SENSOR CALIBRATION AS AN APPLICATION OF OPTIMAL SEQUENTIAL ESTIMATION TOWARD MAINTAINING THE SPACE OBJECT CATALOG

Transcription

1 SENSOR CALIBRATION AS AN APPLICATION OF OPTIMAL SEQUENTIAL ESTIMATION TOWARD MAINTAINING THE SPACE OBJECT CATALOG John H. Seago and James W. Woodburn * As space-object catalogs move from being maintained by general perturbations to special perturbations, and as the number of detectable objects in space increase, sequential (recurrent) estimators have some significant advantages over batch processing that should not be overlooked. A primary advantage of a sequential estimator is that the time evolution of the satellite state error covariance can be more accurately determined. The availability of accurate uncertainty estimates makes other types of operational analyses easier, including sensor calibration. A sequential estimator can also provide estimates in near real-time, which can be very important for sensor calibration. In this paper, an optimal sequential estimator is applied, via simulation, to the problem of detecting and estimating large changes in sensor performance closer to real time. By using statistical significance tests on the estimated biases, a possibility exists to detect substantial changes in sensor performance more quickly and accurately. INTRODUCTION Optimal sequential (recurrent) estimators were first developed for space navigation problems near the beginning of the Space Age. Nevertheless, batch estimators have been used almost exclusively for routine catalog maintenance and sensor calibration. The perception is likely amongst the operational communities that sequential estimators, such as the Kalman filter, may require too much tuning or manual intervention, or that operational tracking data are too sparse, to be useful for maintaining large satellite populations. However, as space-object catalogs move from being maintained by general perturbations to special perturbations, and as the number of detectable space objects increases, sequential estimators have some significant advantages over batch processing that should not be overlooked. A by-product of special-perturbations orbit determination is the satellite state error covariance, which is often used as an estimate of satellite-state uncertainty. A primary advantage of a sequential estimator is that the time evolution of the satellite state error covariance can be more accurately determined. Given a reasonably accurate covariance, * Analytical Graphics, Inc., 220 Valley Creek Blvd, Exton, Pennsylvania,

2 many other operational analyses are made easier, such as probability of correlation association (i.e., the chance that a given measurement was of a particular spacecraft), measurement outlier detection, spacecraft maneuver detection and estimation, and sensor calibration. Covariance is also used to draw inferences about the probability of collision between two objects that closely approach each other in orbit. 1 As more and smaller objects are detected and tracked in space, there will likely be fewer and fewer resources available to track orbiting objects. In order to maintain or increase the accuracy of the space catalog with potentially fewer observations per spacecraft, there will be a growing need to have more accurate tracking data. Sequential estimation can provide error estimates in near real-time, the timeliness of which can be very important for the routine maintenance of space objects. In this paper, the authors revisit a proposal to calibrate the US Space Surveillance Network using an optimal sequential estimation, first presented over two decades ago. A commercially available sequential estimator is used to simulate the problem of reasonably detecting and estimating changes in sensor performance closer to real time. By testing the estimated biases against their computed uncertainty determined from the state covariance, it appears that the possibility exists to detect sensor performance issues more quickly and accurately than the more recently practiced methods. SENSOR CALIBRATION The Air Force Space Command (AFSPC) operates the US Space Surveillance Network (SSN). This network performs the spacecraft surveillance needed to maintain space object catalogs for the US DoD using various dedicated, contributing and collateral radar and optical sensors. For operational orbit determination purposes, the AFSPC Space Control Center (SCC) has remained interested in a level of sensor error assessment limited to a bias (a static offset from truth) and a sigma (a measure of observation uncertainty approximating the population standard deviation). Historically, more complicated descriptions of sensor error have not been required, partly because the sensors are already supposed to be internally calibrated to a high degree, and partly because the special-perturbations least-squares differential correction process cannot utilize more complicated observation-error characterizations. Errors are ordinarily estimated by subtracting computed measurements from the sensor measurements. The computed measurements are based on a satellite reference ephemeris of higher accuracy than the sensor itself. 2 AFSPC uses spacecraft ephemerides derived from SLR and GPS tracking data that are independent of the SSN. The error estimates, in the form of measurement residuals, are averaged to estimate the biases in the SSN sensors. The scatter of calibration residuals are also used to estimate a sample variance that weights the observations during least-squares orbit determination. Together, these calibration values are often described as the sensor s static weight and bias. 3 "INTERNAL" VERSUS "EXTERNAL" CALIBRATION Prior to the mid-1990s, the Space Control Center relied on a weekly "internal" calibration process that used data collected by the SSN sensors on a select group of spacecraft. 3 This method relied on building the precision spacecraft ephemerides from the SSN sensor observations being calibrated. To do this, a subset of the most accurate SSN

3 sensors was used in a special-perturbations (SP) least-squares differential-correction process to produce spacecraft reference ephemerides of moderate accuracy. Although this method was not an independent calibration technique, it was believed to provide adequate results for the majority of space control applications at the time. An additional calibration analysis was performed annually using precise "external" ephemerides from the former US Navy Transit navigation satellites. The "external" calibration provided a check on the "internal" calibration process. In the mid-1980's, Applied Technology Associates, Inc. (ATA) came under contract with AFSPC to develop a more advanced sequential orbit determination algorithm. In October 1987, ATA published the results of an initial study that included an analysis of SSN tracking-data errors. 4 The study was based on a one-year span of SSN tracking data of Defense Meteorological Satellite Program (DMSP) spacecraft. This study used static weights and biases supplied from previous internal calibration by the US SCC. ATA studied the ability of the filtering method to detect SSN sensor errors by estimating DMSP orbital elements, spacecraft drag parameters, and sensor ranging biases. The ATA study concluded that the operational use of batch least-squares methods could interfere with the identification of measurement errors and serious force-modeling problems with orbiting objects. The report concluded that: a) it was possible to identify and estimate significant ranging errors that had not been previously identified, b) improvements in orbit determination accuracy were indicated (based on a reduction in the overall scatter of predicted spacecraft ephemerides relative to an ephemeris reduced from later observations), c) the sequential filter's residuals were in closer agreement to the sensor's design noise level than the residuals based on operational batch least-squares differential-correction. The study suggested that dedicated sensors might not always perform as well as internal calibration indicated. Since the "internal" reference ephemerides were significantly influenced by the observations of the dedicated sensors being calibrated, their own errors could be masked and their performance could appear better relative to other collateral or contributing sensors. Likewise, the performance collateral or contributing sensors could look worse. 3 The conclusions of this study became highly controversial when they were used to support arguments to reallocate or halt funding of some radars. 5 The SSN calibration capabilities of a sequential estimator were not actively investigated after the conclusion of this study, and the ATA development contract was not extended. * Rather, the independent "external" calibration effort was extended to something more frequent and of higher accuracy than a yearly calibration using the US Navy Transit System. AFSPC concluded that data from the international SLR network could be used for US SSN calibration purposes based on efforts by Lincoln Laboratory. By the mid-1990's, the SCC had implemented improved sensor calibration techniques to more accurately calibrate the * Shortly thereafter ATA's filter technology was instead developed into a commercial orbit-determination product known as RTOD (Real-Time Orbit Determination).

4 SSN sensors. The SLR data were later supplemented by a select number of spacecraft ephemerides using space-borne GPS receivers. SENSOR BIAS BEHAVIOR Most analyses to date affirm that the "static weight and bias" model is not necessarily a realistic model of sensor phenomenology. The fact that sensor error estimates are serially dependent during a track has been well established, even after calibration. 6 To compensate for this, a track weighting procedure has been practiced at the SCC to inflate the satellite-state covariance to more realistic levels. The procedure is merited by the empirical evidence that the variance of the estimated satellite state tends to be more correct than without track weighting. 7 Given the multi-parameter linear estimation problem: y = Aβ + ε, (1) where y is an n vector of the multi-component observations taken at l times, and ε is an n vector of random deviates distributed according to E[ε] = 0, E[εε T ] = W -1 = cov[ε], then β = A T WA ( ) 1 A T Wy, (2) is an m vector of the best, linear unbiased estimates of the parameters β. These are the socalled normal equations. 8 Let j represent the total number of separate tracks belonging to y, and n i the number of elements belonging to the i th track. In the method of track weighting: j β = ( (A T WA) i / n i=1 i ) 1 j ( (A T Wy) i / n i =1 i ) (3) The entire i th track sub-matrix (A T WA) i is divided by n i to provide track-weighted normal equations. When W i is a diagonal matrix of constant values (one for each observation component), it is possible to divide the individual measurement weights W i by n i. Barker et al. (1999) stated that the underlying statistical basis of track weighting "was dependent on the assumption of short observation tracks experiencing high autocorrelation". 6 A stationary first-order autoregressive process, also known as the AR(1), may be represented by the following recursion: X(t i ) = ax(t i 1 ) + ε(t i ) ; -1< a < 1 (4) where a = ρ(1) is the first-order autocorrelation coefficient and ε(t i ) is a Gaussian iid variate having zero mean and variance Var(ε(t i )). 9 In words, an AR(1) process implies that the current observation error X(t i ) is equal to a fraction of its previous level X(t i-1 ) plus an additional zero-mean Gaussian innovation ε(t i ). However, in practice, an abbreviated stationary series experiencing "high autocorrelation" (a 1) cannot be easily distinquished from a nonstationary random walk (a 1), also known as Brownian motion. The random-walk process can no longer be considered stationary autoregressive; rather, this type of recursion is the sum of a sequence of iid innovations ε(t i ): X(t i ) = X(t i 1 ) +ε(t i ) (5)

5 Stationarity implies that the mean, variance, and correlation are invariant with time. The apparent need for track weighting raises the question of whether there may be a need to test for significant changes in calibrated values in the short term. Seago et al. (2003a) studied this and found that SSN measurement-error estimates often demonstrated a significant tendency to be "directionally non-random"; that is, the error estimates tended to continue moving in the same direction during a track more often than could be attributed purely to randomness. 10 Seago et al. (2003b) later suggested that the limiting behavior of the track-weighted variance over long observation tracks may be consistent with a randomly walking sensor bias, complementing an earlier suggestion by Alfriend and Wilkins (1999) that a stated tendency toward serial correlation could be alternatively viewed as a slowly varying, random bias. 11. Seago et al. (2003b) also noted that, when tasking is limited to five (5) or less observations per track, (nearly) nonstationary trends may appear like track-specific biases. Finally, exploratory techniques and graphical data analyses appeared to support the conjecture that other nonstationary processes, such as the integrated moving average (IMA), might provide for a more realistic model of sensor error than first-order autoregressive models. 12 Seago et al. (2004) further proposed process-monitoring methods used by quality control engineers to identify statistically significant changes in sensor-performance data, since detecting a mean-shift is equivalent to a test of statistical hypothesis. However, to use such statistical methods required that certain assumptions be met in the data being tested, such as independently and identically distributed Gaussian errors. It was shown that these assumptions could only be met if the estimated errors were smoothed. Unfortunately, the method using smoothed data required about two hundred tracks before a significant determination of bias change could be made. This meant that the method was really only useful for detecting very large change-points retrospectively, and was not good for identifying changes in sensor performance close to real time. THE FUTURE OF SENSOR CALIBRATION In the past, NASA's part of the international SLR network has experienced funding shortfalls. This prompted Devere et al. (2003) to conduct a study to determine how the SSN would calibrate itself if the SLR network stopped producing sufficient data. It recommended that the SSC purchase high-fidelity ephemerides from other agencies and spacecraft owner/operators if necessary to maintain its "external" calibration activities. Devere et al. (2003) also recognized that the former "internal" calibration method was available for use at "no cost"; however, this method was not recommended as a long-term solution because the method was considered to be too inaccurate. However, there was no apparent consideration of using sequential orbit-determination and bias-estimation software in that study. The 1987 ATA study is interesting as it supports the idea that the "no-cost" SSN-only calibration method might be useful if a different estimator was used. Advantages are that results can be obtained more quickly than the "external" calibration methods employed today. Also, there is a potential to greatly expand the handful of independent calibration satellites from the dozen or so now available. 2 The licensing of commercial software for this application would be much less expensive than purchasing non-slr spacecraft ephemerides for one year. Perhaps most importantly, this method is best able to

6 accommodate error behavior that has been noted by various authors: that SSN biases are not really constant. SIMULATION METHOD For this paper, the authors desired to verify the assertions of the original ATA report: that a sequential orbit-determination processor could detect relatively large, systematic biases in SSN sensor. If this were so, there is a potential to improve network quality in the short term, thus leading to improved accuracy in the space-object catalog. A simulation exercise is required since it is necessary to know the size and the time of introduction of a sensor bias to gauge success. Analytical Graphics, Inc. (AGI) offers a commercial software product known as the Orbit Determination Tool Kit, or ODTK, for geocentric spacecraft orbit determination and analysis. It employs an established algorithm for sequential orbit determination, first published by Wright (1981) and was first used by the General Electric Company for spacecraft operations. 13 Its features include a tracking-data simulator, an optimal (Kalman-like) sequential filter, and a matching optimal sequential smoother. 14 The Kalman filter measurement update is used. The time-update equations differ from those of a Kalman filter due to the use of process-noise based on models of the uncertainty of the underlying physical models. This enables the calculation and propagation of a more realistic state error covariance. ODTK's optimum filter can simulate and estimate time-varying biases on the tracking system, as well as time-varying corrections to the force models. The filter can simultaneously estimate the orbits of multiple spacecraft, their ballistic coefficients and/or solar radiation pressure coefficients. It can adjust the uncertain priors in tracking facilities, and solve for time-varying measurement biases, transponder biases, and atmospheric density corrections. 15 ODTK has the ability to model stochastic (random) process noise for state, acceleration, and measurement errors. The stochastic process-noise model primarily adopted in ODTK is a stationary, two-parameter Gauss-Markov sequence, its scalar representation being: x 2 2 ( t ) Φ( t t ) x( t ) + 1 Φ ( t, t ) w( t ); k { 0,1, 2, L} ; w~ N ( 0 σ ) k + 1 = k+ 1, k k k+ 1 k k+ 1, w α t t k + 1 k 2 where Φ( tk + 1, tk ) = e, σ w is the variance of independent (white) Gaussian noise, and α is a constant < 0 prescribing the degree of process autocorrelation. In practice, the analyst defines α through the exponential half-life τ1/2 = ln(1/2) / α *. The half-life regulates how quickly the bias value can be changed during prediction. Most importantly, Eq. (6) is the same error model originally used in the ATA study to calibrate radar ranges. The model expressed in Eq. (6) is Gauss-Markov and demonstrates an exponentially fading autocorrelation function. It is a nonstationary function in the presence of measurement updates; otherwise its behavior is stationary., (6) * For a time interval equal to the half-life τ ½, a Gauss Markov bias will decay by a factor of two (2) during estimation, in the absence of measurements.

7 Table 1. Simulator Measurement Statistics Property Bias Bias Half-Life Bias σ White Noise σ Description Constant offset between the value predicted by the measurement model and the observed value, defined in the sense that observed = predicted + bias. The half-life of the exponentially-decaying Gauss Markov bias. Square root of the variance of the exponentially-decaying Gauss-Markov bias associated with a measurement model. The bias is used to estimate/simulate time varying biases associated with the measurement system. Square root of the variance representing the random uncertainty in the measurements. The purpose of the simulator is to create a set of realistically deviated measurements by varying the initial conditions of satellites and measurement biases, and by adding noise to modeled measurements. A parameter is deviated when its value is perturbed by a Gaussian pseudo-random number drawn from a N (0, σ) distribution. The deviated result defines a randomly defined truth for the simulation. Exercised options included deviating orbits (the initial spacecraft orbit state is deviated against the full orbit covariance in the RIC frame), atmospheric density, ballistic coefficient, solar-pressure coefficient, and measurement biases. White noise was added to simulated measurements, and simulated process noise was further added to density corrections, solar pressure, and measurement biases. The sequences for simulated process noise, are described by Eq. (6). Measurement Modeling A measurement bias was simulated as a combination of an initial (prior) value and some level of noise. The parameters that control the level of process noise for simulation and estimation purposes are the measurement statistics belonging to the tracking system, the most significant of which are listed in Table 1. Ground-based range, azimuth, elevation, Doppler, right ascension and declination were simulated for this demonstration. An ionosphere model can be applied when a spacecraft is tracked at a frequency that is significantly affected by the ionosphere, as can a troposphere model. For the purposes of this study, the authors did not simulate or apply these meteorological effects due to their computational burden. Custom tracking intervals were constructed using AGI's Satellite Tool Kit (STK). This allowed the authors to define inclusion and exclusion intervals and vary the sampling rate by facility and/or satellite to produce a realistic schedule. The schedules were defined so that sensors reported observations ten (10) seconds apart and tracked their satellites up to four (4) minutes. This level of tasking was believed to be typical of what might be experienced for a sensor undergoing calibration. * Optical sensors reported right ascension and declination observations 10 seconds apart and tracked for up to thirty (30) seconds * The level of tracking used for calibration will be higher than that of routine tracking.

8 during favorable lighting conditions, usually resulting in three (3) observations per track. AFSSS Fence sensors were modeled as azimuth and elevation measurements occurring in a narrow plane passing through zenith. Since this was a sensor calibration exercise, all measurements tracked below fifteen (15) degrees elevation were excluded, with the understanding that low elevation tracking would be undesirable for calibration purposes. Outages for weather were not modeled. It was assumed that optical sensors were capable of tracking the same low-earth orbiting objects as radars, although in practice most optical systems are reserved for deep-space surveillance. Measurements were written to and read from the so-called B3 tracking-data format. ODTK has a feature called "Flexible State Space" that allows an analyst to add, remove, or change elements within the filter state space without re-initializing the filter or simulator. This feature was used to add biases into the simulated measurements. The simulator was run to a predetermined time, then, a change was made to the measurement bias state before the simulator was allowed to continue. Force Modeling In our force modeling, the authors randomly perturbed the initial conditions of the orbits, density, ballistic coefficient, solar pressure scale factor, and measurement biases. Process noise was added to the force models where applicable. SSN station locations were assumed fixed, and no maneuvers were added to the satellites. Table 2 summarizes the modeling used for simulation. Gravity. A gravity process noise model was used to account for certain errors associated with integrating through the gravity field. The gravity process noise model accounts for errors of commission (uncertainty in the gravity field coefficients) and/or errors of omission (uncertainty due to truncating the gravity field). Adding gravity process noise increases the orbit covariance to more realistic levels. Third-body accelerations due to the Sun and Moon were also applied. Table 2. Spacecraft Force Modeling Used For Simulation Force Modeling Modeled? Simulated? Estimated? Earth gravity field Degree & Order 21 No w/ Process Noise Ocean tide & solid earth tide perturbations No - - Third body gravity (Solar, lunar) Yes - - Atmospheric density (air drag) Jacchia, 1971 Half-life = 3 hours Time-Varying Drag coefficient Jacchia, 1971 Half-life = 7 days Time-Varying Solar-radiation pressure Perfectly absorbing sphere Half-life = 2 days Time-Varying Earth reflectivity and emissivity No - - Maneuvers No - -

9 Air Drag. Deceleration due to atmospheric drag was modeled using the CIRA 1972 (an empirical model of atmospheric temperature and densities as recommended by COSPAR, and the same as Jacchia 1971). The spacecraft ballistic coefficient is defined as B = C d A / m, where C d is coefficient of drag, A is frontal cross-sectional area, and m is the spacecraft mass. For this simulation, the authors choose to work with B directly as this is available from the NORAD two-line elements. A novel feature of ODTK is its ability to estimate a relative correction to the ballistic coefficient while simultaneously estimating a relative correction to the local atmospheric density. 16, 17 The relative corrections to atmospheric density and ballistic coefficient are each modeled as Gauss-Markov processes, wherein the analyst can define their exponential half-lives. The two states become separable when the half-life of the density correction is significantly different from the half-life of the B correction and when the atmosphere is in an excited state. Except during an initialization period, the root-variance of the density correction is calculated internally from the atmospheric model, while the root-variance of the B correction is analyst defined. 18 The authors adopted the default values for ballistic coefficient half-life (7 days) and density correction half-life (3 hours). Table 3 Relative Sensor Uncertainties Chosen For Simulation Range Range Rate Azimuth Elevation Noise Bias 1/2 Noise Bias 1/2 Noise Bias 1/2 Noise Bias 1/2 σ σ Life σ σ Life σ σ Life σ σ Life I.D. Name (m) (m) (min) (cm/s) (cm/s) (min) (") (") (min) (") (") (min) 211/2/3 Stallion * /2/3 Maui * /2/3 Diego Garcia * 260 MOSS * AMOS * ALCOR ALTAIR TRADEX /5/6 Fylingdales Moor Millstone Hill 382/3 Clear /7 Cape Cod /9 Beale Shemya /5 Thule Cavalier Eglin San Diego Elephant Butte 743 Silver Lake Tattnall Red River Hawkinsville * natively reported in right ascension and declination

10 Radiation Pressure. Solar-radiation pressure (SRP) was modeled using a penumbral cone model for the Earth s shadow and the Moon s shadow. 19 A relative correction to the spacecraft s nominal solar pressure coefficient was randomly applied by the simulator and estimated by the filter, modeled as an exponentially decaying Gauss-Markov process. The authors adopted the ODTK default exponential half-life of 2 days for this exercise. The spherical radiation model was configured to be perfectly absorbing. 20 Eclipsing by lunar shadow was ignored for this demonstration. Earth reflectivity and emissivity models were also unused because of their small effect and added computation burden. Initial State. An initial orbit uncertainty matrix is necessary to compute the satellite s initial filter covariance. This matrix is also used by the simulator to randomly deviate the initial orbit state. The matrix elements represent an error-covariance in the directions of radial (R), transverse (or in-track, I), and normal (or cross-track, C) position and velocity. The authors chose to use the default ODTK values, which were R = 50 m [0.06 m/s], I = 100 m [0.04 m/s], C = 20 m [0.02 m/s]. SSN Network. Table 3 presents the values for sensor measurement statistics adopted for the simulation. Except where noted elsewhere, all biases were assumed to be zero. The variances were based on numbers published by Seago et al. (2002). 21 SIMULATION RESULTS The simulation began with a starting epoch of 1 September, 2007 and spanned up to seven days. Twenty-nine low orbiting spacecraft were included in the simulation. These spacecraft were mostly "microsatellites" that do not maneuver, are known to be tracked by the SSN, and have been in orbit several years. The initial conditions of the spacecraft were based on NORAD two-line elements available in early October, 2007, and provided courtesy of Dr. T.S. Kelso and Celestrak ( Two simulations were performed. First, a time-varying bias for every ranging measurement within the SSN was estimated, as was the atmospheric density correction, the ballistic coefficients, and the orbital elements of all twenty-nine satellites. The total number of simultaneously estimated states was 280. The entire seven-day span was filtered in about 35 minutes using a c personal desktop computer. The estimation of ranging biases alone was consistent with the original ATA calibration study. Next, a time-varying bias for every SSN measurement was estimated (not just range), as well as the atmospheric density correction, the ballistic coefficients, and the orbital elements of the twenty-nine satellites. The total number of simultaneously estimated states in this case was 355. The entire seven-day span was filtered in about 45 minutes. For the purposes of detecting large changes in ranging bias in real time, the results appear to be nearly identical, so presentation of figures is limited to the latter case. During the first two days, all sensors were simulated with no bias, according to Table 3. After the first 48 hours, the range biases of two sensors were changed to be equal to twice their assumed standard deviation: Eglin = +80 m, and Kaena Point = -20 m. This level of bias was chosen since the filter was using a measurement residual rejection threshold of three standard deviations. A two-standard-deviation bias would be seen clearly, but it was not so large that the biased tracking data would be completely ignored.

11 Figure 1. Filtered Tracker Measurement Bias State History for Eglin Range Based on Simulated Measurements (Biased Positively after Two Days) Figure 2. Filtered Measurement Residual and Corresponding Residual Standard Deviation for Eglin Range Based on Simulated Measurements

12 Figure 3. Filtered Tracker Measurement Bias State History for Kaena Point Range Based on Simulated Measurements (Biased Negatively after Two Days) Figure 4. Filtered Measurement Residual and Corresponding Residual Standard Deviation for Kaena Point Range Based on Simulated Measurements

13 Figure 5. Filtered Tracker Measurement Bias State History for TRADEX Based on Simulated Measurements (No Bias Simulated) Figure 6. Filtered Measurement Residual and Corresponding Residual Standard Deviation for TRADEX Range Based on Simulated Measurements

14 Figure 1 shows the results for the Eglin radar. For improved readability, the data plot is limited to displaying only the first three days. It is apparent to the eye of the analyst that a change has occurred after two days, and the level of ranging bias generally exceeds the computed standard-deviation of the measurement residual by a factor of two. The residual plot in Figure 2 shows no significant residual signature that might indicate that the bias has not been accommodated by the filter. Figure 3 shows the results for the Kaena Point radar. It is again apparent to the eye of the analyst that a change has occurred after two days, and the (negative) level of ranging bias generally exceeds the computed standard-deviation of the measurement residual by a factor of two. Again, the residual plot in Figure 4 shows no significant residual signature that might indicate that the bias has not been accommodated by the filter. Figures 5 and 6 show the results for the TRADEX radar, which is typical. These figures illustrate that the bias of the other radars did not adversely effect, or significantly alias into, the ranging of the unbiased radars. Within each SSN measurement, there is immediate information about the behavior of the sensor bias, as well as information about the state of the object being tracked and other estimated parameters. The estimator presumes that this information is distributed according to relative uncertainties assigned to the estimated parameters and the measurement itself, as mapped through the information matrix up to that time. When estimating biases on all sensors, the effect of sensor biases does not systemically alias into other filter estimates as it might for a leastsquares differential corrector. The filtered bias estimate is allowed to move relatively quickly to its new state. As more observations are accrued, and the properly weighted measurements of other sensors will further move estimates of satellite state and/or local atmospheric density closer to their true values. (When there are no measurements from a faulty sensor, of course the system remains unaffected.) If a sensor is performing suddenly and unusually out of specification, its behavior has become much different than what the filter has come to know. Since the estimated sensor bias may not yet be fully observed using the available information, the sensor bias may temporarily affect, or become aliased into, other estimated parameters, such as the satellite state or the local atmospheric density, depending on the situation. Such an effect is usually quite short-lived and damps out quickly when using a properly formulated sequential filter. Otherwise, a filter-smoother consistency test will usually reveal any modeling problems, which is discussed next. For this demonstration, the authors added a significant number of parameters into which a sensor bias might be aliased, to provide for a very demanding case. For example, the authors simulated and estimated both a local change to the atmospheric density at the spacecraft as well as the ballistic coefficient. For an operational application using nonsimulated data, fewer estimated parameters might be preferred. CONSIDERATIONS FOR WORKING WITH NON-SIMULATED DATA A filter is a sequential (recurrent) estimator where the resulting estimate is conditioned or influenced by prior observations available up to the time of the result. A smoother is also a sequential estimator where the resulting estimate is conditioned or influenced by observation information both before and after the time of the result.

15 ODTK makes available a fixed-interval sequential smoother that is a non-linear adaptation of the linear fixed-interval optimal smoother presented by Meditch (1969). 22 A fixed-interval smoother combines filtered state and covariance information in reverse chronological order to calculate the best post-fit estimate of the orbit throughout the interval of interest. As a smoother operates with information backwards through time, it smoothes ephemeris discontinuities caused by state corrections in the filter resulting from measurement processing. The advantages of the smoother are smoothly behaving state estimates and a covariance which is reduced relative to the filter covariance due to the inclusion of two-sided information. McReynolds (1984) proved that the difference between the filtered state and smoothed state is normally distributed in n dimensions, where n is the size of the state-difference vector, and that the variances and correlations of the state-difference vector are equal to the smoother error-covariance subtracted from the filter error-covariance. 23 Therefore, sampling this difference vector over time creates a population that should be normally distributed. If it is abnormal, then this is interpreted as a defect in the filter-smoother model. Filter-Smoother Consistency Theorem: Let the array x f (t) be an n x 1 filtered estimate at time t having n x n error-covariance P f (t), and let the array x s (t) be its smoothed estimate at time t having error-covariance P s (t). Then, assuming the state-estimate errors of x f (t) and x s (t) are multivariate normal: The n x 1 statistic x (s-f) (t) = x s (t) x f (t) is multivariate normal at time t, and has n x n covariance P (f-s) (t) = P f (t) P s (t). The time sequence of z(t) = [x (s-f) (t)] T [P (f-s) (t)] -1 [x (s-f) (t)], t = {t 0, t 1, t 2 } provides an (auto-correlated) sample population over the estimation interval upon which the null hypothesis of multivariate normality can be tested. Filter-Smoother Consistency Test: If the sequence z(t) supports the null hypothesis of multivariate normality, then the hypothesis of consistency between the filter and smoother models is accepted. If the sequence z(t) does not support the null hypothesis of multivariate normality, then the hypothesis of consistency between the filter and smoother models is rejected. The so-called filter-smoother consistency test is very useful for general model validation. Failure of the McReynolds test implies that the filter and the smoother do not give the same answer (within the stated uncertainty of both), and the authors must reject the hypothesis that the total modeling is adequate. It is actually very difficult to pass McReynold s filter-smoother consistency test for all time using real data if the measurement noise level, process noise level, and state parameterization are not physically realistic, or are in some other way deficient. A graphical method of examining filter-smoother consistency of each state element may be used, instead of a multivariate test statistic originally proved by McReynolds. For the single-state case, the state differences are divided by the variance differences. Filtersmoother consistency is generally claimed when this metric stays within ± 3 over the fit interval. This graphical method does not rigorously consider autocorrelations within the univariate sequence, and ignores cross-correlations of state-estimate error; nevertheless, the univariate McReynolds test statistic provides very useful diagnostic information in the presence of erroneous filter inputs or model specifications, and its employment would likely be most necessary to validate models adjusted to process actual SSN tracking data.

16 Figure7. Filter-Smoother Consistency Statistic for Radial, Intrack, and Crosstrack Components of Position for the Sapphire Satellite (355-Parameter Case) CONCLUSIONS For this paper, the authors used commercially available software to perform a simulation using twenty-five (25) SSN sensors and twenty-nine (29) objects in low Earth orbit. The simulation appears to verify an earlier claim that ranging biases in SNN radars are detectable in near-real time using only SSN tracking data. This claim was advanced more than twenty years ago, the result of a study that processed one year of actual SSN measurements. Using a sequential estimator for calibration would allow results to be obtained more quickly than current methods. There is also a potential to greatly expand the calibrationsatellite population in order to improve the method's accuracy. This method would also be much less expensive than previously studied contingencies that solely rely on external ephemerides. This method may be best available to accommodate SSN biases that are not really constant. Improvements in sensor calibration would provide for a more accurate space object catalog. Higher accuracy implies that a greatly increased catalog would be easier to maintain. ACKNOWLEDGMENTS The authors are grateful to former technical staff of Kaman Sciences, who first brought the ATA calibration study to our awareness. The authors also appreciate more recent correspondence with Mr. James R. Wright, the original author of this report. His ability to accurately recall and summarize details of the report, and its ramifications, was invaluable to our understanding of why these methods had not been more fully developed at that time. The authors also thank Dr. T.S. Kelso of Celestrak, who provided realistic initial conditions for our simulation. REFERENCES 1 Alfriend, K.T., M.R. Akella, Lee D J, M. Wilkins, J. Frisbee, J.L. Foster (1998), Probability of Collision Error Analysis. AAS , Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, August 10 12, Boston, MA.

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