 0.50 K, and 0.75 K. In the absence of satellite observation data, it is a reasonable pre-research method used by many researchers (Bobylev et al., 2010).

      Four retrieval algorithms for sea surface temperature retrieval from the measured temperature ${T_{{\rm{B}},{\rm{meas}}}}$ and incidence angle are developed. The algorithms are backpropagation neural network algorithm (BP), random forest algorithm (RF) and two multiple linear regression algorithms (RE1 and RE2) (Aires et al., 2001; Bobylev et al., 2010; Jung et al., 1998; Krasnopolsky et al., 2000). The four retrieval algorithms are all statistical methods. Each of them has its own advantages and disadvantages. BP has a strong nonlinear fitting ability. A three-layer neural network can approach any nonlinear problem infinitely. However, the gradient descent method is employed to train the network in BP; thus, local optimal solutions and gradient disappearances are easily produced. RF utilizes a bagging integration strategy based on decision tree-based learners and further introduces the selection of random attributes in the decision tree training process to enhance its generalization ability. RF can process multidimensional data with a faster calculation speed and no computational cost. Its anti-noise ability is strong, but overfitting occurs when the data are relatively noisy. The multiple linear regression algorithms (RE1 and RE2) are the simplest among all four algorithms. The two algorithms only need regression coefficients to retrieve sea surface temperature with ${T_{{\rm{B}},{\rm{meas}}}}$. However, the relationship between ${T_{{\rm{B}},{\rm{meas}}}}$ and sea surface temperature is not a simple linear relationship. A diagram of the process to retrieve sea surface temperature is shown in Fig. 4. It should be noted that the radiation transmission model is only used to calculate the model brightness temperature ${T_{{\rm{B}},{\rm{mod}} }}$, and it does not participate in the training of the retrieval algorithms.

      Figure 4.  Diagram of the process to retrieve sea surface temperature.

      We use a three-layer neural network, that is, the neural network contains a hidden layer. The number of hidden layer neurons is the same as that in the input layer. The model of the BP is shown in Fig. 5. There are 10 neurons in the input layer, which are incidence angle $\theta $, measured brightness temperatures ${T_{{\rm{B}},{\rm{meas}}}}$ and bias; the output layer is sea surface temperature ${T_{\rm{S}}}$. We use the sigmoid function as the activation function of the BP network. When training the RF, the inputs and output are the same as BP.

      Figure 5.  Schematic diagram of the BP neural network.

      We use logarithmic regression and linear regression to retrieve sea surface temperature (Wentz and Meissner, 2000), which are called RE1 and RE2, respectively. RE1 and RE2 are performed at each incidence angle because the sensitivity of the brightness temperature to the incidence angle is very high. The generalized linear statistical regression algorithm is as follows:

      $${T_{\rm{S}}} = R\left( {{A_0} + \sum\limits_{i = 1}^{2 \times n} {{A_i}} X\left({{T_{{\rm{B}},{\rm{meas}},i}}} \right)} \right),$$ (7)

      where ${T_{\rm{S}}}$ is the sea surface temperature, $R$ and $X$ are linearizing functions, $n$ is the number of frequencies, and ${T_{{\rm{B}},{\rm{meas}},i}}$ is the measured brightness temperature. $R(Y) = Y$ is the linear function.

      When fitting RE1,

      $$ X({T_{\rm{B}}}) = {T_{\rm{B}}},\;\;\;f = 6.9\;{\rm{GHz}}\;{\rm{and }}\;10.65\;{\rm{ GHz}}, $$ (8)
      $$ X({T_{\rm{B}}}) = - \ln (290 - {T_{\rm{B}}}),\;\;\;f = 18.7\;{\rm{GHz }}\;{\rm{ and}}\;{\rm{ 23.8}}\;{\rm{GHz }}, $$ (9)

      which is,

      $$\begin{split} {T_{\rm{S}}} =& {A_0} + {A_1} T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{6.9} + {A_2} T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{6.9} + {A_3} T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{10.65} + \\ &{A_4} T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{10.65} - {A_5} \ln (290 - T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{18.7}) - \\ &{A_6} \ln (290 - T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{18.7}) - {A_7} \ln (290 - T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{23.8}) -\\& {A_8} \ln (290 - T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{23.8}) , \end{split}$$ (10)

      when fitting RE2,

      $$ X({T_{\rm{B}}}) = {T_{\rm{B}}},\;\;f = 6.9\;{\rm{GHz }},\;10.65\;{\rm{GHz }},\;18.7\;{\rm{GHz }}{\rm{and }}\;{\rm{ 23.8}}\;{\rm{GHz }}, $$ (11)

      which is,

      $$\begin{split} {T_{\rm{S}}} = &{A_0} + {A_1} T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{6.9} + {A_2} T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{6.9} + {A_3} T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{10.65} + \\&{A_4} T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{10.65} + {A_5} T_{{\rm{B}},{\rm{mean}},{\rm{v}}}^{18.7} {\rm{ }} + {A_6} T_{{\rm{B}},{\rm{mean}}{\rm{,h}}}^{18.7} + \\&{A_7} T_{{\rm{B}},{\rm{mean}}{\rm{,v}}}^{23.8} + {A_8} T_{{\rm{B}},{\rm{mean}},{\rm{h}}}^{23.8} . \end{split}$$ (12)

      It should be noted that the radiation transmission forward model can calculate radiation in the range of ${0^ \circ }\leqslant\theta \leqslant{65^ \circ }$. In this paper, we study the sea surface temperature retrieval in the range of ${0^ \circ }\leqslant\theta \leqslant{65^ \circ }$ though the incidence angle range of the one-dimensional synthetic aperture microwave radiometer is ${0^ \circ }\leqslant\theta\leqslant{51^ \circ }$.

    • The sensitivity to sea surface temperature determines the retrieval precision at different incidence angles. We assumed a environment scene to research the sensitivity of the brightness temperature to the sea surface temperature, which are shown in Table 2. The sea surface temperature is set range from 273.15 K to 313.15 K, and other parameters are set as constants. We calculated the sensitivity to sea surface temperature at different incidence angles; the equation is as follows:

      Table 2.  Geophysical parameter values for the background field used in the sensitivity analysis

      SceneSTS/KW/(m·s)–1 φ/(°)V/mmL/mm
      Scene235273.15–313.151530300.1
      $${\rm{SE}} = \frac{1}{n} \sum\limits_{i = 1}^n {\frac{{\partial {T_{{\rm{B}},{\rm{mod}},{\rm{p}}}}}}{{\partial {T_{{\rm{S}},i}}}}},$$ (13)

      where ${\rm{SE}}$ is the sensitivity of the brightness temperature to the sea surface temperature. $n$ is the number of the sea surface temperature ${T_{\rm{S}}}$, and ${T_{{\rm{B}},{\rm{mod,p}}}}$ is the model brightness temperature.

      Figure 6 shows the relationship between the sensitivity of the vertical and horizontal polarization brightness temperature to the sea surface temperature and incidence angle. It can be observed that 6.9 GHz provides the best sensitivity, with the consequence that 6.9 GHz has the greatest influence on retrieval precision. The sensitivity to the sea surface temperature is higher at vertical polarization than at horizontal polarization. The sensitivity to sea surface temperature increases with incidence angle at vertical polarization, with the result that vertical polarization increases the retrieval precision at a larger incidence angle. At horizontal polarization of 6.9 GHz and 10.65 GHz, the sensitivity of to the sea surface temperature decreases with the incidence angle. Its influence is the opposite of vertical polarization. However, at 18.7 GHz and 23.8 GHz, the sensitivity of horizontal polarization is almost constant. Figure 7 shows the trend of sensitivity with sea surface temperature at $\theta = {30^ \circ }$.

      Figure 6.  The relationship between sensitivity of brightness temperature to SST and incidence angle.

      Figure 7.  The relationship between sensitivity and SST at $\theta ={30^ \circ }$.

    • There are two major problems that make the one-dimensional synthetic aperture microwave radiometer determination of sea surface temperature a substantial challenge: the calibration error and modeling error. For convenience, we assume that the forward model is accurate. The calibration error can be regarded as random noise. Therefore, we add Gaussian noise to the model brightness temperature.

      To improve the reliability of error statistics, we compute the root mean square (RMS) error and bias:

      $$\sigma = \sqrt {\frac{1}{N}\sum\limits_{k = 1}^N {{{(T_{{\rm{S}},k}^{{\rm{true}}} - {T_{{\rm{S}},k}})}^2}} } ,$$ (14)
      $$B = \frac{1}{N}\sum\limits_{k = 1}^N {(T_{{\rm{S}},k}^{{\rm{true}}} - {T_{\rm{{S}},k}})},$$ (15)

      where $\sigma $ is the RMS error of ${T_{\rm{S}}}$, N is the number of ${T_{\rm{S}}}$, and $T_{\rm{S}}^{{\rm{true}}}$ is true value of ${T_{\rm{S}}}$. $B$ is the mean bias of ${T_{\rm{S}}}$.

    • Figure 8 illustrates the relationships between the RMS error and the mean bias of the four retrieval algorithms and incidence angle. The bold dotted lines, thin dotted lines and dashed lines are the retrieval results with added noises of 0.25 K, 0.50 K, and 0.75 K, respectively. In Fig. 8, the training set results are on the left, and the test set results are on the right. The results of the training set and test set have few differences.

      The RMS errors increase with the Gaussian noise increasing, which means that the more accurate the one-dimensional synthetic aperture microwave radiometer measure is, the more accurate the retrieval of the sea surface temperature. In addition to RF, the RMS error differences between dashed lines (0.75 K) and thin dotted lines (0.50 K) are larger than those between thin dotted lines (0.50 K) and bold dotted lines (0.25 K). The effect of the Gaussian noise on the RMS error is different at different incidence angles. In general, it is larger at ${\rm{0}}^\circ \leqslant\theta \leqslant20^\circ$ than at $\theta > {\rm{20}}^\circ$. In contrast, the effect of the Gaussian noise on the mean bias is small. The mean bias fluctuates around approximately 0 K with the incidence angel.

      From Figs 8i, j, m and n, we can see that the RMS error first decreases with the incidence angle in the range of ${\rm{0}}^\circ \leqslant\theta \leqslant{\rm{55}}^\circ$ and then increases in the range of ${\rm{55}}^\circ \leqslant\theta \leqslant65^\circ$. This trend in Figs 8i, j, m and n can be explained by the sensitivity to the sea surface temperature (Fig. 6). We find that the reason for the RMS error reaching the minimum at ${\theta _{EIA}}\approx55^\circ$ is affected by the horizontal polarization brightness temperature of 6.9 GHz and 10.65 GHz. The minimum of the RMS error in Figs 8i, j, m and n will occur at $\theta =65^\circ$ when we retrieve the sea surface temperature without 6.9 GHz and 10.65 GHz horizontal polarization brightness temperature. Therefore, we conclude that the horizontal polarization brightness temperature of 6.9 GHz and 10.65 GHz has a greater effect on the large incidence angles. In contrast, the vertical polarization has a greater effect on the low incidence angles.

      The relationship between the RMS error and the incidence angle in Figs 8a, b, i and j are similar because the principles of BP and RE are similar. BP updates weights through error back propagation, and RE obtains the regression coefficients through fitting. However, BP presents higher retrieval precision than RE1 and RE2. In general, the RMS error of BP, RE1 and RE2 are consistent with the sensitivity to the sea surface temperature to some extent. In contrast, the RMS error of RF is the opposite of other algorithms.

      Tables 3 and 4 show the average RMS errors and mean biases, respectively, that is, the average of the retrieval result at all incidence angles. The top values are the training set, while the bottom values are the test set. Table 3 shows that the RF has the largest retrieval errors, whereas the BP has the smallest retrieval errors (its accuracy is approximately twice that of the BP). The retrieval errors of the RE1 and RE2 are mostly the same. As shown in Table 4, the average mean biases of the four algorithms are very low.

      Table 3.  Mean RMS errors, where the top values are the training set and the bottom values are the test set

      Noise/KMean RMS error/K
      BPRFRE1RE2
      0.250.300.76 0.60 0.61
      0.30 0.78 0.60 0.61
      0.50 0.42 0.900.68 0.69
      0.42 0.92 0.68 0.68
      0.75 0.50 1.04 0.74 0.75
      0.51 1.07 0.74 0.75

      Table 4.  The average mean biases, where the top values are the training set and the bottom values are the test set

      Noise/KMean bias/K
      BPRFRE1RE2
      0.25−1.0×10−4−6.0×10−42.1×10−4−1.5×10−4
      −1.1×10–4−0.001−1.2×10−3−1.3×10−3
      0.50−3.7×10−4−6.0×10−4−3.2×10−57.1×10−5
      −5.7×10−4−4.7×10−4−7.7×10−4−8.9×10−4
      0.75−0.005−5.4×10−4−1.2×10−5−2.6×10−5
      −0.005−6.4×10−46.6×10−4−6.6×10−4
    • To research the relationship between the four algorithms’ retrieval accuracy and sea surface temperature, this section analyzes the retrieval errors with a noise of 0.50 K and $\theta=30^\circ$ from Section 4.1. We divide the sea surface temperature into 10 intervals and calculate the RMS error and mean bias for each temperature interval, as shown in Fig. 9. The scatterplots of the retrieval sea surface temperature and true sea surface temperature are shown in Fig. 10. Tables 5 and 6 list the sample size N, the RMS error and mean bias of each temperature interval in the training set and test set, respectively.

      Figure 9.  Relationships between the retrieval errors and sea surface temperature when the added noise is 0.50 K and $\theta =30^\circ$.

      Figure 10.  Scatterplots of retrieval sea surface temperature and true sea surface temperature for Gaussian error of 0.5 K and $\theta =30^\circ$.

      Table 5.  SST interval and sample size for each interval in the training set and the corresponding RMS error and mean bias

      SST/KNRMS error and mean bias/K
      BPRFRE1RE2
      271.15–274.8458 4590.29/−0.030.67/−0.320.77/0.470.72/0.50
      274.85–278.5329 0870.28/0.020.74/−0.090.49/−0.050.48/−0.08
      278.54–282.2128 5180.28/0.010.81/−0.050.70/−0.500.72/−0.53
      282.22–285.9024 1940.29/−0.040.89/−0.060.84/−0.710.86/−0.71
      285.91–289.5925 8500.32/−0.090.99/−0.130.83/−0.670.87/−0.69
      289.60–293.2731 8020.32/−0.031.06/−0.150.65/−0.400.70/−0.43
      293.28–296.9644 6250.33/0.051.08/−0.090.49/−0.050.55/−0.06
      296.97–300.6570 2490.36/0.100.97/−0.010.59/0.320.64/0.33
      300.66–304.3352 6580.37/0.0011.05/0.740.66/0.290.71/0.32
      304.34–308.02154 0.36/0.204.09/3.861.01/0.791.13/0.91
      All365 6240.38/−0.010.93/0.0040.67/0.000.69/0.00
      Note: The RMS error and the mean bias are separated by /.

      Table 6.  SST interval and sample size for each interval in the test set and the corresponding RMS error and mean bias

      SST/KNRMS error and mean bias/K
      BP RF RE1 RE2
      271.15–274.8458 0010.29/−0.040.69/−0.340.78/0.470.72/0.50
      274.85–278.5329 0910.28/0.030.76/−0.090.49/−0.060.49/−0.08
      278.54–282.2128 6620.28/0.010.85/−0.070.70/−0.500.72/−0.53
      282.22–285.9024 1390.29/−0.040.92/−0.060.84/−0.700.85/−0.71
      285.91–289.5925 9930.32/−0.091.03/−0.150.82/−0.670.86/−0.69
      289.60–293.2731 7390.33/−0.031.11/−0.160.66/−0.410.71/−0.44
      293.28–296.9644 5340.34/0.051.11/−0.090.50/−0.060.56/−0.07
      296.97–300.6570 5950.36/0.131.01/0.000.59/0.330.65/0.34
      300.66–304.3352 7030.37/0.001.07/0.760.70/0.300.72/0.33
      304.34–308.02167 0.43/0.164.03/3.760.98/0.691.07/0.78
      All365 6240.39/−0.010.97/0.0040.67/0.000.69/−0.001
      Note: The RMS error and the mean bias are separated by “/”.

      As seen from Fig. 9, Tables 5 and 6, although the temperature interval and samples size slightly differ between the training set and the test set, the differences between the RMS errors and mean biases of them are small. The following describes the main results for the training set. Table 5 shows that there are only 154 data points in the interval of 304.34–308.02 K, but the RMS error of BP is only 0.36 K. This confirms that the BP is less affected by the sample size. In addition, the curve of BP in Figs 9c and d is consistently stable at approximately 0 K.

      The retrieval errors of RE1 and RE2 are similar. The RMS error and mean bias are the largest in the temperature range with the smallest sample size, with the consequence that the sample size has a considerable influence on the retrieval precision in multiple linear regression. The curves of RE1 and RE2 in Figs 9c and d first decrease and then increase. In addition, at most temperature intervals, the absolute value of the mean bias exceeds half of the RMS error.

      Compared with the other algorithms, the RF has the highest retrieval error. The RMS error is the largest at each temperature interval (much larger than the RMS error of the BP). The mean bias of the RF is small relative to the RMS error in the sea surface temperature range from 271.15 K to 300.65 K, which indicates that the retrieval sea surface temperature evenly distributes around the true value in this sea surface range. However, the RF performs poorly in the sea surface temperature range from 300.66 K to 308.02 K, and this can be seen in Figs 10c and d. We conclude that the sample size has a substantial influence on the RF retrieval accuracy. Compared with the other three algorithms, RF is not suitable for retrieving sea surface temperature with the one-dimensional synthetic aperture microwave radiometer.

    • In this paper, a one-dimensional synthetic aperture microwave radiometer is proposed to retrieve sea surface temperature from space. We assume that the spaceborne one-dimensional synthetic aperture microwave radiometer is operating at an orbital height of 900 km and the antenna elements of all frequencies have the same spatial resolution. First, we use the ERA5 reanalysis data to calculate the model brightness temperature with the radiation transmission forward model. The measured brightness temperature is simulated by adding different Gaussian noises to the model brightness temperature. Then, the BP neural network, RF algorithm and multiple linear regression algorithms (RE1 and RE2) are used to retrieve sea surface temperature within the incidence angle range from 0° to 65°. The main conclusions are as follows:

      The sensitivity of all frequencies to sea surface temperature increases with the incidence angle at the vertical polarization. At the horizontal polarization of 6.9 GHz and 10.65 GHz, the sensitivity to sea surface temperature decreases with the incidence angle. However, at 18.7 GHz and 23.8 GHz, the sensitivity of the horizontal polarization to the sea surface temperature is almost constant.

      The RMS errors increase with the Gaussian noise increase. The effect of the Gaussian noise on the RMS error is larger at ${\rm{0}}^\circ \leqslant\theta\leqslant20^\circ$ than at $\theta > {\rm{20}}^\circ$. In contrast, the effect of the Gaussian noise on the mean bias is small. The mean bias fluctuates around approximately 0 K with the incidence angel.

      The RMS error of RE1 and RE2 decreases first and then increases with the incidence angle and reaches the minimum at $\theta \approx55^\circ$. The reason is that the influence of horizontal polarization signals of 6.9 GHz and 10.65 GHz on the retrieval precision increases with the incidence angle, and the sensitivity of the horizontal polarization of 6.9 GHz and 10.65 GHz to sea surface temperature decreases with the incidence angle.

      The BP boasts the smallest retrieval errors, while the RF has the largest retrieval errors. Furthermore, the sample size has little effect on the BP retrieval precision. To a certain degree, for example, even if there are only 154 data points, the BP can obtain a sufficiently low retrieval error of 0.36 K. However, the sample size has a great influence on the RF, RE1 and RE2. In particular, the RMS error of the RF reaches 4.09 K with a sample size of 154. Although the BP is an older model, it is most suitable for retrieving sea surface temperature by a spaceborne one-dimensional synthetic aperture microwave radiometer.

      The research in this paper provides a reference for sea surface temperature retrieval by a spaceborne one-dimensional synthetic aperture microwave radiometer. However, our findings are based on clear-sky conditions. In the future, we will work on sea surface temperature retrieval under precipitation conditions.

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