# A Direct Georeferencing Method for Terrestrial Laser Scanning Using GNSS Data and the Vertical Deflection from Global Earth Gravity Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Laser Scanner Geocentric Orientation Parameters

- X
_{0}, Y_{0}, Z_{0}geocentric GRS80 coordinates of the origin P of the laser scanner’s measurement frame (x, y, z), - Σ, ξ, η orientation angles of the measuring frame (x, y, z) with respect to the external reference frame (X, Y, Z).

_{0}, Y

_{0}, Z

_{0}, ξ, η) can be obtained by GNSS measurements providing the coordinates of the origin (X

_{0}, Y

_{0}, Z

_{0}) and the computation of two orientation angles (ξ, η) on a basis of the global Earth gravity model [23]. The sixth parameter Σ can be obtained by solving Equation (2) for given laser scanner measurements (s, α, β, i, j) and GNSS measurements of the coordinates of one georeference point Q(X, Y, Z) (see Figure 1).

_{0}, Y

_{0}, Z

_{0}, ξ, η, Σ are used. Prior to the transformation, the orientation parameters can be adjusted and thus also improved by the least squares method which provides the expected values of estimated parameters as well as standard deviations of estimated parameters and adjusted observations.

#### 2.2. Adjustment of the Laser Scanner Geocentric Orientation

_{0}, Y

_{0}, Z

_{0}, ξ, η, Σ) computed on the basis of the laser scanner measurements in the local frame (x, y, z), the GNSS measurements in the global frame (X

_{0}, Y

_{0}, Z

_{0}, X, Y, Z) and EGM2008 vertical deflection data (ξ, η) can be integrated and improved using the common least squares adjustment including the a priori covariance information of adjusted parameters and observations.

_{ξ}, v

_{η}—random errors of the deflection of the vertical components ξ, η, dΣ—correction of the approximated value of the horizontal orientation angle Σ.

**v**and the covariance matrix of adjusted observations

**l**

_{a}can be expressed as:

## 3. Results of the Field Experiment

_{0}, Y

_{0}, Z

_{0}) denotes the laser scanner reference point position measured by a GNSS receiver; Q(X, Y, Z) is the georeferencing point measured both by the laser scanner and the GNSS receiver; EGM gravity denotes the direction of the plumb line from the global Earth gravity model EGM2008, assuming that the plumb line direction coincides with the vertical laser scanner rotation axis. Points 1, 2, 3, 4, 5 and 6 are remote testing points extracted from the point cloud and measured by the GNSS receiver, as well. The testing points 1 and 2 are located on the building’s roof, 20 m over the scanner’s position. The remaining points are located on the ground. In order to ensure unambiguous identification of the testing points in the point cloud, the identification targets were placed on both the roof points and on the ground points while scanning. The typical vendor’s targets were used as the identification targets and also scanner manufacturer software (Leica Cyclone) was used in order to determine the target center.

**Laser scanner data**: The point cloud coordinates {x, y, z} of the testing points 1, 2, 3, 4, 5, 6 and of the georeferencing point Q are measured in the local reference frame of the laser scanner at the position P using laser scanner Leica P20. The heights of the laser scanner (i) and reflector (j) are also the measured quantities (see Table 1). The standard deviations, provided in Table 1, are the a priori values used for parameter estimation. These values were set up according to the Leica P20’s technical specifications.**GNSS data**: The X, Y, Z coordinates of the laser scanner position P and the georeferencing point Q, as well as the testing points on the roof 1, 2 and in the ground 3, 4, 5, 6 measured by GNSS (see Table 2). The GNSS coordinates of the testing points 1, 2, 3, 4, 5, 6 are not included in the least squares adjustment of the laser scanner position, as they are only used for the assessment of the laser scanner geocentric georeferencing accuracy. For the GNSS measurements, the GNSS Leica Viva GS08plus Smart Antenna receiver was used. The real-time kinematic (RTK) GNSS measurements were carried out using corrections from a single base station WROC (permanent station located in Wroclaw, Poland), provided by the ASG-EUPOS reference GNSS system in Poland [26]. WROC belongs to the global network of the International GNSS Service as well (IGS, [27]). The distance of the mobile GNSS receiver from the reference station WROC was shorter than 100 m. Each point was measured three times in 2 min sessions. The values provided in Table 2 are the mean values.**Global gravity field model data**: We used the northern ξ and the eastern η components of the vertical axis deflection of the laser scanner with respect to the normal to the GRS80 ellipsoid, which are calculated at the point P on the basis of the EGM2008 gravity field model. These values are calculated according to equations (7) and (8) where $\phi $ and $\lambda $ are known from the GNSS positioning and$$\Phi =\mathrm{arctan}((-\partial W/\partial Z)/\sqrt{{(\partial W/\partial X)}^{2}+{(\partial W/\partial Y)}^{2}},$$$$\Lambda =\mathrm{arctan}((\partial W/\partial Y)/(\partial W/\partial X).$$

_{ξ}= σ

_{η}= 1 arcsec. The EGM2008 gravity has been selected as one of the most highly accurate global gravity field models, which is based on a combination of satellite laser ranging data for the longest wavelengths of the gravity field, the Gravity Recovery and Climate Experiment (GRACE) data for the long and mid wavelengths of the gravity field, as well as from the altimetry and terrestrial measurements for the local gravity. Pavlis et al. [16] show that EGM2008 provides very accurate gravity anomaly, disturbance, and vertical deflection data from global to local scales and the model is particularly well-suited for the area of Central Europe with the RMS of differences to the local geoid, e.g., in Germany at a level of 3.0–3.5 cm.

_{0}, Y

_{0}, Z

_{0}) and the georeferencing point Q(X, Y, Z), using the Levenberg–Marquardt method of conjugate gradients. The obtained adjusted corrections ${v}_{x}$ ${v}_{y}$ ${v}_{z}$ ${v}_{{X}_{0}}$ ${v}_{{Y}_{0}}$ ${v}_{{Z}_{0}}$ ${v}_{X}$ ${v}_{Y}$ ${v}_{Z}$ ${v}_{\xi}$ ${v}_{\eta}$ of the parameters x, y, z, X

_{0}, Y

_{0}, Z

_{0}, X, Y, Z, ξ, η are acceptable when comparing to the standard deviations thereof, since the condition $\left|v\right|\le 2{\sigma}_{v}$ is fulfilled for all cases.

## 4. Discussion

## 5. Conclusions

- -
- minimum number of GNSS measurements; only two reference points have to be positioned by GNSS,
- -
- determination of vertical deflections component ξ and η as the scanner orientation parameters; ξ and η can be easily interpolated from a global vertical deflection model based on EGM2008.

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Measurement errors due to neglecting the vertical deflection in georeferencing for selected values of the vertical deflection θ = (ξ

^{2}+ η

^{2})

^{0.5}in lowlands θ = 7.5” and mountainous areas θ = 50” and different distances s as a function of the elevation angle. Errors are calculated using Equation (1) and the error propagation law assuming the error of distance measurements m

_{s}= 2 mm.

Measured Values | |||||
---|---|---|---|---|---|

x (m) | y (m) | z (m) | i (m) | j (m) | |

Georefer. point Q | −13.480 | 3.881 | −0.076 | 1.843 | 1.763 |

Testing point: 1 | 18.612 | −8.379 | 19.041 | 1.843 | 2.065 |

2 | 11.934 | −22.744 | 19.045 | 1.843 | 1.563 |

3 | −12.620 | −10.105 | 0.086 | 1.843 | 1.862 |

4 | 6.315 | −9.557 | −0.014 | 1.843 | 1.567 |

5 | −8.502 | 30.867 | 0.117 | 1.843 | 1.767 |

6 | 34.530 | −8.366 | 0.206 | 1.843 | 1.862 |

Std. deviation | 0.005 | 0.005 | 0.005 | 0.002 | 0.002 |

**Table 2.**The results of the global navigation satellite system (GNSS) field measurements in the global reference frame GRS80 in (m).

Measured Values | ||||||
---|---|---|---|---|---|---|

X | St. Dev. | Y | St. Dev. | Z | St. Dev. | |

Laser scanner point P | 3835659.499 | 0.008 | 1177290.998 | 0.008 | 4941636.307 | 0.008 |

Georefer. point Q | 3835653.453 | 0.008 | 1177303.563 | 0.008 | 4941637.903 | 0.008 |

Testing point: 1 | 3835681.535 | 0.008 | 1177277.573 | 0.008 | 4941646.969 | 0.008 |

2 | 3835691.060 | 0.008 | 1177286.086 | 0.008 | 4941637.608 | 0.008 |

3 | 3835664.478 | 0.008 | 1177304.702 | 0.008 | 4941629.347 | 0.008 |

4 | 3835668.242 | 0.008 | 1177286.195 | 0.008 | 4941630.683 | 0.008 |

5 | 3835633.954 | 0.008 | 1177294.966 | 0.008 | 4941655.200 | 0.008 |

6 | 3835673.791 | 0.008 | 1177258.615 | 0.008 | 4941633.229 | 0.008 |

**Table 3.**Differences between geocentric converted and measured coordinates of the test points of the point cloud.

Coordinates | Test Point 1 | Test Point 2 | ||||

GNSS Measured | Laser Scanner Transformed | Differences | GNSS Measured | Laser Scanner Transformed | Differences | |

X (m) | 3835681.535 | 3835681.530 | −0.005 | 3835691.060 | 3835691.067 | 0.007 |

Y (m) | 1177277.573 | 1177277.574 | 0.001 | 1177286.086 | 1177286.082 | −0.004 |

Z (m) | 4941646.969 | 4941646.964 | −0.005 | 4941637.608 | 4941637.603 | −0.005 |

Coordinates | Test Point 3 | Test Point 4 | ||||

GNSS Measured | Laser Scanner transformed | Differences | GNSS Measured | Laser Scanner Transformed | Differences | |

X (m) | 3835664.478 | 3835664.482 | 0.004 | 3835668.242 | 3835668.244 | 0.003 |

Y (m) | 1177304.702 | 1177304.709 | 0.007 | 1177286.195 | 1177286.193 | −0.002 |

Z (m) | 4941629.347 | 4941629.336 | −0.011 | 4941630.683 | 4941630.687 | 0.004 |

Coordinates | Test Point 5 | Test Point 6 | ||||

GNSS Measured | Laser scanner Transformed | Differences | GNSS Measured | Laser Scanner Transformed | Differences | |

X (m) | 3835633.954 | 3835633.961 | 0.007 | 3835673.791 | 3835673.783 | −0.008. |

Y (m) | 1177294.966 | 1177294.976 | 0.010 | 1177258.615 | 1177258.616 | 0.001 |

Z (m) | 4941655.200 | 4941655.207 | 0.007 | 4941633.229 | 4941633.225 | −0.004 |

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**MDPI and ACS Style**

Osada, E.; Sośnica, K.; Borkowski, A.; Owczarek-Wesołowska, M.; Gromczak, A.
A Direct Georeferencing Method for Terrestrial Laser Scanning Using GNSS Data and the Vertical Deflection from Global Earth Gravity Models. *Sensors* **2017**, *17*, 1489.
https://doi.org/10.3390/s17071489

**AMA Style**

Osada E, Sośnica K, Borkowski A, Owczarek-Wesołowska M, Gromczak A.
A Direct Georeferencing Method for Terrestrial Laser Scanning Using GNSS Data and the Vertical Deflection from Global Earth Gravity Models. *Sensors*. 2017; 17(7):1489.
https://doi.org/10.3390/s17071489

**Chicago/Turabian Style**

Osada, Edward, Krzysztof Sośnica, Andrzej Borkowski, Magdalena Owczarek-Wesołowska, and Anna Gromczak.
2017. "A Direct Georeferencing Method for Terrestrial Laser Scanning Using GNSS Data and the Vertical Deflection from Global Earth Gravity Models" *Sensors* 17, no. 7: 1489.
https://doi.org/10.3390/s17071489