An Empirical Computational Model For The Correction Of Refraction Errors In Photogrammetric Restitution

ABSTRACT

Photogrammetric Refraction is one of the major causes of systematic errors affecting the accuracy of maps compiled from aerial images. It has to be eliminated so as to attain the goal of the best results that the technology of photogrammetric mapping is capable of. Many researchers have devoted substantial efforts to develop a number of analytical formulas to remove the effect of refraction. However, these formulas suffer from certain limitations, such as: (i) inconsistent values for the refraction coefficient (K) for the same flying height (H) and (ii) non-availability of the value for the refraction coefficient for a desired flying height because the formulas do not cover the particular flying heights of interest. These limitations become challenges for photogrammetrists in deciding which of the formulas to use for various projects. This research approached the solution of the problem in an innovative way by using the numerical approach to derive an Empirical Computational Model (ECM) for solving the problem. This new model will provide consistent values for the refraction coefficient (K) for all values of the flying height (H) from zero (0) to twenty-five (25) kilometres. Three different data sets were used for the Model Derivation. The “best fitting model” was chosen by subjecting all the equations obtained during the derivation process to three different criteria which were satisfied simultaneously by the chosen model. The accepted Derived Model was a third order polynomial function. The Derived Model was then validated by using six different data sets. The computations were carried out on Pentium IV Personal Computer using Computer Programs written in MATLAB, for this research. The Hypothesis Testing of the ECM was carried out and it was concluded that the individual parameters of the Model made statistically significant contributions at the 5 % significance level.The Derived Model was thereafter applied to compute the radial image displacement (∆r) caused on the film by Photogrammetric Refraction on an Aerial Photograph of Tundunwada Area of Kaduna Township, with flying height of 1.52 km. The minimum and maximum effects of ∆r were 0.51 µm and 3.47 µm respectively. The larger the value of r (the radial distance from the nadir point to the actual image point), the larger the effect of ∆r. The results obtained from the Derived Model were better than those from the existing models. The effect of the radial image displacement (Δr) on ground coordinate values was also computed for the same area. The minimum effect of ∆r on the ground coordinate values was zero near the nadir point and the maximum effect was 3.61 cm at a point near the corner of the model. The model was applied to two more areas with flying heights of 0.80 and 4.80 km respectively. The minimum effect of ∆r was 0.04 and 0.329 µm and the maximum effect was 0.69 and 10.563 µm for the two flying heights respectively on the film. In respect of the effect of ∆r on the ground coordinates, the minimum effect was zero near the nadir points and the maximum effect was 0.80 and 28.28 cm for the two flying heights respectively at the corners of the model. The higher the flying height (H), the larger the effect of the radial image displacement (∆r) on both the film and ground coordinates. These radial image displacement (∆r) errors must be corrected for in order to obtain the best possible accuracy in a photogrammetric project.