**4. Data processing methods**

### **4.1 Photon geolocation**

The basic principle of satellite-based laser measurement and geometric positioning is that the laser beam is transmitted by the satellite and received by the satellite after reflection from the ground, and the time interval between the laser transmission and

reception *t* is calculated. The propagation speed of light is*c*, and the one-way transmission distance of the laser is *ρ* ¼ *ct=*2, and the three-dimensional coordinates of the laser footprint can be obtained by combining the satellite position and attitude information obtained from the GNSS positioning instrument and the star sensor on board the satellite [44]. The rigorous geometric model is shown in **Figure 7**, where *P*laser is the reference point of laser emission, *P*GNSS is the GNSS phase center, *O*Body is the satellite center of mass, and *P*Ground is the laser ground footprint (Bounce Point Location).

Laser footprints were precisely geolocated and point clouds were generated using onorbit calibrated laser ranging parameters, laser pointing parameters, and precision attitude/orbit data, taking into account geophysical corrections such as atmospheric delays and tides. The rigorous geometric location equation for satellite-based LiDAR is as follows:

$$
\begin{pmatrix} \mathbf{X}\_{\text{spot}} \\ \mathbf{Y}\_{\text{spot}} \\ \mathbf{Z}\_{\text{spot}} \end{pmatrix}\_{\text{ITRF}} = \begin{pmatrix} \mathbf{X}\_{\text{s}} \\ \mathbf{Y}\_{\text{s}} \\ \mathbf{Z}\_{\text{r}} \end{pmatrix}\_{\text{ITRF}} + R\_{\text{ICRF}}^{\text{ITRF}} \mathbf{R}\_{\text{BOD}}^{\text{ICRF}} \begin{bmatrix} \boldsymbol{\Delta\mathbf{X}\_{\text{ref}}} \\ \boldsymbol{\Delta\mathbf{Y}\_{\text{ref}}} \\ \boldsymbol{\Delta\mathbf{Z}\_{\text{ref}}} \end{bmatrix} + \rho \begin{pmatrix} \sin\left(\boldsymbol{\theta} + \boldsymbol{\Delta\theta}\_{\text{1},i}\right)\cos\boldsymbol{a} \\ \sin\left(\boldsymbol{\theta} + \boldsymbol{\Delta\theta}\_{\text{1},i}\right)\sin\boldsymbol{a} \\ \cos\left(\boldsymbol{\theta} + \boldsymbol{\Delta\theta}\_{\text{1},i}\right) \end{pmatrix}. \tag{1}
$$

where ð Þ *Xs Ys Zs <sup>T</sup>* ITRF is the coordinate of the satellite in the ITRF coordinate system, determined by the precision orbiting system; ð Þ *ΔX*ref *ΔY*ref *ΔZ*ref *<sup>T</sup>* is the fixed offset from the laser emission reference point to the phase center of the GNSS antenna; *θ* is the laser exit axis pointing angle, i.e., the angle between the projection of the laser emission direction and the XOY plane of this system and the *Z*-axis negative direction,

### *Spaceborne LiDAR Surveying and Mapping DOI: http://dx.doi.org/10.5772/intechopen.108177*

*α* is the angle between the projection of the laser in the XOY plane of this coordinate system and the *X*-axis positive direction; *ρ* is the laser range value, corrected by the calibration system error and atmospheric delay correction.

Based on the satellite-based LiDAR measurement of the rigorous geometric model, the positioning and elevation calculation of the laser footprint is achieved by the following process:


Taking ICESat-2 data as an instance, the nominal 6.5 m planimetric positioning accuracy and better than 1 m elevation accuracy were achieved after processing [45]. As shown in **Figure 8**, the evaluation results compared with the airborne DSM data by the iterative least z-difference method in Hanzhong region showed that the horizontal biases of the test data were �0.1 m (east) and �4.1 m (north), and the elevation bias was 0.6 m, which reached the nominal accuracy specifications.
