## Earth curvature and other additional corrections

## Background

Earth curvature corrections are applied to minimize distortions resulting from the usage of a non-cartesian coordinate system, 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 to use 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. For higher resolution datasets, undulating terrain like mountains or cities and oblique imagery the common equations should not be applied during Aerotriangulation.

Besides that, earth curvature correction models are due to traditional reasons often carried out in image space as a correction on the image. Some software solutions use depth depending terms (the actual required way) however with varying definitions, others solutions even 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 nowadays 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 global coordinate cartesian coordinate system to a Cartesian camera coordinate system. Additionally, various distortion models are supported. When using SURE, it should always be ensured that a consistent camera model definition is in use. Please make sure, that the input data should be derived without earth curvature compensation and other additional corrections, in order to get correct data.

Otherwise, shifts in the data of up to multiple decimeters can occur in particular in undulating terrain. When having an Aerotriangulation result to be used with such corrections, please re-perform the bundle adjustment step without the corrections. Remaining effects resulting from the usage of non-cartesian GPS orientations or ground control points will be shifted to the exterior orientation – leading to a consistent result when using SURE. As shown in the following alternative, a fully consistent result can be achieved by carrying out the Aerotriangulation using Cartesian coordinates only from the beginning.

## Solutions

Solution | Simplicity | Accuracy | Recommended for | |
---|---|---|---|---|

1 | Exterior Orientation | +++ | ++ | if projected AT is delivered already or to keep it simple |

2 | Cartesian System | + | +++ | for highest accuracy requirements |

### Solution 1: Exterior Orientation

This option just deactivates the correction terms while still using a projected coordinate system.

Through this, the effects from the earth curvature will be sufficiently approximated as they are pushed/contained in the exterior orientation.

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

If an *AT is already given* with corrections activated, we recommend to just re-run the Adjustment without the corrections and feed the result to SURE.

This way, just the adjustment is repeated but the tie point collection does not have to be repeated.

### Solution 2: Cartesian 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:

- Choose 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 or 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) - Transfer all GPS/INS as well as all gound control point measurements to this coordinate system
- Set up the Aerotrianguation specifying the used coordinate system
- Execute Aerotriangulation
- Pass results of Aerotriangulation to SURE
- 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 softwares), 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. Cesium export.

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. Cesium export.