# Estimating Retrieval Errors From Neural Network Inversion Schemes-Application to the Retrieval of Temperature Profiles From IASI

Filipe Aires, Victor Pellet

► To cite this version:

Filipe Aires, Victor Pellet. Estimating Retrieval Errors From Neural Network Inversion Schemes-Application to the Retrieval of Temperature Profiles From IASI. IEEE Transactions on Geoscience and Remote Sensing, 2021, 59 (8), pp.6386-6396. 10.1109/TGRS.2020.3026944. hal-03880941

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# Estimating Retrieval Errors From Neural Network Inversion Schemes—Application to the Retrieval of Temperature Profiles From IASI

Filipe Aires* and Victor Pellet*

***Abstract***—Neural networks (NNs) are becoming increasingly more popular to interpret remote sensing observations in many contexts. Their flexibility and accuracy have been a true advantage for the retrieval of many geophysical variables in the atmosphere, land, and ocean, over the last three decades. Uncertainty of the retrieved products is important to assess the quality of the retrievals but also to combine products *a posteriori*, combine them in a complex way to study for instance the water cycle, or assimilate them in numerical weather prediction (NWP) centers. However, no easy solution has been proposed so far to estimate the retrieval uncertainties of NN inversion schemes. A simple, pragmatic, and easy-to-implement scheme based on the input space clustering is proposed here to perform this task. Tests are conducted using an application aiming at retrieving atmospheric profiles based on improved atmospheric sounding in the infrared (IASI) high-spectral resolution observations in the infrared.

***Index Terms***—Atmospheric temperature, high-spectral infrared observations, improved atmospheric sounding in the infrared (IASI), neural networks (NNs), retrieval uncertainties, satellite remote sensing.

## I. INTRODUCTION

SATELLITE observations are utilized in two contexts. First, they can be used as inputs to retrieval algorithms in order to build geophysical data sets describing the state of the atmosphere [21] or the ocean [22] and continental [10] surfaces. In this case, users of such data sets are generally in need of uncertainty estimates to assess where these satellite estimates are reliable or not or to decide if the data set is good enough for a particular application. Users might also need uncertainty assessment in order to do data fusion of several data sets to improve the quantification of a geophysical variable [12], [13] or to mathematically combine several estimates to better describe complex processes [15], [39].

Another way to utilize these satellite measurements is given by the “variational assimilation” [26], [32], [41], [48], where observations are assimilated into a numerical weather prediction (NWP) model. In this case, the information provided by the observations (either raw radiances *R* or retrieved

products *P*) and the model is weighted based on their respective uncertainties [42]. Therefore, an uncertainty estimate for the satellite information is necessary for this context too. Since reliable uncertainties on retrieved products are difficult to obtain (i.e., they are state-dependent and sensitive to many factors), the assimilation of raw radiances *R*, instead of retrieved products *P*, has been privileged in the past. It is often assumed that observation errors can be represented by the sum of: 1) a constant instrument error and 2) a constant radiative transfer (RT) error whatever the surface or atmospheric conditions. This simple assumption is correct in many cases, but it can also be misleading since instrument and RT errors might be state-dependent [17] and very difficult to model (e.g., [16] for an investigation of more realistic assumptions for cloudy situations with microwave observations). Furthermore, the assimilation of *R* is possible only when the RT to simulate *R* is of good quality. This has been the case, for instance, for atmospheric applications (e.g., sounding of temperature and humidity atmospheric profiles) where RT calculations are reliable, at least in the clear case (i.e., cloudy and precipitation cases are still an issue [16]). The RT calculations are less accurate for surface simulations because the required auxiliary parameters (soil texture, surface emissivities, and so on) are often difficult to obtain at a global scale. For instance, the assimilation of retrieved soil moisture seems to be more pertinent than the assimilation of the raw radiances *R* [9], [44]. Therefore, the estimation of retrieval uncertainty is also necessary for this context of the NWP variational assimilation. This would greatly facilitate the assimilation of many retrieved products. Estimating retrieval uncertainties is a difficult task, but it should be noted that *a priori* errors (i.e., the error of the *a priori* information before the satellite observations are used in the assimilation) need to be characterized too. This was not an easy task for the NWP community: first, estimates were global at first and then latitudinally dependent, and they now became state-dependent [30].

Neural networks (NNs) have been widely used in the field of satellite remote sensing, for more than 20 years, for the atmosphere [4], [28], or the land [4] and oceanic [40] surfaces. However, estimating retrieval uncertainties for an NN inversion scheme is not an easy task. These retrieval uncertainties need to take into account all the sources of uncertainty: instrument error, retrieval-model error, representation errors, database limitations (RT, if the database is simulated,

Manuscript received September 25, 2019; revised January 21, 2020 and May 12, 2020; accepted May 12, 2020. (Corresponding author: Filipe Aires.)

The authors are with LERMA, Observatoire de Paris, CNRS, Sorbonne University, 75002 Paris, France (e-mail: filipe.aires@obsm.fr).

Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2020.3026944
space/time coincidence errors, if the database is empirical, and lack of samples in the database), and so on Furthermore, these error estimates need to be state-dependent and should be available in real time (i.e., simultaneously with the retrieval).

The estimation of the NN retrieval uncertainties is generally estimated after/during the learning stage over a data set that is as independent as possible from the learning data set used to train the NN. Often, this “generalization” data set is built in the same way as the training data set, so even if it is not used during the learning, the independency is not always satisfied because it shares many features with the training data set. The retrieval errors can also be estimated in a truly independent data set gathering for instance in situ measurements or radiosondes. This a posteriori (i.e., once the NN is trained) comparison of the retrieval with a desired solution is generally used to obtain a global and static NN-quality statistics that allows for instance comparing the quality of different algorithms. In order to better understand the errors of the retrieval, it is necessary to analyze the state-dependence of the retrievals as some states are more difficult to retrieve: very complex states (e.g., atmospheric temperature profile with several inversions), extreme states (e.g., very cold or very warm temperature profile), states that are not well represented in the learning data set for lack of availability, and so on. Performing such a posteriori uncertainty assessment has been used to specify uncertainties and compare satellite instruments or retrieval schemes, but they can hardly be used to estimate a real-time retrieval uncertainty.

One proposed approach to estimate the errors in the NN outputs has been to propagate errors from the NN inputs to its outputs. For instance, a standard multilayered perceptron (MLP) is a simple mathematical expression [45], so it is possible to use the standard approximation named “propagation of uncertainties” often used in physics to quantify uncertainties throughout a model [24]. An example in the remote sensing domain can be found in [43]. Since an MLP has a simple structure, it is actually possible to find a closed-form solution and then numerically integrate probability distributions [33]. If input errors are Gaussian, the closed-form solution is easier to obtain. For more complex input uncertainties or for bigger and more complex NNs (such as in deep learning and convolutional NNs), computing an analytical expression is unrealistic [47], and the Monte Carlo or extended Kalman filter methods have been used to take a set of samples from the input distribution and propagate them through the NN to approximate the first and second moments of the output distribution [1]. However, the propagation of error approach only finds the output errors that originate directly from the input uncertainty, not accounting for the inherent errors of the NN itself or the configuration (e.g., representation errors or missing auxiliary data). This means it assumes that NN is a perfect model, which is rarely the case.

A complex theoretical framework has been proposed in [34]; this was updated very recently for deep learning NNs in [29]. These approaches are based on the Bayesian models. The principle is to have randomness parameters in the NN (for instance, a distribution on the weights of the NN) so that this randomness translates into a dispersion of the NN outputs.

A theoretical framework was proposed by [5]–[8] in the context of remote sensing applications. The idea is to learn this internal randomness to represent the errors present during the learning. However, this type of approach is very complex theoretically and practically too difficult to implement due to numerical instabilities. As a consequence, no technique has really caught on. The NN user community, and in particular, in the satellite remote sensing field, is in need of a pragmatic, simple, and efficient method to estimate the NN error uncertainties. It is the objective of this article to propose such a framework.

When the NN learning data set is built from real observations (the “empirical database” discussed in the following), most of the sources of uncertainty of the remote sensing problem are present in this learning data set: input noise (as in the previous propagation of errors), output noise (due to the limited quality of the data set), collocation errors, missing information, model error, and so on. Adding noises in the learning data set has been a very fruitful approach to improve the quality of the NN. This includes the introduction of noise in the inputs (this is equivalent to increase the number of samples, but it also regularizes the NN model [18]) or noise in the outputs. Since most of the sources of uncertainty are included in the learning data set, it actually makes sense to use this information to estimate uncertainties too.

Section II introduces the data sets used in this study. The methodology is introduced in Section III. Experiments using improved atmospheric sounding in the infrared (IASI) observations are presented in Section IV. Finally, Section V concludes this study and provides some technical and application perspectives.

## II. DATA SETS

### A. Real IASI Brightness Temperatures ($T_B$)

The IASI sounder is an infrared Fourier transform spectrometer based on a Michelson interferometer [20]. It is a part of the Metop-A (launched in 2006), Metop-B (2012), and Metop-C (2018) missions. It will also be a part of the IASI-NG (Next Generation) mission that should fly in 2022. The main IASI purpose is to support the NWP centers [25]. Its spectral range goes from 645 to 2760 cm⁻¹, with a spectral resolution of 0.25 cm⁻¹. In the following, we will focus on the retrieval of the temperature atmospheric profiles *T* in the troposphere and the lower stratosphere [3], [31]. Channel selection algorithms have been proposed to isolate the most pertinent information in a high-dimensional observation, such as an IASI spectrum [42]. Collard [25] used such a channel selection in a NWP center. In [38], the perturbation effect of contaminating variables (i.e., geophysical variables impacting the RT, not constrained or retrieved in the retrieval) was used with state-dependent covariance matrices of the *a priori* errors to obtain a channel selection of 97 channels for the retrieval of the temperature profile *T*.

Fig. 1 shows a sample of a Brightness Temperatures ($T_B$) IASI spectrum. At a given wavenumber, $T_B$ results from the emission and absorption of the different atmospheric components along the atmosphere at this wavenumber. For instance,
<figure><img src="image_2.png" /><figcaption>Fig. 1. Typical IASI $T_B$-spectrum. Temperature profile information is exploited here in the 645–800-cm$^{-1}$ spectral range.</figcaption></figure>

up to the tropopause, the hotter the $T_B$ is, the less the signal has been absorbed through its path in the atmosphere (no absorption band of atmospheric components at this wavenumber) and the deeper, in the atmosphere, the IASI channel probes. For instance, the hottest $T_B$'s (280 K between 770 and 980, 1080 and 1150, and 2420 and 2700 cm$^{-1}$) correspond to the window channels, characterizing the surface properties. Above the tropopause, the $T_B$ will start getting larger again as the temperature increases with increasing altitude.

## B. ARSA Radiosonde Database

The construction of an optimal learning and testing database for the training of an NN inversion scheme is a delicate process [11], [23], [37]. In this study, the emphasis is not on the retrieval itself but on a simple method to estimate the retrieval uncertainties. We then need a data set that is not perfect, which includes as many sources of uncertainties as we can encounter in real-world problems. The objective is then to show that the NN can estimate errors that are close to what is observed in this database.

The databases can be built using reanalysis describing the state of the Earth's surface and atmosphere, and a radiative transfer can be used to simulate the corresponding TBs. An instrumental noise could be added to these TBs in order to specify a priori the instrument uncertainties. We refer to this approach as a “physical database.” If we were to use this type of simulated database, we would certainly obtain in return uncertainties that would be equal to what was specified a priori. Instead, it is preferred here to use a data set with real uncertainties in order to measure if our NN uncertainty method can estimate them.

Another approach is to use a so-called “empirical database” where observations of the geophysical variables are put into space/time coincidence with real satellite observations. The geophysical variable measurements can originate, for instance, from radio-occultation data, e.g., from the Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) data sets. We decided, instead, to use data from radiosonde measurements because they are used operational in NWP centers, and it is the usual way to validate and calibrate satellite retrievals. In this article, the objective is not to develop and use a perfect database but, instead, to illustrate the retrieval error estimation on a simple application with several, large, and realistic sources of uncertainties. Therefore, we considered the radiosondes over the ocean only because the retrievals would be more complex over land.

The Laboratoire de Météorologie Dynamique (LMD) IASI Composite Observatory (LMD_ICO) data set is generated at the LMD. This data set is a composite of real IASI $T_B$'s put into coincidence with radiosonde measurements around the world. The data set starts in July 2007 and is updated on a monthly basis. Radiosonde information is from the Analysed RadioSoundings Archive (ARSA) database provided by the Atmospheric Radiation Analysis Group.<fnref n="1" /> ARSA is based on observations from worldwide distributed stations. Validation of ARSA data has been performed using numerous comparisons between the simulated and observed IASI $T_B$'s. The profiles are provided on 43 atmospheric levels for temperature, specific humidity, and ozone. The current ARSA database (about six million elements) starts in January 1979 and is extended onward on a monthly basis.

The coincidence of these radiosondes with the real $T_B$'s of Section II-A is performed using a maximum time difference of 30 min and a spatial distance of less than 100 km. For each radiosonde measurement, only the nearest IASI measurement is used, so the radiosonde is used only once in the final database. Since the closest IASI measurement is chosen, it is actually much closer to the radiosonde than the threshold of 100 km. Only oceanic cases during the night were chosen during the year 2014. Many radiosondes are launched from coasts, but the radiosondes at less than 20 kms from the coast have been suppressed to avoid radiosondes potentially contaminated by the land. Two filters were added to select data.

1) The clear-sky case was chosen with a cloud flag equal to zero (this flag is based on thresholding using a combination of IASI and advanced microwave sounding unit (AMSU) channels) [19], [27].

2) The nadir geometry was selected using an incidence angle between -10° and +10°.

The total number of samples is equal to 27.721 diverse situations. This is not a huge number, but it was shown in [4] that a lower database can be enough to train a simple NN as the one we will be using in the following. Note that space/time collocation between IASI $T_B$'s and radiosondes introduces uncertainties in the database, but this is actually a good thing for our experiment.

# III. METHODOLOGY

## A. NN Retrieval

Among the NN techniques that can be used to perform satellite remote sensing, the MLP is certainly the most widely used one. We pose

$$ \hat{T} = \text{NN}_1(T_{B\epsilon}) \qquad (1) $$

the first classical retrieval model, where $\hat{T}$ is a retrieved geophysical variable vector of dimension $n$ (e.g., temperature), and $T_{B\epsilon} = T_B + \epsilon$ is the observed radiance vector of dimension $m$, including the instrument noise $\epsilon$. The $NN_1$ function is a mapping model composed of parallel processors called “neurons.” These processors are organized in distinct layers:

<footnote n="1">¹http://ara.abct.lmd.polytechnique.fr/index.php?page=arsa</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Traditional NN retrieval scheme. (b) New framework for NN estimation of the retrieval errors.</figcaption></figure>

the first layer represents the input $T_B$'s (dimension $m$) of the mapping. The last layer represents the output of the mapping $T$ (dimension $n$). The intermediate layers are the “hidden layers.” These layers are connected via neuronal links: two neurons $i$ and $j$ between two consecutive layers have synaptic connections associated with a synaptic weight $w_{ij}$ (we pose $W$ as the set of all synaptic weights $w_{ij}$). A neuron executes two simple operations: first, it makes a weighted sum of the inputs and then transfers this signal to its output through a so-called transfer or activation function. Fig. 2(a) represents such a classical NN inversion scheme. Examples of $NN_1$ retrieval algorithms for the IASI instrument can be found, for instance, in [2]–[4].

Once the NN architecture is defined, a training data set, including input–output couples, is used to find the right parameters $W$ of the model. The basic optimization algorithm to perform this task is the backpropagation algorithm from [45]. It is based on the minimization of a quality criterion (such as the least-squares errors) that specifies how good the NN is on the learning database.

## B. Monovariate Case and Sources of Uncertainties

Let us first start with a simple case for illustrative purposes. In this section, only one input and one output are considered in the retrieval. Fig. 3(a) represents, in red, the scatterplot of the $T_{43}$ (i.e., temperature at level 43, i.e., 1013 hPa) versus its most correlated IASI $T_B$-channel at 740.25 cm$^{-1}$. The link between these two variables is very strong: the relationship is almost linear. The goal of a retrieval algorithm is to fit $T_{43}$ based on the information of $T_B$. It should be noted here that even if more samples were to be added in the database, the structure of this

scatter of points would not change. This means that the link between input and outputs would not be modified and that the retrieval would not be significantly impacted. It shows that the number of samples required in a training data set does depend on the complexity of the function that we intend to represent with the NN. However, adding more points in some of the bins would improve the robustness of the estimation of the error statistics (bias, standard deviation, and correlation of matrices). This makes sense that, to assess second-order statistics, there is a need for more samples in the database. It shows that, in a data-driven approach, such as with NNs, if fewer samples are available to describe the input space, then the NN fit will not be as good in that part of the input space.

An $NN_1$ retrieval scheme, such as in (1), was developed in order to reach this goal. We use here a single-input/single-output architecture, with five neurons in the hidden layer (other architectures were tested but with no significant impact). Once the NN architecture is defined, an optimization algorithm is used in order to find the optimal parameters of the NN model (i.e., synaptic weights $W$). A backpropagation algorithm with a conjugate gradient optimization is used for this learning stage [18]. Considering the limited number of samples in the database, the NN is trained using 90% of the samples, and its generalization ability is tested on the 10% remaining ones. This process is performed ten times in order to have a robust generalization error estimate. Results for the retrieval of $T_{43}$ are presented in Table I. Learning and testing statistics are almost identical meaning that the training does not suffer from overfitting.

For a given $T_B$ observation in the horizontal axis, there is only one retrieved temperature [in blue in Fig. 3(a)] because
<figure><img src="image_4.png" /><figcaption>Fig. 3. Some statistics for the monovariable case: (a) $T_B$ at 740.25 cm$^{-1}$ versus $T_{43}$, the temperature at level 43 (i.e., 1013 hPa). Red dots are the real data, and the NN inversion scheme is represented in blue. The horizontal axis has been divided into eight bins representing 3 K $T_B$-ranges. In each one of the bins, $T_{43}$ averages are represented in black squares ± one STD, also indicated in a rectangular window. (b) Horizontal axis: $T_B$ at 740.25 cm$^{-1}$; in the vertical axis, the red dots represent the absolute value of the retrieval errors abs($T_{43} - NN_1(T_B)$), black squares are the bias error per bin, and blue squares are the STD error per bin. The line of green dots is the NN estimate of the bias errors, and the line of cyan dots is the NN estimate of the STD errors.</figcaption></figure>

<table><caption>TABLE I
RETRIEVAL STATISTICS FOR THE $T_{43}$ NN MODELS: $NN_1$ IS THE RETRIEVAL MODEL WITHOUT UNCERTAINTIES, AND $NN_2$ IS WITH UNCERTAINTY ESTIMATE</caption><thead><tr><th rowspan="2">Network</th><th colspan="2">Bias Error</th><th colspan="2">STD Error</th></tr><tr><th>Learning</th><th>Testing</th><th>Learning</th><th>Testing</th></tr></thead><tbody><tr><td>NN<sub>1</sub></td><td>0.00</td><td>0.00</td><td>2.91</td><td>2.93</td></tr><tr><td>NN<sub>2</sub></td><td>0.00</td><td>0.00</td><td>2.91</td><td>2.92</td></tr></tbody></table>

the NN is a function with a single determinist output. However, there is a vertical dispersion on the true temperatures in the $T_{43}$ vertical axis (in red). This dispersion represents the “ambiguities” of the $T_B-T_{43}$ relationship (i.e., systematic differences and any kind of uncertainties). The NN inversion can be optimized by improving: the NN structure, the number of samples in the learning data set, or the learning algorithm. However, the limiting factor is often not the quality of the database or the complexity of the NN but instead the limitation of the

input information. The vertical dispersion here is real uncertainty in the $T_B-T$ relationship. The dispersion surrounding the $T_B$ versus $T_{43}$ relationship present in the learning data set cannot be overcome unless additional information is added in the NN inputs, such as an additional $T_B$ channels or auxiliary information, in order to decrease this vertical dispersion. This is independent of the retrieval scheme or even the number of samples.

The horizontal axis has been divided into eight bins representing $T_B$-ranges of 3 K. In each one of these bins, the $T_{43}$ averages are represented in black squares ± one STD Deviation (STD). The STD values are also indicated in the plot. The NN retrieval follows well these bin-averaged values, but some oscillations in less populated bins<fnref n="2" /> indicate that there is an insufficient number of learning samples in these input-space regions.

Fig. 3(b) is similar to the plot in (a) except that the vertical axis represents the absolute retrieval errors. The averaged retrieval bias errors for each bin are represented by black squares, and the STD errors are represented in dark blue. There are some small bias oscillations for a lower $T_B$ bins, but they are only due to a lack of robustness in the less populated bins. Contrarily to the plot in (a), (b) is dependent on the retrieval scheme. In addition to the “natural sources of uncertainties” in the $T_B-T_{43}$ relationship, the retrieval uncertainties are also included here.

Several sources or retrieval errors can cause this vertical dispersion of the red dots.

1) **Observation Errors:** $T_B$ inputs of the NN can be corrupted by instrumental noise: $T_{B\epsilon} = T_B + \epsilon$, this means that the true $T_B$ in Fig. 3(a) has been perturbed in the horizontal axis by a random error $\epsilon$.

2) **Structural Errors:** The architecture of the NN might not be adequate (e.g., not enough or too complex mapping function); the learning scheme used to train the NN might not be perfect (e.g., bad initial conditions and insufficient optimization algorithm); the learning data set might be deficient (i.e., insufficient number of samples or lack of diverse data). All these reasons cause the NN to be unoptimal for the retrieval.

3) **Insufficient Input Information:** Additional $T_B$'s might be necessary to constrain enough the inverse solution. Performing satellite remote sensing is solving an inverse problem and not enough information can cause the nonunicity of the solution (see [14] for the exploitation of multispectral synergy to reduce this type of difficulty).

4) **Missing Parameters:** Some parameters impacting the observation $T_B$ might be missing, e.g., surface emissivity might be necessary auxiliary information, in particular, for retrievals performed over land (see [36]).

5) **Representation Errors:** They are linked, for instance, to the inherent smoothing of a discrete description of profiles (i.e., atmospheric layers instead of an infinity of levels) [42].

<footnote n="2">²Bins could be chosen with varying sizes so that the same number of samples would be included in each one of them.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 4. (a) Averaged temperature profiles. (b) Bias retrieval error. (c) STD retrieval error for the ten clusters. These statistics are computed on real data with the NN₁ retrieval, and they are not estimates.</figcaption></figure>

6) **Measurement Errors:** Since the geophysical variables (e.g., temperature profiles) are radiosonde data in our case, measurement errors are also included in the horizontal dispersion.

7) **Collocation Errors:** Radiosonde measurements are put into coincidence with the IASI measurements, temporal delays, or spatial distances that introduce errors too in the database.

These nonexhaustive lists of sources of uncertainties describe real physical causes for retrieval errors. They have been listed here for an NN retrieval in the context of satellite remote sensing, but most of the listed sources are general and pertinent to many problems. It is important to estimate the impact of all these sources of uncertainty because they are used in the database used to train our retrieval. The best source of information to estimate them is the learning data set, so our approach will use the observed dispersions to estimate the retrieval uncertainties.

The important question is: are all sources of uncertainties represented in the learning database? If yes, then the goal in the next sections of this article is to estimate the uncertainty represented by this dispersion in the vertical axis of the retrieved variable. One constraint is that this uncertainty estimation should be state-dependent.

### C. Characterization of the Uncertainties

Three types of information about the uncertainties can be considered for an NN retrieval:

1) **Bias or Mean Error:** $ME = mean(\hat{T} - T)$, with $\hat{T} = NN(T_{BE})$.

2) **STD of Errors:** $STD = STD(\hat{T} - T)$.

3) $n \times n$ CORrelation Matrix of the Retrieval Errors: $COR = COR(\hat{T} - T)$.

To estimate these error statistics, the retrieval of (1) is updated with these quantities added to $T$ in the outputs of the NN

$$ (\hat{T}, \widehat{ME}, \widehat{STD}, \widehat{COR}) = NN_2(T_{BE}). \quad (2) $$

Fig. 2 compares a regular NN retrieval scheme (A) and the scheme that is proposed here to estimate the uncertainties (B). The two $NN_1$ and $NN_2$ retrievals are represented in green; the $T_B$-space clustering process is represented in orange; and the two learning processes are represented in blue.

An $NN_2$ retrieval scheme is tested in the monovariate example of Section III-B. Each sample in the training data set is associated with the retrieval uncertainty of its bin [i.e., the black squares for the bias and the dark blue squares for the STD errors in Fig. 3(b)]. The line of green dots represents the $NN_2$ estimate of the bias error, and the line of cyan dots represents the estimate of the STD error. $NN_2$ results are provided in Table I. Again, there is no sign of overfitting at the global scale as learning and testing rates are very similar. This is probably due to the fact that we limited the number of neurons in the hidden layer, this is called structural stabilization, and it is a form of regularization. Comparison between the $NN_2$ and $NN_1$ rates show that $NN_2$ reproduces well the $NN_1$ retrievals, so the errors measured with $NN_1$ can be used to train $NN_2$. We actually produced the scatter plots (not shown) between the errors of the $NN_1$ and $NN_2$, and they are almost identical. NNs are generally unbiased estimators at the global scale (when merging all the samples of the several bins), but they can be biased for some of the bins. The bias estimate (in green) can be considered
to be flat close to zero. The STD estimate (in cyan) fits well the dark blue squares, but some oscillations are present. These oscillations result typically from a limited number of samples in the training for some of the bins. This retrieval and its uncertainty estimate are, therefore, not perfect, but it is normal to have higher errors in the less populated parts of the input space. Furthermore, this preliminary test of the methodology is performed for illustrative purposes using only one $T_B$.

In order to obtain a more realistic uncertainty estimate for real-world applications, the NN should have multiple $T_B$ inputs (i.e., channels), so the “binning strategy” of Section III-B needs to be extended to the multivariate case.

### D. Partitioning of the Input Space

To estimate the uncertainties linked to the retrieval $T_B \rightarrow \hat{T}$, it is necessary to define sets of samples in which we can estimate the three uncertainty diagnostics ME, STD, and COR of the previous section. The question is: how to define these sets of samples that share common features, with a definition based on the information that is available at the time of the retrieval, namely, the satellite observations $T_B$? For the test in Fig. 3, the solution was to define bins in the horizontal axis of the satellite observations $T_B$. In each one of these bins, the ME, STD, and COR of errors were estimated based on $NN_1(T_B) - T$. However, the problem becomes more complex when the $T_B$ dimension increases.

To partition the satellite observation space, we propose here to perform clustering, i.e., an algorithm that will gather the samples of the database into coherent clusters. Several clustering algorithms exist, and we chose here the traditional k-means approach [35]. The database samples ($T_B$'s) are gathered into “clusters” based on their similarity. The similarity is based on a distance on the input space, and the Euclidean distance is generally chosen for simplicity. The number k of clusters (i.e., the equivalent to the number of bins in Fig. 3) needs to be specified a priori. This number should be high enough to describe well the variability of the $T_B$-space, but it needs to be small enough so that quantifiable differences between them can be observed.

It could be argued that other auxiliary information could be used in order to help the clustering to find the right classification. For instance, the latitude of the samples could be added, the level of humidity, and so on. It is correct that such information could be useful to define better clusters, facilitating the goal of estimating error statistics. However, if this information is useful for the estimation of the uncertainties, it should be useful too for the retrieval, so we would expect it to be part of the NN inputs. In this article, the choice was made to use only a selection of channels as inputs, and the temperature profile as the output, for illustration purpose. Our application framework should be further improved in the future.

### E. NN-Uncertainty Characterization

Once the clustering is done on the $T_B$-space, it is proposed here to estimate the NN retrieval error statistics (ME, STD, and COR). It is then asked the NN to simultaneously perform

<figure><img src="image_6.png" /><figcaption>Fig. 5. Five retrieval examples from clusters C3, C4, C8, C10, and C6. True profile is in green, retrieval in black, and retrieval $\pm 2 \cdot \text{STD}$ of errors is in dashed red (estimated from $NN_2$).</figcaption></figure>

the retrieval of the temperature profile $T$ and the retrieval of the error statistics.

The algorithm to perform the retrieval of $T$ and simultaneously estimate the retrieval error statistics is defined as follows.

1. Partition the $T_B$-input space into $k$ clusters.

2. Train $\hat{T} = NN_1(T_B)$ using $\mathcal{B} = \{(T_B^e, T^e); e = 1, \dots, E\}$, where $E$ is the number of sample.

3. Estimate errors ($ME_T$, $STD_T$, and $COR_T$) for the $T$-retrieval, for each $k$ clusters using $\hat{T} - T$ statistics, and add them in the outputs to the temperature profile, to retrieve $(\hat{T}, \hat{b}_T, \hat{\sigma}_T, \hat{c}_T)$.

4. Train $(\hat{T}, \hat{b}_T, \hat{\sigma}_T, \hat{c}_T) = NN_2(T_B)$.

5. Go back to [3] until convergence (i.e., errors in [4] and [3] are close enough) is reached.
<figure><img src="image_7.png" /><figcaption>Fig. 6. STD of the retrieval as measured in the ten clusters (in continuous line), as shown in Fig. 4(c), and its corresponding estimated by the NN retrieval scheme (in dashed line).</figcaption></figure>

Errors in step [3] are based on the previous version of the NN. The iterative process in step [5] means that errors of the previous NN version are used to train the NN of the next version in step [4]. In the applications of this article, we noticed, however, that the convergence is reached in one iteration only (errors of $NN_2$ are almost identical to errors of $NN_1$). We still mention the convergence of step [5] for potential applications that would require such iterations.

The NN is not asked to reproduce the classification of the clustering (i.e., errors would be stepwise in the transition from one cluster to the other) but rather to model a continuous, smooth, and state-dependent mapping between the inputs and the retrieval errors (see Fig. 2).

# IV. ATMOSPHERIC TEMPERATURE PROFILE RETRIEVAL

## A. NN Retrieval of the Temperature Profile

This section corresponds to the step [2] in the algorithm of Section III-E. A regular NN retrieval, specified as $NN_1$ in (1), is used to retrieve the temperature profile: $\hat{T} = NN_1(T_{BE})$. The LMD IASI Composite Observatory (including $T_B$'s/radiosondes) database is used to train and test this retrieval scheme. The number of inputs is equal to 97, and it corresponds to the IASI channels that have been selected for the retrieval of the temperature profile [38]. The hidden layer is chosen with 20 neurons; this number was tested to be big enough for this input configuration. The number of outputs is 43 corresponding to the 43 atmospheric levels for temperature.

The training of the NN is performed using a gradient descent backpropagation optimization algorithm. This retrieval scheme performs satisfactorily, similar to what was found for instance in [38] for the IASI instrument.

## B. Partitioning of the $T_B$ Input Space

This section represents the step [1] of the algorithm of Section III-E. The number of clusters $k$ is chosen equal to ten so that ten clusters are identified in the $T_B$-space, and the number of samples in each cluster is, respectively, 6332, 1345, 2910, 1443, 4517, 1572, 1919, 2491, 1784, and 3408. Some of the clusters are more populated: they are generally stable and frequent situations in the database. Other clusters have a lower number of samples (e.g., clusters 2 and 4): they are generally representative of extreme cases.

The clustering is performed in the $T_B$-space, but the results are presented in Fig. 4(a) in the temperature profile space. The mean temperature profile is computed in each one of the ten clusters. Some of these profiles are tropical (e.g., C1 in continuous red with elevated surface temperature), and others are polar (e.g., C2 in continuous green with low surface temperature). Other characteristic features are the tropopause level and the steepness of its inversion.

The error statistics of Section III-C are computed for the $NN_1$ retrieval for each cluster. Fig. 4(b) represents the retrieval bias for the ten clusters and Fig. 4(c) is for the retrieval error STD. There are no significant systematic bias errors for the
clusters. Some differences and departure from zero can be observed, but they are not statistically robust. Therefore, in the following, the retrieval scheme $NN_2$ will not attempt to retrieve these ME bias errors.

On the contrary, the error profile STDs have significant differences. For instance, cluster C2 (in yellow) has lower STD errors in the stratosphere than cluster C8 (dashed green). In general, atmospheric profiles for tropical situations with a very smooth and standard shape are easier to retrieve. Colder or more rare/complex situations are more difficult to estimate.

Correlation of errors has also been estimated for each one of the ten clusters (not shown), but no robust differences could be observed. Therefore, in the following, the retrieval scheme $NN_2$ will not attempt to retrieve them.

As a conclusion, the $NN_2$ retrieval in the following will attempt to estimate two quantities: the temperature profile $T$ itself and $\sigma_T$, and its associated retrieval STD: $(\hat{T} \ \hat{\sigma}_T) = NN_2(T_B)$. $NN_2$ was trained, and it was checked that its errors are similar to $NN_1$, so only one iteration is necessary for the algorithm of Section III-E. If this was not the case, iterations would be required until convergence of the errors was achieved.

### C. Estimation of the Retrieval Uncertainty

Fig. 5 represents five $NN_2$ retrieval examples from clusters C3, C4, C8, C10, and C6. The estimated retrieval STD errors are also represented in this figure. The retrieval in black properly estimates the true profile in green. As expected, bigger errors occur in the complex parts of the temperature profile, namely, in their inversion at the tropopause, stratopause, and surface layers. The retrieval is mostly inside the ±2 STD uncertainty envelope provided by $NN_2$. Furthermore, the estimated uncertainty is lower in simple parts of the profile (e.g., smooth profile in the troposphere), and they increase over the tropopause and stratopause, which is a natural and expected behavior.

Fig. 6 compares the measured (continuous line) and estimated (dashed line) STD of the retrieval errors when averaged over each one of the ten clusters. The agreement is very satisfactory and shows that the estimated STD of the retrieval errors is reliable and state-dependent. Note that $NN_2$ is not using the direct classification of the clustering, so the second part of $NN_2$ has been able to represent well a continuous and smooth mapping between $T_B$'s and the retrieval uncertainties.

If the covariance matrices would be required, we could propose using the general (i.e., over the ten clusters) correlation matrix of the retrieval errors and then combine the correlation matrix and STD of errors to build a state-dependent covariance matrix of the retrieval. This will be investigated in the future.

## V. CONCLUSION

This article introduced a novel, simple, and pragmatic framework to estimate the retrieval uncertainties of NN algorithms. Instead of estimating variables, the new NN model estimates a distribution. This method is general and valid for any user of artificial NNs, but we focused here on the remote sensing application where this framework should become very

useful in the coming years. In this community, NNs are increasingly been used, and the need for retrieval uncertainties is really important, especially for the NWP community.

In order to easily test if this approach is valuable or not for a particular application, a single and easy test can be conducted: first, perform clustering on the NN inputs; then, investigate if the retrieval errors vary with these clusters. If this is the case, then there is a true relationship between the inputs and the retrieval errors, and the NN should be able to represent this mapping.

Our retrieval could certainly be improved by using a better learning data set and deal with ocean and continental cases, using a priori information, with a specification for surface emissivities, and a cloud filtering and classification. This will be investigated in the future. We will also consider the impact of long-term CO2 variations on the IASI observations [27].

There are several technical perspectives for this work.

1. This technique will be tested in a larger database to test the possibility of estimating the bias and the correlations of errors. The ultimate goal is to retrieve the error covariance matrices to facilitate the variational assimilation of retrieved products in the NWP forecasting scheme.

2. The retrieval of humidity profiles should be tested, and this should be an interesting test because this retrieval is more nonlinear than the retrieval of temperature, so the state-dependence of our retrieval errors should be stronger.

3. The estimation of error statistics should help analyze the efficiency of the NN retrieval models: This should help answering questions about the quality of the learning data set or the complexity of the NN.

4. We would like to implement this type of simple strategy into more complex NN architectures, such as “deep-learning” models or convolutional NNs [29].

A very important application would be to test the assimilation of NN-retrieved products with the help of the retrieval error specifications. It would be challenging to test this framework on temperature profiles (the assimilation scheme is already well constrained in NWP centers for this parameter), but we could test simpler variables, such as the soil moisture [9], [44].

## ACKNOWLEDGMENT

The authors would like to thank the Centre National d'Études Spatiales and, in particular, C. Deniel for supporting our analysis of the IASI observations. They would also like to thank the Atmospheric Radiosonde Analysis/Atmosphere Biosphere Climate (télédétection) Group for providing the ARSA (free access) and the LMD_ICO (private communication) databases. Filipe Aires would like to thank Samuel Favrichon for interesting discussions about NNs.

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of several projects for NASA, National Oceanic and Atmospheric Administration (NOAA), European Space Agency (ESA), or Centre National des Etudes Spatiales (CNES). He has been part of the Science teams of several satellite missions such as Megha-Tropiques, Global Precipitation Mission (GPM), or Surface Water and Ocean Topography (SWOT). He is involved in climate impact studies for socioeconomic activities, such as agriculture, and analysis of the water cycle using optimized satellite observations. He has more than 100 publications in international peer-reviewed journals. His research interests focus on satellite remote sensing of the Earth and statistical analysis of the climate.

<figure><img src="image_8.png" /></figure>

**Filipe Aires** received the Ph.D. degree in statistics from the University of Paris-Dauphine, Paris, France, in 1999.

He was an Associate Research Scientist with the Goddard Institute for Space Studies, NASA, New York, NY, USA, for five years. He then worked as a Research Scientist with the Laboratoire de Météorologie Dynamique (LMD), Palaiseau, France. He is a Senior Scientist with the Laboratoire d'Etudes du Rayonnement et de la Matière en Astrophysique et Atmosphères, Centre National de

la Recherche Scientifique (CNRS), Paris, and an Adjunct Research Scientist with Water Center, Columbia University, New York. He is also the Co-Founder of Estellus/Paris, an Institut Pierre-Simon Laplace (IPSL) spin-off specialized in Earth observation. In earlier works, he analyzed climatic variability using sophisticated artificial intelligence and learning theory. He has developed multi-instrument and multiparameter remote sensing algorithms to retrieve atmospheric variables such as temperature or water vapor profiles, and surface variables such as surface skin temperature, wetland, soil moisture, or infrared/microwave emissivities. He has been the Principal Investigator

<figure><img src="image_9.png" /></figure>

**Victor Pellet** received the B.E. degree in signal processing from the Grenoble Institute of Technology, ENSE3, Grenoble, France, in 2015, the Ph.D. degree in environmental sciences from Pierre et Marie Curie University/Sorbonne University, Paris, France, in 2018, and the Ph.D. degree from LERMA/Paris Observatory, Paris.

He is a fellow with Oki Laboratory, Japan Society for the Promotion of Sciences (JSPS), Tokyo University, Tokyo, Japan. His research focuses on geosciences with an emphasis on the use of applied mathematics and statistics. He developed a particular channel selection algorithm for hyperspectral satellite observation. He also analyzed the water cycle over the Mediterranean region by optimizing the satellite data for water cycle monitoring. His research focuses on the lateral underground water exchanges at a regional scale over the Amazon basin.