40 BALLISTIC PHOTOGRAMMETRY, SCHMID
enter the photogrammetric measuring procedure, but which are not compatible with the
concept of the condition of collinearity.
In this respect, two distinct areas of influence must be considered. Obviously, one
source of disturbance is the atmosphere bending the line of sight between the center of
projection and the object. Therefore, refraction corrections become necessary in all photo-
grammetric precision measurements, with the exception of special applications, when the
influence of refraction is either negligible, owing to the limited overall dimensions, as
e.g. in work shop measurements or where there is no atmosphere, a condition which may
be encountered, e.g. when mapping the moon from photographs taken outside of the at-
mosphere. In general, it is possible to apply the necessary corrections to the correspond-
ing object coordinates, which are either known as given parameters or which are com-
puted during the computational procedure.
The other source of disturbance on the condition of collinearity is caused by the
shortcomings of the physical camera and the associated photographic process. The dis-
placement of the image on the plate from its ideal position is caused by quite a few per-
turbations, the major source being what is conventionally understood as radial distortion.
But a host of secondary perturbations, such as coma and chromatic aberrations, and
probably more disturbing, the so-called tangential distortion caused by manufacturing
flaws, combine with the influence of lack of flatness of the emulsion and its dimensional
instability.
Either during the process of performing an independent camera calibration or while
evaluating a specific triangulation problem, some of these perturbations can be deter-
mined quantitatively by merely adding corresponding algebraic expressions to the afore-
mentioned basic formulas. In Ballistic Photogrammetry, it is common practice to deter-
mine in this way the elements of interior orientation, including radial distortion. Obvi-
ously, the influence of perturbations which act consistently can be eliminated, once they
are determined, by corresponding corrections on the measured plate coordinates. Finally,
the residuals as obtained with a rigorous least squares solution will give evidence of any
unresolved biases.
Thus, a potential evolves which can support one of the most important efforts in
photogrammetry, concerned with the testing and analyzing of basic components in the
photogrammetric procedure.
Furthermore, the simplicity of the basic geometrical model leads directly to the deri-
vation of significant results in regard to error theoretical studies.
The rigorous requirement in Ballistic Photogrammetry for geometrical significance
of the result of triangulation as well as for the orientation parameters, requires addi-
tional computation of corresponding expressions for precision. In other words, the most
probable values of the unknowns must be computed together with the corresponding mean
errors. Thus, the problem to be studied is that which is associated with the propagation
of the uncertainty of the image positions on the individual photographs into the con-
figuration of the spatial triangulation.
In principle, the problem is the one which in classical geodesy is solved with the
adjustment of each precision triangulation net work. In a rigorous least squares adjust-
ment, the mean errors of the unknowns are computed by multiplying the mean square
error of an observation of unit weight with the corresponding propagation factors, which
are obtained from the square roots of the corresponding diagonal terms in the matrix of
the inverted normal equation system. Admitting the necessity of somewhat intricate
manipulations in establishing the corresponding computing procedures, these propoga-
tion factors can be obtained even for complex photogrammetric triangulation schemes,
as e.g., strip and block triangulations, by applying available computing programs already
in use, on various high speed computers. The significance of these results is obviously
not limited to the determination of the precision with which the results of a specific trian-
