We seek to estimate the column-integrated thermospheric temperature from the top-of-atmosphere LBH band emission sensed by the GOLD instrument. Specifically, photon counts from radiometrically calibrated, geolocated L1C GOLD science data products are considered . The column-integrated (effective) temperature Teff can be given as a function of the wavelength λ as 2Teff(λ)=∫0∞V(r,λ)exp⁡(-τ(r,λ))Tn(r)dr∫0∞V(r,λ)exp⁡(-τ(r,λ))dr where r (cm) is the slant path distance, V(photons×s-1cm-3) is the LBH volume emission rate, τ the optical depth, and Teff (K) the neutral temperature . This represents a weighted average of the neutral temperature along the line of sight, with the weight determined by the product V(r,λ)exp⁡(-τ(r,λ)). The LBH volume emission rate depends on neutral temperature, collisional quenching by O₂ and other species, and the local excitation rate from electron impact or solar radiation . Since LBH emissions derive from the same electronic transition of N₂ and there is only a weak wavelength dependence in absorption, effective temperature is nearly independent of wavelength in the LBH system. As shown in Fig. 4 of , the most weight is assigned to altitudes around 120–200 km. Since these observations are integrated quantities, they cannot be assigned to a specific altitude without a-priori knowledge of the temperature structure e.g.,. Despite this limitation, the NASA GOLD mission data product of column-integrated temperature (TDISK) has broad scientific applications and has been widely used in research .

The forward modeling of the neutral temperature is based on the vibrational-rotational band model (), which supplies laboratory LBH spectra across a range of neutral temperatures given vibrational populations of N₂. For the purposes of the study, we use the populations in , which are derived from laboratory data. The modeled spectra are convolved with the spectral point-spread function of the instrument to approximate the intensities as seen by the detector. Principal component analysis of simulated GOLD data has shown that under typical geophysical conditions the intensity in the upper portion of the LBH (2,0) band, from wavelength 138.56–139.2 nm, is positively correlated with temperature, while the intensity in the lower portion from 138–138.56 nm is negatively correlated with temperature . A similar structure is shown to hold in the (1,1) and (2,3) bands as well . So, as the column-integrated temperature increases, the observed photon counts in the upper portion of the band increases while the observed counts in the lower portion of the band decrease. This leads to the approximately linear relationship between the ratio of the long wavelength to short wavelength portions of the band and the neutral temperature , which is used in the study. As noted in and , two-channel ratio approaches have several potential advantages. These include the ease of calculating a ratio compared to fitting a full spectral model and the traceability of uncertainty by not requiring knowledge of variation in instrument performance across the whole band, rather just a small section of it ().

The first step is to model the two-channel intensity ratio as a random process. To motivate the choice of model we first consider the case where the inversion is done without consideration to spatial or temporal correlation, as in and . Throughout the remainder of the paper, the notation X∼F means that the random variable X has the distribution F. If we assume that the count data from the channels are independently Poisson distributed with different mean parameters Λa and Λb, so that 3{ai}i=1na|Λa∼Poisson(Λa){bi}i=1nb|Λb∼Poisson(Λb) then the posterior distribution of the ratio Z=Λa/Λb given the data is a generalized Beta-Prime distribution BP(α,β,p,q), which has a density of the form 4pZ(z)=pqB(α,β)(z/q)αp-1(1+(z/q)p)α+β if z>0, and 0 elsewhere, where α>0,β>0,p>0 and q>0 are parameters and B(α,β) is the Beta function. In this parameterization, α and β are shape parameters that control behavior near 0 and ∞ respectively, and p and q are respectively “peakedness” and location parameters . A proof of this is given in Appendix A, and further discussion of this distribution along with applications can be found in and .

We extend this to incorporate spatial information by utilizing Poisson point processes . A Poisson point process is a random measure defined on a space S, which for our application becomes a spherical cap domain, such that, for all measurable subsets R of S, we have that N(R) the number of points in R is Poisson distributed with 5EN(R)=Λ(R)=∫Rλ(s)dw(s) for some intensity λ(s), where w(s) is the ambient measure on S and s is a point in S, for example a spatial location. These are extended by the Cox process model, where λ(s) is itself a random function . We are interested in a special kind of Cox process known as the permanental process, where the assumption is that λ(s)=12cf(s)2, with f(s) a Gaussian process . The estimation of λ(s) then becomes a problem of estimating the latent process f(s). In our specific case, the underlying point process has been binned, losing knowledge of the emission latitude and longitude. For this reason, instead of estimating λ(s) we estimate Λ(Ri) for each bin. However, our model can be generalized to the model of if, instead of binned data, the raw point process data are available.

Consider a region S divided into disjoint bins Ri such that S=⋃i=1dRi. Given a point process on S, our data become pairs of counts ai in each bin Ri, i=1,…,d. We assume each ai is Poisson distributed with intensity Λ(Ri)=c2fi2 and attempt to recover the latent vector f=f1,f2,…,fdT. The log-likelihood of f is given by 6ℓ(f|{ai}i=1d)=∑i=1dailog⁡c2fi2-c2f,f. where ⋅,⋅ is the standard inner product on Rd. Adding the prior f∼N(0,1γK) for some positive definite Hermitian matrix K, we have that the log-posterior is 7ℓ(f|{ai}i=1d)=∑i=1dailog⁡c2fi2-c2f,f-γ2f,K-1f. Since K is a covariance matrix, it can be written as K=ΦHΦT with Φij the value of the jth eigenvector at location i and H=diag(ηj), with ηj the eigenvalues of K. Then the inner products can be combined to generate an equivalent norm as 8c2f,f+γ2f,K-1f=12f,cI+γK-1f. This new norm defines a norm on an equivalent kernel space with kernel K̃ which we can write as 9K̃=ΦH̃ΦT,H̃=diagηjcηj+γ. As in the infinite-dimensional case, we can say that the solution has the form f^=K̃ψ^ for some coefficients ψ

This is not strictly necessary for the problem we are solving of recovering f in a binned point process, however it is critical for the feasibility of the method in recovering a random function, which is the case with unbinned point process data. This allows the estimation to be reduced to a finite dimensional problem.

From Eq. () we know that Z≈mTeff+z0. Since Z∼BPα^a,α^b,1,q, the temperature Teff=1mZ-z0 has the distribution and MAP and posterior mean estimates 14Teff∼-z0m+BPα^a,α^b,1,qmT^eff(MAP)=-z0m+qmα^a-1α^b+1,α^a>10,elseT^eff(PM)=-z0m+qmα^aα^b-1,α^b>1∞,else. The MAP and posterior mean estimates rely on the estimated parameters α^a and α^b respectively being greater than 1. For all cases examined in the study, the minimal estimated values are considerably larger than this limit, suggesting that it does not present an obstacle for either estimate. This allows estimation of the posterior of the neutral temperature without any sampling, leading to fast inference once the intensities are known. It also allows extension to problems where Z≈mTeff+z0p for some exponent p, in which case Teff∼-z0m+BPα^a,α^b,p,1mq1/p.

Due to the construction following that of , this method inherits many of the sources of error described therein, and additionally in and . We divide these errors into two types: measurement error, such as variations in detector performance, and model misspecification error in addition to the effects of shot noise which the model is designed to account for.

Since the process we are interested in is observed on a section of a spherical shell by GOLD, we choose a kernel given by 15k(s1,s2)=∑l,m1lν(1+l)νVlm(s1)Vlm(s2) where the Vlm are spherical cap harmonics (SCHAs), which are eigenfunctions of the Laplacian on the spherical cap . Then the kernel matrix is given by Ki,j=k(si,sj) and si is the reference latitude and longitude of Ri. These functions are well-studied in geophysics, especially in inverse problems that involve estimating the gradient of a process from incomplete measurements, where only a small portion of the globe can be observed at a time e.g.,. Further details on the construction of the spherical cap harmonics are included in and . The parameter ν, which we call the smoothness parameter, controls the decay of the coefficients in the eigenvector expansion, leading to smoother fields with suppressed high-frequency variation when ν is large. In the context of random function estimation, for example when the data come from a realization of an unbinned point process, f(s) is in Hν(S) the Sobolev space of order ν when ν>1, and the Sobolev embedding theorem implies that f(s) has n derivatives if ν>1+n (). For this reason, we focus our analysis on the cases ν=0.5,1+10-8,2+10-8, with the latter two corresponding to a continuous and differentiable random field, respectively. When applying this method operationally, a cross-validation procedure should be used to select the best value of ν. This can be done as a function of Kp index, Dst index, solar activities, or other geophysical variables since we expect that the optimal value of ν will change during disturbed periods. It is also desirable to include other kernels in cross-validation analysis. A more in depth discussion of the SCHA kernel along with a comparison with other common choices is included in Appendix . For the purposes of the simulation study we chose γ=1. Since there is no ground truth to compare to, this should be chosen via a cross-validation procedure or by kriging the latent vector f onto held out spatial locations and selecting the value that minimizes some error measure.

We evaluate the model performance primarily using the continuous ranked probability score (CRPS) and RMS error. The CRPS is given by 16CRPS(F|y)=∫RF(x)-H(x-y)2dx where F is the predictive cumulative distribution function (CDF) of the neutral temperature, y the true neutral temperature, and H the Heaviside step function. This is a strictly proper scoring rule, meaning its unique minimizer is the deterministic CDF H(x-y) . Since it incorporates the entire distribution rather than just a point estimate it allows us to assess the quality of the posterior beyond the error in estimation. CRPS is negatively-oriented in that smaller values indicate superior predictive models.

As a first pass, we test how our model performs in estimating the neutral temperature at 15:00 UTC, or local noon at satellite nadir. At this time the largest portion of the instrument's viewing area is sunlit, meaning that the algorithm is expected to perform best in this situation. The retrieved fields from JD 306 – JD 310 are shown in Fig. , along with the estimates derived using the method of (top row, CM21) and the “true” field from WAM simulations (bottom row). The algorithm closely tracks the true temperature field for all cases, with variations in ν leading to smoother or rougher estimates of the underlying field. It also appears to be much more robust to shot noise than the algorithm of .

In addition to performance at local noon, we are interested in the performance of our algorithm in low-light conditions. Currently, the GOLD mission does not report TDISK values at an observing zenith angle (OZA) greater than 75° or SZA greater than 80°, which are conservative limits designed to avoid biases in the data products that may arise in these situations (). This partially motivates our investigation, as we believe that including spatial structure can allow us to obtain accurate estimates of the temperature even at high SZAs. To investigate this, we processed data from 12:00 and 18:00 UTC as well. In this situation, a portion of the viewing area has a solar zenith angle higher than the nominal observation boundary of 80° used by the GOLD mission, however there is still a large portion of the viewing area that is sunlit. At UTC 12:00 and 18:00 approximately 30 % of the viewing area has less than 50 expected counts in the upper band, and about 14 % less than 25. In the lower band, these numbers are about 15 % and 10 % respectively. In this regime, the assumption that counts are Gaussian breaks down, introducing biases and inhibiting uncertainty quantification. Additionally, there are portions of the viewing area where the solar zenith angle exceeds 80° at local noon. These results for the average CRPS at each time are shown in Fig.  along with the Kp index. What we see is that the CRPS of the models for ν=0.5 and ν=1+10-8 are roughly constant with time and show only a slight increase with the Kp index, never exceeding a CRPS of 15 K. However, the CRPS of the model with ν=2+10-8 increases significantly, especially in the afternoon, and exceeds 20 K at 18:00 UTC on JD 308. This is because the onset of even this relatively minor storm generates structure in the field that the model is unable to capture.

We have plotted the RMS error in estimation over latitude, longitude, OZA, and SZA in Fig. . The permanental process model reduces error by half or more relative to the model of and maintains errors less than 6 % up to nearly 90° SZA. This suggests that the method may be able to extend the scientific utility of FUV disk emission data by allowing analysis of data collected from areas with high SZA.

Our model provides a full posterior distribution of the intensity ratio, and therefore the temperature. In this section, we examine how accurately our model captures uncertainty. This exercise is important to ensure that the model handles uncertainty induced by shot noise properly and that the estimates of the neutral temperature are reliable. We quantify the uncertainty captured by the model by determining the highest posterior density sets with coverage rate r (r-HPD set) for varying values of r. The r-HPD set is the set Ir such that ∫Irf(x)dx=r and that ∀x∈Ir,y∈IrC,f(x)≥f(y) . If the posterior distribution accurately quantifies the uncertainty, then the true parameter lies in the r-HPD set with probability r. To determine how well the posterior HPD matches the nominal coverage for varying coverage parameters r, we compare the probability that the true value is within Ir for varying values of r from 0.05 to 0.95. These results are shown in Fig.  for the SCHA kernel with ν=0.5,1+10-8,2+10-8, and have been divided into times where Kp<3, indicating calmer conditions, and Kp≥3, indicating higher levels of geomagnetic disturbance.

The coverage probabilities are estimated from 10 retrievals for each simulated field, and then compiled according to Kp index. The plots in Fig. 4 show the median coverage probability over all applicable times and spatial locations from the simulations, and error bars show the 10th and 90th percentiles of coverages. The mean uncertainties at nominal coverage of 0.95 are shown in Table 1 in units of percent of the estimated temperature. Since the intervals are asymmetric, we provide upper and lower uncertainties. The model with ν=0.5 provides the widest uncertainty intervals, and we can see from Fig. 4 that these intervals are too wide, since the achieved coverage greatly exceeds the nominal coverage in all conditions. Practically, this model leads one to believe that estimates of Teff are more uncertain than they actually are. In a data assimilation scheme, for example, these inflated observation errors can cause the estimated state to artificially biased toward the model, leading to poorly constrained model forecasts. In calm conditions corresponding to a true temperature of 600 K, this model would lead to approximate uncertainties of approximately (-44,+45) K, while the ν=1+10-8 model would report uncertainties of about (-32,+33) K and the ν=2+10-8 model would report uncertainties of about ±27 K, which Fig. 4 suggests are more accurate representations of the uncertainty due to shot noise. Generally, we see that the ν=1+10-8 and ν=2+10-8 models appear to be the best at quantifying uncertainty, especially in geomagnetically calm conditions, and their performance degrades in geomagnetically active time periods. The ν=2+10-8 model appears especially affected by this transition. This is due to the fact that setting ν higher makes the retrieved field smoother, so it cannot properly recover the structured temperature fields due to geomagnetic disturbances.

As a first demonstration of the method on real GOLD data, we apply it to background corrected GOLD L1C data from 2–6 November 2018. This is the same time period studied using WAM simulations in Sect. , and includes a minor geomagnetic storm on 4–5 November (JD 308-309).

Since the GOLD instrument scans each hemisphere independently, the inversion incorporates two consecutive scans into each retrieval. The start times of these scans are separated in time by 12 min, which is short enough relative to the nominal thermospheric timescales to ignore the time difference between the scans. Since the scans overlap, we average the data in the overlap region, which reduces (but does not completely eliminate) the bias in the retrievals caused by the varying sensitivity of the detector along the slit . Again we selected γ=1.

The results of this inversion are shown in Fig. . While we obtain similar results to TDISK at high latitudes, as well as on average over SZA and OZA (Fig. ), there is a visible discrepancy near the equator, corresponding to the lower portion of the scan of the northern hemisphere and the upper portion of the scan of the southern hemisphere. Upon examination of the L1C data, we can see a corresponding enhancement in the count ratio at the same location that is present in the scan of the northern hemisphere, but not the southern hemisphere. This, coupled with the fact that similar artifacts are not seen in the results on simulated data (Fig. ) suggests that this variation is due to the varying sensitivity of the detector along the slit. Similar artifacts can also be found in earlier versions of the TDISK data (e.g. Fig. 7 of ). These artifacts lead to a significant bias in the estimates relative to TDISK at the equator (Fig. 6). It is therefore important to have an understanding of the variations in the detector sensitivity and eliminate artifacts in the retrievals.

The second demonstration example includes a major geomagnetic storm that occurred in May 2024 (also known as the Gannon Storm, ). With the Dst index reaching below -400 nT and a peak Kp index of 9, this storm resulted in a major expansion of the upper atmosphere causing the neutral temperature to elevate, leading to orbital decay of low Earth orbit satellites due to increased atmospheric drag.

The GOLD mission detected previously unseen structure in the thermosphere during this event, simultaneously seen in both LBH and 135.6 nm radiances, as well as retrievals of neutral temperature, atmospheric composition, and total electron content. The detected equator-to-pole differences in neutral temperature of over 400 K, with high-latitude temperatures in excess of 1400 K are well beyond a typical FUV observation and retrieval scenario under nominal conditions. Similar structures have since been observed in other storms as well . To investigate the performance of our algorithm with the corresponding GOLD L1C data under extreme conditions, we compare our results to the TDISK data product using both the SCHA kernel and a Wendland kernel function given by 17K(d;r)=131-dr635dr2+18dr+3,|d|<r0,|d|≥r Note that in the model evaluation with simulated data for November 2018 the Wendland kernel is found to perform similarly to the SCHA kernel, albeit with slightly worse uncertainty quantification (see Appendix  for details). The results of the retrieval for May 2024 are shown in Fig.  using r=1/3 and ν=1+10-8 in the Wendland and SCHA kernels respectively. Due to the extreme nature of the storm, we selected a slightly smaller γ=1/5, corresponding to a less informative prior distribution. We see that both models are able to recover spatial structure in the field, especially earlier in the day, however the SCHA kernel tends to oversmooth the estimated field. The retrievals can be improved by more careful selection of the parameters of the kernel, allowing capture of smaller scale structure such as the vortices that are apparent in TDISK, or traveling ionospheric disturbances in more typical scenarios. Our method primarily improves upon by allowing rejection of uncorrelated noise, however small spatial structures can be washed out as well if kernel parameters are not chosen carefully.

We have also included a plot of the relative biases between our two estimates and TDISK during the Gannon storm (Fig. ). We see that the Wendland kernel results are biased slightly high on average over OZA and SZA, with varying bias with latitude and a low bias at the equator. The SCHA result is generally biased low. The largest bias is less than 8%, and the average 95% random uncertainties in TDISK across the plotted estimates are approximately 8.6%,7.6%,7.8%, and 8.7%. The uncertainties in the SCHA and Wendland estimates due to shot noise are approximately ±3% and ±6% respectively. The uncertainties from the SCHA appear to be too low, as seen in Fig.  when the prior is too smooth. However, using the Wendland kernel leads to uncertainties due to shot noise that are more in line with the estimates provided by TDISK, which incorporate other sources of uncertainty as well .

One of the drawbacks of Bayesian methods as practical retrieval algorithms is their computational load. Often, these methods require generating samples from the posterior distribution using an algorithm such as Markov Chain Monte Carlo, which can be computationally expensive. This work overcomes this problem with the use of a closed form approximation to the posterior distribution, shifting the computational load to maximizing the likelihood in Eq. (). Other options, such as a variational inference approach, are possible (e.g. ); however, as shown in , variational inference is much less efficient than the techniques described in this paper.

The algorithm presented in the previous section is timed on a small runtime analysis study to recover the intensity function: 18Z(x)=25sin⁡π2x2+108cos⁡π2x2+10,-1≤x≤1 a plot of which is shown in the upper portion of Fig. , using logarithmically spaced numbers of bins from 10 to 5600. A Wendland kernel with r=3/4 is used for the timing study. The minimization is performed with the implementation of conjugate gradient in the optim function in base R using the analytic gradient of Eq. () as an input. The results of the timing study are shown in Fig. .

With 1500 bins, the average time to retrieve the ratio function is about 40 s on a desktop computer with 16 GB RAM and an Intel i7 CPU. For problems involving less than 500 bins, the algorithm converges in an average of about 3 s, and in about 30 min for problems involving 5600 bins. This suggests that the model can be made fast enough to run in an operational capacity if desired, especially for problems involving data sets in the low to mid thousands of points or less. Due to the eigenvalue decomposition necessary to determine K̃ the algorithm scales asymptotically as O(n3), which can be mitigated by using a low-rank decomposition, sparse kernel matrices, or precomputing the eigendecomposition of the kernel. Additionally, if the kernel has an explicit Mercer expansion, the need for the eigenvalue decomposition can be avoided altogether. Precomputing the eigendecomposition reduces the runtime for problems involving 1500 bins to about 15 s. The rest of the algorithm does not require any matrix inversions and scales approximately as O(n2). For problems with 1500 bins the peak usage is approximately 2 GB, making it easily small enough to run on commercially available laptops.

The LBH (1,1) and (2,3) bands are not used in this study in part because they are not considered to have sufficient SNR for temperature retrievals on their own using the two channel ratio method (). However, both of these bands may be beneficial to consider in the future. While the (2,0) band is not isolated from other LBH emissions, with the lower portion of the band overlapping with the (5,2) band, the (1,1) and (2,3) bands are. This means that, in principle, retrievals using these bands are independent of the specified populations of molecular nitrogen, removing a significant source of model specification error . Although several works, such as , , and , have attempted to determine these populations, any errors in the relative populations of the v′=2 and v′=5 species will contribute to errors in the temperature estimation. For this reason, extending these techniques in a way to allow estimates of the neutral temperature using the (1,1) and (2,3) bands, which have too low of SNR to get accurate Teff estimates with the band ratio technique , or developing a technique that incorporates uncertainty in the vibrational populations may be of interest to practitioners.

One drawback of the two channel ratio method is that it is not robust to wavelength specification errors . This can be partially overcome by using multiple bands, such as the (1,1) and/or (2,3) bands in conjunction with the (2,0) band, to do estimation and weighting observations by their inverse variance, as observed in . Under our model the posterior distribution no longer takes on a closed form, being given by the Lauricella D function . One avenue of future research is to determine whether there are suitable approximations to these distributions that would allow incorporation of additional spectral information into the retrieval approach while maintaining efficient computational performance.

One significant advantage of spatial models is the ability to leverage the correlation structure for optimal estimation of the underlying field at unobserved locations, for example via kriging (). By kriging the latent vector f onto unobserved locations we are able to derive statistically optimal estimates of the neutral temperature at these locations given the model, as well as covariances between observed and unobserved locations. This would allow estimation of the quantity of interest at arbitrary resolution, and additionally allows simulation of spatial fields given observations via conditional simulation (). The spatial correlation may also allow estimates of the quantity of interest to be derived without having to perform binning, which is sometimes necessary due to low SNR. The conditions under which such estimation can be done with this model are yet to be studied, but provide a potential avenue for future research.

While we have developed this method with the assumption that the data contain counts generated by a single process, that is not necessarily true. Photons from other sources inhibit retrieval, especially when the source counts are low, and the GOLD instrument's geostationary orbit means it can be subjected to high energy particles from outside sources . While the GOLD mission L1C data product used mitigates most of the effect of the background , an algorithm that can account for it independently is desirable for further improvement and application to other domains, as uncertainty in the background measurement will necessarily propagate forward into the estimation of the quantity of interest. For instance, the model of handled this by adding a separate term for the background count intensity and estimating both jointly. It is feasible to adopt such an approach to extend this work in the future.