We present a newly developed algorithm for simultaneously inferring the peak height and peak density of the O+ ions in the nighttime ionosphere. The technique relies on the simultaneous observation of the emissions of atomic oxygen at 130.4 and 135.6 nm that are primarily produced by radiative recombination, a natural decay process of the ionosphere. The 135.6 nm emission has become the workhorse for sensing O+ distribution from space from low-Earth platforms where it has been used to infer the peak electron density. A previous study showed that the line ratio of the intensity of the 130.4/135.6 nm radiances is sensitive to the peak height of the ionosphere, as the ratio of the two radiances is dependent on the overlap of the O+ distribution with the thermospheric O layer. We present a new parametric study of these emissions using a new algorithm that permits the retrieval of the peak electron density and the peak height of the F region ionosphere from the measured radiances of the 135.6 and 130.4 nm emissions. We examine the sensitivity of the retrievals to the ionospheric and thermospheric state and to the signal-to-noise ratio of the observations. This new technique enables the determination of the peak height and peak density of the nighttime F region ionosphere as functions of latitude and longitude from nadir-viewing geostationary satellites.
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 The 135.6 nm emission has become a workhorse for global sensing of the nighttime ionosphere from space. Since 2000, it has been used to study the nighttime ionosphere by the Global Ultraviolet Imager (GUVI) on the NASA Thermosphere, Ionosphere, and Mesosphere Energetics and Dynamics (TIMED) satellite [Christensen et al., 2003]; by the Low-resolution Airglow and Aurora Spectrograph (LORAAS) on the Advanced Research and Global Observing Satellite (ARGOS) [Dymond et al., 2001]; by the Far-Ultraviolet (FUV) instrument on the NASA IMAGE satellite [Immel et al., 2006]; and is being used by the Tiny Ionospheric Photometers (TIP) on the Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC/FORMOSAT-3) [Dymond et al., 2000; Anthes et al., 2008]. Meier  contains an excellent review of the use of the 135.6 and 130.4 nm emissions for ionospheric sensing prior to 1990. When viewing in the nadir, the 135.6 nm emission intensity has been used to infer the F region peak electron density, the vertical total electron content, and to study the morphology and temporal evolution of the Appelton Anomaly. The 135.6 nm emission has a companion emission at 130.4 nm that is produced by the same mechanisms, which is not normally used because it has a large optical depth and is difficult to interpret.
 The O I 130.4 and 135.6 nm emission had been observed by several satellite and rocket experiments but Gérard et al.  showed that the 130.4/135.6 nm line ratio was sensitive to the peak height of the nighttime ionosphere using data from the Orbiting Geophysical Observatory-4 (OGO-4) satellite. Gerard et al. argued that the 130.4 nm emission was optically thick and therefore did not vary strongly with altitude while the optically thin 135.6 nm emission varied strongly with altitude. The varying altitude of the OGO-4 satellite, which was in an elliptical orbit, moved the satellite with respect to the ionospheric peak causing the 135.6 nm brightness to change while the 130.4 nm brightness varied weakly; a correlation was shown between the 130.4/135.6 intensity ratio and a model of the ionosphere. Tinsley and Bittencourt  discussed using the line ratio of the 135.6 nm to 630 nm as a means for the sensing the F region peak height. They recognized that the 630 nm emission is produced by dissociative recombination of O2+ and electrons and that the O2+ is produced by charge exchange between F region O+ ions and O2. The production of O2+ and therefore the 630 nm emission brightness is modulated by the overlap between the O2 layer and the F region ionosphere. These studies were followed by a study by McCoy and Anderson , who developed a technique for using the 130.4/135.6 line ratio to infer the peak electron density in the F2 ionosphere, nmF2, and the peak height, hmF2. However, their work appeared in the 1984 Proceedings of the Ionospheric Effects Symposium but was not subsequently published in the archival literature. In 1997, the Office of Naval Research commissioned a study [Gladstone, 1997] to reevaluate the use of the 130.4/135.6 line ratio for ionospheric sensing and to refine the remote sensing technique developed by McCoy and Anderson . This study concluded that, although the line ratio was sensitive to the peak height, there were too many uncertainties including poor knowledge of the background neutral atmospheric oxygen density and instrumental measurement uncertainty to accurately untangle the various contributions to the line ratio and produce a reliable ionospheric sensing algorithm.
 We have reinvestigated the use of the line ratio for ionospheric sensing and using a nonlinear least squares technique have demonstrated that we can remove the ambiguities induced by inaccurate knowledge of the neutral atmosphere, by nonlinearities due to photon transport, and by uncertainties due to measurement noise to accurately retrieve the ionospheric peak height and peak density from the line ratio. We limit our study to consideration of observations from geostationary platforms only, as considerable interest has recently developed in deploying an ionospheric sensing system on a communications satellite.
2. Photophysics of the 130.4 and 135.5 nm Emissions
Both of these reactions contribute to the daytime emission as well, but during the daytime the 130.4 and 135.6 nm emissions are dominated by photoelectron impact excitation and are not useful for ionospheric sensing [Meier, 1991]. The 135.6 nm radiative recombination rate coefficient (reaction (1)) can be found in the work of Melendez-Alvira et al. , and the coefficient for the 130.4 nm emission can be found in the work of Julienne et al. . The most current rate coefficients for the neutralization reaction (2) can be found in the work of Melendez-Alvira et al. . The fraction of neutralization reactions that result in a 135.6 nm photon, β = 0.56, is taken from Tinsley and Bittencourt , but this fraction for the production of 130.4 is not found therein; however, it can be determined from atomic physics. The ground state of the oxygen atom is the 3P state, while the upper state of the 135.6 nm emission is the 5S state [Tinsley, 1972; Meier, 1991]; thus the 135.6 nm transition is spin forbidden for electric dipole emission. The upper state of the 130.4 nm emission is the 3S and the 3S–3P transition is electric dipole allowed. Recombination of ionospheric O+ ions and electrons can populate all of the electronic states above the 5S and 3S states, which then decay via cascade to the 5S and 3S and then to the ground state with emission of the 135.6 nm and 130.4 nm photons, respectively. Recombination into the singlet states of atomic oxygen is forbidden as angular momentum cannot be conserved during the recombination process. (The nighttime emission at 630 nm, in the singlet manifold, is produced by dissociative recombination in which angular momentum can be conserved.) Thus, because the singlets cannot be produced, following Gladstone  we take the fraction of neutralizations producing the triplets, β = 0.44, to be the complement of the fraction producing the quintets.
 The 130.4 nm emission has a large optical depth while the 135.6 nm emission is essentially optically thin. The optical depth of the 135.6 nm emission approaches ∼1 at 100 km and is negligible at ionospheric altitudes. The 135.6 nm photons are essentially produced and then escape to be observed. As discussed above, the 130.4 nm emission is allowed by electric dipole selection rules and therefore the upper state of the emission can be directly populated from the ground state via the absorption of a photon. The vertical optical depth for this emission is several thousands at 100 km [Meier, 1991] but is of order 100 at ionospheric altitudes. Atomic oxygen essentially entraps the 130.4 nm photons by resonant scattering, which enhances the volume emission rate or the rate that photons are emitted per unit volume. The amount of enhancement depends upon the O density and upon the amount of overlap between the ionosphere, where the photons are initially produced, and the thermospheric oxygen layer where they are entrapped.
 We must model the radiation transport to properly interpret the 130.4 nm emission. We do so in the Complete Frequency Redistribution (CFR) approximation [Meier, 1991] which is appropriate at the optical depths seen at ionospheric altitudes. We model the photon transport by inverting the integral version of the photon transport equation [Dymond et al., 1997; Meier, 1991; Anderson and Meier, 1985]. Once the volume emission rate is calculated, we calculate the radiance by integrating the volume emission rate along the instrument's line of sight through the ionosphere taking into account absorption of the photons by molecular oxygen and scattering from the line of sight by atomic oxygen. The scattering cross sections and atomic constants were taken from Meier . Photon transport calculations were carried out in the plane parallel approximation, where the curvature of the atmosphere is neglected. The atmosphere was assumed to be horizontally stratified with no gradients in latitude or longitude. Figure 1 shows how the volume emission rate of the 130.4 and 135.6 nm emissions vary with the peak height and peak density; the entrapment of photons that enhances the 130.4 nm emission intensity is clearly seen. The peak seen at ∼130 km in all plots is caused by photons generated in the ionosphere being entrapped by the O layer and enhancing the volume emission rate.
 The neutral thermospheric densities were calculated using the NRLMSIS-00 model [Picone et al., 2002]. The O density was used in the calculation of the neutralization rate and in the radiation transport calculations. The O2 density was used in the radiation transport calculations as molecular oxygen absorbs the 130.4 and 135.6 nm emissions.
 The ionospheric O+ ion and electron density distribution was modeled using the Chapman altitude distribution [Chamberlain and Hunten, 1987], namely:
where nmF2 is the peak electron density, z is the altitude, hmF2 is the peak height of the ionosphere and H is the O scale height, which is one-half the plasma scale height in this formulation. Although this approximation is not expected to hold perfectly, especially in the Appelton anomaly region, we have found that it works well in practice.
 The inversion algorithm uses the forward model of the 130.4 and 135.6 nm emissions described above and a nonlinear least squares fitting routine based on the Levenberg-Marquardt (LM [Press et al., 1992]) algorithm. The LM technique is an iterative procedure that uses the local partial derivatives of the function to estimate where the minimum of the function lies. The LM algorithm begins with an initial guess for the parameters. It then evaluates the forward model and the fit to the data at the current values and calculates the local gradients. We use one-sided numerical partial derivatives to estimate the local gradients. These gradients are used to estimate the location of the best fit and the parameters are updated and the forward model is evaluated and the local gradients are calculated and a new estimate of the location of the minimum is calculated. This process is repeated in an iterative fashion until the χ2 statistic for the fit changes by less than 0.1% during a step. The LM algorithm traverses the parameter space along the steepest gradient, like the Steepest Descent algorithm, when the parameters are far from optimal and smoothly transitions to the Inverse Hessian method, which converges quadratically when the parameters are nearing optimality [Press et al., 1992]. The LM algorithm has excellent global convergence properties.
3. Results and Discussion
3.1. Demonstration of Sensitivity to hmF2
Figure 2 shows how the ratio of the 130.4/135.6 intensities varies with the peak height of the ionosphere at various levels of solar activity and at a variety of atomic oxygen concentrations. The 10.7 cm solar flux and its 81-day average are used to drive the NRLMSIS-00 model that produced the oxygen densities. The 10.7 cm flux essentially determines the exospheric temperature and the density of O in the F region ionosphere. These plots were created assuming that the ionosphere was Chapman-like with a peak electron density, nmF2, of 1 × 106 cm−3 and a plasma scale height, Hp, of 120 km (the scale height in the Chapman layer formula above is 1/2 the plasma scale height) and only the peak height was varied. The satellite/observer was assumed to be at geostationary altitude above the equator over the prime meridian (0°E, 0°N) and the local time was assumed to be midnight.
Figure 2 shows that the intensity ratio is sensitive to the height of the ionosphere. Note that the large entrapment of the 130.4 nm photons shown in the volume emission rate plots in Figure 1 does not show up as a large enhancement of the line ratio. On the basis of Figure 1, one would think that the line ratio curves in Figure 2 should decrease as the height of the F region peak increases, but instead the ratio increases with peak altitude. This is a result of 130.4 nm photons being scattering out of the line of sight dominating the increased photon entrapment. An F layer that is closer to the observer is observed through a lower O column density and therefore the number of photons scattered out of the line of sight is decreased. The altitude distribution of O atoms is affected by the exospheric temperature. NRLMSIS-00 uses the 10.7 cm solar flux and the 81-day average of the 10.7 cm flux to parameterize the exospheric temperature; so, we adopted the 10.7 cm solar flux as a proxy for the exospheric temperature. At mid to high 10.7 cm flux, the line ratio is seen to vary monotonically with peak height. At the lowest 10.7 cm flux, 80 SFU (1 SFU = 1 × 10−22 W m−2 Hz−1), the ratio is seen to saturate at peak heights above ∼470 km. This saturation is caused by the thermosphere becoming optically thin as the O density is low due to low exospheric temperatures. Thus, the 130.4 nm photons are not scattered from the observer's line of sight and the 130.4/135.6 radiance ratio becomes insensitive to the ionospheric peak height. At higher 10.7 cm fluxes the ratio tends to vary more weakly with peak height and the curves tend to become more clustered at peak heights below ∼350 km.
Figure 2b shows the effect of scaling the O profile while the 10.7 cm flux was held fixed at 160 SFU. The curves in Figure 2b should be compared to the yellow curve in Figure 2a. The 130.4/135.6 nm varies with the peak height but the amount of variation decreases as the O scalar increases. When the O scalar is low (0.1 and 0.3 in Figure 2), the curves are nonmonotonic; this is caused by the opposing effects of scattering out of the line of sight and entrapment of the photons increasing the volume emission rate. On the basis of the volume emission rates in Figure 1, we expected the line ratio to decrease as the F region peak height increased due to increased photon entrapment; this is seen in the curves for low O scalar. But, the entrapment effect is mitigated at lower altitudes when scattering from the observer's line of sight again begins to dominate the line ratio.
 From a practical point of view, the O density is not perfectly known. However, the NRLMSIS-00 climatology is empirical, so that it is expected to be fairly accurate under geomagnetically quiet conditions. The 10.7 cm flux is usually assumed to be known to about 10%. So that in principle, it is possible to select one of the curves from the family of curves in Figure 2a and use it infer the F region peak height from the line ratio. To better understand the effect of imperfect knowledge of the O density on the retrieved peak height and peak density, we performed an ensemble simulation where a large number of test cases were generated and the inversion results were compared to the input values to assess the fidelity of the retrievals under more realistic conditions.
3.2. Ensemble Simulations
 We tested the retrieval algorithm and its sensitivity to our imperfect knowledge of the O density by simulating a large ensemble of ionospheres and neutral atmospheres and calculating the 135.6 and 130.4 nm intensities. The satellite/observer was assumed to be at geostationary altitudes above 0°E, 0°N at midnight. The ionospheric parameters were chosen using separate uniform random deviates to select peak densities in the range, 1 × 105 < nmF2 < 2.1 × 106 cm−3, peak heights in the range, 200 < hmF2 < 600 km, and plasma scale heights in the range, 80 < Hp < 160 km. The neutral atmosphere was also varied to assess the sensitivity of the algorithm to the underlying thermosphere. We scaled the scale the entire O density profile using normally distributed random deviates with mean of 1 and a standard deviation of 0.5; the absolute O density is one of the least certain density values used in the NRLMSIS-00 formulation. To select the 10.7 cm solar fluxes, we used normally distributed random deviates with a mean of 170 SFU and standard deviation of 21 SFU; this range was selected as the measurement of the 10.7 cm solar flux is believed to be accurate to approximately 10%. In our simulations, we assumed the 10.7 cm flux on the previous day (an NRLMSIS-00 input) and the 81-day average of the 10.7 cm solar flux (an NRLMSIS-00 input) were equal. The geomagnetic ap index, used to drive NRLMSIS-00 when modeling geomagnetic storms, was set to 4 nT, or geomagnetically quiet conditions. Once the radiances were calculated, we converted the radiances to counts and then superimposed realistic instrumental Poisson noise and reconverted back to radiances to create the “data” and then inverted these “data” using our forward model of the emissions and the LM fitting routine. We assumed the instrument sensitivity was ∼100 counts Rayleigh−1 based on the NRL design of the Regional Scale Imager recently proposed to the National Aeronautics and Space Administration as a Mission of Opportunity [Dymond et al., 2002; Kalmanson et al., 2005]. In the retrieval algorithm, the 10.7 cm flux was fixed at 170 SFU, the O scalar was fixed at unity, and the plasma scale height was fixed at 120 km. We retrieved the ionospheric parameters: nmF2 and hmF2.
Figure 3 shows the results of the ensemble simulations and inversions. Figure 3a shows a scatterplot of the retrieved peak density versus the input peak density. A slope of unity in Figure 3 is indicated by the dashed line. The data points cluster around the unit slope line indicating good agreement. However, there is some scatter, which increases with peak density. The 135.6 nm radiance is proportional to the square of the peak density times the plasma scale height. When the data are broken out versus plasma scale height as shown by the different symbol colors in Figure 3a, the reason for the scatter becomes clear: the algorithm cannot determine the plasma scale height so that the fixed value of 120 km used in the inversion has led to systematic errors. When the input plasma scale height was larger than 120 km, the algorithm underestimated the peak density. When the input plasma scale height was smaller than 120 km, the algorithm overestimated the peak density. In practice, the accuracy of the retrieved peak densities could be improved through the incorporation of a priori information such as vertical total electron content or the slab thickness derived from data assimilation using the Global Assimilation of Ionospheric Measurements (GAIM [Schunk et al., 2004]) model.
Figure 3b shows the scatterplot of the retrieved peak height versus the input peak height. Clearly, the algorithm is able to accurately determine the peak height from the line ratio and is able to resolve the ambiguity due to poor knowledge of the O density, as evidenced by the data points clustering about the unity slope line (dashes). However, the scatter about the unity slope is somewhat larger than expected. This scatter is caused by the sensitivity to the O density, which varied during the radiance simulation by varying the O density scalar and the 10.7 cm solar flux but which was fixed in the retrieval at the nominal values of 1 for the O scalar and 170 SFU for the 10.7 cm flux. The data points colored green in Figure 3b indicate cases when the O scalar was above 1.5 while the data points colored blue indicate cases when the O scalar was below 0.5. As expected, imperfect knowledge of the O density led to increased scatter about the line of unity slope and led to systematic errors in the retrieved peak height. In the simulation, the retrieved peak height was more affected by inaccurate knowledge of the O scalar than it was by inaccurate knowledge of the 10.7 cm flux, this behavior is consistent with the wider range of variation of the line ratio seen in Figure 2b, when the O scalar is varied, than is seen in Figure 2a, when the 10.7 cm flux is varied.
Figure 4 shows the scatterplot of the fractional error between the retrieved and input hmF2 values versus the input O density scalar. When the O density scalar is unity, the retrieval error is ∼0 km. The scattergram shows a negative correlation between the hmF2 and the O density scalar, which makes perfect sense given that the sensitivity to the ionospheric peak height comes through the scattering of the 130.4 nm photons out of the observer's line of sight by the O layer. When the O scalar is below unity, the O density is lower and the 130.4 nm photons undergo less scattering and by Figure 2a the 130.4/135.6 radiance ratio is larger, which the algorithm accommodates by moving the peak height upward to decrease the photon scattering and increase the radiance ratio. When the O scalar is greater than unity, the 130.4 nm photons undergo increased scattering by the O layer and the 130.4/135.6 radiance ratio decreases causing the algorithm to move the peak height downward to increase the scatter from the line of sight.
 The desired instrument performance can be assessed by examining the precision of the retrieved parameters versus the signal-to-noise ratio of the 130.4/135.6 radiances. The precision (uncertainty) of the retrievals is calculated from the covariance matrix generated by the LM inversion process. This covariance matrix depends upon the curvature of the χ2 hypersurface in the vicinity of the best fit solution. The curvature of the χ2 hypersurface depends upon the signal-to-noise ratio and the partial derivatives of the forward model with respect to the model parameters. In a photon counting instrument, the signal-to-noise ratio is dependent on the count rate. Photon shot noise is a Poisson process and the noise is proportional to N1/2 where N is the number of counts and, as a result, the signal-to-noise ratio is proportional to N1/2. As the radiances of the 130.4 and 135.6 nm emissions are proportional to the square of the peak density and the count rates are proportional to the radiances, the signal-to-noise ratio is then proportional to the peak density. Because the peak densities were varied in the ensemble simulations the signal-to-noise ratio was varied and the ensemble simulation results can be used to assess the precision of the retrieved parameters versus signal-to-noise ratio. The signal-to-noise ratio for the 130.4/135.6 nm intensity ratio was calculated from the simulated radiances and their uncertainties using conventional propagation of errors techniques [Bevington, 1969]. Figure 5 shows the how the precision of the retrieved parameters is affected by the signal-to-noise ratio of the 130.4/135.6 radiances. As the signal-to-noise ratio increases, the uncertainty of the retrieved parameters asymptotically approaches zero. If a 5% or better uncertainty is desired for determination of the peak density (shown by the dashed line in Figure 5b), then the signal-to-noise ratio must be greater than ∼7. Because most commonly used ultraviolet instruments are photon counters, the desired number of counts must be >49. The 130.4 and 135.6 nm radiances are roughly equal (within a factor of ∼2 from Figure 2). The radiance produced by an ionosphere with a peak density of ∼106 cm and plasma scale height of 120 km is ∼20 Rayleighs (R). The desired instrument responsivity (number of counts generated per unit input radiance = sensitivity × exposure time) must be 49 counts pixel−1/20 R or ∼2.5 counts R−1 pixel−1. The sensitivity required to determine the peak height to better than 20 km uncertainty is much higher, because, as is seen in Figure 2, the intensity ratio is a weak, nonlinear function of the peak height. In Figure 5b, the signal-to-noise ratio required to determine the peak height to better than 20 km uncertainty is ∼20 (intersection of curve with dashed line) and therefore the number of photons needed is 202 = 400. The radiance is ∼20 R, so the desired responsivity is 400 counts pixel−1/20 R or ∼20 counts R−1 pixel−1, which is 8 times higher than the sensitivity required to accurately determine the peak density.
 The ensemble simulations have demonstrated that it is possible to accurately determine the ionospheric peak height and peak density from the 130.4/135.6 nm intensity ratio in the presence of realistic measurement noise and given reasonable assumptions about our knowledge of the O density.
3.3. GAIM Simulations
 We performed additional testing of the algorithm by using output from the GAIM model as the input ionosphere and simulated “data” which were then inverted. The Kalmann filter version of GAIM assimilated ionospheric measurements over the American Sector during a magnetic storm on 1 April 2005. We used the GAIM electron profiles over the Caribbean Sea and the southern United States and the NRLMSIS-00 model to simulate the 130.4 and 135.6 nm radiances, using our forward model. Using the instrument characteristics for the Regional Scale Imager (discussed above), we then superimposed photon shot (Poisson) noise on the radiances and then inverted them to determine the ionospheric parameters: nmF2 and hmF2. An image of this region would be acquired the Regional Scale Imager [Dymond et al., 2002; Kalmanson et al., 2005] in approximately 36 min. The electron density profiles output from GAIM, which were used in the “data” simulation, were not Chapman layers. But, the inversion was performed assuming Chapman layer profiles. Figure 6a shows the peak density from the GAIM model and Figure 6b shows the retrieved peak density. The agreement between the images is excellent, but this is not surprising as many experiments have demonstrated and validated the use of the 135.6 nm emission for nighttime ionospheric sensing. The effects of the photon shot noise are seen as small-scale fluctuations of the retrieved peak density. Figure 6c shows the peak height from GAIM and Figure 6d shows the retrieved peak height. The gross structure of the peak height variation is well captured. The fluctuations are due to the photon shot noise superimposed on the radiances. The mean difference between the retrieved peak height and the GAIM peak height was ∼2 km, which is excellent agreement in this fairly stressing simulation.
4. Concluding Remarks
 We have described and demonstrated a new algorithm for determining the ionospheric peak height and peak density from the nighttime 130.4/135.6 nm intensity ratio. These lines are bright and easily observed at high sensitivity with instruments already under development. The algorithm we presented is a refinement and extension of a technique tested and developed in 1984 [McCoy and Anderson, 1984]. Our refinements include the use of updated algorithmic techniques to invert nonlinear least squares problems and more current knowledge of the physical constants. The algorithm was shown to accurately retrieve the peak density and peak height in an ensemble simulation over a wide range of input conditions. We also used the output from the GAIM model to simulate a more realistic observation scenario, then superimposed realistic instrument noise, and inverted the noisy images and showed that the algorithm was able to accurately reproduce the input ionosphere in a fairly stressing simulation.
 We also determined the signal-to-noise ratio required to precisely infer the peak height and peak density. Our simulations showed that a signal-to-noise ratio >7 is required to determine the peak density to within 5%. We also showed that a signal-to-noise ratio >20 is required to precisely determine the peak density to within 20 km. When combined with spatial and temporal resolution requirements and a desired minimum observable peak density, the instrument sensitivity can be determined from these required signal-to-noise ratios.
 This work was supported by the Office of Naval Research. We are grateful for useful discussions with R. P. McCoy, S. A. Budzien, and J. M. Picone and to C. Coker for providing the GAIM model output.