The computation of a rigorous math model is often referred to as a bundle adjustment. The math-model solution calculates the position and orientation of the sensor—the aerial camera or satellite—at the time when the image was taken. After the position and orientation of the sensor is identified, it can be used to accurately account for known distortions in the image.
With the aerial-photography math model, the geometry of the camera is described by six independent parameters, called the elements of exterior orientation. The three-dimensional coordinates, x, y, and z of the exposure station in a ground-coordinate system, identify the space position of the aerial camera. The z-coordinate is the flying height above the datum, not above the ground. The angular orientation of the camera is described by three rotation angles: omega, phi, and kappa.
In the radar or optical satellite math model, the position and orientation of the satellite is described by a combination of several variables of the viewing geometry reflecting the effects due to the platform position, velocity, sensor orientation, integration time, and field of view.
During the math-model calculation, ground control points (GCP) and tie points (TP) are combined with the knowledge of the rigorous geometry of the sensor to simultaneously calculate the best fit for all images in the project.
With aerial-photography projects, the model calculation can only be performed with a minimum number of GCPs and TPs. When you are using data from a Global Positioning System (GPS) with or without inertial navigation system (INS) data, the math-model calculation can be performed without GCPs and TPs. Due to the ephemeris data, the math-model calculation for satellite orbital projects is performed with or without GCPs or TPs.
You can add GCPs and TPs to refine the math-model solution. Not all the GCPs in your project will have the same reliability. When the math-model calculation is performed, the GCPs, TPs, GPS data, and INS data are automatically weighted inversely to their estimated error. The most accurate GCPs or TPs affect the solution the most, and the least reliable affect the solution the least. Using many GCPs and TPs provides redundancy in the observations, so a few bad points do not greatly affect your model and bad points are easier to identify.
After the sensor orientation is calculated, it is used to drive all the other processes, such as digital elevation model (DEM) extraction, editing in three-dimensional stereo, and orthorectification. You must obtain an accurate math-model solution before continuing with other processes.
© PCI Geomatics Enterprises, Inc.®, 2026. All rights reserved.