First inversions of observed submillimeter limb sounding radiances by neural networks

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. D24, 4791, doi: /2003jd003826, 2003 First inversions of observed submillimeter limb sounding radiances by neural networks Carlos Jiménez, Patrick Eriksson, and Donal Murtagh Department of Radio and Space Science, Chalmers University of Technology, Göteborg, Sweden Received 2 June 2003; revised 8 September 2003; accepted 29 September 2003; published 26 December [1] First inversions of observed limb sounding data by a very fast neural network technique are presented. The technique is based on training a set of multilayer perceptrons. The training set was built by random generation of atmospheric states followed by forward model calculations to derive the corresponding radiances. A novel spectral reduction technique was used to reduce the dimensionality of the spectral space, allowing the setting up of multilayer perceptrons with a practical topology. For training, an algorithm that relies on Bayesian techniques was applied in order to favor generalization. The technique was tested by inverting measured radiances from the Odin-SMR submillimeter radiometer and judged by comparing with similar inversions from an optimal estimation code. The neural network technique resulted in much faster inversions than the optimal estimation code, but biases between the optimal estimation and neural network retrievals were found. The problem was tracked down to the presence of spectral artifacts in the measured radiances. The optimal estimation inversions were also affected but seemed less sensitive. Nevertheless, the results were promising and opened the possibility of implementing a very fast processing algorithm for the Odin-SMR data. INDEX TERMS: 0340 Atmospheric Composition and Structure: Middle atmosphere composition and chemistry; 0394 Atmospheric Composition and Structure: Instruments and techniques; 3260 Mathematical Geophysics: Inverse theory; 6969 Radio Science: Remote sensing; 6994 Radio Science: Instruments and techniques; KEYWORDS: neural networks, limb sounding, inversions Citation: Jiménez, C., P. Eriksson, and D. Murtagh, First inversions of observed submillimeter limb sounding radiances by neural networks, J. Geophys. Res., 108(D24), 4791, doi: /2003jd003826, Introduction [2] Monitoring the atmosphere has become essential to understanding today s changing environment. This is done by a handful of observing techniques, including different combinations of spectroscopic techniques and measurement geometries. One of them is microwave limb sounding from space, where the instrument is placed on board an orbiting platform and measures atmospheric emission while scanning the Earth s limb. As most of the emission originates from the atmospheric layer around the tangent point, this observing geometry measures atmospheric emission with good vertical resolution. In addition, the long observation paths allow the detection of species with weak emission or low abundance. Existing instruments measuring in the microwave region are the Microwave Limb Sounder on board the Upper Atmosphere Research Satellite (UAR-MLS) [Barath et al., 1993], the Millimeter- Wave Atmospheric Sounder (MAS) that flew on board the Space Shuttle as part of the ATLAS mission [Croskey et al., 1992], and the Submillimeter Radiometer on board the satellite Odin (Odin-SMR), the first satellite to use submillimeter frequencies for atmospheric limb sounding Copyright 2003 by the American Geophysical Union /03/2003JD observations from the space [Murtagh et al., 2002]. The next instrument scheduled is the follow-on UARS-MLS, the EOS-MLS experiment, to be flying on board the satellite Aura with a expected launch in January 2004 [Waters et al., 1999]. [3] Radiometric measurements of the atmosphere are ill posed in the sense that the solution is not well defined by the measurement. Solving the inversion problem becomes then an estimation problem, out of all possible solutions a single solution has to be selected following some criteria. As many other estimation problems, the radiometric inversion problem can be placed into the framework of Bayesian theory. This means that the inversion problem is solved by first determining the a posteriori distribution of the atmospheric state given the measurement, and then selecting a state and some error estimate following some criteria. This typically leads to an optimization of the error variance of the solution. If Gaussian statistics and a linear inversion problem are at hand, the solution that minimizes the error variance can be derived by applying the Bayes theorem and expressed as an analytical expression relatively easy to compute. This solution is commonly known as optimal estimation (OEM) [Rodgers, 1976] and it is possibly the most used inversion method for atmospheric radiometry. If the inversion problem is ACH 23-1

2 ACH 23-2 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS nonlinear then an explicit expression cannot be obtained and the solution has to be derived by solving numerically an implicit equation [e.g., Marks and Rodgers 1993], with the correspondingly large computational burden. [4] As processors become faster and faster the associated cost of iterative inversion schemes is decreasing, but this is also making possible the use of other numerical methods also connected with Bayesian techniques. They typically select as the solution the mean state averaged over the a posteriori distribution, either by numerical integration or by regression. Both approaches require a database of atmospheric states and radiances to be available. For instance, Monte Carlo integrations of the average a posteriori solution are reported by Evans et al. [2002] for a randomly generated database. Tamminen and Kyrölä [2001] use a Markov chain to build a database containing a set of sampled points approximating the a posteriori distribution for each measurement. The first approach needs very large and dense databases due to the very localized inputs of the multivariate Gaussian distribution used in the model, the second is computationally very intensive. To model the distribution by a regression is the approach taken in this paper, using artificial neural networks (NNs) and overcoming some of the limitations of the integration approaches. [5] NNs is a generic term for a group of algorithms that have proved efficient at addressing problems where elements from two different data sets have to be associated, such as classifying tasks or statistical estimation. Their use for atmospheric radiometry from space started with regressions for temperature or water vapor profiling [Escobar- Muñoz et al., 1993; Motteler et al., 1995; Butler et al., 1996; Shi, 2001], but latest developments are also beginning to address the retrieval of trace gases [Hadji-Lazaro et al., 1999; Del Frate et al., 2001]. This effort is driven by the fact that for nonlinear situations NNs can provide very fast inversions. This makes them very attractive alternatives to the traditional iterative implementations of the OEM method, and this is precisely the objective of the NN technique to be presented here. [6] A NN inversion technique to retrieve atmospheric profiles from Odin-SMR radiometric data has already been presented by Jiménez and Eriksson [2001] and Jiménez et al. [2003]. The idea was to construct a model of the a posteriori distribution of the retrieval species given the measured spectra with a set of NNs. The NN adaptive parameters, or weights, were determined from a set of randomly generated atmospheric states and the corresponding spectra calculated by a forward model calculation. The results on simulated data were promising, but no tests on real data were performed. The objective here is to present the first application of the NN algorithm to radiances measured by Odin-SMR and to discuss the practical difficulties encountered. [7] The paper is organized as follows. First, the Odin- SMR instrument is introduced. A description of the OEM operational inversions follows. Thereafter the NN algorithm is described. Next is to compare the performance of both techniques. This is done by retrieving O 3 and N 2 O from sets of simulated spectra and spectra recorded by Odin-SMR. Then the practical difficulties of applying the technique on real data are discussed. Finally, the main conclusions are stated. 2. Odin-SMR Instrument [8] Odin is a Swedish initiative for a small satellite built in cooperation with Canada, Finland, and France. It was launched in February Both astronomical and aeronomical measurements are conducted by the Odin-SMR submillimeter radiometer, but there is also a complementary instrument combining an optical spectrometer and an infrared imager (OSIRIS) for the aeronomy part of the mission. At the moment of writing Odin has survived its expected 2 year lifetime and continues providing global distributions of different atmospheric species. The main species currently measured are grouped in three observational modes, with most of the observational time allocated to the stratospheric mode. The work here deals with measurements of the species O 3 and N 2 O in this mode. For more details about the scientific part of the mission, the reader is referred to Murtagh et al. [2002]. [9] Odin-SMR is a radiometer Odin-SMR receiver with 4 tunable single-band receivers operating in the submillimeter region. Different frequency ranges within the GHz band are covered by each mixer. The mixers are followed by two hybrid auto-correlators and one acoustooptic spectrometer. The measurements presented here use one of the auto-correlators configured to cover 800 MHz with a resolution of 1 MHz. More technical considerations are given by Eriksson et al. [2002b], Baron et al. [2002], and Merino et al. [2002]. [10] This paper illustrates the performance of the NN technique by presenting inversions of spectra measured in the GHz band. For these observations the atmosphere is scanned at a speed of 750 m/s between 10 and 70 km. Spectra are recorded in steps of 1.5 km up to an altitude around 50 km, where the step is increased due to data storage constrains. The atmospheric signal is recorded together with the signal from cold and hot loads and a load calibration scheme follows, resulting in a batch of spectra calibrated in brightness temperature. 3. Optimal Estimation Inversions [11] Odin-SMR inversions are performed by a processing chain that converts the calibrated radiances into estimates of vertical species composition. The processing algorithm is based on an iterative OEM scheme. If the parameters to be retrieved are grouped in a vector x, and a vector b contains the remaining parameters defining the atmospheric state and any other parameters needed to specify the radiative transfer calculations and the sensor characteristics, the forward model relating the radiance vector y and x can be expressed as y ¼ Fðx; bþþ; ð1þ where refers to measurement errors not covered by b, typically the spectral thermal noise. If the distribution of the state vector given a certain measurement is calculated by applying the Bayes theorem, the most probable state of the distribution, normally called the maximum a posteriori

3 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS ACH 23-3 Table 1. A Priori Variability for the OEM Retrieval Parameters a Parameter Altitude, km Species all 0.5 km 4 Temperature 0 4 K K K K 4 Offset Tangent Altitude, km s Baseline all 2 K Pointing all 0.5 km solution, is obtained by minimizing the cost function [Rodgers, 2000] ½y Fðx; b a ÞŠ T S 1 e ½y Fðx; b a ÞŠþðx x a Þ T S 1 x ðx x a Þ; ð2þ where x a and b a are the a priori state vector and forward model parameters, S x is the covariance matrix giving the uncertainty of the state vector, and S e is the covariance matrix giving the observation uncertainty, calculated as S e ¼ K b S b K b T þ S ; where K b is the weighting function matrix for the forward model parameters, S b the covariance matrix giving their uncertainty, and S the covariance matrix of the measurement errors. For non-linear situations the minimization of equation (2) gives an implicit equation that cannot be solved analytically. Instead, the equation has to be solved numerically by gradient descent techniques. [12] For the practical implementation a general radiative transfer model called ARTS (S. A. Buehler et al., ARTS, the Atmospheric Radiative Transfer Simulator, submitted to Journal of Quantitative Spectroscopy and Radiative Transfer, 2003), a spectroscopic database called Verdandi (P. Eriksson, The Verdandi database, available at ), and a Matlab environment for retrievals and error characterization called Qpack (P. Eriksson et al., Qpack, a tool for instrument simulation and retrieval work, submitted to Journal of Quantitative Spectroscopy and Radiative Transfer, 2003; hereinafter referred to as Eriksson et al., submitted manuscript, 2003) are used. The retrieved state vector x consist of the volume mixing ratio of the species of interest and other atmospheric and instrumental parameters where the measurement can contribute with information. For these inversions they are the temperature distribution, a pointing offset (a common angular offset for all the spectra in one scan) and a baseline offset (a constant brightness temperature offset for each of the spectrum recorded at the different tangent altitudes in one scan). [13] The a priori statistics are described now. For the species a priori, a climatology derived from the UARS-MLS measurements is used. For both species and temperature the s l c, km a Intermediate values for temperature are obtained by a linear interpolation in altitude. The covariance matrix is build by using equation (4). Pointing offset is completely correlated in tangent altitude, baseline offset completely uncorrelated. ð3þ covariance matrix is set by giving a standard deviation and a correlation length as function of altitude. The values are given in Table 1. The correlation lengths are used to model the off-diagonal elements of the covariance matrices by assuming linearly decreasing tent functions. This is done by applying the expression Sði; jþ ¼ max 0; sðþs i ðþ: j 1 1 e 1 2jzi ðþ zðþ j j l c ðþþl i c ðþ j where z is the vertical altitude, s is the standard deviation, and l c is the correlation length, all indexed by i and j. Notice that the species standard deviations given in Table 1 are relative to the mean state, that is, it is the variability of the mixing ratios normalized to the mean state. [14] The Odin-SMR inversions are non-linear and the operational code uses the Marquardt-Levenberg algorithm [Rodgers, 2000] to minimize equation (2). Even in our band, where linearity is a valid assumption for the mapping between spectra and species abundance, the inclusion of the other retrieval quantities in the state vector makes the inversion problem nonlinear. For convergence the following c 2 test is used [Rodgers, 2000]. d ; ð4þ ðbx i bx Þ T S 1 ðbx i bx Þ n; ð5þ where n is the length of the state vector and S d the estimated retrieval error calculated as S d ¼ K T x S 1 e K x þ S 1 1: x ð6þ [15] The iterations are stopped when the c 2 <0.1 n. Typically, 3 4 iterations are needed for convergence. The computational time in a standard PC (2.4 GHz processor) is around one and a half minutes per scan inverted. 4. Neural Network Inversion [16] Neural network inversions for Odin-SMR are produced by a processing algorithm based on feed-forward multilayer perceptrons (MLP). In essence, the MLPs are used to construct a model of the distribution underpinned by a set of training data. Here the training data represent the mapping between measured radiances and atmospheric states. An introduction to MLPs in the context of atmospheric inversions is already given by Jiménez and Eriksson [2001], here we concentrate on the specific details of the practical implementation. First the generation of a training set is presented, then a feature extraction technique to decrease the dimensionality of the input space is described, followed by a description of the topology of the MLPs and the training algorithm Training Set [17] Previous data for the species, altitude range and geographical coverage observed during atmospheric satellite missions are presently very sparse. This means that a training set has to be built either based on a chemistry model or based on random realizations. The latter is the approach taken here. To generate the species profiles, first covariance matrices reflecting atmospheric variability are

4 ACH 23-4 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS Table 2. A Priori Variability Used to Build the NN Training Sets a Parameter Altitude, km Species all 0.2 km 5 Temperature 0 4 K K K K 4 Offset Tangent Altitude, km s Baseline all 1 K Pointing all 0.75 km initialized. Log-normal statistics for the species distribution, linearly decreasing tent functions for the correlation lengths, and hydrostatic equilibrium between pressure and temperature profiles are assumed. The log-normal statistics assure that no negative species realizations are generated. A set of different mean states is obtained by taking all profiles from the Odin-SMR climatology. A few extra profiles have also been included to account for observed low and high values of N 2 O and ClO. In total around 250 different mean states are included in the training set. The species standard deviation s is set to 0.2, relative to the normalized mean state, and the correlation length l c is set to 5 km. These values are in the range of values given by Eriksson and Chen [2002] for O 3, they are also used for N 2 Ointhe absence of real statistics. In both cases they generate data sets fulfilling the objective of covering a large range of atmospheric states. Then the Choleski decomposition method [Cressie, 1993] is applied to each of the mean profiles. A similar procedure is used to generate temperature realizations, but with a normal distribution. The statistics used are summarized in Table 2. [18] Next is to set a tangent altitude grid. Odin-SMR spectra are measured every 1.5 km in tangent height, but the exact altitudes change from scan to scan. When preparing the inversion algorithm, a nominal tangent height is required. The nominal tangent heights will be a set of fixed altitudes separated by 1.5 km, covering an altitude range of km. Here the presence of spectra every 1.5 km is guaranteed. [19] When measurements are presented to the inversion algorithm, the spectra within the scan are assigned to the closest tangent altitude of the nominal altitudes. That means that any measured spectrum from the scan should lie within ±750 m from one of the nominal tangent heights. Correspondingly, a tangent altitude offset from an uniform distribution with a half width of 750 m has to be added to the nominal tangent altitude of each spectrum before running the forward model. This will set MLPs trained to expect this feature in the spectral input. [20] The ARTS forward model is then run on the set of atmospheric states to generate a set of synthetic spectra (see Eriksson et al. [2002b] and Eriksson et al., submitted manuscript, 2003, for details). The final step is degrading the spectra in order to have a realistic representation of the s l c, km a Interpolation, covariance matrices and correlations are defined as in Table 1. For species and temperature, different mean estates are used; these are the statistics for each mean state. The pointing variability given corresponds to half width of a uniform distribution symmetric around zero. measured spectra. Thermal noise of the same magnitude of the expected noise from the Odin-SMR observations, that is, normally distributed with a s value of around 3 K, is added. A random offset in brightness temperature is also added to each spectrum within a scan to simulate offsets observed in the spectra. They are taken from a Gaussian distribution with a s value of 1 K. These offsets are part of a series of spectral artifacts that the present calibration scheme fails to remove, and in order to mimic the measured spectra as much a possible, they are included in the training set. See later the discussion in section 5.4. Typical size of the training sets is 4000 pairs of spectraprofiles. The computation of the set takes around one hour in a standard PC Feature Extraction [21] For the measurements inverted here the Odin-SMR radiance vector has around channels when using the whole GHz range. This is because the set of spectra measured at the different tangent altitudes have to be appended in the radiance vector y. In order to setup MLPs of practical dimensions a feature extraction technique is used. Here the reduction technique presented by Eriksson et al. [2002a] is applied for first time to real data. The practical implementation is detailed by Jiménez et al. [2003]. Thirty eigenvectors for each MLP was judged to be a sufficient number to faithfully transform the spectral space Topology [22] The retrieval altitude grid covers 15 to 45 km in steps of 1.5 km. For each of the retrieval altitudes a MLP is set up. All the MLPs have the same topology, although the input data are different, as explained in section 4.2. The number of input nodes is of course 30, then there is one hidden layer with 10 neurons with hyperbolic tangent activation functions, followed by an output neuron with a linear activation function. The number of hidden neurons is decided based on considerations related to the training algorithm, see section 4.4. The weights of the MLPs are initialized following the Nguyen-Widrow method [Nguyen and Widrow, 1990]. Before training, inputs and outputs to the MLPs are linearly transformed into the range [ 1, 1] to make the initialization of the weights more effective Training Algorithm [23] The MLPs are trained by minimizing the error function b XL l¼1 k ux l ; w t l k 2 þa k w k 2 ¼ be D þ ae W ; ð7þ where {x l,t l } l=1...l is the training set of radiances and profiles, w is a vector with the weights of the MLP, u represents the output of the MLP, b and a are parameters setting a trade-off between both terms of the error function, and kkis the Euclidean norm. [24] This error function is derived by a Bayesian calculation of the maximum a posteriori solution with an a priori distribution for the weights favoring small values. The practical implementation follows Foresee and Hagan [1997]. In this algorithm the Bayesian analysis is carried

5 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS ACH 23-5 [25] Training takes place during 100 epochs. Typical training time for the 21 MLPs needed to give a retrieved species profiles is around one hour on a standard PC. Figure 1. Statistics of the O 3 and N 2 O realizations included in the training set. Plotted are the means, standard deviations and minimum and maximum values of the generated profiles. The training set includes all means of a global climatology organized by months and latitudes. out further to also provide an estimation for the distribution parameters a and b. This is combined with the Marquardt- Levenberg algorithm for minimization of equation (7). After each iteration the new estimates for a and g are calculated as 5. Results [26] To illustrate the performance of the NN technique retrievals of O 3 and N 2 O in the km altitude range are presented. The GHz band is inverted for the O 3 retrievals, the GHz band is inverted for the N 2 O retrievals. The bands selection will be discussed in section 5.5. The performance of the NN technique is judged with respect to similar inversions from the OEM processing chain. As study case an inversion of around 2500 scans is performed. These scans were measured during the VINTERSOL campaign [Harris and Amanatidis, 2003] in January 2003 and correspond to 3 days of continuous global observations. Spectra was produced by the calibration scheme Version 1.5 and the operational processing chain Version 1.3 is used for the OEM inversions Training [27] The mean states and variability used for the atmospheric realizations were intended to provide a training set covering a large range of atmospheric states, as the MLPs are trained to deal with global inversions. The means, standard deviations and minimum and maximum values for the generated O 3 and N 2 O generated profiles are given in Figure 1. The objective seems achieved, as it is difficult to think of possible species values outside the ranges spanned in the set. An example of generated spectra for one of these atmospheric realizations is given in Figure 2. For comparison measured spectra are also plotted. The resemblance between simulated and measured spectra is a ¼ g ; 2E W ð8þ b ¼ N g 2E D ; ð9þ where N is the total number of adaptive parameters, or weights, and g is the effective number of parameters calculated as g ¼ N 2aTrH 1 ; ð10þ with H a Gaussian-Newton approximation to the Hessian matrix, the second derivatives of the error function respect to the weights. The effective number of parameters gives an indication of how many weights are effectively used to reduce the error function. A number close to N might be an indication that the MLP is not complex enough. This has been the criteria used to select the number of hidden neurons, 10 neurons give values with g sufficiently smaller than N. Notice then that generalization and model complexity are regulated by the training algorithm and no other methods such as structural stabilization or cross-validation are used here. Figure 2. Typical Odin-SMR spectra measured in the bands and GHz. Spectra are not recorded around GHz due to an instrument problem. The left panel shows measured spectra, the right panel measured spectra. Only selected altitudes are plotted. For clarity the spectra are shifted in brightness temperature, with the zero levels plotted as dotted lines.

6 ACH 23-6 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS Figure 3. Histograms of the first eigenvalue of the transformed spectral space at three different altitudes. Plotted are the eigenvalues corresponding to spectra in the training set and in the measured data. good. The auto-correlator used as spectrometer gives a significant correlation between the noise in neighboring channels, but this fact was neglected for the simulated spectra. This was done in order to keep a common approach with the OEM inversions, as they assume completely uncorrelated thermal noise. However, some tests were performed by adding some correlation in the thermal noise added to the training spectra without any noticeable improvement, so this matter was nor pursued further. [28] The validity of the training sets was checked further by comparing the ranges spanned by the eigenvalues for both the training spectra and the measured spectra. Ideally all eigenvalues of the observed spectra should be inside the range spanned by the eigenvalues of the training set. This is normally the case, as illustrated by Figure 3. The distributions of the first eigenvalue at three different altitudes are plotted. It can be observed that only very few eigenvalues of the measured spectra are not found inside the range covered by the training eigenvalues. It can be noticed also that the shape of the histograms when transforming training and simulated spectra differ, but this is expected as the training set covers all seasons while the measurements inverted here only represents a few days of the year. [29] A typical training session is illustrated in Figure 4. The error function for the training and a validation set is given as a function of the number of training epochs. The error on the validation set remains nearly constant once the minimum is reached, an indication that the MLP is not over-fitting the training set. The effective number of parameters g is also plotted. The final g value is lower than the total number of parameters N. This indicates that the MLP is large enough for the given problem, as not all parameters are constrained by the training set. The MLP is over-dimensioned but the error function controls the complexity of the model and no over-fitting takes place Inversion of Simulated Spectra [30] Before attempting the inversion of real spectra, first conditional simulations (P. Eriksson et al., submitted manuscript, 2003) were used to test the NN technique. A set of 250 profiles were generated around one mean state by using the statistics given in Table 1. For the OEM inversions a mean species state slightly different from the state used for the realizations is used as a priori state, to make the simulation more realistic, while the MLPs are trained with the global set of states. The mean and the standard deviation of the whole set of retrieved profiles are plotted in Figure 5. The curves are very similar, saying that on the average both techniques perform similar inversions. There is theoretical grounding to expect this. The maximum a posteriori solution for OEM is equivalent to the solution that minimizes the error variance of the solution [Rodgers, 2000]. It can be seen that the NN technique also tries to minimize the variance of the solution, see equation (7). Of course, the representativity of the training set, the practical implementation of the MLP algorithm, the penalty terms added to the MLP error function, the numerical method used to find the OEM solution, all are factors affecting the solutions obtained. Figure 4. Typical training. The top panel plots the error function for the training set and a validation set. The bottom panel shows the effective number of parameters. The total number of parameters was 321.

7 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS ACH 23-7 Figure 5. Mean and standard deviation of a set of profiles retrieved from simulated spectra by both inversion techniques. This means that nonidentical but close solutions will be obtained in practice Inversion of Measured Spectra [31] Now we examine the inversions of the real spectra. In Figure 6 the mean and standard deviation of the whole set of retrieved profiles are plotted. The agreement between the means is not as good as for the case of simulated spectra. At some altitudes there is a bias of around 10 percent between the methods, specially for O 3 at high altitudes and N 2 O at low altitudes. As the inversions correspond to a set of global measurements zonal means can also be produced. They are given in Figure 7. The main structure of the fields is similar, but at some latitudes and altitudes differences are noticeable. If compared with climatologies, the OEM inversions look more reasonable. Scatter plots at two altitudes are also given in Figure 8, including the correlation coefficients and the best linear fit to the data. To get an idea about the expected dispersion of the data, similar plots for the inversion of simulated data are also given. Although the number of retrieved values is different in the two cases, the correlation coefficients show that for O 3 a larger dispersion is observed for the measured case, an indication that the retrieval errors are larger for the measured data. For N 2 O the dispersion is more similar for measured and simulated data, but the best linear fit shows a bias for the measured case, the NN retrievals under-estimate the abundance of N 2 O respect to the OEM values. This confirms the results of Figures 6 and 7, there are some differences in the retrieved values from both techniques Practical Difficulties [32] The results for simulated spectra showed that the NN technique is able to invert spectra. The NN retrievals compared well with the results from the OEM inversions. As the technique works, failures should be connected to the representativity of the training set. This implies that either the generated atmospheric states do not represent the true states, or for some reason the measured spectra are different from the forward model calculation. [33] Jiménez and Eriksson [2001] demonstrated that a training set with a large number of mean states could invert spectra corresponding to the states generated by an independent chemistry model. The training sets used here include all mean states of a large climatology, and the variability of the sets looks reasonable and covers most of possible atmospheric states as discussed in section 5.1.

8 ACH 23-8 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS Figure 6. Mean and standard deviation of a set of profiles retrieved from observed spectra by both inversion techniques. This was also confirmed by checking the eigenvalues of the training and measured spectra, also discussed in section 5.1. It has also been checked that extreme spectral cases could be inverted. For instance, inversions of spectra showing very little N 2 O emission were reasonably inverted by the NN technique. Perhaps some specific inversions might failed because some extreme cases are well not represented, but on an average of a large number of inversions the effect should be minimal. This leaves us with the second possibility, there is something in the measured spectra that is not reproduced in the training spectra and deteriorates the MLP regression. [34] Unfortunately, it is known that the present calibration algorithm fails to produce spectra without artifacts, such as baseline offsets and ripples. The spectral artifacts seem related to different instrumental issues. For instance low level standing waves have been detected in the receivers and are presently subject to further analysis for a possible correction in a new calibration scheme. The artifacts clearly affect the inversions, an example is given in Figure 9 to illustrate this. The plots show one case considered as good spectra and one case judged as bad spectra. Only the part of the spectra where the artifacts are more visible is shown. It can be observed that the N 2 O line for the bad case exhibits some features that even the OEM inversion cannot fit. The corresponding retrievals of O 3 and N 2 O differ here more, compared to the retrievals for the good case. The largest differences are found for N 2 O just where the measured spectra look more problematic. Notice that the artifacts are also affecting the OEM inversions, the OEM retrieval for N 2 O looks suspiciously oscillatory at low altitudes. This illustrate the fact that the sensitivity to this problem is different. [35] It can be expected that the inversion methods react to spectral artifacts differently. For OEM the measured spectra are part of the cost function to be minimized, see equation (2). This guarantees that the solutions obtained are somehow consistent with the measured spectra. For instance, the MLP solution for the bad case would not have been selected by the OEM inversion as the fitted spectra would have differed too much from the measured spectra and consequently the cost function would have not been minimized. There is no guarantee that the OEM solution is correct, but at least if convergence is reached the solution corresponds to spectra that are fairly consistent with the measurement. The situation for the MLP inversions is different. The MLP inversion uses a set of regression coefficients derived without any interaction with the now

9 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS ACH 23-9 Figure 7. Zonal means of the profiles retrieved by OEM and the NN technique from observed spectra. different measured spectrum. When the regression coefficients are applied to spectra there is no mechanism to guarantee consistency between fitted and measured spectra. In principle this should make MLP inversions more sensitive than iterative OEM solutions Processing Aspects [36] Now we asses the computational burden of the techniques. Figure 10 illustrates this. The inversions of spectra by both the OEM and MLP schemes have been done on a standard PC. Each scan was inverted by OEM in about one and a half minutes, so nearly 3 days of computations were needed. The NN technique took only around three hours, including building the training set and training the MLPs. For this concrete set of inversions the NN technique is more than 20 times faster than the OEM scheme. Of course, if the number of OEM inversions were doubled, the MLPs would be double as fast as before. This is because the computing time for the MLPs would have been nearly the same for the now larger data set. Once the set of MLPs is at hand, the only processing time is related to reading the radiances, the matrix multiplications needed to convert from radiances to retrieved profiles by the MLP algorithm, and the storage of the data. This is extremely fast for a large number of inversions, in fact it would be a matter of hours to process all the data gathered during the whole Odin-SMR mission with a standard PC. The OEM scheme would take more than a year of computations. It is also worth mentioning that the other Odin-SMR bands present a more nonlinear inversion problem. For these other bands the number of iterations needed for convergence is typically double, so the MLP scheme will be relatively even faster. [37] A final comment regarding the bands selected for the inversions. In general it is favorable to use as broad frequency bands as possible for the inversions. This is particularly important for lower altitudes where the spectral line shape is broader. To use the whole range GHz was found to be the best choice for N 2 O, but not for O 3. For O 3 the range GHz was found to give more reasonable inversions. The missing spectrometer subband around GHz is one of the reasons explaining this different behavior between both inversions. For the N 2 O inversions the region around GHz can partly compensate for the missing information around GHz, while the range at GHz is of less interest for the O 3 inversions. The fact that spectral artifacts are most notable around the N 2 O transition has a similar effect. For N 2 O the inclusion of the GHz region helps to decrease the effect of spectral artifacts around GHz, while the retrieval of O 3 is affected negatively by expanding

10 ACH JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS Figure 8. Scatterplots of the O 3 and N 2 O retrievals at 35 km from the 250 simulated spectra and 2500 measured spectra. The values are normalized to the corresponding O 3 and N 2 O a priori values. The figure also shows the correlation coefficient between NN and OEM retrieved values and the best linear fit to the data. The latter is plotted as a solid line. the considered frequency range upward, as the instrumental problems then become more accentuated. 6. Conclusions [38] First inversions of observed submillimeter limb sounding radiances by a NN technique have been presented here. The technique builds a model of the a posteriori distribution of the atmospheric state given the measurement by using a set of MLPs. One MLP is set up for each retrieval altitude and species. As the measured radiance vector has a very large dimensionality, a feature extraction technique is also needed. A novel technique based on deriving the eigenvectors of the spectral space from the weighting functions of the observation has been used. The technique reduces the dimensionality of the spectral space by more than two orders of magnitude, permitting setting up MLPs of reasonable dimensions. This is the first reported application of the technique on real data. Concerning the training, a set of randomly generated atmospheric states and forward model calculations are used to build the training set, and the training algorithm uses Bayesian techniques to produce MLPs able to generalize well to new data. [39] A set of 2500 limb sounding scans from the Odin- SMR radiometer has been inverted by the NN technique and the operational code based on OEM. The inversions are not perfectly linear and time consuming iterative schemes are needed for the operational inversions. The NN technique focuses on processing speed, trying to offer a fast alternative to the operational chain. First inversions of simulated spectra showed that the NN technique could do inversions that compare well with the OEM inversions, but much faster. The inversions of real spectra proved more problematic and some biases between both inversions were observed. The problems were tracked to the presence of spectral artifacts in the measured spectra that the present calibration scheme fails to removed. The OEM seems to be less sensitive to these artifacts, due to the different inversion approach. [40] Notice that the practical difficulties are related to the representativity of the training set, not to the technique itself. For this type of measurements either the current approach or methods relying on chemistry models to build the set of atmospheric states are the only possibilities, as there are no previous data to build a regression set. This is very demanding and perfect knowledge of the instrument

11 JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS ACH Figure 9. Examples of individual retrievals by both techniques. The left panels correspond to a case judge as good spectra, the right panels correspond to a case where spectral artifacts were identified. The top panels show part of the spectra inverted. The dashed curves in the top panels show the fitted spectra by the OEM inversions.

12 ACH JIMÉNEZ ET AL.: LIMB SOUNDING INVERSION BY NETWORKS Figure 10. Illustrating the computational burden from both techniques. The NN requires some time for building a training set and training the MLPs, but after that the computational time is minimum. The OEM requires a fixed time for each scan inverted. A number of 2500 scans were inverted here, the plot zooms in the first 500 for clarity. and very accurate forward modeling are needed, this compared to more typical application of neural networks where a set of measured spectra and some co-located data can be used as training set. This is the main challenge, but a challenge that has to be confronted if the application of NN algorithms has to be extended to atmospheric satellite data. In this sense this work is a large step forward in this direction. [41] Future work is related to the improvement of the calibration schemes in order to decrease the impact of the spectral artifacts in the inversions. 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Murtagh, Department of Radio and Space Science, Chalmers University of Technology, SE Göteborg, Sweden. (patrick@rss.chalmers.se; jimenez@rss.chalmers.se; donal@rss. chalmers.se)