Presented at the NASA workshop on Video Meteor Astronomy and Analysis, Pisgah Astronomical Research Institute, Rosman, North Carolina, 3–5 August, 2011, and the International Meteor Conference in Sibiu, Romania, 15–18 September, 2011.
Abstract– A new approach has been formulated to solve for the straight line trajectory of a meteor through the atmosphere when given multiple camera views of the meteor’s luminous track. Using a motion propagation model in 3-D space plus time, and iteratively solving for all free model parameters simultaneously, one can obtain a fully coupled solution to the apparent radiant direction, 3-D begin position, atmospheric entry speed, deceleration terms, and timing offsets when using data from unsynchronized video cameras. A Monte Carlo component adds empirical error estimation for each of the key model parameters computed. This multiparameter fitting method extends the allowable collection geometries for meteor trajectory estimation to lower convergence angles between camera-meteor-camera lines of sight and smaller site separation distances.
The fundamental goal of multistation meteor observations is to accurately estimate the meteor’s trajectory through the Earth’s atmosphere and ultimately derive its Keplerian orbital elements in the solar system. With that information, one can then associate a particular meteoroid with a parent body such as a comet or asteroid, provide insight into the dynamical evolution of meteoroid streams, deduce the characteristics of the parent bodies, and perform both validation and refinement of shower streams in the IAU data base. The basic computation involves the combination of multiple views of the meteor’s luminous track to determine its atmospheric path. Existing methods to solve for meteor trajectories have been well documented in several papers and include the intersecting planes method (Davidson 1936; Porter 1942; Whipple and Jacchia 1957; Wray 1967; Ceplecha 1987; Bettonvil 2005), as well as a straight least squares approach (Borovička 1990). In addition, the image processing literature addresses this problem under the more generic title of “Trajectory Triangulation,” some papers of which are applicable to meteor flight when employing multiple cameras (Kaminski and Teicher 2002, 2004).
The conventional trajectory estimation methods that have been typically used in the meteor community solve for a meteor’s path through the atmosphere using a two-step boot-strapping technique. First the orientation and position of the track is obtained, followed by a solution for the velocity and deceleration along the track. The new approach, as described in this paper, can best be described as a “multiparameter fit” algorithm, and attempts to estimate all unknown parameters simultaneously. The algorithm iteratively solves a fully coupled motion propagation model, converging to the most likely 4-D trajectory in 3-D space plus time that best matches the measurements. The algorithm is designed to handle multiple cameras from two or more disparate sites as well as additional camera tracks obtained from the same site. The method provides greater robustness and accuracy and permits the processing of station geometries with smaller inherent convergence angles to the meteor. The algorithm has been applied routinely on video meteor data from the Cameras for All-sky Meteor Surveillance (CAMS) as described briefly at the 2010 International Meteor Conference (Gural 2011) and then again in another paper with significantly more details that covered the collection, archiving, detection, coincidence, trajectory, and orbital estimation for multistation meteors (Jenniskens et al. 2011).
To estimate meteoroid orbits, it is now common practice to deploy multiple video cameras at widely separated sites and use triangulation techniques to first determine the trajectory through the Earth’s atmosphere. That trajectory can then be converted into a Solar System orbit using standard techniques (Ceplecha 1987; Clark 2010; Jenniskens et al. 2011). The camera systems possess a number of known quantities such as GPS site coordinates, the date and time within approximately one second, the camera’s field of view (FOV), an astrometric solution that relates focal plane pixel coordinates to stellar right ascension and declination, and finally, the sample rate of the video sequence. An assumption that the frame rate of the cameras is a fixed value during the course of the meteor’s flight is an important characteristic that makes the proposed solution possible without frame-by-frame time stamping. It provides the relative timing between frames reducing the temporal parameter estimation to just the timing offsets between cameras.
The unknowns to be solved for describe the meteor’s trajectory, which include the apparent radiant direction (i.e., before zenith attraction correction), the 3-D position of the meteor’s begin point, the velocity magnitude at the begin point (atmospheric entry velocity), deceleration parameters due to atmospheric drag, and the camera-to-camera timing offsets. Essentially, the meteor’s initial position and velocity state vectors and their change over time. The camera timing offsets are an integral part of this solution because video cameras deployed at separated (or even the same) sites are typically not well synchronized to each other. This may occur due to the timing drift of either the cameras or their video time stamping devices, human imprecision in the setting of clocks, and the milliseconds timing offset between the vertical sync signals that border the start of the frame integration period of each camera.
Additional challenges arise from the current proliferation of video meteor cameras where there are many systems employed in a measurement. For example, it is not uncommon for more than two sites to have observed the same meteor and with different focal plane resolutions. There may be two or more partial tracks from cameras at the same site containing separately the beginning, middle, and/or ending portion of the track. An example, which is what drove the development of the multiparameter fit solution to begin with was the geo-location configuration of the CAMS system. CAMS deployed forty video cameras, split between two mountain-top sites, plus individual amateur cameras randomly set up across the San Francisco Bay area. Meteors can potentially traverse across two cameras at a given mountain station because of the camera battery’s closely packed all-sky coverage of many narrow fields of view. In addition, for triangulation to occur, the meteor must nominally be seen at two or more separated sites. Thus, a solution that could merge all these measurements into a single trajectory solution was a requirement for a robust answer, which can be addressed by various approaches.
Trajectory Solution Approaches
The classic trajectory estimation technique in the meteor community is commonly referred to as the intersecting planes solution. This was initially formulated and used for the data reduction of multisite visual observations (Davidson 1936) and can be more formally described as follows. First take the sequence of a single camera’s measurements (line-of-sight, angle-angle) and estimate the best geometric plane passing through the station’s position on the Earth’s surface and the fan of measurement rays emanating from that point. Next, a second station’s observation fan and associated plane estimation is computed. The cross-product of the normal vectors to both planes provides the orientation of the meteor path in 3-D space. However, if there are more than two stations involved, the solution requires an ad-hoc weighted average of the individually paired solutions based on the convergence angle and meteor angular extent (Ceplecha 1987). Secondly, only the orientation and position are obtained with this algorithm, and thus a second separate fit is done to estimate the velocity and deceleration terms for the propagation along the path once the orientation is defined.
An alternative solution was proposed that more naturally handled more than two camera tracks (Borovička 1990). The alternative solution was formulated as a least squares minimization problem that tries to find the best 3-D line that minimizes a cost function. The cost function is the sum of spatial distances between each measurement ray’s closest approach to that of the hypothesized 3-D line. Borovička (1990) reports that this approach appeared to be no more accurate than the intersecting planes method, but simulations shown below will indicate that this least squares method is in fact more robust for lower resolution cameras. Again, however, the solution only provides orientation and position, and a separate fit through the data is done for the velocity and deceleration estimation. As a side note, this author would suggest that minimization of the angle (not distance) between the measurement ray and the closest point on the line may be preferable, as the spatial distance separation is dependent on range with the impact that stations with a longer range to the meteor will have an effective overweighting of their measurements.
This straight least squares approach is what inspired the algorithm developed herein. The multiparameter fit technique is an extension of the measurement-to-model minimization concept by including the temporal nature of the measurements as well. Thus, rather than minimize each measurement ray relative to the closest spacing to a 3-D line, one should attempt to minimize relative to the propagating points uniformly stepped in time along the 3-D line. This creates a more tightly coupled solution to all the available information.
After a presentation of this algorithm and its results at a NASA video meteor workshop at the Pisgah Astronomical Research Institute in August 2011, it was discovered that Wayne Edwards of the University of Western Ontario had also attempted such an approach. One difference was that his method did not deal with the unsynchronized camera issue as their cameras were GPS time stamped and any frame start timing offsets in their all-sky cameras were of little consequence due to the large spatial resolution per pixel available (Edwards, private communication). The algorithm was abandoned as the deep penetrating fireballs they were studying at the time had velocity profiles that were not well matched to the standard velocity deceleration models available in the literature. These included the constant, linear, and exponential models shown in the algorithmic formulation below. The multiparameter fit algorithm described in this paper also requires that the underlying velocity model adhere to the observed propagation changes of the meteor. Thus, this technique is most applicable for smaller mass meteors (<5 g) of short duration (<3 s) as observed in typical moderate field of view video meteor systems, where the conventional propagation formulae options presented herein are usually applied. A more realistic propagation model would need to be formulated for long duration fireballs, which if such a model were to become available, could be easily integrated into the multiparameter fitting framework. One option chosen by many researchers when the straight line assumption or deceleration models break down along the entire luminous track is to segment the track into pieces, applying a solution technique that is valid over a shorter stretch of the measurements (Ceplecha 2000). The multiparameter fitting technique can be used in this way just as the other approaches have been applied in the past.
Multiparameter Fit Formulation
The new trajectory solution algorithm recognizes the fact that there is really only one trajectory to solve for and that by employing the timing information (uniform sampling rate per camera plus camera-to-camera timing offsets), the result yields a more robust solution as all measurement information is included simultaneously, both spatially and temporally. This multiparameter fit technique requires that a motion model be defined. The geometry is as depicted in Fig. 1 with measurement rays from an observation site passing close to a hypothesized linear track in the atmosphere for a single camera. This is effectively repeated for each camera, but always defined relative to the same 3-D straight line track.
The camera for the kth observation site is geo-located typically via a GPS device and has a geocentric vector from Earth’s center given by rk. The meteor has a hypothesized reference begin point xb and starting velocity vector Vb also defined in geocentric coordinates that is assumed the same for all cameras but are unknowns to be solved for. The time at the begin point tb and the initial guesses for position and velocity are typically chosen to be for the earliest measurement of all the cameras and defines the absolute reference time for all measurements. Clock setting errors, clock drift, and nonsynchronization of each camera is accounted for with a fixed relative timing offset Δtk. Thus, the jth line-of-sight vector for camera k has a time shift δtjk from the begin point that is defined by its measurement time tjk minus the reference begin time tb plus the timing offset Δtk. The measurement times are not required to be evenly spaced and can have missing measurements due to video frame dropouts, but for video cameras, the frames are assumed to be originally collected at a uniform sampling time step, thus avoiding the need to obtain precise time stamping of every frame. The relative timing is thus well defined with video and only the timing offset between either the cameras or the reference starting point needs to be estimated.
The motion model itself is generically defined in Equation 1 and represents the physical motion of the meteor along the linear track. As the motion model is general enough to account for deceleration, the spatial coordinate along the track can produce an uneven positional spacing despite a temporally uniform sampling rate. Note that the 3-D straight linear path assumption has been assumed herein and previously verified even for long duration fireballs (Borovička 1990). Thus, the assumption is made that a linear path holds for at least the first seconds of luminous flight, and accounts for nearly all meteors other than the brightest fireballs that penetrate deeply into the atmosphere. This effectively covers a mass range up to 50 g or equivalent magnitude of −2 (Jenniskens 2006), below which most meteors are observed in video meteor camera systems.
The position propagation model X(t) is obtained trivially from the integral of a user selectable velocity motion model, which has been based on several published in the meteor community. The current software assumes that the meteor’s propagation model follows one of three choices. The analyst would need to pick a model for the meteor under investigation or run all the models and down select the best fitting parameterization. Those models implemented include a constant velocity along the track, linearly decreasing velocity in time, and an exponentially dependent deceleration (Whipple and Jacchia 1957; Jacchia et al. 1967). Each derived position model is defined in Equations 2, 3, and 4, respectively.
The propagation model thus hypothesizes where the meteor should be observed along the track and the trajectory solver attempts to minimize a cost function with respect to actual measurements. The cost function formulated for the best trajectory solution is the sum of angles between the measurement rays relative to the modeled rays at each time step as depicted in Equation 5. This is for a given set of unknown fit parameters. Each angle is computed from the arc-cosine of the dot product between the measurement mjk and camera-to-model-ray unit vectors.
The unknown parameters include the three position and three velocity components, optionally up to two deceleration coefficients, and the timing offsets of all the included cameras. Once the cost function reaches a minimum by iteratively adjusting the free parameters, the result is a fully parameterized, single trajectory solution. This requires a multiparameter minimization routine as the cost function to be minimized is not linear and does not lend itself to a simple linear least squares solution. The algorithm has been implemented with an iterative Nelder-Mead algorithm, also known as downhill simplex or amoeba minimization, and was chosen for its robustness and ease of implementation. The danger with such an approach and the Nelder-Mead technique in general is that the solution can get caught in a local minimum rather than the true global minimum of the cost function and yield a poor trajectory. To avoid this problem, the solution proceeds via a boot-strapping technique somewhat reminiscent of the existing meteor community’s method of trajectory estimation. First, the intersecting planes solution is obtained for the camera pair with the best convergence angle and the position and orientation are refined with the least squares method. The timing offsets are estimated next with all other parametric terms held constant. The velocity and deceleration terms are then computed, again with all other terms held constant. And finally, only on the last function minimization are all the parameters allowed to be freely variable to get the best final fit solution to the coupled trajectory and propagation.
An interesting benefit obtained from applying this method to the CAMS multicamera measurements was that the timing offsets helped to properly align the light curves between disparate cameras. This was of significant importance because due to the delay in retrieving data from the remote sites and no onsite time stamp resetting possible for the video collection servers, there were often relative timing drifts between the cameras of several seconds. Thus, while allowing for the timing offsets to be a free parameter to provide coupling of the measurements into a single solution, it was also a modeling feature that helped to provide robust trajectory solutions to what would otherwise have been non-coincidental observations.
An important part of any parameter estimation technique is to be able to provide error estimates on each of the parameters computed. For the intersecting planes and least squares techniques, the implementations of those algorithms have included closed form expressions for the error estimates. Unfortunately, the multiparameter fit technique involves a non-linear solution, and thus closed form expressions would require deriving them via linear approximations. Instead, what was chosen was a Monte Carlo approach to empirically estimate errors in each of the parameters.
This is achieved by taking the nominal parameter fit as a baseline solution and then perturbing the measurements with a 2-D Gaussian random distribution of noise added to the measurement angles. The amount of noise is controlled by the measurement standard deviation provided by the user based on how well the meteor positions have been measured. A value one can use is the error estimate of reported stellar positions from the astrometry for each camera as it includes both centroid and astrometric fitting errors, but this is usually an underestimate of the measurement noise of a blob-like and elongated meteor. At this time, a fixed one sigma value is specified by the analyst and is used for all the Gaussian distributed error draws added to the measurements of any given camera. It is also independent of direction, but the sigma value is currently allowed to vary from camera to camera to account for focal length differences (resolution differences between cameras).
In the future, a more general noise model will have a unique measurement error value for each and every measurement, for example making it dependent on magnitude or signal-to-noise ratio and direction of motion. It is also recognized that a meteor’s measurement error is significantly different in the along-track versus cross-track directions due to the elongation of a meteor in a video frame from its motion, brightness changes during the frame integration, fragmentation spread along the track, and the presence of wake signatures. Thus, it is planned that a 2-D (along and cross-track) sigma value per measurement will become a required input from the user, and will effectively weight each measurement relative to others and also relative to the propagation direction. This more elaborate error model would have to be appropriately tied into the cost function formulation. Note that the algorithmic formulation herein does not address best practices when it comes to making measurements of meteor positions in each frame. For example, it is quite often better to measure the leading edge of the meteor’s position rather than its centroid.
For each Monte Carlo trial, the noise perturbed measurements are used to solve for a new trajectory when given the baseline solution as the starting point. This permits fast iterative convergence as the cost function is already close to a minimum. This process is repeated for N trials (N is typically 100) and the standard deviation computed for each parameter that makes up the trajectory solution. Thus, the baseline fit parameters plus/minus a standard deviation from the Monte Carlo trials is the reported output from the estimator. The resultant state vector can then be used to feed an orbit estimator with appropriate error estimates on the key input parameters.
To validate the improved performance expected of the multiparameter fit algorithm, a simple but geometrically realistic meteor simulation was built in Matlab and statistically run using Monte Carlo trials. A meteor begin point was situated over the surface of the Earth at latitude 0°, longitude 90°, and height of 100 km. The meteor motion was restricted to always move in a northerly direction along a 3-D linear track with uniformly distributed random draws on its radiant elevation between 10 and 80° and entry velocity between 12 and 72 km s−1. The observing sites were placed randomly around the meteor sub-point to include all possible viewing angles to the meteor. Those site positions were determined by taking a uniformly distributed random draw of sight-lines on a hemispherical dome with an elevation limitation. Those are projected onto the spherical Earth to populate the latitude and longitude of each observation site k. Thus, the meteor could be observed from any direction and the geometry for convergence angle and site placement was not always optimal. The number of sites chosen for most of this analysis was two, but toward the end of this performance section is an example of three site results.
The number of measurements was also varied between Monte Carlo trials and was modeled on an analysis of the duration of meteors seen in the Japanese meteor database (Sonotaco 2010). For a typical meteor population index, 99% of meteors will fall in magnitude bins no higher than three magnitudes above the meteor limiting magnitude (lm) of the sensor system. For each sensor resolution and its associated meteor limiting magnitude based on typical field deployed meteor cameras (Gural 2011), the duration histogram of the Sonotaco data was found to be shaped like a Rayleigh distribution. Sensor resolution is defined here to be the angle subtended on the sky by a single pixel, which is a function of focal length and pixel pitch. A random draw for the number of measurements was taken from a Rayleigh probability distribution with a coefficient σray specified below. The durations were restricted between 0.1 and 3.0 s.
The site observing lower elevation limit was set based on typical observing geometries deployed in the field: the all-sky was limited to 10° by ground horizon obscuration, whereas moderate, narrow, and telescopic were based on both common practice and optimal flux counting results (Gural 2000).
The measurement interleaved frame rate was chosen to be 60 fps representing NTSC type cameras, but the results are equally applicable to PAL (50 fps) systems. The equivalent measurement count per camera thus ranged from 6 to 180 for the frame rate and durations modeled. To evaluate performance under various focal length systems from telescopic to all-sky, the pixel resolution and measurement error were two inputs that were parameterized for the study. Random noise was added to each ideal measurement ray as a 2-D Gaussian random draw in angle–angle space with one-sigma values of 0.3 pixels, which is scaled by the sensor resolution in radians per pixel to provide a noise standard deviation in radians. This value may be considered a lower limit to the typical noise seen when measuring actual meteor positions in the focal plane depending on the pre-processing methods used to measure meteor positions. Measurement errors can be influenced by the blob-like appearance and elongation of the meteor from the finite time exposure. However, the errors applied in the simulation represent a fixed measurement noise level for which all the methods can be consistently compared.
For all the scenarios explored, 50,000 Monte Carlo trials of meteor orientation, speed, measurements, and site random draws were run to build the statistics for analysis. The three different algorithms that were compared included the intersecting planes, least squares distance minimization, and multiparameter fit approaches. The performance curves for each algorithm will be discriminated in the following plots through the use of stars, open circles, and closed circles, respectively. For most of the results presented, a constant velocity model of meteor propagation was used in the simulation. Not until the last part of this section is there a discussion on the performance of the algorithm in terms of velocity estimation with a decelerating meteor model present.
Because there were a large and random range of site locations relative to the meteor involved in the simulation, many of the site-to-meteor-to-site geometries were less than ideal, but this was done to ensure that the full range of geometric encounters was considered. Thus, when the first results were plotted of the angle error between the true and estimated radiant positions, versus a commonly used parameter such as convergence angle, the points looked like a very dense and messy scatter plot. There was no easily discernable distinction between the intersecting planes set of trials versus the multiparameter fit. However, on closer inspection, many more of the multiparameter fit points had low error and only the few outliers were grabbing one’s visual attention in both scatter plots. Thus, a median calculation of the radiant error is presented using small bin extents across the plotted abscissa and will comprise the plots shown herein. Note that the outlier cases for the multiparameter fit were predominately due to situations in which the angular extent as seen from one of the cameras was <3°. That is, the geometry was such that the meteor was almost heading directly toward the camera. To a lesser extent, having only a small number of available measurements was also a characteristic of those cases which tended toward poorer performance.
It is not surprising that some performance expectations are borne out in the simulations. For example, if using a 2.8 arcminute per pixel camera resolution (30° FOV), an improvement in radiant estimation error of factors of four or more is clearly seen in Figs. 2, 3, and 4 for all the algorithms when there is:
1 An increase in the number of measurements of the meteor track from ten to thirty.
2 An increase in the site separation between cameras from a few kilometers to fifty.
3 An increase in the entry velocity of the meteor from 12 to 72 km s−1.
The first two are obvious as longer meteors and large baseline separation between cameras make the geometry more favorable for direction estimation. An equivalent behavior to increasing the number of measurements was to see the same radiant estimation improvement when evaluated against greater physical length of the meteor or wider apparent angular extent as viewed from a given site. The third bulleted result is most likely due to the greater induced angular extent of the meteor across the focal plane and, just like increasing the number of measurements, makes it easier for a 3-D line estimation algorithm to determine the orientation in space. A needle-shaped object’s orientation is easier to discern than a short blob. In general, the multiparameter fit performs somewhat better than the other two methods. When the simulation was run with varying levels of measurement noise, it was also found that the radiant error was typically proportional to the measurement error (independent of the sensor resolution). Thus, only the results for the one sigma measurement noise of 0.3 pixels will be shown from here on out, but if a preprocessing approach to obtaining meteor positions has worse error than this measurement standard deviation, comparative results should be appropriately scaled.
Other, somewhat unexpected, results were also found. For example, the trajectory estimation performance for all the methods was fairly insensitive to radiant elevation or the relative site positioning. The curves in Figs. 5 and 6 are restricted to cameras that were well separated and had a convergence angle to the meteor of greater than 20°. The steepness of the meteor entry angle in the atmosphere (radiant elevation angle) was found to not be a limiting factor. Also, all the algorithms could estimate the trajectory just as accurately if the camera sites were aligned parallel to the meteor’s motion (north–south), perpendicular to it (east–west), or at any angle between those extremes.
The most significant difference between the three trajectory estimation methods was found with respect to the convergence angle. As seen in Fig. 7, for a typical video meteor camera of 30° field of view (2.8 arcminute per pixel), the median radiant estimation errors of below one-quarter degree are achieved for all methods when the convergence angle exceeds 25°. However, the multiparameter fit solution gives consistently better answers for very low convergence angle geometries far below 25°. This was first noticed in the CAMS data reduction results and is what initially spurred doing the performance simulation study. Up until this realization, it was assumed that good trajectories were usually only obtained for convergence angles >30°. But results from both this simulation and the processing of actual collected video meteors show the advantage of the coupled solution method in extending trajectory estimation into more difficult geometries of small convergence angles, short site separation distance, and fewer available measurements.
The improved result of the multiparameter fit at low convergence angles is conjectured to be due to the implicit addition of range to the angle–angle measurements, arising from the arc-length subtended over the video time step. This temporal component coupled to the spatial geometry effectively adds an extra degree of freedom not available in the intersecting planes or least squares solution. This is further borne out in the wider field of view systems and especially the all-sky results as seen in the plots of Figs. 8 and 9. In these two wide angle cases, the best performance versus convergence angle was found for the multiparameter fit, next came the least squares solution, and finally the intersecting planes. Note the reduced quality of the intersecting planes solution for almost all geometries at the all-sky pixel resolution, resulting in errors of a few degrees in radiant position. Thus, trajectory estimation using intersecting planes, as is very often used due to its ease of implementation, could be improved for all-sky cameras by using a multiparameter fit algorithm at all convergence angles or the least squares solution for moderate convergence angles.
At the other extreme, for cameras with extremely narrow fields of view that are on the order of 5°, it was found that the sites could be physically spaced much more closely together than usually recommended. The idea of very short baseline trajectory estimation was made apparent to this author after a serendipitous collection of a meteor from closely spaced video cameras measuring the diameter of the asteroid Metis (Degenhardt and Gural 2010). This concept was also independently evaluated and processed for 5 km spaced zenith pointing cameras at the University of Western Ontario (Kikwaya et al. 2009). The simulation herein bears this out, as it shows that as long as the observed elevation angle of a meteor is above 35°, the sites could be as close as 1 km apart for the multiparameter fit to function as well as seen in Fig. 10. Equivalently, the convergence angle could be as low as 1° from the results shown in Fig. 11. The higher sensor resolution of 20 arcsecond per pixel was the reason the algorithm works so well in these special narrow field of view cases.
One additional simulated result, as shown in Figs. 12 and 13, was that by adding a third camera site to the solution, the least squares method and intersecting planes method had greatly improved performance as compared with the two station results in Figs. 7 and 9. The multiparameter fit also showed improvement as well (by a factor of two). Clearly adding further degrees of freedom through new viewing perspectives helps with all the trajectory estimation approaches. There is not much difference between the intersecting planes (where more than one solution’s linear path must be combined into a weighted average) and the least squares approach that minimizes distance to a single line. But the temporal component of the multiparameter fit adds additional information to constrain the solution more robustly with a resultant smaller residual error. The general conclusion is that in all-sky trajectory estimation, one could best compensate for the lower camera resolution by using the new coupled parameter solution in two or more station data reduction.
The previous performance demonstrations had concentrated on the radiant estimation error results from the meteor simulation. However, an important attribute of the multiparameter fit algorithm is the ability to simultaneously estimate the velocity and deceleration terms depending on the propagation model selected. In Fig. 14 is shown the velocity error histogram when a constant velocity simulated meteor data set is fit with a constant velocity parametric model (closed dots). For the 50,000 meteors simulated using a 2.8 arcminute per pixel sensor resolution, the velocity error distribution looks Gaussian with a one-sigma value of 0.13 km s−1. Also shown in the plot as open circles is the result when the simulated meteors had a random deceleration component included (Equation 4) and the measurements were fit with the same deceleration propagation model containing three unknowns. The one-sigma value increased to 0.16 km s−1 and still provided the same quality fit to the radiant estimation. The distribution worsened by 25% due to those cases where there were too few measurements (shorter duration meteors) to fit the deceleration. When the meteor duration was restricted to longer than one-half second, the velocity estimation one-sigma value for the deceleration parameter dropped by one-half. Also note that fitting a deceleration velocity model to a constant velocity meteor showed the same performance level and effectively fit the deceleration coefficients to zero, thus the algorithm is not ill-conditioned to having too many velocity fitting parameters as long as there is measurement sample support.
A last simulation result involved adding a different level of noise to the measurements in two of the measurement directions for a 2.8 arcminute resolution camera. Gaussian random draws were taken for the along-track and cross-track directions with the along-track standard deviation twice the level of the cross-track sigma. This attempts to simulate the measurement issue of estimating centroids in the propagation direction of varying intensity, blob-like, wake producing, fragmenting, and elongated meteors. As the cross-track error level was the same as the case presented in Fig. 7, the performance for radiant direction estimation came out as expected, essentially matching that previous result. However, one would expect the along-track velocity estimation to be degraded. This was indeed seen with a one sigma velocity residual error rising from 0.13 to 0.23 km s−1. This did not quite scale as a doubling of the error because recall that the velocity fit is part of a multimeasurement minimization and the errors tend to partially cancel out.
Performance on Video Data
The final tests for measuring the multiparameter fit’s performance were performed using actual field-collected data sets that involved multiple video cameras and site deployments. The test and evaluation were first performed on some of the earliest multistation video meteors recorded in the fall of 2010 by the CAMS system. A single representative case for November 1, 2010 is shown and involved recordings from two sites in the United States located at Mountain View and Fremont Peak, California. Each site had two cameras that were sufficiently overlapped in sky coverage to have recorded the same meteor. Thus, a total of four cameras from two stations 90 km apart were fed into the new trajectory estimation algorithm. The analysis compared the measurement positions of the meteor with the trajectory fit, which are presented in Fig. 15 as O-C errors for both the along-track and cross-track directions.
The best fit was obtained for the position propagation model of Equation 4, the exponential deceleration model, which improved the alignment of the measurements to the trajectory’s fit positions at the very ends of the track. Note the consistent quality of the fit errors along the entire track except at the extreme start point where the meteor’s original measurement position was poorly estimated because it had low signal-to-noise ratio. Note also that the cross-track errors are lower than the along-track errors, which are related to the meteor image centroid measurements used rather than a leading edge position estimator. The solution produced a geocentric radiant and velocity of α = 39.7°, δ = 0.77°, and Vg = 25.0 km s−1. The orbital elements derived from that trajectory appeared to be similar to a weak minor shower, the Southern October Delta-Arietids (Jenniskens 2006), but as there is little observational evidence for that shower, the association could just as likely be a sporadic meteor. The point to be made, however, is that higher quality trajectories can be obtained using this algorithm to help validate major and minor meteoroid stream parameters. There are now more than 40,000 trajectories that have been computed during CAMS’s first year of operation, all using the multiparameter fit methodology.
The second case presented herein was obtained from a more distributed camera network operating in the southern European country of Croatia. Unlike CAMS, which concentrates many cameras at two to three sites, the Croatian Meteor Network (CMN) operates single cameras at over 30 disparate sites with sufficient density and volumetric sky coverage to usually have many more than two separated cameras capture any given meteor. The example shown is potentially an early Leonis Minorids meteor observed on October 8, 2011 from four separate camera sites. The computed geocentric parameters were α = 138.0°, δ = 48.2°, and Vg = 62.3 km s−1. The fit reproduced in Fig. 16 shows very good alignment to the measurements for nearly the entire length of all four camera tracks. Given that the resolution of the CMN sensors is twice as large relative to the CAMS system, the fit residual errors are proportional to the sensor resolution when compared to Fig. 15, thus verifying the earlier simulation results. Note that there is a bias in one of the cameras (diamond symbol) in the along-track residual. This was further analyzed and determined NOT to be due to timing offset estimation error, but instead, the meteor for that camera was at the edge of the field of view. As such, for the lens employed, the optical aberrations induced a measurement bias in right ascension which was mostly projected in the along-track direction and thus shows up as an O-C error bias in the plot. The derived timing offsets of the clocks were 1.297, 1.328, 1.293 seconds relative to the first camera’s time, thus 3 of the 4 cameras were actually fairly well synchronized. These examples demonstrate the ability to combine many cameras from multiple sites into a single trajectory solution.
Finally, how well does the trajectory estimator do during a major meteor shower in terms of tightness of the radiant produced? A subset of one night’s data from CAMS for the peak night of the Perseid meteor shower in 2011 was chosen for its statistical strength in numbers of shower members triangulated. To separate Perseid meteors from other showers and sporadics, the Southworth-Hawkins D-criteria (DSH) was employed (Southworth and Hawkins 1963). Each meteor was autonomously processed through the CAMS data reduction stream and produced not only a trajectory but also a multistation solution for the Keplerian orbital parameters. DSH was computed for all the meteors, given their five orbital elements, and the standard approach of setting a cutoff where the DSH histogram just merges with the average background count was made when binned in 0.02 steps. This yielded a maximum DSH for the Perseids of 0.22 and resulted in 274 meteors associated with the Perseid stream.
As shown in Fig. 17, the radiant positions of the Perseid-only meteors are plotted that were collected and processed with the CAMS system using the three methods described in this article. The spread in angle corresponds to the expected large radiant diameter for this shower, which masks some of the visual benefits seen of the multiparameter fit over the other two solutions. It should be noted that for the plot limits chosen, nearly 5% of the Perseids were outside the plot bounding box for the intersecting planes and 1% for the least squares approach. All 274 Perseids are within the given bounding box plot limits for the multiparameter fit (the plot limits were chosen to more readily see the detail in the central core of the radiant). Visually, the multiparameter fit method appears to have the tightest radiant spread. A more meaningful statistic is the standard deviation relative to the mean radiant position computed as a radial offset. For the intersecting planes method, the standard deviation came out to 3.0° due to a wider central spread and several large outliers. The least squares method was better with a standard deviation of 1.8°. Finally, the multiparameter fit resulted in the smallest and tightest radiant distribution with a value of 1.4°. Note that tossing out the 5% of Perseids outside the plot box for the intersecting planes method reduces its standard deviation to 2.0°. This still demonstrated at least a 30% improvement in radiant direction estimation with the multiparameter fit and a dramatically improved solution for many of the intersecting planes’ extreme outliers. Note that in all three methods, the mean positions were very close to the known radiant for the Perseids.
While the radiant is improved with the multiparameter fit method when estimating the spatial orientation of the trajectory, it is also worth examining the simultaneous solution of the entry velocity of the propagating meteor. Figure 18 is a histogram of the meteoroid atmospheric entry velocities converted into geocentric values as obtained from the 12 August 2011 Perseids. A mean value of Vg = 59.03 km s−1 was found with a standard deviation of 0.95 km s−1 using the multiparameter fit technique. The error measure is within the expected errors of the CAMS system and represents 2% of the measured speed. By way of comparison, this falls just above the accuracy of photographic meteor orbit surveys conducted by the Dutch Meteor Society (DMS), and the Harvard Super-Schmidt program (Jacchia et al. 1967), which had greater angular measurement precision, and is similar to or better than other video camera programs carried out by the DMS, Ondrejov, and the Sonotaco network in Japan (Jenniskens et al. 2011). Thus, the multiparameter fit technique yields better accuracy for both radiant direction and entry velocity when applied to video meteor measurements and could potentially improve results as well for photographic systems that may still be in operation.
There are several potential improvements that can be made to the existing software implementation of the multiparameter fitting algorithm. The first is to replace the cost function minimization algorithm with one that is more likely to converge toward the global solution. A promising approach is the simulated annealing minimization algorithm (Kirkpatrick et al. 1983; Cerny 1985) that finds a global optimum in a large parameter search space such as encountered with this algorithmic approach. Second, adding a measurement-by-measurement error estimate may provide more robust solutions by de-emphasizing measurements that have the worse error. This can occur for example when the meteor’s integrated counts per frame are close to the image background noise level, or when extremely bright meteors produce saturation or blooming and make it difficult to locate a centroid. Thus, it would be best to avoid the current practice used in the existing software implementation that applies an equal weight to all the meteor measurements. Instead, one should apply an error that is a function of the meteor’s direction of flight, signal-to-noise ratio, magnitude, and/or saturation in a given frame. To do so, however, would require reformulating the cost function to account for each measurement’s individual error. Deviation from a straight linear trajectory can also be built in to account for increasing zenith attraction from slower moving decelerated meteors of long duration. Finally, the current implementation assumes that the input quantities for the measurements are in right ascension and declination from any given site. Meteor researchers have typically used azimuth and elevation above the local horizon as the common inputs to trajectory solver programs, so the software could be set up to do the coordinate conversions internally rather than expect the user to make the necessary transformations prior to calling the estimator function.
A new trajectory estimation technique has been formulated and implemented, which involves a simultaneous spatial–temporal solution to a set of multicamera angle–angle–time measurements. It works by combining several disparate site cameras that can be mixed together with same site cameras, all of which may not be synchronized, and that works with a total camera count greater than or equal to two. Both Borovička’s least squares method and the multiparameter fit are well suited to this deployment scenario. An additional result uncovered during the simulation analysis was that all the trajectory estimation methods could handle meteors propagating parallel to the line connecting two stations, but only if the convergence angle were at least 20°.
An attractive feature of video cameras is that they run at a well-defined frame rate and thus the relative time step between measurements is assumed known rather than directly measured on a per-frame basis. This provides a good relative time base for the measurements. By including the temporal information with the spatial (angular) measurements, the new method effectively adds an extra degree of freedom to permit calculations in more difficult collection geometries. The multiparameter fit algorithm was found to provide improved trajectory determination accuracy in geometries with low convergence angle and/or smaller site separation as well as being beneficial at all-sky camera resolutions. The multiparameter fitting technique adds a new method of analysis in the meteor researcher’s tool kit during an era of proliferating deployment of video meteor camera systems of various resolutions. The software will be made available as a single self-contained C function file to interested users.
Acknowledgments— This work was sponsored through a grant by NASA’s Planetary Astronomy Program administered by the SETI Institute under the direction of Dr. Peter Jenniskens. The author thanks Dr. Jenniskens for considering the development of this algorithmic alternative as part of the CAMS project, as well as for the use of CAMS data for this study. The author also thanks Damir Segon of the CMN for also providing data sets used in this study and for taking on the challenge of presenting this material at the 2011 International Meteor Conference. The comments and suggestions by both the anonymous reviewer and Jiri Borovička were most helpful in improving the quality of this work and their time spent is greatly appreciated.