Impact of Orbit Determination Accuracy and System Biases on Geolocation System Performance with a Molniya Orbit Collector

Impact of Orbit Determination Accuracy and System Biases on Geolocation System Performance with a Molniya Orbit Collector | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Impact of Orbit Determination Accuracy and System Biases on Geolocation System Performance with a Molniya Orbit Collector Gerald Lerner This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9633022/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract The performance of a geolocation system consisting of three collectors: one in a Molniya high-eccentricity orbit (HEO) and two in low-inclination geosynchronous orbits (IGEO) is evaluated. Bias error sources for the collector in the Molniya orbit are described and quantified: orbit determination accuracy, tropospheric delay, relativistic Doppler shift, light-transit-time (LTT) delays in the signal reaching the collectors, and the integration time used to process the communications signal and obtain time- and frequency-difference-of-arrival measurements. An analysis approach to mitigate the impact of these biases on geolocation data is presented. Monte Carlo (MC) and covariance analysis are used to validate the data and to assess the performance of the conceptual geolocation system in terms of mean geolocation position and uncertainty. Within ± 2 h of apogee, twice daily, the addition of a Molniya collector can provide a performance advantage over a 3 IGEO collector constellation. Still, that advantage is very sensitive to orbit determination errors for most of the orbit, particularly near the rise and set of the Molniya collector. Geolocation Geosynchronous and Molniya orbits Orbit Determination Considered Covariance analysis Figures Figure 1 Figure 2 Figure 3 Figure 6 Figure 8 1 Introduction Satellites in a Molniya orbit for terrestrial communications were first proposed by Arthur C. Clarke in Wireless World magazine in 1945. Russia used that orbit in 1961 for communications and weather monitoring, and the United States followed in 1971 with its own Molniya orbit surveillance and communications satellites. A Molniya Orbit Imager for high-latitude weather observations [ 1 ] has been proposed. In support of these and other applications, NASA/GSFC has developed a Navigator GPS receiver to enable precision orbit determination for satellites at high altitudes [ 2 – 3 ]. The high- and medium-latitude long-dwell coverage of Earth from a satellite in this orbit suggests that, with high-accuracy orbit determination, Molniya collectors could also support the precision geolocation of RF signals. Global navigation systems (GNSSs), such as GPS, support the real-time navigation requirements of terrestrial and low-Earth-orbit (LEO) satellite users, but are not accessible to GEO and Highly Elliptical Orbit (HEO) users beyond GPS orbits at 20,200 km altitude. Combining GPS coverage with other existing global and regional systems such as GLONASS (Russia), Galileo (ESU), BeiDou (China), QZSS (Japan), and GAGAN (India) can substantially improve orbit determination and thus the performance of IGEO and HEO geolocation collectors. Use of these global and regional systems, coupled with extended Kalman Filter (EKF) processing [ 4 – 8 ], has demonstrated simulated accuracies below 10 m in position and 0.01 m/s in velocity (3dRMS) for IGEO users. For HEO users, GPS plus BDS can achieve real-time accuracy below 10 m and 2 mm/s [ 9 ]. However, the Molniya orbit introduces additional complications in modeling drag at perigee, and, at least at lower altitudes, one would expect substantially poorer orbit determination performance in that orbit than in IGEO or HEO. Geolocation system performance depends on signal characteristics, signal-to-noise ratio (SNR), frequency, bandwidth, and modulation; position and velocity errors on the collector platform; the gain of the collector antennas; the signal environment; measurement types; processing models; and hardware system biases. Low Earth Orbit (LEO) satellites, drones, and aircraft have significant advantages over those at high altitudes, specifically in signal strength and observation geometry, particularly for mobile signals. LEOs have limited geographic coverage, despite their advantage of sub-meter precision orbit determination [ 10 ]. Drones provide targeted coverage but are vulnerable to attacks, and as powered aircraft, their performance is highly sensitive to real-time navigation errors [ 11 ]. In contrast, high-altitude spacecraft can collect signals over a much larger area, are not visible to the target, and provide more precise navigation data than powered collectors. These high-altitude systems, however, are limited in their ability to acquire weak signals, are subject to signal interference, and generally have poorer collector geometry. A pair of IGEO collectors using time- and frequency-difference of arrival (T/FDOA) measurements has poor geolocation geometry twice each day, when the lines of constant TDOA and FDOA position cross at small angles [ 12 ]. 2 Bias Sources on Molniya and Geosynchronous Orbit Collectors Adding a third collector, either IGEO or in a 12-h, high-eccentricity Molniya orbit, can eliminate those poor-geometry times, thereby substantially improving coverage and accuracy. However, Molniya collectors are subject to sources of error and bias that require mitigation because of their high relative collector-emitter velocity: the relativistic Doppler impact on collector frequency of arrival (FOA) and thus the FDOA measurements, the need to compensate for variable LTT from the emitter to the collector, and the change in the differential measurements during the collected signal integration time. Additionally, Molniya collectors' ephemerides are less accurate than those of IGEO collectors due to the difficulty of modeling atmospheric drag at perigee and the need for frequent burns to maintain the desired orbit [ 13 ]. These sources of system bias on the measurement data can significantly degrade the anticipated geometrical performance gain from the HEO collector. Consider the example of a collector acquiring a signal from a cell phone to locate the (assumed stationary) observer. Cell phone uplink frequencies range from 600 MHz (T-Mobile) to 850 MHz (AT&T and Verizon), with Starlink’s 4G service at 2 GHz. The space-borne receiver acquires that signal with an atmospheric and hardware processing-time delay bias and a frequency measurement bias error. Assuming that the transmitter is fixed on Earth's geoid and solving only for latitude and longitude, there is an altitude bias due to terrain variations near the signal and the mobile phone's height above the terrain. Calibration can mitigate these biases [ 14 ], but a residual bias remains. Navigation errors on the collector platform are a dominant source of geolocation error. They must be included in the geolocation covariance analysis to avoid large errors in providing uncertainty (error ellipses) to consumers of the data [ 11 ]. High-quality tracking and processing of GEO and IGEO spacecraft can provide avigation errors of below 10 m and 5 cm/s in position and velocity, respectively [ 4 ]. However, the routine operational performance can be substantially worse. Orbital station-keeping burns can result in large navigation errors that persist until post-burn tracking data are obtained and processed. HEO collectors with a low perigee are subject to atmospheric drag errors, and their ephemerides cannot be computed with the same high precision as those of GEO and IGEO collectors. Those navigation errors must be included in the geolocation covariance data as described in [ 11 , 12 ]. Geolocation measurements, TDOA, and FDOA are obtained by acquiring signals from two or more collectors and using the complex-ambiguity function (CAF) to correlate the data [ 15 ]. The measurement noise depends on the frequency, signal-to-noise ratio (SNR), processing signal bandwidth, and integration time. In this analysis, the assumed measurement noise, using the optimistic Cramer-Rao (CR) lower bound, is 100 ns (1-sigma) for TDOA and 10 parts-per-trillion (ppT) (1-sigma) for FDOA. For a typical 1 GHz uplink signal from the target emitter, the FDOA noise is 10 mHz. 2.1 Integration time bias Typically, an integration time of one or more seconds is required to achieve an acceptable level of measurement noise, and the time and frequency measurements vary over that time. This time variation introduces bias uncertainty in the T/FDOA data via the time tag. Digital resampling of the data, based on an estimate of the time- and/or frequency-measurement rate of change, can reduce noise and bias, but some error remains. This bias error arises from the need to associate the data time tag with the collector's position and velocity, and it increases with the integration time required to achieve adequate SNR in the signal environment. If the SNR varies during the integration time at either receiver, the time-tag error, and thus the collector position and velocity uncertainty, will increase. For an IGEO collector, the environment and measurements vary little over the integration, but this is not the case for the higher-velocity Molniya collector at lower altitudes. 2.2 Light transit time error The collector position and velocity are required at the time of signal detection, which, for the Molniya collector, is 40 to 150 ms after the emitter transmission time, as the Molniya range to the emitter varies from less than 10000 km near perigee to 45000 km at apogee. For the IGEO, at a 35000 km range, the transmission time is nearly constant, at about 130 ms. These LTT delays, however, are easily addressed in processing. 2.3 Atmospheric impact on TOA and FOA measurements The major atmospheric error source in RF transmissions from the emitter to the Molniya or IGEO collector is tropospheric refraction delays [ 16 ]. This delay depends on the path through the troposphere and thus on the elevation angle to the collector. At apogee, for high-latitude emitters, the Molniya collector views the signal at higher elevation angles and thus is less sensitive to tropospheric delays than IGEO collectors. However, at rise and set, the Molniya collector’s tropospheric errors are large. At zenith, a typical signal delay or TOA bias through the troposphere is 2–3 m or a 10 ns error; however, it increases with decreasing elevation and can exceed 100 ns at 5 deg elevation. A discussion of these errors is beyond the scope of this work, but there are many approaches to reduce them by 90% or more for GNSS and other applications. Still, the uncertainty in these weather-dependent corrections increases at the low elevations encountered in geolocation and must be appropriately addressed in the processing [ 17 – 18 ]. 2.4 Special relativity The Doppler-shifted emitter frequency observed by a collector is: Frequency (collector) = γ (1 – β cos θ) Frequency (source) (1) with γ = 1 /(1 – β 2 ) 1/2 , β is the relative emitter-collector relative velocity divided by the speed of light, and θ is the angle between the relative velocity and line-of-sight vectors, as observed in the source (emitter) frame of reference. For most applications, β is small, and the non-relativistic equation for Doppler, with γ = 1, will suffice; however, for a Molniya collector near perigee at 500 to 600 km, its velocity can exceed 8 km/s, and the relativistic Doppler equation must be used, as can be seen if we write the FDOA measurement model equation for two collectors as: FDOA 12 / Frequency(source) = (γ 2 - γ 1 ) + γ 1 β 1 cos θ 1 - γ 2 β 2 cos θ 2 (ppT). (2) The FDOA model measurement error that results from neglecting the relativistic Doppler on Molniya (collector 2), assuming that for the IGEO (collector 1), γ 1 is near unity, and η = γ 2 − 1, is: δFDOA 12 /Frequency = η (1- β 2 cos θ 2 ) ≈ η (ppT). (3) For Molniya collectors, η can exceed 100 ppT or 100 mHz for a 1-GHz signal, which may be compared with the much smaller receiver bias or CR bound measurement noise of 10 ppT (10 mHz). The FDOA error, Eq. (3), is a maximum near perigee for the collector and at rise and set for the emitter. In contrast, the relativistic time-delay error for the Molniya collector is δTOA/TOA = (γ-1), which, for the peak value η = 100 ppT and a 15,000 km range, is less than a nanosecond. 3 Example: Collector and Emitter Model We first describe an analysis model to quantify the impact of these known bias sources on a hypothetical geolocation collection system consisting of two collectors in low IGEO (GEO1 and GEO2), and a third collector in a 12-h Molniya orbit. The emitter is stationary on the geoid at 30º N latitude and 10º W longitude. The Molniya collector has an apogee (43000 km) at 82º E and W longitude and 63.6º N latitude. Perigee is 530 km. For comparison, a second constellation consisting of three IGEO collectors (GEO3, GEO4, and GEO5) is used in the analysis. Table 1 and Fig. 1 provide the geometry for the collector platforms. The constellations were chosen to provide similar coverage and visibility of the emitter, but with no attempt to optimize the geolocation performance of either constellation. Table 1 Molniya and IGEO Collector Orbits After the rise of the emitter to the Molniya collector in the Western Hemisphere (time = 0 in Fig. 1), it reaches apogee after 5 hours, near (63º N, 82º W), with the emitter elevation of 27º. The emitter sets about 6 hours later, then rises to 12º elevation as the collector reaches apogee in the Eastern Hemisphere before setting and repeating this orbital track. The corresponding relativistic Doppler impact on the Molniya FOA measurement is shown in Fig. 2. The collector velocity (relative to the emitter) at rise at 0.7 hours induces a (γ-1) = 43 ppT bias in FOA, which increases to 48 ppT before decreasing to near zero at apogee as the relative velocity to the emitter approaches a minimum. It then increases as the collector falls to perigee, reaching 62 ppT before the emitter sets. At rise, in the Eastern hemisphere, 3 hours later, the FOA error is 22 ppT, decreasing to zero as the collector reaches apogee in the East, then increasing to 28 ppT when the collector sets, and the cycle repeats. The emitter is below the Molniya collector horizon for about 6 hours each day. The emitter elevation to the IGEO collectors over 24 hours is illustrated in Fig. 3. For most of the data, the elevation angles are below 10º, and are below 5º for the IGEO at 173 E longitude. The resultant IGEO tropospheric delays are large, and compensation is required. The residual error in the tropospheric correction is included in the 50 ns path delay bias. 4 Results The analysis approach here is the same as that described in references [ 11 – 12 ]. Data are simulated for a 10-s collection with a 1-s integration time, obtaining ten sets of T/FDOA12, T/FDOA13, and T/FDOA23 measurement pairs from the three collectors. Biases are applied using the 1-sigma values in Table 2 for each MC draw. The assumed signal path delay and frequency errors for TOA and FOA are 50 ns and 10 ppT, respectively, for all three collectors. The terrain error, for the emitter assumed on the geoid, is 5 m, and incorporates the terrain variation in the vicinity of the emitter as well as the altitude of the transmitter. The 1-s integration time required to produce the T/FDOA measurements results in an assumed time-tag error of 50 ms. Orbit determination errors for the IGEO collectors are taken from the EKF results [ 4 ] as 7 m RMS in position and 40 mm/s RMS in velocity for each axis. For the Molniya collector, the assumed orbit determination errors are significantly larger: 20 m in position and 60 mm/s in velocity for each axis [ 9 ]. As described in [ 12 ], a proper analysis for an operational geolocation system must include the navigation covariance data in the considered covariance analysis to obtain reliable estimates of geolocation uncertainty. Table 2 Simulated Bias Parameters A Gauss-Newton filter is used to compute the mean miss distance and a covariance matrix for the noise and biases in the table. The truth model used to generate the measurements for the MC trials includes the relativistic and LTT corrections. The simulation model corrects for LTT and relativistic errors and uses the 1-sigma biases in the table to obtain the covariance matrix and the containment, based on the Mahalanobis distance (MD)- the percentage of MC trials with an MD less than the 0.95p theoretical value of 2.448 for a 2-parameter state vector- geodetic latitude and longitude. Table 3 Impact of noise and biases on geolocation accuracy at T=6 h for 10000 MC trials, with 60 measurements The impact of noise alone and noise plus each bias source (treated individually) on the geolocation data is described in Table 3. The mean miss distance on the geoid, the geolocation, with the 0.95p uncertainty over 10,000 MC trials, is given in columns 2 and 3. The geolocation 95p circular-error probability (CEP95) describes the covariance uncertainty, and the containment is the percentage of the MC trial geosolutions that lie within the covariance error ellipse. The noise-only simulation performance, at 6 h, about 30 min before apogee in the Western Hemisphere, is 7.6 ± 0.1 m, increasing to 40.1 ± 0.5 m with noise and all biases applied. The containment values, all near 95%, confirm that the MC and the considered covariance analyses are consistent. The biases in Table 3 were selected to provide a performance budget near 50 m and ensure each bias source has an observable impact on performance. Figure 4 illustrates the geolocation sensitivity to the individual bias sources. The ordinate is the mean geolocation miss distance obtained, assuming only noise, and each bias is considered independently. The abscissa axis scale at zero corresponds to the noise-only case of 7.6 m; the abscissa at unity defines the performance at the Table 3 bias level, i.e., 50 ns for path-delay bias on all three collectors. At an abscissa of 3, the large 150 ns path-delay bias results in a mean geolocation error of 63 m, as illustrated by the red curve. A position (blue) or velocity (green) bias on the HEO collector follows the curve and shows that a three-fold increase in the baseline position error of 20 m, or a velocity bias of 60 mm/s, increases the miss distance to 55 m, or 37 m, respectively, at 6 h- a good geolocation geometry about an hour before the Molniya apogee in the Western Hemisphere. Away from the Molniya apogee at six hours, the geolocation performance is far more sensitive to the collector velocity error, as shown in Fig. 5 . Near 1.5 h., each 1 mm/s increase in the velocity error from the baseline assumption of 40 mm/s increases the geolocation error by 7 m. In contrast, the impact of the Molniya collector position errors on geolocation error is far less, only 1 m for every 1 m increase (or decrease) from the baseline value of 20 m. The need for proper relativistic Doppler treatment for Molniya collectors is illustrated in Fig. 6 . The gray and orange curves illustrate the mean miss distance and CEP95, respectively, for the 2 GEO + 1 HEO constellation in Table 1 , assuming only noise, no payload biases, or predicted orbit determination errors. The blue curve shows that ignoring the impact of special relativity on Doppler can result in biases of 110 m near Molniya rise and set in the Western Hemisphere. The Molniya collector is below the horizon for the first 40 min, from 5.8 to 7.0 h and 22 to 24 h, and thus, there is no relativistic bias. The CEP95 data also illustrate (the orange spikes near 0, 12, and 24 h) that occur when the emitter is below the Molniya horizon, the remaining 2-collector IGEO constellation has poor geometry, and the crossing angle between the TDOA isochrone and the FDOA isotone approaches zero. These singularities in 2-IGEO collector geolocation systems also lead to large, high eccentricity, error ellipses with the major axis along the nearly parallel TDOA isochrone and FDOA isotone. A comparison of the geolocation performance for the two collector constellations is illustrated in Fig. 7 . The blue curve shows the mean miss distance for the 2 IGEO + Molniya constellation from the MC analysis (1000 trials at 300 s intervals), with the Table 2 measurement noise and bias. The red curve is the corresponding data for the 3 IGEO constellation. Near apogee, at 6 and 18 h, the 2 + 1 Molniya constellation provides better performance for about half the day. The mean miss distance for the 3 IGEO constellation varies far less over the day, 100 to 200 m. The illustrated 2 + 1 geolocation performance is heavily dependent upon the accuracy of the Molniya orbit, particularly as the collector approaches and leaves apogee in the Western and Eastern Hemispheres. At these times, the assumed orbit determination accuracy is optimistic, as is the illustrated geolocation performance. The full 6x6 orbit-determination covariance matrix over 24 h is required for a realistic geolocation covariance matrix and is the performance driver [ 12 ]. Near apogee, the model's expected orbit determination errors are pessimistic, and the Molniya collector can obtain excellent geographical coverage for both detection and geolocation with a mean miss distance below 100 m for 8h/day, and below 40 m at apogee. Figure 8 compares the CEP95 performance for five different geolocation constellations. The blue-dashed curve shows the simulated 3 IGEO constellation geolocation performance at 30º N x 10º W. The mean CEP95 and miss distance, over 24 h, are 281 m and 117 m, respectively The best performance, CEP95 = 60 m, is near the western hemisphere apogee at T = 6 h. The other four curves illustrate the performance of 2 IGEO + Molniya constellations under various assumptions about the orbit-determination error, expressed as 1-sigma uncertainties in each axis. The solid blue curve (bottom of figure) assumes, optimistically, that the velocity error, the performance driver, can be reduced to 10 mm/s for both the IGEO and HEO collectors. The position accuracy is assumed to be 10 m. The grey curve assumes that the errors for both orbits can be reduced to 7 m position and 40 mm/s velocity, approximately the level achieved in the Reference [ 4 ] analysis for the GEO case. The orange curve is the baseline described in Table 3, and the yellow curve assumes significant improvements in orbit determination accuracy, which may be achievable at GEO and near apogee for the HEO collector. 5 Summary A Molniya collector can provide excellent Geometric Dilution of Precision (GDOP) and long-dwell geolocation coverage for high-latitude emitters near apogee. However, meeting the orbit determination accuracy requirements to extend that performance throughout the orbit at lower altitudes will remain challenging, despite ongoing improvements in GNSS systems and receivers [ 19 ]. It will likely eliminate the perceived benefit in geolocation performance from the emitter position's sensitivity to the large Doppler at lower altitudes. A detailed covariance analysis of performance in orbit determination accuracy is required to fully understand the benefits and risks of the 2 + 1 constellation.. A proper accounting of relativistic errors, time-tag errors, and low-elevation atmospheric errors on the time and frequency measurements is also required. Declarations Competing Interests: The author declares that he has no financial or nonfinancial interests to disclose. Funding: No funding was received to assist in the preparation of this manuscript. Author Contribution GML wrote the manuscript and performed the analysis Data Availability The data underlying this article, in the form of EXCEL spreadsheets and simulation summaries, are available on the Figshare repository. The code for implementing and evaluating the covariance filter, along with the Monte Carlo analysis, is in the repository. References Kidder, S. Q. and T.H. Vonder Haar, On the use of satellites in Molniya orbits for meteorological observation of middle and high latitudes. J. Atmos. Ocean. Technol., 7, 517–522 (1990) Bamford, W, Naasz, B, Moreau, M; Navigation Performance in High Earth Orbit Using Navigator GPS Receiver, 29th Annual AAS Guidance and Control Conference, Breckenridge, CO., GSFC Report AAS-06-045 February 2006 Bamford, W., Heckler, G., Holt, G., & Moreau, M. (2008). A GPS receiver for lunar missions. 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Nievinski (2017), Tropospheric delays in ground-based GNSS multipath reflectometry— Experimental evidence from coastal sites, J. Geophys. Res. Solid Earth, 122, 2310–2327, https://doi:10.1002/2016JB013612 Timothy H. Kindervatter; Fernando L. Teixeira, "Predictive Models of the Troposphere," in Tropospheric and Ionospheric Effects on Global Navigation Satellite Systems , IEEE, 2022, pp.129–161, https://doi:10.1002/9781119863069.ch4 Parker, J.J.K., Dovis, F., Konitzer, L., Esantsi, N., Ashman, B., Minetto, A., Nardin, A., Vouch, O., Zocca, S., Bernardi, F., Boschiero, M., Fantinato, S., Miotti, E., Facchinetti, C., Musmeci, M., and Varacalli, G. (2026). GNSS reception at the Moon: First results of the lunar GNSS receiver experiment (LuGRE). NAVIGATION, 73. https://doi.org/10.33012/navi.756 Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9633022","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":639506058,"identity":"79898cbf-0c22-425f-a53e-cc3126372ee9","order_by":0,"name":"Gerald Lerner","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYFAC5gYwxQ8iEojTwgjRItmA0AIVIqTF4ACGEA4gH3aw8XHlDpt84xvJxz48+GOTxyDdfPwBPi2GtxObDc+eSbPcdiMteUYCT1oxg8yxRLy2GM5ObJNsbDtsYHbmjDFDgsThxAaJHENCWtp/Nrb9NzDuOf+ZIcEApCX/I36/SCe2MTa2HTAwYO9hZkhIANuC3/sG0onNQIclG0gcbwM67EBaYptEmuEMvLbMTj74sbHNzoC/mfkx448/Non9EskPPuC15QC6CBs+5WBb8Dp7FIyCUTAKRgEIAACQykxw/443dAAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Gerald","middleName":"","lastName":"Lerner","suffix":""}],"badges":[],"createdAt":"2026-05-06 15:53:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9633022/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9633022/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109405345,"identity":"a868142e-234d-4382-a02f-d3d7e14bf612","added_by":"auto","created_at":"2026-05-17 13:17:21","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":293392,"visible":true,"origin":"","legend":"\u003cp\u003eMolniya Ground Track and Elevation to Emitter at 30º N latitude x 10º W longitude\u003c/p\u003e","description":"","filename":"Onlinefloatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-9633022/v1/c3e11d48f48f59e95cde0d62.png"},{"id":109333809,"identity":"981a16a4-5d05-4a91-88b6-560eb1064337","added_by":"auto","created_at":"2026-05-15 16:36:53","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":201968,"visible":true,"origin":"","legend":"\u003cp\u003eMolniya elevation (deg) and relativistic FOA bias (ppT) to emitter\u003c/p\u003e","description":"","filename":"Onlinefloatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-9633022/v1/1891deb7d6e0c66ca97a73db.png"},{"id":109333810,"identity":"1d888803-b75e-47f9-bc59-49e1837e0a02","added_by":"auto","created_at":"2026-05-15 16:36:53","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":204955,"visible":true,"origin":"","legend":"\u003cp\u003eIGEO collector elevation to emitter\u003c/p\u003e","description":"","filename":"Onlinefloatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-9633022/v1/3b35d70861c86e0f12540855.png"},{"id":109333811,"identity":"36e021b6-901f-4444-9289-a2f388b25e45","added_by":"auto","created_at":"2026-05-15 16:36:53","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":323253,"visible":true,"origin":"","legend":"\u003cp\u003eImpact of the neglect of special relativity on geolocation performance (noise-only)\u003c/p\u003e","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-9633022/v1/eb5c9cb25e338eb940979a81.png"},{"id":109333816,"identity":"2ab89485-2412-4f8d-93af-aa07e01e98c4","added_by":"auto","created_at":"2026-05-15 16:36:53","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":535430,"visible":true,"origin":"","legend":"\u003cp\u003ePerformance of Molniya+2 IGEO and 3 IGEO Geolocation constellations as a function of orbit determination performance\u003c/p\u003e","description":"","filename":"Onlinefloatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-9633022/v1/d6eebae9311d3751262ab573.png"}],"financialInterests":"No competing interests reported.","formattedTitle":"Impact of Orbit Determination Accuracy and System Biases on Geolocation System Performance with a Molniya Orbit Collector","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eSatellites in a Molniya orbit for terrestrial communications were first proposed by Arthur C. Clarke in \u003cem\u003eWireless World\u003c/em\u003e magazine in 1945. Russia used that orbit in 1961 for communications and weather monitoring, and the United States followed in 1971 with its own Molniya orbit surveillance and communications satellites. A Molniya Orbit Imager for high-latitude weather observations [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] has been proposed. In support of these and other applications, NASA/GSFC has developed a Navigator GPS receiver to enable precision orbit determination for satellites at high altitudes [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The high- and medium-latitude long-dwell coverage of Earth from a satellite in this orbit suggests that, with high-accuracy orbit determination, Molniya collectors could also support the precision geolocation of RF signals.\u003c/p\u003e \u003cp\u003eGlobal navigation systems (GNSSs), such as GPS, support the real-time navigation requirements of terrestrial and low-Earth-orbit (LEO) satellite users, but are not accessible to GEO and Highly Elliptical Orbit (HEO) users beyond GPS orbits at 20,200 km altitude. Combining GPS coverage with other existing global and regional systems such as GLONASS (Russia), Galileo (ESU), BeiDou (China), QZSS (Japan), and GAGAN (India) can substantially improve orbit determination and thus the performance of IGEO and HEO geolocation collectors. Use of these global and regional systems, coupled with extended Kalman Filter (EKF) processing [\u003cspan additionalcitationids=\"CR5 CR6 CR7\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], has demonstrated simulated accuracies below 10 m in position and 0.01 m/s in velocity (3dRMS) for IGEO users. For HEO users, GPS plus BDS can achieve real-time accuracy below 10 m and 2 mm/s [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. However, the Molniya orbit introduces additional complications in modeling drag at perigee, and, at least at lower altitudes, one would expect substantially poorer orbit determination performance in that orbit than in IGEO or HEO.\u003c/p\u003e \u003cp\u003eGeolocation system performance depends on signal characteristics, signal-to-noise ratio (SNR), frequency, bandwidth, and modulation; position and velocity errors on the collector platform; the gain of the collector antennas; the signal environment; measurement types; processing models; and hardware system biases. Low Earth Orbit (LEO) satellites, drones, and aircraft have significant advantages over those at high altitudes, specifically in signal strength and observation geometry, particularly for mobile signals. LEOs have limited geographic coverage, despite their advantage of sub-meter precision orbit determination [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Drones provide targeted coverage but are vulnerable to attacks, and as powered aircraft, their performance is highly sensitive to real-time navigation errors [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn contrast, high-altitude spacecraft can collect signals over a much larger area, are not visible to the target, and provide more precise navigation data than powered collectors. These high-altitude systems, however, are limited in their ability to acquire weak signals, are subject to signal interference, and generally have poorer collector geometry. A pair of IGEO collectors using time- and frequency-difference of arrival (T/FDOA) measurements has poor geolocation geometry twice each day, when the lines of constant TDOA and FDOA position cross at small angles [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e"},{"header":"2 Bias Sources on Molniya and Geosynchronous Orbit Collectors","content":" \u003cp\u003eAdding a third collector, either IGEO or in a 12-h, high-eccentricity Molniya orbit, can eliminate those poor-geometry times, thereby substantially improving coverage and accuracy. However, Molniya collectors are subject to sources of error and bias that require mitigation because of their high relative collector-emitter velocity: the relativistic Doppler impact on collector frequency of arrival (FOA) and thus the FDOA measurements, the need to compensate for variable LTT from the emitter to the collector, and the change in the differential measurements during the collected signal integration time. Additionally, Molniya collectors' ephemerides are less accurate than those of IGEO collectors due to the difficulty of modeling atmospheric drag at perigee and the need for frequent burns to maintain the desired orbit [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. These sources of system bias on the measurement data can significantly degrade the anticipated geometrical performance gain from the HEO collector.\u003c/p\u003e \u003cp\u003eConsider the example of a collector acquiring a signal from a cell phone to locate the (assumed stationary) observer. Cell phone uplink frequencies range from 600 MHz (T-Mobile) to 850 MHz (AT\u0026amp;T and Verizon), with Starlink\u0026rsquo;s 4G service at 2 GHz. The space-borne receiver acquires that signal with an atmospheric and hardware processing-time delay bias and a frequency measurement bias error. Assuming that the transmitter is fixed on Earth's geoid and solving only for latitude and longitude, there is an altitude bias due to terrain variations near the signal and the mobile phone's height above the terrain. Calibration can mitigate these biases [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], but a residual bias remains.\u003c/p\u003e \u003cp\u003eNavigation errors on the collector platform are a dominant source of geolocation error. They must be included in the geolocation covariance analysis to avoid large errors in providing uncertainty (error ellipses) to consumers of the data [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. High-quality tracking and processing of GEO and IGEO spacecraft can provide avigation errors of below 10 m and 5 cm/s in position and velocity, respectively [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. However, the routine operational performance can be substantially worse. Orbital station-keeping burns can result in large navigation errors that persist until post-burn tracking data are obtained and processed.\u003c/p\u003e \u003cp\u003eHEO collectors with a low perigee are subject to atmospheric drag errors, and their ephemerides cannot be computed with the same high precision as those of GEO and IGEO collectors. Those navigation errors must be included in the geolocation covariance data as described in [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eGeolocation measurements, TDOA, and FDOA are obtained by acquiring signals from two or more collectors and using the complex-ambiguity function (CAF) to correlate the data [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. The measurement noise depends on the frequency, signal-to-noise ratio (SNR), processing signal bandwidth, and integration time. In this analysis, the assumed measurement noise, using the optimistic Cramer-Rao (CR) lower bound, is 100 ns (1-sigma) for TDOA and 10 parts-per-trillion (ppT) (1-sigma) for FDOA. For a typical 1 GHz uplink signal from the target emitter, the FDOA noise is 10 mHz.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Integration time bias\u003c/h2\u003e \u003cp\u003eTypically, an integration time of one or more seconds is required to achieve an acceptable level of measurement noise, and the time and frequency measurements vary over that time. This time variation introduces bias uncertainty in the T/FDOA data via the time tag. Digital resampling of the data, based on an estimate of the time- and/or frequency-measurement rate of change, can reduce noise and bias, but some error remains. This bias error arises from the need to associate the data time tag with the collector's position and velocity, and it increases with the integration time required to achieve adequate SNR in the signal environment. If the SNR varies during the integration time at either receiver, the time-tag error, and thus the collector position and velocity uncertainty, will increase. For an IGEO collector, the environment and measurements vary little over the integration, but this is not the case for the higher-velocity Molniya collector at lower altitudes.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Light transit time error\u003c/h2\u003e \u003cp\u003eThe collector position and velocity are required at the time of signal detection, which, for the Molniya collector, is 40 to 150 ms after the emitter transmission time, as the Molniya range to the emitter varies from less than 10000 km near perigee to 45000 km at apogee. For the IGEO, at a 35000 km range, the transmission time is nearly constant, at about 130 ms. These LTT delays, however, are easily addressed in processing.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Atmospheric impact on TOA and FOA measurements\u003c/h2\u003e \u003cp\u003eThe major atmospheric error source in RF transmissions from the emitter to the Molniya or IGEO collector is tropospheric refraction delays [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. This delay depends on the path through the troposphere and thus on the elevation angle to the collector. At apogee, for high-latitude emitters, the Molniya collector views the signal at higher elevation angles and thus is less sensitive to tropospheric delays than IGEO collectors. However, at rise and set, the Molniya collector\u0026rsquo;s tropospheric errors are large. At zenith, a typical signal delay or TOA bias through the troposphere is 2\u0026ndash;3 m or a 10 ns error; however, it increases with decreasing elevation and can exceed 100 ns at 5 deg elevation. A discussion of these errors is beyond the scope of this work, but there are many approaches to reduce them by 90% or more for GNSS and other applications. Still, the uncertainty in these weather-dependent corrections increases at the low elevations encountered in geolocation and must be appropriately addressed in the processing [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Special relativity\u003c/h2\u003e \u003cp\u003eThe Doppler-shifted emitter frequency observed by a collector is:\u003c/p\u003e \u003cp\u003eFrequency (collector) = γ (1 \u0026ndash; β cos θ) Frequency (source) (1)\u003c/p\u003e \u003cp\u003ewith γ\u0026thinsp;=\u0026thinsp;1 /(1 \u0026ndash; β\u003csup\u003e2\u003c/sup\u003e)\u003csup\u003e1/2\u003c/sup\u003e, β is the relative emitter-collector relative velocity divided by the speed of light, and θ is the angle between the relative velocity and line-of-sight vectors, as observed in the source (emitter) frame of reference.\u003c/p\u003e \u003cp\u003eFor most applications, β is small, and the non-relativistic equation for Doppler, with γ\u0026thinsp;=\u0026thinsp;1, will suffice; however, for a Molniya collector near perigee at 500 to 600 km, its velocity can exceed 8 km/s, and the relativistic Doppler equation must be used, as can be seen if we write the FDOA measurement model equation for two collectors as:\u003c/p\u003e \u003cp\u003eFDOA\u003csub\u003e12\u003c/sub\u003e / Frequency(source) = (γ\u003csub\u003e2\u003c/sub\u003e - γ\u003csub\u003e1\u003c/sub\u003e ) + γ\u003csub\u003e1\u003c/sub\u003e β\u003csub\u003e1\u003c/sub\u003e cos θ\u003csub\u003e1\u003c/sub\u003e - γ\u003csub\u003e2\u003c/sub\u003e β\u003csub\u003e2\u003c/sub\u003e cos θ\u003csub\u003e2\u003c/sub\u003e (ppT). (2)\u003c/p\u003e \u003cp\u003eThe FDOA model measurement error that results from neglecting the relativistic Doppler on Molniya (collector 2), assuming that for the IGEO (collector 1), γ\u003csub\u003e1\u003c/sub\u003e is near unity, and η\u0026thinsp;=\u0026thinsp;γ\u003csub\u003e2\u003c/sub\u003e \u0026minus;\u0026thinsp;1, is:\u003c/p\u003e \u003cp\u003eδFDOA\u003csub\u003e12\u003c/sub\u003e /Frequency\u0026thinsp;=\u0026thinsp;η (1- β\u003csub\u003e2\u003c/sub\u003e cos θ\u003csub\u003e2\u003c/sub\u003e ) \u0026asymp; η (ppT). (3)\u003c/p\u003e \u003cp\u003eFor Molniya collectors, η can exceed 100 ppT or 100 mHz for a 1-GHz signal, which may be compared with the much smaller receiver bias or CR bound measurement noise of 10 ppT (10 mHz). The FDOA error, Eq.\u0026nbsp;(3), is a maximum near perigee for the collector and at rise and set for the emitter.\u003c/p\u003e \u003cp\u003eIn contrast, the relativistic time-delay error for the Molniya collector is δTOA/TOA = (γ-1), which, for the peak value η\u0026thinsp;=\u0026thinsp;100 ppT and a 15,000 km range, is less than a nanosecond.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Example: Collector and Emitter Model","content":"\u003cp\u003eWe first describe an analysis model to quantify the impact of these known bias sources on a hypothetical geolocation collection system consisting of two collectors in low IGEO (GEO1 and GEO2), and a third collector in a 12-h Molniya orbit. The emitter is stationary on the geoid at 30\u0026ordm; N latitude and 10\u0026ordm; W longitude. The Molniya collector has an apogee (43000 km) at 82\u0026ordm; E and W longitude and 63.6\u0026ordm; N latitude. Perigee is 530 km. For comparison, a second constellation consisting of three IGEO collectors (GEO3, GEO4, and GEO5) is used in the analysis.\u003c/p\u003e\n\u003cp\u003eTable\u0026nbsp;1 and Fig.\u0026nbsp;1 provide the geometry for the collector platforms. The constellations were chosen to provide similar coverage and visibility of the emitter, but with no attempt to optimize the geolocation performance of either constellation.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1\u003c/strong\u003e\u0026nbsp; Molniya and IGEO Collector Orbits\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/69519_bce2c0439cd956a6/69519_custom_files/img1778862730.png\" style=\"width: 373px;\"\u003e\u003c/p\u003e\n\u003cp\u003eAfter the rise of the emitter to the Molniya collector in the Western Hemisphere (time\u0026thinsp;=\u0026thinsp;0 in Fig.\u0026nbsp;1), it reaches apogee after 5 hours, near (63\u0026ordm; N, 82\u0026ordm; W), with the emitter elevation of 27\u0026ordm;. The emitter sets about 6 hours later, then rises to 12\u0026ordm; elevation as the collector reaches apogee in the Eastern Hemisphere before setting and repeating this orbital track.\u003c/p\u003e\n\u003cp\u003eThe corresponding relativistic Doppler impact on the Molniya FOA measurement is shown in Fig.\u0026nbsp;2. The collector velocity (relative to the emitter) at rise at 0.7 hours induces a (\u0026gamma;-1)\u0026thinsp;=\u0026thinsp;43 ppT bias in FOA, which increases to 48 ppT before decreasing to near zero at apogee as the relative velocity to the emitter approaches a minimum. It then increases as the collector falls to perigee, reaching 62 ppT before the emitter sets. At rise, in the Eastern hemisphere, 3 hours later, the FOA error is 22 ppT, decreasing to zero as the collector reaches apogee in the East, then increasing to 28 ppT when the collector sets, and the cycle repeats. The emitter is below the Molniya collector horizon for about 6 hours each day.\u003c/p\u003e\n\u003cp\u003eThe emitter elevation to the IGEO collectors over 24 hours is illustrated in Fig.\u0026nbsp;3. For most of the data, the elevation angles are below 10\u0026ordm;, and are below 5\u0026ordm; for the IGEO at 173 E longitude. The resultant IGEO tropospheric delays are large, and compensation is required. The residual error in the tropospheric correction is included in the 50 ns path delay bias.\u003c/p\u003e"},{"header":"4 Results","content":"\u003cp\u003eThe analysis approach here is the same as that described in references [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Data are simulated for a 10-s collection with a 1-s integration time, obtaining ten sets of T/FDOA12, T/FDOA13, and T/FDOA23 measurement pairs from the three collectors. Biases are applied using the 1-sigma values in Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e for each MC draw.\u003c/p\u003e\n\u003cp\u003eThe assumed signal path delay and frequency errors for TOA and FOA are 50 ns and 10 ppT, respectively, for all three collectors. The terrain error, for the emitter assumed on the geoid, is 5 m, and incorporates the terrain variation in the vicinity of the emitter as well as the altitude of the transmitter. The 1-s integration time required to produce the T/FDOA measurements results in an assumed time-tag error of 50 ms.\u003c/p\u003e\n\u003cp\u003eOrbit determination errors for the IGEO collectors are taken from the EKF results [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] as 7 m RMS in position and 40 mm/s RMS in velocity for each axis. For the Molniya collector, the assumed orbit determination errors are significantly larger: 20 m in position and 60 mm/s in velocity for each axis [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. As described in [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], a proper analysis for an operational geolocation system must include the navigation covariance data in the considered covariance analysis to obtain reliable estimates of geolocation uncertainty.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 2\u003c/strong\u003e Simulated Bias \u0026nbsp;Parameters\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/69519_bce2c0439cd956a6/69519_custom_files/img1778862766.png\" style=\"width: 256px;\"\u003e\u003c/p\u003e\n\u003cp\u003eA Gauss-Newton filter is used to compute the mean miss distance and a covariance matrix for the noise and biases in the table. The truth model used to generate the measurements for the MC trials includes the relativistic and LTT corrections. The simulation model corrects for LTT and relativistic errors and uses the 1-sigma biases in the table to obtain the covariance matrix and the containment, based on the Mahalanobis distance (MD)- the percentage of MC trials with an MD less than the 0.95p theoretical value of 2.448 for a 2-parameter state vector- geodetic latitude and longitude.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 3\u003c/strong\u003e Impact of noise and biases on geolocation accuracy at T=6 h for 10000 MC trials, with 60 measurements\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/69519_bce2c0439cd956a6/69519_custom_files/img1778862791.png\" style=\"width: 437px;\"\u003e\u003c/p\u003e\n\u003cp\u003eThe impact of noise alone and noise plus each bias source (treated individually) on the geolocation data is described in Table\u0026nbsp;3. The mean miss distance on the geoid, the geolocation, with the 0.95p uncertainty over 10,000 MC trials, is given in columns 2 and 3. The geolocation 95p circular-error probability (CEP95) describes the covariance uncertainty, and the containment is the percentage of the MC trial geosolutions that lie within the covariance error ellipse.\u003c/p\u003e\n\u003cp\u003eThe noise-only simulation performance, at 6 h, about 30 min before apogee in the Western Hemisphere, is 7.6\u0026thinsp;\u0026plusmn;\u0026thinsp;0.1 m, increasing to 40.1\u0026thinsp;\u0026plusmn;\u0026thinsp;0.5 m with noise and all biases applied. The containment values, all near 95%, confirm that the MC and the considered covariance analyses are consistent.\u003c/p\u003e\n\u003cp\u003eThe biases in Table 3 were selected to provide a performance budget near 50 m and ensure each bias source has an observable impact on performance. Figure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e illustrates the geolocation sensitivity to the individual bias sources. The ordinate is the mean geolocation miss distance obtained, assuming only noise, and each bias is considered independently. The abscissa axis scale at zero corresponds to the noise-only case of 7.6 m; the abscissa at unity defines the performance at the Table 3 bias level, i.e., 50 ns for path-delay bias on all three collectors. At an abscissa of 3, the large 150 ns path-delay bias results in a mean geolocation error of 63 m, as illustrated by the red curve.\u003c/p\u003e\n\u003cp\u003eA position (blue) or velocity (green) bias on the HEO collector follows the curve and shows that a three-fold increase in the baseline position error of 20 m, or a velocity bias of 60 mm/s, increases the miss distance to 55 m, or 37 m, respectively, at 6 h- a good geolocation geometry about an hour before the Molniya apogee in the Western Hemisphere.\u003c/p\u003e\n\u003cp\u003eAway from the Molniya apogee at six hours, the geolocation performance is far more sensitive to the collector velocity error, as shown in Fig. \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Near 1.5 h., each 1 mm/s increase in the velocity error from the baseline assumption of 40 mm/s increases the geolocation error by 7 m. In contrast, the impact of the Molniya collector position errors on geolocation error is far less, only 1 m for every 1 m increase (or decrease) from the baseline value of 20 m.\u003c/p\u003e\n\u003cp\u003eThe need for proper relativistic Doppler treatment for Molniya collectors is illustrated in Fig. \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. The gray and orange curves illustrate the mean miss distance and CEP95, respectively, for the 2 GEO\u0026thinsp;+\u0026thinsp;1 HEO constellation in Table \u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, assuming only noise, no payload biases, or predicted orbit determination errors. The blue curve shows that ignoring the impact of special relativity on Doppler can result in biases of 110 m near Molniya rise and set in the Western Hemisphere. The Molniya collector is below the horizon for the first 40 min, from 5.8 to 7.0 h and 22 to 24 h, and thus, there is no relativistic bias.\u003c/p\u003e\n\u003cp\u003eThe CEP95 data also illustrate (the orange spikes near 0, 12, and 24 h) that occur when the emitter is below the Molniya horizon, the remaining 2-collector IGEO constellation has poor geometry, and the crossing angle between the TDOA isochrone and the FDOA isotone approaches zero. These singularities in 2-IGEO collector geolocation systems also lead to large, high eccentricity, error ellipses with the major axis along the nearly parallel TDOA isochrone and FDOA isotone.\u003c/p\u003e\n\u003cp\u003eA comparison of the geolocation performance for the two collector constellations is illustrated in Fig. \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The blue curve shows the mean miss distance for the 2 IGEO\u0026thinsp;+\u0026thinsp;Molniya constellation from the MC analysis (1000 trials at 300 s intervals), with the Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e measurement noise and bias. The red curve is the corresponding data for the 3 IGEO constellation. Near apogee, at 6 and 18 h, the 2\u0026thinsp;+\u0026thinsp;1 Molniya constellation provides better performance for about half the day. The mean miss distance for the 3 IGEO constellation varies far less over the day, 100 to 200 m.\u003c/p\u003e\n\u003cp\u003eThe illustrated 2\u0026thinsp;+\u0026thinsp;1 geolocation performance is heavily dependent upon the accuracy of the Molniya orbit, particularly as the collector approaches and leaves apogee in the Western and Eastern Hemispheres. At these times, the assumed orbit determination accuracy is optimistic, as is the illustrated geolocation performance. The full 6x6 orbit-determination covariance matrix over 24 h is required for a realistic geolocation covariance matrix and is the performance driver [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Near apogee, the model\u0026apos;s expected orbit determination errors are pessimistic, and the Molniya collector can obtain excellent geographical coverage for both detection and geolocation with a mean miss distance below 100 m for 8h/day, and below 40 m at apogee.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e compares the CEP95 performance for five different geolocation constellations. The blue-dashed curve shows the simulated 3 IGEO constellation geolocation performance at 30\u0026ordm; N x 10\u0026ordm; W. The mean CEP95 and miss distance, over 24 h, are 281 m and 117 m, respectively The best performance, CEP95\u0026thinsp;=\u0026thinsp;60 m, is near the western hemisphere apogee at T\u0026thinsp;=\u0026thinsp;6 h.\u003c/p\u003e\n\u003cp\u003eThe other four curves illustrate the performance of 2 IGEO\u0026thinsp;+\u0026thinsp;Molniya constellations under various assumptions about the orbit-determination error, expressed as 1-sigma uncertainties in each axis. The solid blue curve (bottom of figure) assumes, optimistically, that the velocity error, the performance driver, can be reduced to 10 mm/s for both the IGEO and HEO collectors. The position accuracy is assumed to be 10 m. The grey curve assumes that the errors for both orbits can be reduced to 7 m position and 40 mm/s velocity, approximately the level achieved in the Reference [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] analysis for the GEO case. The orange curve is the baseline described in Table\u0026nbsp;3, and the yellow curve assumes significant improvements in orbit determination accuracy, which may be achievable at GEO and near apogee for the HEO collector.\u003c/p\u003e"},{"header":"5 Summary","content":"\u003cp\u003eA Molniya collector can provide excellent Geometric Dilution of Precision (GDOP) and long-dwell geolocation coverage for high-latitude emitters near apogee. However, meeting the orbit determination accuracy requirements to extend that performance throughout the orbit at lower altitudes will remain challenging, despite ongoing improvements in GNSS systems and receivers [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. It will likely eliminate the perceived benefit in geolocation performance from the emitter position's sensitivity to the large Doppler at lower altitudes. A detailed covariance analysis of performance in orbit determination accuracy is required to fully understand the benefits and risks of the 2\u0026thinsp;+\u0026thinsp;1 constellation..\u003c/p\u003e \u003cp\u003eA proper accounting of relativistic errors, time-tag errors, and low-elevation atmospheric errors on the time and frequency measurements is also required.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003cstrong\u003eCompeting Interests:\u003c/strong\u003e \u003cp\u003eThe author declares that he has no financial or nonfinancial interests to disclose.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eFunding:\u003c/h2\u003e \u003cp\u003eNo funding was received to assist in the preparation of this manuscript.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eGML wrote the manuscript and performed the analysis\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data underlying this article, in the form of EXCEL spreadsheets and simulation summaries, are available on the Figshare repository. The code for implementing and evaluating the covariance filter, along with the Monte Carlo analysis, is in the repository.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eKidder, S. Q. and T.H. Vonder Haar, On the use of satellites in Molniya orbits for meteorological observation of middle and high latitudes. J. Atmos. Ocean. Technol., 7, 517\u0026ndash;522 (1990)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBamford, W, Naasz, B, Moreau, M; Navigation Performance in High Earth Orbit Using Navigator GPS Receiver, 29th Annual AAS Guidance and Control Conference, Breckenridge, CO., GSFC Report AAS-06-045 February 2006\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBamford, W., Heckler, G., Holt, G., \u0026amp; Moreau, M. (2008). A GPS receiver for lunar missions. 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NAVIGATION, 73. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.33012/navi.756\u003c/span\u003e\u003cspan address=\"10.33012/navi.756\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":true,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":false,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"ceas-space-journal","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ceas","sideBox":"Learn more about [CEAS Space Journal](http://link.springer.com/journal/12567)","snPcode":"12567","submissionUrl":"https://submission.nature.com/new-submission/12567/3","title":"CEAS Space Journal","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Geolocation, Geosynchronous and Molniya orbits, Orbit Determination, Considered Covariance analysis","lastPublishedDoi":"10.21203/rs.3.rs-9633022/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9633022/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe performance of a geolocation system consisting of three collectors: one in a Molniya high-eccentricity orbit (HEO) and two in low-inclination geosynchronous orbits (IGEO) is evaluated. Bias error sources for the collector in the Molniya orbit are described and quantified: orbit determination accuracy, tropospheric delay, relativistic Doppler shift, light-transit-time (LTT) delays in the signal reaching the collectors, and the integration time used to process the communications signal and obtain time- and frequency-difference-of-arrival measurements.\u003c/p\u003e \u003cp\u003eAn analysis approach to mitigate the impact of these biases on geolocation data is presented. Monte Carlo (MC) and covariance analysis are used to validate the data and to assess the performance of the conceptual geolocation system in terms of mean geolocation position and uncertainty. Within \u0026plusmn;\u0026thinsp;2 h of apogee, twice daily, the addition of a Molniya collector can provide a performance advantage over a 3 IGEO collector constellation. Still, that advantage is very sensitive to orbit determination errors for most of the orbit, particularly near the rise and set of the Molniya collector.\u003c/p\u003e","manuscriptTitle":"Impact of Orbit Determination Accuracy and System Biases on Geolocation System Performance with a Molniya Orbit Collector","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-15 16:36:48","doi":"10.21203/rs.3.rs-9633022/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"164977393258794123627018267420132390610","date":"2026-05-13T07:22:10+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-05-11T06:08:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-07T17:13:53+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-05-07T07:54:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"CEAS Space Journal","date":"2026-05-06T15:48:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"ceas-space-journal","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ceas","sideBox":"Learn more about [CEAS Space Journal](http://link.springer.com/journal/12567)","snPcode":"12567","submissionUrl":"https://submission.nature.com/new-submission/12567/3","title":"CEAS Space Journal","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"c70da97d-9dcd-47e7-8fcb-da42ff9afee6","owner":[],"postedDate":"May 15th, 2026","published":true,"recentEditorialEvents":[{"type":"reviewerAgreed","content":"164977393258794123627018267420132390610","date":"2026-05-13T07:22:10+00:00","index":6,"fulltext":""},{"type":"reviewersInvited","content":"2","date":"2026-05-11T06:08:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-07T17:13:53+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-05-07T07:54:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"CEAS Space Journal","date":"2026-05-06T15:48:45+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-15T16:36:49+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-15 16:36:48","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9633022","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9633022","identity":"rs-9633022","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}