Earth curvature and other additional corrections

Background

Earth curvature corrections are applied to minimize distortions resulting from the usage of a non-cartesian coordinate systems, such as a tangential coordinate system like UTM. In many Aerotriangulation solutions, a shift is estimated in image space to correct this effect. This allows the use of ground control points in UTM coordinates without transferring them to a local Cartesian coordinate system. The correction equations in image space were designed for Nadir images with relatively flat terrain and traditional lower ground resolution. The common equations should not be applied during Aerotriangulation for higher resolution datasets, undulating terrain like mountains and cities, or oblique imagery.

Besides that, earth curvature correction models are traditionally carried out in image space as a correction on the image. Some software solutions use depth depending terms (the actual required way) however, the definition of these terms vary. Other solutions only use a radial correction which creates issues in non-flat terrain, high resolution, or oblique data. Using an image-space correction term was traditionally useful with mechanic devices or limited processing power, as only individual point measurements have to be updated - and no coordinate transfer is technically required. However, for current 3D datasets with undulating terrain or varying ground resolution, it is more reasonable to avoid an adjustment in a projected coordinate system or with different axis scale and rather project the 2D products derived. 

Impact on SURE

Thus, SURE relies on a Cartesian coordinate system, where the classical pinhole camera model maps from a Cartesian global coordinate system to a Cartesian camera coordinate system. Additionally, various camera distortion models are supported. When using SURE, it should always be ensured that a consistent camera model definition is applied. Please make sure that the input data is derived without earth curvature compensation and other additional corrections, in order to generate correct data. Otherwise, shifts in the data of up to multiple decimeters can occur, particularly in undulating terrain.

If the Aerotriangulation solution has been generated with earth-curvature or other corrections, please refer to the following section explaining the available options in order to use the orientation data as input into SURE.

AT Processed with Earth-Curvature Corrections: Solutions

Recommendation

Solution

Simplicity

Accuracy

1

The simpler way. This is a viable option in case AT is already delivered

Reprocess Bundle Adjustment without earth-curvature corrections

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2

For highest accuracy requirements

Reprocess AT using a Cartesian Coordinate System

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Solution 1: Reprocess Bundle Adjustment step of AT without Earth-curvature corrections

If an Aerotriangulation result has been generated with such corrections, you can re-perform the bundle adjustment step with this corrections disabled. This way, just the adjustment is repeated but the tie point collection does not have to be repeated.

The remaining effects resulting from using non-cartesian GPS orientations or ground control points will be shifted to the exterior orientation – leading to a consistent result when using SURE.

This solution satisfies typical accuracy requirements and is thus useful when Solution 2 is either too complicated or an AT is already given.

Solution 2: Reprocess AT using a Cartesian coordinate system

In order to get an optimal result with full consistency, a cartesian coordinate system should be used through all processes including the following steps:

  1. Choose a Cartesian coordinate system (recommended: local tangential coordinate system approximating the target system if 2.5D products such as a DSM or True Orthophoto are desired. Geocentric coordinates EPSG:4978 for 3D products only (point clouds, meshes etc.), which is supported in all SURE functions that perform a coordinate system transformation and works with Cesium)

  2. Transfer all GPS/INS as well as all gound control point measurements to this coordinate system

  3. Set up the Aerotrianguation specifying the used coordinate system

  4. Execute Aerotriangulation

  5. Pass results of Aerotriangulation to SURE

  6. If local coordinate system is close to desired result (e.g. an approximating tangential coordinate system), it can be used right away

Additional remarks and recommendations

  • If the bundle adjustment was carried out without such corrections, but the Inpho/Trimble Match-AT project file is created with having the respective check boxes being activated (e.g. when importing from BINGO or other software), please edit the first lines of the Inpho/Trimble Match-AT project file with these options by using a text editor ($REFRACT_CORR_DEFAULT : off | $CURV_CORR_DEFAULT : off). If the adjustment was done with corrections, it has to be repeated firstly without corrections.

  • General note: the earth curvature shift correction is depth depending. Consequently, no camera-constant image correction methods such as distortion grids or radial distortion parameters should be used to compensate effects – in particular in undulating terrain and for oblique imagery. 

  • By specifying an EPSG coordinate system (e.g. in the Inpho Match-AT project file), the coordinate system Information WKT will be transferred to point clouds, orthos and DSMs. This is also required for SURE functions that require a coordinate system transformation, e.g. SLPK and Cesium export.

  • If an optimal transfer to a projected coordinate system like UTM is desired for the DSM, you have the option to transform the raw point cloud (3D_Points) with tools like lastools before executing a custom DSM step using the 2.5D tool or the command line. Please note, that for the True Orthophoto also the orientations need to be in the same coordinate system to yield a correct result.

  • Using ellipsoidic heights generally simplifies transformation steps to the tangential coordinate system and for compatibility with subsequent software packages. This is also required for SURE functions that require a coordinate system transformation, e.g. SLPK and Cesium export.