Handling Data Processing

Up to this point we have considered how the data transmitted between an object and the measurement device is modeled, and how this is used to construct the measure- ment reduction function. Now we turn to consider the matter of the generation of the observational data itself. Any substantive issues concerning the measurement of time are obviated at the level of precision we are concerned with for NEO modeling due to the availability of atomic clocks, which ensures that measured time is effectively dynamical time, i.e., time in post-newtonian mechanics. Any remaining issues to do with treatment of time are effectively changes in coordinate system for time. Thus, we will just focus on the measurement of location on the celestial sphere.

While we continue to think of data in terms of transmission of information, we are now concerned with the transmission of information through the measurement device to generate the raw data, which is then processed into a data model. At this stage we are comfortably within a standard picture of the hierarchy of models at the level of models of data. The feature of this part of the process that is of the greatest epistemological interest in this case is the manner in which measurement error is handled. Rather than being handled purely in terms of some statistical analysis, measurement is handled in terms of an error model, which can be solved in a very similar manner to that in which the orbit determination problem is solved, as we shall see.

Before considering this case, let us consider briefly the matter of transmission of information through the measurement device. Since we are considering optical mea- surements, the measurement devices of interest are optical telescopes. Consequently, the incoming light is processed by a system of refracting lenses and/or reflecting mir- rors and is projected onto an image plane where some kind of photographic data is collected. In order to calculate an observed location on the celestial sphere for in- cident light, it is necessary to know how the telescope processes the incident light before it reaches the image plane. At the level of precision we are concerned with the light dealt with in the measurement process effectively constitutes rays undergoing rectilinear propagation. Thus, geometric optics is used for all of the calculations of

how the telescope processes the light. In order to know how the telescope processes the rays of light, it is therefore necessary to construct a geometric optics model of the lens/mirror system of the telescope.

A fully detailed model of the imaging system would require a quite complex optical model together with a model of the rest of the telescope, including the mounting, drive system, camera and any other objects that are effectively coupled to the telescope. Simply, a fully detailed model of the imaging system would require a complete model of the device. In the interests of physics avoidance, it is desirable to ignore the full detail and focus simply on a simple model of the optical processing. And provided that the telescope has been well designed and is functioning properly, this is all that is required. Thus, a central design aim for a telescope is to create a device that will collect data that can be analyzed using only a simple optical model. We will consider briefly this design phase in the next subsection. Thus, we will work with the assumption that the effect of the device on the light travelling through it is well below the level of precision ν0 that has been identified for the modeling task.

In this case, we are considering a charge coupled device (CCD), which is used to take an image from the image plane of the telescope. CCD are useful in optical astrometry because they have a high quantum efficiency, i.e., they register a high percentage of incident photons, and they have a fixed geometry that is easily interfaced with computers. After an image of the sky is taken, it is necessary to reduce this raw image data to a location on the celestial sphere. Ideally, we could align the optical axis of the telescope with a known point on the celestial sphere and a known pixel on the CCD, which was made to lie precisely perpendicular to the optical axis in the image plane. These conditions are not feasible, however, so it is necessary to have a reliable and very precise means of correcting for error in positioning and in the imaging process. This is accomplished by the aforementioned ideal model of the geometric optics of the imaging system, together with a model of the relationship between locations on the photographic image and locations on the celestial sphere. In fact, the need to explicitly consider the optical model of the imaging system can be eliminated through the use of numerical methods.

In general, the precise details of how image reduction is done vary depending on the structure of the imaging system of the telescope and the method of image re-

duction, and need not concern us here. What is of interest is the procedure used to establish the effectively correct relationship between locations on the photographic image and locations on the celestial sphere. Effectively the way that this is handled is by modeling the combined error in the imaging system and in the positioning of the camera as an affine transformation (i.e., a linear transformation together with a translation) of a pair of cartesian coordinates of a point on the photographic image. This reduces the task of analyzing the error in the computed celestial location to cal- culating the required affine transformation needed to correct for the error, followed by the propagation of error from photographic measurements to the computed celestial location.

The affine transformation of the error model is specified by six numbers, so at least six compatible constraints are required to solve the error model to find these six numbers (for details on the following reduction process see (Green, 1985)). In astrometry, this is achieved by ensuring that the image includes at least four known stars that have available ephemerides and that are, ideally, widely separated. With four stars appearing in the image, their location in the image coordinates can be computed (by computer for a CCD image), which gives a system of eight or more constraints, rendering the error model problem overdetermined and a least squares solution to be sought. The more stars that are included in this calculation, the more accurate the solution of the error model. Once the parameters of the error model have been computed, the celestial location of any point on the image can be calculated. In a very similar way to the least squares method in the orbit determination problem, the solution is found by minimizing a sum of squares of measurement residuals. These residuals can then be used to estimate the measurement error for the location of the NEO.

To briefly clarify how this is accomplished, consider that the celestial sphere is curved and the image plane flat. The locations of the known stars obtained from ephemerides can be projected onto a tangent plane to a point on the celestial sphere. The tangent plane and the (ideal) image plane will then be related by a simple linear transformation. The locations of the stars in this tangent plane are used as constraints on the error model, since they can be related to measured locations of the corresponding stars on the photographic image. Using numerical methods, a

vector method can be used to model the required affine transformation for the error model, allowing the location of the NEO in the tangent plane to be computed and the measurement error to be estimated. These two quantities are then converted into the location of the NEO on the celestial sphere and the measurement error there using the inverse of the projection onto the tangent plane. Thus, in this way, what could be an enormously complex process of error analysis and statistics is reduced to the solution of an overdetermined six-parameter model and simple error propagation. And outside of the estimation of the measurement error on the basis of the measurement residuals, no statistical analysis is required.