Antecedentes nacionales

The organization, realization, and reduction of photographic satellite observations with cameras is extremely time consuming, and requires much effort. Only some basic considerations, and the fundamental steps are explained in this chapter. For a full treatment of this subject see e.g Schmid (1977), or Seeber (1972). The basic principles and algorithms, in particular of the plate reduction process, are also valid for the reduction of CCD images [5.2].

Camera observation

Figure 5.3. Visibility conditions for passive satellites

An appropriate observation epoch has to be determined, based on predictions of the satellite orbits, and on a computa-tion of the satellite visibility for all par-ticipating stations. At least two stations have to observe simultaneously in order to contribute to the geometric solution [1.2] in satellite geodesy. The amount of valuable data increases with more than two participating stations. When pas-sive, sunlight reflecting, satellites are used (like AJISAI), the satellite must be outside of the Earth’s shadow (Fig. 5.3) and the Sun must be more than 18^◦ be-low the horizon of the observation sta-tion (astronomical darkness). The re-lated areas of satellite intervisibility are

a function of the geographical locations of the observation stations, and of the respec-tive orbital height.

Fig. 5.4 shows for three stations B1, B2, B3 the visibility areas (zenith angles z < 60^◦), together with the sub-satellite track (cf. Fig. 3.25, p. 127). It becomes evident that only a small portion of the satellite orbit (S_AtoS_E) is simultaneously visible from all three stations. The visibility conditions become more favorable with increasing orbital height and decreasing station separation. The best intersection conditions are

B3

Figure 5.4. Visibility areas for three ground stations

present, from geometrical considerations, when the distance between the ground sta-tions equals approximately the orbital height of the satellite used [5.1.4]. A successful ob-servation from at least two stations is called an event.

The above-mentioned conditions ex-plain the difficulties that arise with the real-ization of an observation project, in partic-ular because fine weather conditions must be present simultaneously at all participat-ing stations. These are the reasons for the

long duration (several years) of all large projects which have been conducted in the past.

Coordinate measurements and corrections

The exposed and developed plates or films (photograms) are measured on a comparator.

The results are the rectangular coordinatesx, y of the stars and of the satellite images, defined in the plane of the photographic plate; thus the analogous information of the

photogram is digitized. For well-defined images of stars and satellites the precision of the coordinate measurement is about±1 µm, corresponding to an angular resolution of about±0.^5.

Within the further reduction process the measured coordinates of the fiducial stars are compared with the star positions from an appropriate star catalog, representing the fundamental reference system [2.1.2]. One such classical catalog is the SAO Star Catalog, that was compiled for the purposes of satellite geodesy and has been used for the adjustment of the BC4 satellite network. Fore more information on star catalogs see Eichhorn (1974), and for modern developments Walter, Sovers (2000).

Before starting the plate reduction process the measured coordinates may be cor-rected for

− radial and tangential distortion,

− astronomical refraction,

− satellite refraction,

− satellite aberration, and

− satellite phase.

The correction for distortion is possible when the coefficientsK_i of a polynomial, which describes the distortion of the particular camera objective lens, are known. The effect of the astronomical refraction (z is the zenith distance, P : atmospheric pressure [HPa],t: temperature in centigrade)

,z = R = A tan z + B tan³z, is not uniform for the whole photogram when a wide field camera is used. It is not the absolute amount of the refraction influence that is important but the variation within the

ρ field of view. Formulas for this

differen-tial refraction can be taken from astro-metric literature (e.g. Seeber, 1972; Ko-valevsky, 1990). For small field obser-vations as in CCD astrometry [5.2], see e.g. Schildknecht (1994).

Stars are at infinite distance; the satellite, however, is often passing within the outer limit of the effective at-mosphere. The problem is schematically pictured in Fig. 5.5. The astronomical refraction ,z_∞ and the portion of the refraction,z which influences the light from the satellite, differ by the so-called satellite refractionσ (Schmid, 1977)

σ = ρs,z

d cos z (5.2)

where

ρ the geocentric distance of the observation station, s constant value = 0.00125, and

d observer – satellite distance.

A new discussion of satellite refraction, after the advent of CCD astrometry, is given by Bretterbauer (2001).

The correction for aberration reduces the observation epochs for all participating stations to a common epoch defined at the satellite. The phase correction reduces the reflected images of the sun to the center of the satellite. Explicit formulas can be taken from Schmid (1977). Note that the last two geometrical corrections are also important for laser ranging to satellites [8.4].

Plate reduction

Figure 5.6. Tangential coordinatesξ, η The photograph of the star field is

noth-ing else but the projection of the astro-nomical sphere into a plane. If we as-sume ideal conditions, i.e. a rigorous central perspective projection without distortion, refraction etc., we can com-pute plane tangential coordinates ξ, η from the equatorial star coordinatesα, δ with respect to a known camera orienta-tionα0, δ0 (Fig. 5.6). The ideal tangen-tial coordinatesξ, η differ from the mea-sured coordinatesx, y on the photogram only by random observation residuals v_x, v_y.

In practice such ideal conditions do not exist. Within the plate reduction process we try to find an adequate model for the relation between tangential star coordinatesξ, η and measured coordinatesx, y. Once the parameters of this model are identified, they can be used to transform the measured satellite coordinatesx_S, y_S via the tangential coordinatesξ_S, η_S into equatorial satellite directionsα_S, δ_S.

Usually the plate reduction models are subdivided into astrometric methods and photogrammetric methods, because they are based on developments from both fields.

The differences are, however, more in the formulation than in the results.

Within the astrometric plate reduction model the tangential coordinatesξ, η and the plate coordinatesx, y are related through polynomials. The tangential coordinates ξ, η are determined with the formulas of the gnomonic projection (Green, 1985; Smart, 1977). Following Fig. 5.6 we introduce a quantityq with

q = cot δ cos(α − α0), and obtain

ξ = tan(α − α0) cos q

cos(q − δ0) and η = tan(q − δ0). (5.3)

When the camera orientation is already well known we can use the simple linear relations:

v_x = Ax + By + C − (ξ − x),

v_y = A^x + B^y + C^− (η − y). (5.4) A more general form with quadratic terms, which allow for corrections in the camera orientation, is:

vx = Ax + By + C + Dx²+ Exy + Fy²− (ξ − x),

v_y = A^x + B^y + C^+ D^x²+ E^xy + F^y²− (η − y). (5.5) A, B, C, . . . are parameters in the adjustment process. In astrometry, the equations (5.4) and (5.5) are often named Turner’s formulas (e.g. Smart, 1977).

Within the photogrammetric plate reduction model an attempt is made to formulate analytically as many influences as possible within the functional model. The relations

y

Figure 5.7. Perspective projection and camera orientation

are based on the general formulas for the perspective projection. From Fig. 5.7, with the approximative values the tangential coordinates ξ, η of the stars can be expressed as

ξ = f1(t0, δ0, κ0, f0, x_H0, y_H0) and η = f2(t0, δ0, κ0, f0, x_H0, y_H0). (5.6) The basic model contains six parameters in the observation equations:

v_x = ∂ξ derivatives in (5.7) and the explicit formulas (5.6) can be taken from Schmid (1977).

Because of the non-linearity in (5.7), the adjustment process requires several iterations.

The model can be refined with additional parameters, e.g. for comparator biases, radial

and tangential distortion, or refraction biases. Equations with more than 20 parameters were used in the adjustment of the BC4 geometric world network (Schmid, 1977).

Not all individual satellite images on the photogram are required for further com-putations; the total information is condensed with a smoothing function into one or more selected points. Usually a polynomial inx and y is formulated as a function of time:

x^= a0+^k

i=1

a_itⁱ, y^= b0+^k

i=1

b_itⁱ. (5.8)

The orderk of the polynomial representation depends on the particular satellite orbit and on the camera type. For PAGEOS and ZEISS BMK the order 5 to 6 was appropriate.

The interpolated fictitious satellite points are converted into tangential coordinates and then into topocentric equatorial directions α_S, δ_S to the satellite using the adjusted model parameters (5.7) and the reciprocal formulas of (5.6).

The accuracy of a single direction in the photogram was found to be±1.^6, and for the smoothed central direction±0.^35, based on analysis of more than 1000 ob-servations in the BC4 worldwide network. With improved star catalogs, improved comparator techniques, and careful analysis of all existing error sources, the accuracy of adjusted directions to satellites based on photographic observations may be in the order of±0.^1 to±0.^2. This corresponds to a position accuracy of±0.7 m to ±1.5 m for a satellite at 1500 km altitude (e.g. AJISAI).