1 Introduction

Meteor analysis software can be separated onto two categories: software for auto-detection of meteors in real time, and post-detection software for determination of meteor parameters from double-station observations. One of the most widely used software of the first type is MetRec (Molau 1998). Algorithms for both meteor trajectory and orbital elements determination and photometry have been developed by practically each meteor group carrying out double-station meteor observations, for instance (Hawkes 2002; Hawkes et al. 1993) and (Koten and Borovicka 2001; Koten 2002). This paper provides a brief overview of new software named “Falling Star” which has been developed in our department over the past ten years. The main idea of the software development was to create a complex program allowing complete astrometric, kinematic and photometric processing of double-station meteor observations. A goal of the software was to create a program that would be independent of the observational system used.

2 Software Goals and Features

In order to achieve universality of the program we had to solve two fundamental problems: input of any digital format representing meteor images sequence, and using methods for astrometry and photometry applicable to wide range of modern video systems.

In order to avoid the limitation of a fixed file format we separate the software onto two applications: the program itself and the preliminary converter from file formats being used (in most cases AVI-files from video conversion, or bitmap data written directly from the detector to the hard drive) into TVS-file. Transformation of input data into TVS assumes the following: transfer of main file parameters (size, bits per pixel, number of frames, frame rate for standard PAL/SECAM or NTSC video or any other value if necessary, progressive or interlaced scanning type etc.) to the header of new file; extracting and transfer of intensity component for each pixel of the image if the frame-grabber doesn’t provide for generation of purely monochrome picture, significantly decreasing in such a way the file size; separation of full frames onto odd and even half-fields in the case of interlaced scanning; providing TVS file with necessary comments; creation of additional subheader including date and time (UT) of observations; geographical coordinates of the observational point (latitude, longitude, altitude); orientation of the camera in horizontal coordinate system (azimuth and zenith angles); approximate angular size of FOV for the lens being used. We plan to provide the final version of the program with a description of the inner format of TVS file, which will allow the user to design a converter for other video formats.

Universality in methods for measurements and data processing of meteor images implies their correctness in results independently on the observational system being progressive or interlaced scanning, linear or non-linear signal response, presence or absence of afterimage and blooming, etc. For instance, we use super-isocon TV systems being of interlaced type, significantly non-linear and having long afterimage for dynamical objects, but most of modern video systems are assumed to be linear and relatively free of afterimages. How this approach is realized we will describe below in corresponding sections.

3 Processing of Meteor Image Sequences

The program consists of a series of procedures, which are consecutively called at meteor processing. Each subroutine saves the results of its calculations in separate files, which can be analyzed later. Processing of a meteor can be carried out manually, when each subroutine is called by a user, or automatically, when the application calls all subroutines consecutively. Execution of each procedure is accompanied by visualization. The time increase is not significant due to visualization, and the user can check visually the correctness of automated processing.

All operations are carried out separately over two TVS-files representing odd end even half-fields, supposing we use a system with interlaced scanning. The results of calculations for each file can be averaged after, or one higher resolution file created similar to the case of progressive scan. We will use the term “frames” for both half-fields for interlaced systems and full frames for progressive ones.

3.1 Mathematical Operations with Frames

Averaging is an operation over a sequence of N ≤ 100 ÷ 120 frames (N given is for our TV system, and will vary with integration time, resolution and field of view). The goal of this averaging is to reduce the background noise \( \sim \sqrt N \) in order to improve the precision of astrometry and photometry of star images. Number of frames for such an operation depends on system noise, but should be limited to prevent star shape deformation due to stellar drift. We recommend to average frames centered around the meteor (neither earlier nor later) to avoid problems with drift in measured coordinates of reference stars. The resultant frames are used only for star images processing.

Subtraction of frames is used to create frames where only the meteor image is present (star images disappear) in a similar manner to Hawkes et al. (2001). In the simplest case one can subtract a single frame taken just before the meteor appearance from each frame with a meteor. In this case the resultant fluctuations increase \( \sim \sqrt 2 \), so we subtract the frame averaged on 40–50 frames before the meteor appearance.

The last main operation is the summing of frames where the meteor or its afterimage still exists. This frame is used for additional meteor measurements for radiant precise calculation and also can be used for photometry.

In addition to these basic operations there are many other mathematical and statistical procedures realized in the program: calculation of statistical distributions of intensities inside selected rectangular zone or entire frame; work with “photometer slit”; detaching and measuring of any photometric profiles etc.

3.2 Measurements of Stars and Meteor Image in the Frame

For measurements of rectangular coordinates of reference stars and their photometrical volumes the reductive method (Kozak 2001) or two-dimensional Gaussian fit can be used. We also recommend using the Gaussian fitting for the profiles on the meteor image. The position of a point corresponding to the meteor head is selected by a user (in current version of the program) using a collection of empirical rules. Supposing the meteor is a point object, and its PSF being of Gaussian type with known half-width determined from star images for the given lens, and considering the equations of charge accumulation and scanning we can generate the motion of PSF over the light detector (over the frame) with different meteor velocities and brightness. Additionally, we can change the brightness during the “meteor flight” to make the simulation more realistic. Using such empirical rules for processing of real meteors we can reach a precision of ±0.5 pixels, which corresponds to the precision of star image measurements using the reductive method. In order to raise the precision we should try to find an exact solution of the resultant integral equation being an improperly posed problem. More detailed description of such an approach will be given elsewhere.

In order for star identification we have chosen two star catalogues: Tycho ACT RC 1997 (I/246) and AS CC 2 (I/280A) containing stars up to 12m, which corresponds to our TV system sensitivity. The second one is preferable since it includes the spectral classes for some bright stars which are important for photometry. Taking data from the astronomical subheader of the TVS-file the program selects stars from the catalogue around the optical center, completely including the entire frame. The star map is plotting automatically using first the ideal projection of the sphere onto the plane and then adding necessary distortions. Identification of stars is realized automatically by means of comparison of their coordinates in observed and star map frames (Kozak 2001; Kozak et al. 2001).

3.3 Astrometry

The polynomial reduction models are used for the astrometric processing: linear polynomial, Deutsch 8 constants method (Deutsch 1965), full square polynomial and truncated cubic polynomial (12 constants), and can be found in (Kozak et al. 2001). The quality of each model was checked using stars of known position but assumed to be unknown objects. We evaluated the precision of astrometry in this way for variation of such parameters as reduction model, number of reference stars in some zone around the object, dimension of the zone, asymmetry of reference stars sample relatively the object, method for measurements of rectangular coordinates of the object and reference stars. The best reduction model for our system has been found to be linear one, and the reference stars zone being local. Precision of equatorial coordinates determination at pixel size of 4 arc min is 2–2.5 arc min, and decreases down to 1.5 arc min when Gaussian fitting of star shapes is used. The method of astrometric processing is presented in detail in (Kozak 2002).

3.4 Kinematics

Kinematics processing of a meteor is realized with the vector method developed by the author and described in (Kozak 2003). First the classic triangulation scheme is used, then, with the help of vector-matrix operations, we calculate all meteor trajectory parameters in the atmosphere proceed to a heliocentric coordinate system and compute the orbital elements of a meteor.

The input parameters for the kinematic processing are the equatorial coordinates of points on the meteor image calculated from both stations. In the express method we use a range of points (N ≈ 3 ÷ 20) corresponding to the meteor head position on the image and respective relative time moments (α; δ; t) i , \( i = \overline {1,N} \). Another approach which allows an increase in the precision of radiant determination, and therefore all other parameters, makes additional use of an array of points along the meteor trajectory obtained from the summed frame (α; δ) j , \( j = \overline {1,M} \) (the image is similar to photographic one) where number of points is much higher (M≈ 50 ÷ 350). The Fig. 1 demonstrates the results of application of the two approaches.

Fig. 1
figure 1

Probability distribution p of errors for equatorial coordinates of geocentric radiant calculation (a Leonid 2002 meteor): (a) right ascension α R (°); (b) declination δ R (°); (1) express method using array of meteor head image points, (2) calculation using array of all points along the meteor trail image from the summed frame

Full size image

3.5 Determination of Errors for Kinematical Parameters

Calculation of errors for all kinematical parameters of each individual meteor, using classic methods, is a very difficult problem due to a long chain of common non-linear transformations. Instead we use Monte-Carlo methods to statistically estimate probable errors in quantities. Application of regression analysis to the astrometric processing (Kozak 2002) provides us with mean values and standard deviations of equatorial coordinates for each kth measured point on the meteor image: \( (\bar \alpha ;\sigma _\alpha )_k ,(\bar \delta ;\sigma _\delta )_k \). The use of the astrometric test described above has shown near Gaussian error distribution, which can be completely described by these two parameters. Thus, we have a possibility to use as input parameters for kinematical processing not arrays of (α; δ) k as it was declared in section on kinematics but arrays of their statistical distributions with known parameters. Generating random values for each point in accordance with its distribution and sending them to kinematical processing procedure we will obtain respective random values for each investigated parameter (trajectory parameters, orbital elements, etc.). About 10–20 thousand steps are required to fully implement this and to obtain statistical distributions of errors for all calculated parameters. Then, having the statistical distribution for each parameter we can use not only its standard deviation as an error of the parameter, but also its average (or modal, median, if the distribution is asymmetrical) value as the physical value of the parameter itself. Detailed description of the method will be given elsewhere.

3.6 Photometry

The method for photometry has been developed using both theoretical and empirical techniques. The empirical approach consists in the use of results from well known experiment “artificial meteor”, which is realized by means of the camera rotation with different angular velocities. Images of stars of different magnitudes moving on the frame (photo-cathode or CCD surface) with different linear velocities draw meteor-like trails being of different intensities. In the simplest approach we can construct a table of measured trail parameters as a function of stellar magnitude and image motion velocity. These results can be used to determine meteor magnitudes from stellar reference. One can do the measurements in single frames with meteor/star image motion or in summed frame, similarly to (Hawkes et al. 2001). This method is partly described in (Kozak et al. 2001). The method of pixel integration (Hawkes et al. 2001; Koten 2002), being applicable to linear response detectors, can also be realized in the program in manual mode of processing.

The theoretical approach, which is currently in testing mode and not included in the release of the software, includes the creation of the model for TV system functioning at the phases of charge accumulation and scanning, modeling meteor PSF motion across photo-cathode or CCD, etc. Possible non-linear response onto input signal and afterimage existence are input as model parameters.

4 Software Application Examples and Prospects

The quality of processing results depends on parameters of the meteor (angular length, velocity, distance of the radiant from optical center etc.), and on observational system parameters (focal length being responsible for spatial resolution, noise level), and ultimately on methods used. For demonstration of influence of these factors we will consider the trajectory parameters for 28 Leonids from 2002 calculated by means of this software (Kozak et al. 2007). During the first hour of the observations the Leonids radiant had been drifting consecutively through view fields of both TV systems, which adversely effected the triangulation processing precision: standard deviations were near 1° (the highest value being 2°.58) for radiant coordinates and 0.6–1 km (maximal value being 4.35 km) for the beginning heights. The situation radically changed when the radiant had left the fields of view: radiant determination errors decreased down to 0°.1–0°.4 (the lowest value being 0°.06), altitude errors down to 0.1–0.3 km (the lowest one being 0.07 km).

We expect to complete the theoretical approach to photometry and to complete automation of all procedures in the near future. Since the software is still developing, the author will be thankful for any recommendations and notes concerning its improvement.