Retrieval of the Atmospheric Temperature and Humidity Proﬁles Using a Feed-Forward Neural Network †

: Using a feed-forward neural network, an inverse algorithm was developed to proﬁle the vertical structure of temperature and speciﬁc humidity. The inverse algorithm (inverse model) was used to calculate temperature and humidity proﬁles, which were then compared with other existing methods. The inverse model is found efﬁcient in proﬁling the vertical structure of temperature and humidity as compared to other existing methods. For example, the statistical methods notorious for their high computational cost, altitude-dependent error, and inability to accurately retrieve the vertical temperature and humidity proﬁles, are enhanced with an inverse model. The inverse model’s diurnal and seasonal cycle proﬁles are also found superior to those of other existing methods, which could be useful for assimilation in numerical weather forecast models. We suggest that incorporating such an inverse model into the ground-based microwave radiometer (GMWR) will enhance the accuracy of the vertical structure of temperature and humidity proﬁles, and so the improvement in weather forecasting. The developed inverse model has a resolution of 50 m between the surface to 500 m and 100 m between 500–2000 m, and 500 m beyond 2000 m.


Introduction
The three-dimensional structure of the atmosphere has been widely studied using numerical weather prediction (NWP) models, networks of in situ radiosonde (RS), and satellite remote sensing [1][2][3][4]. Although, RS measurements are prohibitively expensive and biased due to horizontal drifts in the balloon path during high wind conditions [5]. Satellite measurements were found to overcome these limitations to some extent, but their horizontal and temporal resolution is coarse near to the earth's surface, and so the measured data are not helpful for present-day high-resolution NWP models. Additionally, laser radars and Fourier-transform infrared spectrometers can profile the atmospheric states, but they do not work in the presence of clouds [6].
In the late twentieth century, researchers reported that a ground-based microwave radiometer (GMWR) can profile the atmospheric temperature, humidity, and cloud liquid water content with high accuracy and perform continuous measurements at an approximate interval of 2-min in all weather conditions [7][8][9]. GMWR measurements were beneficial in studying several atmospheric processes and improving the predictability of NWP models. For example, over the last two decades, this was used in severe weather prediction and deriving the diurnal variability of atmospheric stability indices [10], boundary layer studies [11][12][13], understanding cloud properties [14], and lightening studies [7,8]. In addition, Kadygrov et al. (2013) reported that GMMR is preferable over the other existing methods for studying complex processes such as turbulence fluctuation and changes in heat transfer rate in the atmospheric boundary layer during a solar eclipse [15].
GMWR uses two built-in functions to profile the vertical structure of atmospheric states. For instance, function-1 calculates the brightness temperature (radiance) in 20-75 GHz band-width using atmospheric water vapor resonance at 22.235 GHz and oxygen resonance at 60 GHz. Function-2 transforms these radiances into atmospheric states. The brightness temperature calculated surrounding the atmospheric water-vapor resonance is preferred for humidity profiling, and the brightness temperature calculated surrounding oxygen resonance is preferred for temperature profiling [16]. The brightness temperature obtained from function-1 is converted to atmospheric states using an inverse algorithm in function-2 (also called as inverse modeling). Such inverse transformation is needed due to the presence of nonlinearity and non-Gaussian caused by the temperature dependence of atmospheric transmission, the Plank function's wave number dependence, clouds, and nonlinear constraints. Inverse techniques used in function-2 in GMWR to convert the measured brightness temperature from function-1 to atmospheric states include the Newtonian iteration method (NEM), regression retrieval method (RRM), and neural network method (NNM). NEM performs the mapping of n-dimensional atmospheric state vectors (e.g., temperature, humidity, cloud liquid content, etc.) to the m-dimensional measurement vectors (i.e., brightness temperature; [17]). However, the primary drawback of NEM is the large number of iterations required for an optimal solution and the large error in atmospheric state vectors at higher altitudes. RRM is considered a well-established traditional method in earth and atmospheric science, where atmospheric state vectors are calculated linearly from measurement vectors (i.e., brightness temperature at different frequencies [18,19]), but it produces slightly worst results than NEM due to a linear relationship between dependent and independent vectors. Further, NNM has received global attention by the scientific community to solve the nonlinear problems in atmospheric sciences in the last two decades. For example, Acciani et al. (2003) reported that neural network outperforms other retrieval techniques. Vivekanandan et al. (1997) showed that neural network-based algorithms could incorporate multiple measurements into the retrieval algorithm; however, their application to GMWR (in function-2) has received lesser attention than other traditional methods [20].
In this work, we develop an inverse algorithm (inverse model) using a feed-forward neural network framework to derive the atmospheric temperature and humidity profiles from calculated (and/or radiometric measured) brightness temperatures at the different frequencies (see Table 1). The paper is structured as follows: Section 1 provides an introduction, and Section 2 describes the data and Inverse model framework. In Section 2 of the inverse model framework, the description of a feed-forward neural network, the structure of the inverse model, and the sensitivity to learning rate are provided. Section 3 presents the results of this work, and Section 4 summarizes the conclusions of this work. We used a total of 825 radiosonde profiles for temperature and 750 radiosonde profiles for humidity over 11 years during January to March, as the remaining profiles were found to have some drastic deformation at higher levels and were, therefore, excluded from this analysis. The missing data in vertical levels were interpolated using a cubic-spline interpolation method (as the information of all possible heights was not available). Further, to generate the brightness temperatures at the different frequencies (see Table 1) for 11-year radiosonde observation, a forward model (see Figure 1 and Rambabu et al. 2013 for more details) was used. Seventy percent of the total number of temperature and humidity profiles was used to train the algorithm. Meanwhile, the remaining 30% of profiles were used to validate and test the algorithm, each by 15%. In addition, to validate the derived temperature and humidity profile from the developed inverse model, the existing GMMR data over the Mahbubnagar during the CAIPEX IGOC campaign 2011 was used.  Table  for 11-year radiosonde observation, a forward model (see Figure 1 and Rambabu et 2013 for more details) was used. Seventy percent of the total number of temperature a humidity profiles was used to train the algorithm. Meanwhile, the remaining 30% of pr files were used to validate and test the algorithm, each by 15%. In addition, to validate t derived temperature and humidity profile from the developed inverse model, the existi GMMR data over the Mahbubnagar during the CAIPEX IGOC campaign 2011 was use

Feed-Forward Neural Network
A standard feed-forward neural network (FNN) was used for constructing an inverse model that transforms the forward model calculated (and/or GMWR radiometric) brightness temperatures at the different frequencies (see Table 1 for the list of frequencies) to the atmospheric temperature and humidity profiles. The FNN structure used in this case is composed of three distinct layers: input, hidden, and output ( Figure 2). Additionally, each layer comprises several compute nodes arranged parallelly. All compute nodes in the output layer use a linear function, while all compute nodes in the hidden layer use a nonlinear function (also called an activation function). The activation function specifies a range for the output calculated at hidden nodes, which can be used as an input to other hidden layers arranged parallelly or as an input to the output layer (see Figure 2 for more details on FNN structure). The output at a particular hidden node (y), was obtained by the weighted sum of all inputs of the brightness temperature at the different frequencies (see Equation (1)), while the output at a particular node in the output layer (z), is obtained by the weighted sum of all the hidden node's output (see Equation (2)).
where x i is the ith input, w i,j is the associated weight between the ith input node and jth hidden node, w j,k is the associated weight between the jth hidden node and kth output node of the output layer, b and c are the correction factors for hidden and output layers; i, j, and k indices refer to nodes of inputs, hidden, and output layers, respectively. Function f(•) is a nonlinear function (see Equation (3)) and the first derivative of Equation (3) ) is used as a sigmoidal activation function in hidden nodes, and function g(•) is an identity function used in output nodes (see Figure 2). The schematic diagram of a neural network with an example of 5 compute nodes at the hidden layer, 4 input nodes at the input layer, and 6 output nodes at the output layer. The weights between the input and hidden layer are given by W1(i,j), and the weights between the hidden and output layer are given W2(j,k).

Sensitivity to Learning Rate
To initialize an optimal value of learning rate (η), we performed a sensitivity analysis of η for the estimated value from FNN at the output layer against the number of iterations. The variation of root-mean-square error (RMSE) against the number of iterations at the three different learning rates (i.e., 0.01, 0.001, and 0.0001) is shown in Figure 3 For the first iteration of FNN, the weights in the hidden and output layers are randomly initialized so that the sum of weights in the hidden layer and the sum of weights in the output layer is less than or equal to 1. The gradient descent back-propagation method is then used to assign these weights to some fixed values. The gradient descent backpropagation method is one of the most straightforward and most efficient methods used in supervised learning of a multilayer FNN. The network weights and correction factors in each iteration are updated by propagating errors back to the network (and so-called back-propagation) until the cost function (see Equation (4)) is minimized.
thus, the weights w i,j in the hidden layer and the weights w j,k in the output layer are updated in each iteration by computing the error δ k and δ j at the output and hidden nodes, respectively (see Equations (5) and (6)). The overall learning (or training) of FNN using Equations (5) and (6) for fixing the network weights are controlled by a learning rate (η).

Sensitivity to Learning Rate
To initialize an optimal value of learning rate (η), we performed a sensitivity analysis of η for the estimated value from FNN at the output layer against the number of iterations. The variation of root-mean-square error (RMSE) against the number of iterations at the three different learning rates (i.e., 0.01, 0.001, and 0.0001) is shown in Figure 3. The RMSE variation against the number of iterations at the other η values between 0.01 and 0.0001 is also calculated but is shown only for the above said η values. The learning rate η specifies the relative size of the weight that needs to be changed during the weight updating process. The RMSE is found to be very sensitive to the learning rate: at a learning rate of η = 0.01, a large increase in RMSE with an increase in the number of iterations is seen, and at a learning rate of η =0.001 and 0.0001, the variation of RMSE against the number of iterations is seen to be optimized. The RMSE, however, is optimized at a slightly lesser number of iterations at η = 0.001 than at η = 0.0001. As a result, we use a learning rate of 0.001 for the inverse model in our subsequent analysis of this paper.

Sensitivity to Learning Rate
To initialize an optimal value of learning rate (η), we performed a sensitivity analysis of η for the estimated value from FNN at the output layer against the number of iterations. The variation of root-mean-square error (RMSE) against the number of iterations at the three different learning rates (i.e., 0.01, 0.001, and 0.0001) is shown in Figure 3. The RMSE variation against the number of iterations at the other η values between 0.01 and 0.0001 is also calculated but is shown only for the above said η values. The learning rate specifies the relative size of the weight that needs to be changed during the weight updating process. The RMSE is found to be very sensitive to the learning rate: at a learning rate of η = 0.01, a large increase in RMSE with an increase in the number of iterations is seen, and at a learning rate of η =0.001 and 0.0001, the variation of RMSE against the number of iterations is seen to be optimized. The RMSE, however, is optimized at a slightly lesser number of iterations at η = 0.001 than at η = 0.0001. As a result, we use a learning rate of 0.001 for the inverse model in our subsequent analysis of this paper.   The structure of the proposed FNN to profile temperature and humidity from measured/estimated brightness temperature is shown in Figure 4. We used eight parallel FNN structures arranged in a vertical direction for increasing the accuracy of retrieved vertical profiles. Each FNN is made of five output nodes, with each node corresponding to a different altitude (or height). Thus, we use a total of 40 vertical levels (i.e., 8 FNN * 5 output nodes) with a resolution of 50 m up to 500 m, 100 m between 500 m and 2 km, and 500 m beyond 2 km up to 10 km. The resolutions of vertical levels are divided so that many levels occur close to the earth's surface. Most of the other existing methods are deteriorate/coarser to examine the processes occurring closure to the earth's surface.
In addition, we also use a large number of hidden nodes in the FNN structure that are close to the earth's surface (see Figure 4). Therefore, the complete FNN framework is used as an inverse model to estimate the temperature and humidity value at the various vertical levels.

Structure of the Inverse Model
The structure of the proposed FNN to profile temperature and humidity from measured/estimated brightness temperature is shown in Figure 4. We used eight parallel FNN structures arranged in a vertical direction for increasing the accuracy of retrieved vertical profiles. Each FNN is made of five output nodes, with each node corresponding to a different altitude (or height). Thus, we use a total of 40 vertical levels (i.e., 8 FNN * 5 output nodes) with a resolution of 50 m up to 500 m, 100 m between 500 m and 2 km, and 500 m beyond 2 km up to 10 km. The resolutions of vertical levels are divided so that many levels occur close to the earth's surface. Most of the other existing methods are deteriorate/coarser to examine the processes occurring closure to the earth's surface. In addition, we also use a large number of hidden nodes in the FNN structure that are close to the earth's surface (see Figure 4). Therefore, the complete FNN framework is used as an inverse model to estimate the temperature and humidity value at the various vertical levels.  6 show vertical profile of temperature and humidity, respectively, from the inverse model and radiosonde (RS) observation over Mumbai on three different days during winter season (December to February; or DJF). The brightness temperature at different frequencies (see Table 1) over Mumbai (an input to inverse model) is calculated using the feed-forward model because radiometric brightness temperature estimated from GMWR was not available (see Rambabu 6 show vertical profile of temperature and humidity, respectively, from the inverse model and radiosonde (RS) observation over Mumbai on three different days during winter season (December to February; or DJF). The brightness temperature at different frequencies (see Table 1) over Mumbai (an input to inverse model) is calculated using the feed-forward model because radiometric brightness temperature estimated from    The vertical profile of temperature and humidity calculated from the inverse model varies very similarly to that obtained from the RS observation on all three days. The inverse model based on a feed-forward neural network well captures the temperature inversion seen in the RS observation. The temperature inversion occurs when a layer of cool air at the surface is overlain by a layer of warmer air, causing a reversal in temperature structure within the boundary layer (i.e., air temperature increases despite decreasing with an increase in height). Further, in both the RS observation and the inverse model, the maximum variation in humidity is seen near the earth's surface ( Figure 6). This maximum variation in the humidity profile close to the earth's surface is most likely due to precipitation, dew, and fog. Furthermore, if we consider the temperature and humidity vertical profiles obtained from RS observation as true profiles, we find that the inverse model overestimated/ underestimated (at some levels) by about ±2 • C for temperature, ±4 g/kg for humidity in the lower troposphere, and about +4 • C for temperature and ±0.1 g/kg for humidity in the upper troposphere for all three successive days, as compared to the RS observation. However, this overestimation and underestimation in the vertical profile of temperature and humidity cannot be considered as 100% accurate as the RS observation can itself have some measurement bias due to the drift in balloon path during high horizontal wind flow and in fog or rain conditions). , from an inverse model (in black) and radiosonde observation (in red), as well as the corresponding differences (d-f). The brightness temperatures that are fed as an input to the inverse model, are calculated from a feed-forward model using radiosonde observation. Figure 6. Same as Figure 5 but for specific humidity profile. Figure 6. Same as Figure 5 but for specific humidity profile.

Figures 5 and
In addition to validating the inverse model's result with RS observation, we also validate the inverse model with an existing GMWR along with RS observation for two different days over Mahbubnagar, India, during the CAIPEX IGOC campaign in 2011, using Figures 7 and 8 for temperature and humidity, respectively. In this case, the brightness temperature measured by an existing GMWR over Mahbubnagar is used as an input to the inverse model instead of calculating it from RS observation using the forward model. We find good agreement between existing GMWR, RS, and inverse models in the lower troposphere. The large difference in vertical temperature and humidity profile overestimation/underestimation seen between RS observation and GMWR is seen to be reduced between the RS observation and inverse model. For example, the temperature profile calculated from inverse model shows a much smaller overestimation/underestimation than that seen in existing GMWR with respect to RS observations at all levels. The profiles calculated from the inverse model agree better with RS observation than the existing GMWR, not only for temperature but also for humidity, as evidenced by the reduced bias at all heights compared to the existing GMWR. Further, we investigate how the inverse model estimates the diurnal cycle of humidity and temperature with respect to existing GMWR during the CAIPEX IGOC 2011 over Mahbubnagar for three successive days from 29 to 31st October 2011. Figure 9 shows the time-height plots for the diurnal variation of temperature and humidity from the inverse model and existing GMWR and their corresponding differences. We find that the temperature and humidity profiles retrieved from an existing GMWR and inverse model are close at all levels except minor irregularities near the surface. The difference between the existing GMWR and inverse model is noted in the range of (+3, −1) • C for temperature, ±1 g/kg for humidity in the lower troposphere, and in the range of (+1.5, −4) • C for temperature and a minimal difference for humidity in the upper troposphere. The temperature inversion can be seen in diurnal variation as well with both methods (Figure 9).    The diferent color scale have unit of degree celsius and g/kg for temperature and humidity Where horizontal color scale is error scale for temperature 10    Furthermore, Figure 10 shows daily annual variation of temperature and humidity vertical profiles over Mumbai from the inverse model and RS observation during 2014, with an input of brightness temperature to Inverse model computed using a forward model from RS observation. The difference in temperature and humidity profiles between the inverse model and RS observation are also shown in Figure 10e,f. The rapid fall in temperature and increase in humidity during the transition of the pre-monsoon to monsoon period and the increase in temperature and decrease in humidity during the transition of the monsoon to the post-monsoon period, seen in the RS observation, are well captured in magnitude and pattern by the inverse model. In both the RS observation and inverse model, we find a stable humidity variation during winter, and the maximum variation of temperature and humidity is located near the earth's surface regardless of seasons. From the difference between the inverse model and RS observation, we find the difference in range of (−1 to −4) • C below 1.5 km, 2 to 4 • C above 7 km, and ±1 • C for the rest of the heights of temperature during the monsoon period. However, in the case of humidity, we notice the large differences during the monsoon periods and lesser difference during the rest of the periods, with a variation of ±1.5 g/kg for all heights in all seasons except during the monsoon season where it is about 3 to 5 g/kg.
iation of temperature and humidity is located near the earth's surface regardless of seasons. From the difference between the inverse model and RS observation, we find the difference in range of (−1 to −4) °C below 1.5 km, 2 to 4 °C above 7 km, and ±1 °C for the rest of the heights of temperature during the monsoon period. However, in the case of humidity, we notice the large differences during the monsoon periods and lesser difference during the rest of the periods, with a variation of ±1.5 g/kg for all heights in all seasons except during the monsoon season where it is about 3 to 5 g/kg.

Conclusions
An inverse algorithm (inverse model) was developed to profile the vertical structure of temperature and humidity in the troposphere, using a feed-forward neural network. The inverse model was used to calculate several temperatures and humidity profiles and

Conclusions
An inverse algorithm (inverse model) was developed to profile the vertical structure of temperature and humidity in the troposphere, using a feed-forward neural network. The inverse model was used to calculate several temperatures and humidity profiles and compared with different existing techniques/methods. The inverse model is found more efficient in profiling the temperature and humidity vertical structure than other existing methods. The statistical methods used in the existing ground-based microwave radiometer (GMWR), known for their high computational cost and altitude-dependent inaccuracy, and inability to estimate the vertical temperature and humidity profile, are enhanced when an inverse model is used. The diurnal and seasonal cycle of temperature and humidity vertical profiles calculated by the inverse model outperformed other existing methods, which could be helpful for assimilation in numerical weather prediction models. We suggest that introducing such an inverse model into GMWR will improve/increase the accuracy of temperature and humidity retrieval and so the improvement in weather forecasting. Data Availability Statement: Data used in this research are freely available in the community, for example, the used radiosonde data can be found from the Upper-Air-Sounding Division of University of Wyoming (http://weather.uwyo.edu/upperair/sounding.html, accessed on 14 August 2019) and the data from CAIPEX IGOC campaign 2011 in India can be found at Indian Institute of Tropical Meteorology Pune (https://www.tropmet.res.in/~caipeex, accessed on 23 November 2019). However, the developed inverse model code can be shared on request to the corresponding author.