## Abstract

This paper aims to establish and develop a calibration model for two time-of-flight terrestrial laser scanners (TLS): Trimble GX200 and Riegl LMS-Z390i. In particular, the study focuses on measurement errors and systematic instrumental errors to compile an error model for TLS. An iterative and robust least squares procedure is developed to compute internal calibration parameters together with a TLS data set geo-reference in an external reference system. To this end, a calibration field is designed that performs as an experimental platform that tests the different laser scanner methods. The experimental results show the usefulness and potential of this approach, especially when high-precision measurements are requires.

© 2011 OSA

## 1. Introduction

The recent emergence of terrestrial laser scanners (TLS) and their variability in shapes and models has greatly popularised this technology in multiple applications: from disciplines related to industry, metrology and reverse engineering to those related to architectural heritage, engineering geodesy and even forensic engineering. In spite of the fact that the most brilliant characteristic of this technology is its massive acquisition of information, once in a while, the process itself produces systematic errors that could make its application useless in those cases that require high-quality measurements. In fact, in other disciplines, such as photogrammetry and engineering geodesy, the estimation and modelling of possible errors is established before starting any measurement. More precisely, this is similar to knowing, e.g., the internal camera calibration parameters in the case of photogrammetry or a coping method for systematic instrumental errors that allows us to correct angular and distance observations for engineering geodesy. Curiously, this situation does not happen in the case of TLS because no standard calibration procedure besides awareness of systematic trends exists. Furthermore, the methods employed by the individual manufacturers on laboratory are based on specialized tools (e.g. calibrated baselines, oscilloscopes, etc), are expensive and are applied following strictly secret protocols, thus the benefit of the presented self-calibration approach over manufacturer’s laboratory calibration is that the scanner does not need to be dissembled, no specialized equipment is required and the self-calibration can be performed as frequent as the user desires without the need to ship the TLS away for weeks.

Therefore, the need to calibrate TLS is clear and has been widely stated in the last years. In photogrammetry, camera calibration protocols are intended to estimate the camera constant deviation and lens distortion, for example. However, for active sensors, standards for error evaluation have not been established yet. With the publication of ISO standard 17123 part 8 (GNSS field measurement systems in Real Time Kinematic –RTK–) in September 2007, TLS are the only remaining geodetical measuring systems without standardized field test procedures. As a result of the absence of standards, the accuracy specifications given by TLS manufacturers in their publications and pamphlets are not comparable [1]. Boehler et al. [2] suggest that sometimes these should not be trusted and highlight that the accuracy of these instruments, which are built in small series that vary from instrument to instrument, depend on the individual instrument calibration and the care of that instrument.

#### 1.1 Previous work

The first suggestions for calibrations, system tests and accuracy checks for TLS were provided by [3]. The literature that addresses error evaluation and calibration in the last decade focuses on two main themes:

- - Quantitative description of the accuracy achievable with a particular instrument. The performance evaluation is both important and essential in understanding the limitations and characteristics of the scanners, as well as to compare equipment. However, performance evaluation does not analyze systematic and methodological errors in detail, so investigations and calibrations are required.
- - Identification of the significant systematic errors (calibration parameters) in the instrument. These parameters constitute a calibration model, which can be used to correct the systematic instrumental errors. The calibration procedure can be carried out by system calibration (self-calibration) or a component calibration [4].

The approaches for TLS self-calibration developed recently can be divided into two groups according to [11]: point-based self-calibration (i.e., using targets whose centroids can be extracted and used as input to the self-calibration) and plane-based self-calibration (i.e., using the laser return coordinates on the surface of planar targets as input to the self-calibration). Rietdorf et al. [12] introduced a system calibration by using a test field of planes to estimate the additive constant for range measurements, vertical index error, and horizontal collimation error. It is stated that using planes guarantees laser scanner system and software independence as well as high redundancy. Similar calibration procedures were performed by [13,14]. They designed calibration rooms using different types of targets [13]. reported four- and five-term error models for three different TLS (Callidus 1.1, Leica HDS 2500 and Leica HDS 3000) [14]. used the FARO 880 laser scanner and presented a seven term error model that significantly improved residuals. Lichti [15] presented the full mathematical model for a point-based photogrammetric approach to FARO 880 TLS self-calibration. Boehler et al. [2] developed a test procedure to assess the quality of measurements obtained by different laser scanners, e.g., noise and systematic offsets in range measurements. Schulz and Ingensand [9,10] obtained the parameters that represent the mechanical-optical stability, such as the geometry of the axes, eccentricity, and addition constant, for certain instruments.

In this paper, we address the self-calibration problem with special attention to time-of-flight laser systems: Trimble GX200 and Riegl LMS-Z390i, for application to structural health monitoring systems. In particular, based on the identification of systematic instrumental errors, a self-calibration model is proposed and tested using our own software, CalibTLS. To this end, a laboratory calibration field has been designed that consists of a network of artificial targets homogeneously distributed over the whole volume of the laboratory. More recently, Chow et al. [16,17] applied a self-calibration model for Trimble GS200 and Trimble GX. In particular, the point-based approach presented uses a self-calibration model with different additional parameters and applied over two different indoor calibration fields. However, although different systematic trends seem to remain, in the end, the percentage of improvement before and after modeling is not very significant. Focusing the attention to the GX model, it is important to remark the consideration of sinusoidal trends for angles, as well as the no consideration of horizontal and vertical offsets in the mathematical models.

This paper is organized as follows: Section 2 describes the error sources in the measurements with both lasers; Section 3 develops the self-calibration model with special attention to the functional and stochastic model; Section 4 presents the experimental results and a technical discussion; and Section 5 provides the concluding remarks and future works.

## 2. Analysis of instrumental errors

The instrumental errors that have been considered in the calibration process primarily belong to the study of systematic instrumental errors in engineering geodesy [18,19]. Specifically, the following error sources are proposed initially:

#### 2.1 Errors in the Measurement of Ranges (Laser Rangefinder)

*Range offset error (k _{0}),* represents a discrepancy between electrical and mechanical zero position at the scanner.

*Range scale error (m),* represents a scale factor in the measured range, which depends on the measured range.

Influence of the offset and scale errors on the measured range are expressed as follows:

where*Δr*is the error in the measured range

*r*, and

*k*and

_{0}*m*are the range offset and scale errors, respectively.

#### 2.2 Errors in the measurement of angles (deflection and rotation units)

Two different angular errors, vertical and horizontal, should be considered in the study of TLS angular errors. More precisely, the vertical angular error provides one of the main differentiable characteristics relative to other topographic equipment, such as reflector-less total stations, because it is generated from a deflection unit. Likewise, the horizontal angular error derived from a rotation unit should be considered as well, especially in those laser systems that use this unit to obtain full azimuthal coverage.

*Vertical and horizontal offset errors* are an additive constant to the measured vertical or horizontal angle. These errors are similar to the eclimeter error (vertical offset error) and the horizontal limb error (horizontal offset error) in topographic equipment. These errors could be caused by mirror and encoder mechanical misalignment or zero offset within the analogue-to-digital converter.

*Vertical and horizontal scale errors* are linearly dependent on the measured angle. The reason for these errors may be a false gain-control within the analogue to digital converter or faults of the encoder. This error, in the case of vertical angles, could be explained as follows: because the angular position of the scanning mirror is sampled by the encoder in fixed increments, the vertical angle is recorded as a sum of these increments. If the actual value of such an increment differs from the nominal value, the scale error appears. Something similar could be extrapolated to the case of horizontal angles because the horizontal directions in Trimble GX200 and Riegl LMS-390i are derived with the aid of a servomotor from mechanical increments of the scanning head.

Both angular errors (offset and scale) affect the vertical and horizontal angle measurement as follows:

*θ*and

_{scan}*θ*are the vertical angle measured by the laser and the corrected vertical angle, respectively, and

_{corr}*θ*and

_{0}*δθ*are the offset (eclimeter) and vertical scale errors, respectively. Furthermore,

*φ*and

_{scan}*φ*are the horizontal angles measured by the laser and the corrected horizontal angle, respectively, and

_{corr}*φ*and

_{0}*δφ*are the offset (horizontal limb) and horizontal scale error, respectively.

#### 2.3 Axes errors

Prior to defining the axes errors in a TLS, the main axes of this instrument should be defined. Therefore, the following axes for the scanner may be defined:

- - Vertical axis. For panoramic TLS, such as Trimble GX200 or Riegl LMS-Z390i, this is the rotation axis of the scanning head. It is possible that this axis lies in the vertical scanning plane, i.e. the plane in which the laser beam moves in the vertical direction to scan the object.
- - Collimation axis. Assuming the divergent laser beam is conical, this is the axis that passes through the centre of the scanning mirror and the centre of the laser spot on the object surface. Roughly speaking, the collimation axis of a TLS coincides with the laser beam.
- - Horizontal axis. The rotation axis of the scanning mirror that passes through its centre.

*Collimation axis error (c)* is the angle between the collimation axis and the normal to the horizontal axis, which is measured in the plane containing the collimation and horizontal axes.

*Horizontal axis error (i)* is the angle between the horizontal axis and the normal to the vertical axis, which is measured in the plane containing the horizontal and vertical axes.

*Vertical axis error (v)* is the orthogonality offset of the vertical axis, i.e., the vertical axis and the rotation base are not perfectly orthogonal. This error could generate the classical topographic error: *precession vertical* axis error, which causes a rotation variation of the laser scanner around its vertical axis, which normally oscillates sinusoidally. The reasons for this error are the mechanical characteristics of the laser scanner. Likewise, this error could be affected and even increased by the instability of the base of the scanner, especially when the scanner is stationed on a survey tribrach and a wobbling of the scanner could appear. In particular, in the case of the Trimble GX200 this error could be considered insignificant because a dual axis compensator is incorporated. However, this error should remain in the Riegl LMS-Z390i because this equipment does not incorporate this compensator.

The following figures (Fig. 1 and Fig. 2 ) illustrate for both scanners, Trimble GX200 and Riegl LMS-390i, the scanner axes together with their corresponding axes errors. Special attention is focused on the scanning mirror because it does not always clearly follow from the manufacturer’s specifications which type of mirror is used in a particular TLS, and no technical descriptions of the scanning mechanism are provided. The Trimble GX200 scanner presents an oscillating planar mechanism (Fig. 1(b)). It is assumed that the origin of the scanner coordinate system is at the intersection of the instrument axes and the centre of the scanning mirror.

The laser beam after leaving the rangefinder unit is guided towards an optical system constituted by a divergent lens and a fixed concave mirror. In particular, the longitudinal position of the lens can be modified in order to get laser beam focalisation at different distances. Thus, it is guaranteed that the spot size is as small as possible over the scan object. The concave mirror remains in a fixed position. This mirror guides the laser beam toward the oscillating planar mirror, which oscillates at a constant frequency between two angular positions (maximum and minimum) and is driven by a galvanometric motor. To provide precise positioning of the mirror, it is attached to a mount that is fixed without deformation to the shaft of the scanner. Thus, the vertical angle is obtained by providing vertical coverage. This vertical angle can take values between −20° and + 40°. However, the oscillating planar mirror only has to rotate through half of these angles, i.e., between −10° and + 20°, because the optical lever effect rotates an angle *2α* (Fig. 1(b)).

The Riegl LMS-390i laser scanner uses a polygonal rotating system (Fig. 2(b)). This kind of scanning mirror consists of a prismatic optical polygon with parallel mirror facets that are equidistant and face away from the central rotational axis. The advantages of this mirror are its high speed, the availability of wide scan angles, and velocity stability. The polygon is fastened directly to the rotor shaft. Once emitted from the rangefinder, the laser pulses are directed toward the objects by the rotating optical polygon. For a facet inclination angle*θ*, the laser beam azimuth is *2α* (Fig. 2(b)).

Therefore, any source of error in polygonal or oscillating mirrors, such as the imperfections of the mirror, as well as their performance, has been considered as a systematic trend in the angular offset and scale errors.

## 3. Self-calibration model

Based on the preceding study, a calibration model for TLS has been developed. A TLS calibration model should express the relationship between the observables (distance, horizontal and vertical angles) and the systematic instrumental errors. Although a laser scanner system is based on the same polar measurement principle as the reflector-less total station, its performance is not exactly the same because these scanners incorporate the deflection unit and rotation platform to measure the vertical and horizontal angles, respectively.

Therefore, the proposed calibration model should consider these additional components. Likewise, to provide a more effective and integral approach, the calibration model is transformed into a self-calibration model, so the external parameters of the scanner (geo-referencing) with respect to a reference coordinate system and the internal parameters (calibration) corresponding to systematic instrumental errors can be estimated simultaneously.

According to this proposal, the TLS self-calibration model should meet with the following conditions:

- - Geo-referencing all data sets to the same reference system based on high-precision engineering geodesy, which performs as “ground truth”. Thus, a three-dimensional transformation reinforced with a robust estimator and a stochastic test is implemented.
- - Integrating the systematic TLS instrumental errors in the same way that the collinearity model integrates the internal camera parameters.
- - Allowing system feedback such that once the calibration parameters have been estimated and the laser scanner observations have been corrected, a new adjustment will be performed to detect possible systematic errors not considered initially in the self-calibration model.
- - Estimating precision, accuracy and reliability for each one of the observations and self-calibration parameters of the TLS. Thus, statistical approaches are incorporated that allow us to analyze the precision of the observations and parameters, as well as estimating their reliability and detecting possible gross errors. Particularly, the accuracy of observations and adjusted unknowns are of prime interest when analyzing quality in an adjustment procedure. On the contrary, the precision of observations and unknowns are directly related with the calculated stochastic values which provide information about the quality of the functional model with respect to the input data. Therefore, the precision describes an internal quality of the adjustment process, while the accuracy performs as an external validation which should only be used if a comparison to reference data of higher accuracy is performed.

#### 3.1 The functional model

The functional model proposed for TLS self-calibration has been developed using 14 parameters: 6 for the external parameters (three rotations and three translations) and 8 for the internal parameters (those corresponding to the additional parameters of systematic instrumental errors). In particular, the corrections for the horizontal (*Δφ _{corr}*) and vertical (

*Δθ*) angles are estimated as follows:

_{corr}*c*,

*i*,

*φ*,

_{0}*δφ*,

*θ*and

_{0}*δθ*are the errors corresponding to the collimation axis, horizontal axis, horizontal offset and scale, and vertical offset and scale, respectively, and

*φ*and

_{scan}*θ*are the horizontal and vertical angle measurements, respectively.

_{scan}The self-calibration model proposed with 14 parameters is the following:

*X Y Z*]

^{T}are the coordinates of the target centres in the external coordinate system (engineering geodesy), which are also referred to as “true” coordinates; [

*ΔX ΔY ΔZ*]

^{T}is the vector of translation parameters;

*R*(

*α*) is the rotation matrix between the two systems, which is a function of the rotation angles (

_{1}, α_{2}, α_{3}*α*) about the

_{1}, α_{2}, α_{3}*x*,

*y*and

*z*coordinate axes, respectively. While the coordinates of the target centres in the laser scanner coordinate system [

*x y z*]

^{T}are defined by the observations

*r*,

_{scan}*φ*and

_{scan}*θ*as distances, horizontal and vertical angles, respectively; and the systematic instrumental errors

_{scan}*k*,

_{0}*m*,

*c, i*,

*θ*,

_{0}*φ*,

_{0}*δθ*and

*δφ*, as distance offset and scale errors, collimation and horizontal axes errors, vertical and horizontal offset errors, and vertical and horizontal scale errors, respectively.

Furthermore,

*m*, has been considered in the functional model (5). Although, some authors [20] assert that the presence of this factor could negatively affect the final solution and suggest rejecting it; other authors, such as [21], advice including it in the adjustment because it could verify the results. Therefore, in order to test this controversial parameter, we have decided to include it in the adjustment.

The functional model (5) developed is linearised using the Taylor linearisation approach and computed through a re-weighted least squares adjustment. The linearised model is as follows:

Note that the vector *F ^{0}* contains the laser scanner observations

*r*,

_{scan}*φ*and

_{scan}*θ*. The approximate values for the angles

_{scan}*α*and

_{1}*α*are set to zero because these scanners were leveled during the experiments. Likewise, the approximate values for all the calibration parameters are set to zero because it is reasonable to assume that the manufacturer tries to minimize the instrumental errors as much as possible.

_{2}#### 3.2 The stochastic model

The stochastic model proposed for the TLS self-calibration starts with the least squares basic principle and is reinforced by the addition of robust and statistic techniques: more precisely, the Danish modified robust estimator [22] and the Pope statistical test [23].

*v*is the residual vector,

*W*is the weight matrix and

*x*is the unknown vector.

To this end, this stochastic model takes the law of propagation of the variance-covariance matrix as a “guideline”, so the a priori variance-covariance for TLS observations *Σ _{ll}* has been computed as follows:

*R*(

*α*) is the rotation matrix,

_{1}, α_{2}, α_{3}*J*is the Jacobian matrix of the derivates of TLS coordinates [

*x y z*]

^{T}with respect to

*r*,

_{scan}*φ*and

_{scan}*θ*;

_{scan}*Σ*is the diagonal variance-covariance matrix of the “true” topographic coordinates of the targets; and

_{XYZ}*Σ*is computed as follows:

_{inst}*σ*,

_{r}*σ*and

_{φ}*σ*are the accuracies of the TLS observables

_{θ}*r*,

_{scan}*φ*and

_{scan}*θ*, respectively, including the beam divergence

_{scan}*σ*, which are provided by the manufacturers. Nevertheless, it is important to remark that these technical specifications are due to individual point measurements, while the extraction of the centre of the targets is performed considering many points around the targets.

_{beam}Finally, the statistical and robust techniques are implemented following a twofold strategy:

- - First, the modified Danish robust estimator is applied while supported by a variable exponential weight function (12), which updates the weight matrix
*W*iteratively by taking the residual*v*of the previous iteration. This estimator is only applied in the first iteration to detect the most unfavourable gross errors. - - Second, the normalized residuals
*v*are computed in the following iterations through the T-Student distribution and the Pope test, so based on the data snooping strategy (Kraus [24], ), the gross errors can be detected._{Pope}

*√C*contains the redundancy number that is determined from the diagonal of the residual cofactor matrix (

_{VV}*C*), and $\widehat{\sigma}$ is the a posteriori deviation. Note that the Pope test does not require knowing the a priori deviation of the observations; thus, this is sometimes really useful for studies in which the manufacturer does not provide them.

_{vv}As a result, the functional and statistic model developed for the self-calibration of TLS is applied within an iterative process based on a Gauss-Markov model.

## 4. Experimental results

The protocol developed for the calibration of the TLS Trimble GX200 and Riegl LMS-Z390i has been divided into three steps: 4.1) design and signalling of the calibration field (network design); 4.2) data acquisition (field work); and 4.3) data processing (laboratory work).

#### 4.1 Design and signalling of calibration field

A laboratory calibration field, 14m long, 10m wide and 3m high, was designed that consists of a network of artificial targets homogeneously distributed over the whole volume of the laboratory using different heights and depths (Fig. 3 ). To provide durable targets, none of them was placed on the floor. Assuming that each laser scanning system uses a specific type of flat target, which is detected automatically and is guaranteed the highest precision by the manufacturer, two different arrays of targets were used. Specifically, 20 self-adhesive flat targets that are 15 cm x 15 cm were used for the TLS Trimble GX200, while 20 retro-reflective circular targets with a diameter of 5 cm were used for the TLS RieglLMS-390i. The scanners were levelled with the aid of a bubble level for the Riegl LMS-390i and with an electronic level reinforced with the dual-axis compensator for the Trimble GX200. Furthermore, both scanners were stationed over a stable surface in the centre of the laboratory.

Note how the distribution of the targets on the laboratory was performed to guarantee different incidence angles, as well as different horizontal and vertical angles. Likewise, a homogeneous distribution of targets has been applied in order to obtain a more favourable geometric design. In addition, the laboratory acts as a workspace to test different TLS.

It should be noted that the calibration was performed under stable temperature, pressure and humidity conditions. Besides, the targets are completely distributed by 360° around the scanner in horizontal. However, regarding vertical angle the targets considered show an elevation angle between 0° and 30°, since targets placed on ceiling provided incidence angles really unfavourable. This problem had been already reported by [15] who detect that for planar targets, the signal-to-noise ratio of the measured target centroid drops significantly when the incidence angle is greater than 60°.

#### 4.2 Field work

Once the calibration field has been designed and signalised, data acquisition is executed considering two survey types:

- - Scan-based surveying of the special targets, using the maximum TLS resolution (2-3 mm) with automatic centre extraction.
- - Topographic-based surveying of the centre of the special targets, which establishes an external reference frame using high-precision topographic equipment: Leica TCA2003. In this sense, the engineering geodesy performs as “ground truth”, which is a reference for all TLS measurements. The horizontal angles are observed by the directional method: reading the horizontal circle in both the hindsight and foresight directions.

#### 4.3 Laboratory work

Laboratory work provides the previously gathered information. It is carried out in three stages: the first involves computing the input data set (raw observables), i.e., distances and angles; the second establishes a statistical analysis of the different TLS observables in order to find possible correlations and systematic trends; the third estimates and analyses the different external (geo-referencing) and internal (instrumental errors) calibration parameters.

In particular, an iterative self-calibration model is applied that is supported by Danish modified robust estimator and the Pope statistical test. As a result, a precision and an accuracy control are performed following a twofold approach: firstly controlling the model, analyzing the adequacy of the functional model and its geometric configuration based on the correlations between adjusted additional parameters coming from the covariance matrix; and secondly, validating the model assuming that the topographic reference system is the “ground truth”.

A specific tool, CalibTLS, which incorporates a Matlab library was developed for testing the different experiments.

### 4.3.1 Computation of input data set

The raw laser and topographic observables that constitute the input data are:

TLS target observations are composed of spherical (*ρ,θ,α*) and Cartesian (*X,Y,Z*) coordinates that correspond to the geometric target centres that define the calibration network. Likewise, the point cloud of each one of these targets is obtained, which could be used as an alternative process (planar target modelling) in those cases in which automatic target extraction does not work [26].

Topographic target observations are composed of spherical (*ρ,θ,α*) and Cartesian (*X,Y,Z*) coordinates that correspond to the geometric target centres that define the calibration network. These observations constitute the external reference system (data) and are considered to be “ground truth”.

### 4.3.2 Analysis of possible systematic trends

Prior to proposing a TLS calibration model, a statistical analysis based on the different observation errors is performed. This analysis is based on the computation of errors that correspond to distances and angles, using the topographic data as “ground truth”. Table 1 presents the most relevant results.As one can see (Table 1), it seems clear that the laser Trimble GX200 is more accurate than the Riegl LMS-Z390i for all observations (angles and distance). Also, the target survey is performed more accurately by Trimble with a standard deviation less than one millimetre. Note that minimum and maximum errors are computed for absolutes values.

To reinforce the study of systematic trends, several graphics (Fig. 4 ) were obtained based on the analysis of observable errors. A twofold strategy was applied: first, an individual analysis of the observable errors in distance and angles was performed; second, to determine possible correlations, several bivariant graphics were established. Only those graphics that show significant correlations are illustrated.

As one can see (Fig. 4), it seems that the errors do not follow a rigorous normal distribution, especially in the case of Riegl LMS-Z390i. In particular, the following must be addressed:

- - The laser Trimble GX200 presents low dispersion (deviation) in the distribution of errors, with the errors approximately normally distributed with zero means for the case of all observations. However, the horizontal observation errors are two times higher than the vertical angle errors, which could indicate a possible systematic trend (offset or scale error).
- - The laser Riegl LMS-Z390i presents a higher dispersion (deviation) in all errors, with the distance error as the most favourable according to the range specifications provided by the manufacturer. Again, it should be remarked the dispersion obtained by the vertical angle and especially the horizontal angle, which could be caused by another systematic trend.
- - It is interesting to note how particular observations could follow a different distribution, probably due to the presence of gross errors. Therefore, the above mentioned robust statistical approaches will be required in order to detect and reject these observations.

As can be seen in Fig. 5 , the Z coordinate error is approximately one half that of the X and Y errors for both scanner systems. The distribution of these errors follows an approximate normal distribution with zero means. Nevertheless, the Riegl LMS-Z390i (Fig. 5(b)) presents a higher coordinate error dispersion, which could be due to the presence of gross errors, as well as the intrinsic precision of the scanner system and the method used to extract the targets.

The more significant error in the X and Y coordinates is directly related to the greater error in determining horizontal angles, which matches the histogram results in Fig. 5. This may be explained by the lower accuracy of the rotation platform, which measures horizontal angles with the aid of a servomotor; the servomotor has greater inertia than the scanning mirror.

Analysis of the following figures (Fig. 6 ) shows how several systematic trends remain in TLS observations.

Analysis of Fig. 6 shows two different systematic trends in the horizontal angle measurements. While the correlation between the horizontal angular error and the horizontal angle is clear for the Trimble GX200, this is not the case for the Riegl LMS-Z390i because the correlation is less clear due to the randomly of data points. In particular, in the case of the Trimble GX200, the magnitude of the horizontal angular error tends to be inversely proportional to the horizontal angle until the angle reaches a minimum value (zero-cross) halfway across the horizontal limb (180°). Afterwards, the magnitude of the horizontal error grows linearly as the horizontal angle increases. These results confirm the high values obtained for the horizontal angular error, which exceed those provided by the manufacturer. One hypothesis about this performance it could be related with a horizontal encoder circle eccentricity error. For the Riegl LMS-Z390i, the systematic trend is different because the correlation is not clear at all, describing more or less randomly the data points. Therefore, the systematic trend of the horizontal angle error is not significant.

On the other hand, from the analysis of Fig. 7 , a systematic trend is observed in the vertical angular error for both laser scanner systems. More precisely, in the case of Trimble GX200 (Fig. 7(a)), when plotting the vertical angular errors versus the vertical angle: the magnitude of the errors tends to change inversely proportionally as the vertical angle increases up to 18°. Note that afterwards the errors start to increase linearly as the vertical angle increase. Regarding on Riegl LMS-Z390i, an interesting systematic behaviour is observed since the vertical angular error decrease linearly as the vertical angle increases. Therefore, these results could confirm a systematic trend, the possible presence of scale and offset angular errors.

The remaining bivariant analyses between vertical angular error and observations do not show any systematic trend; the errors are random.

### 4.3.3 Self-calibration results

After the statistical analysis of the TLS observation errors, a self-calibration model with 14 parameters is developed that incorporates robust statistical techniques. In particular, Trimble targets T2 and T9 were rejected in the first iteration based on the modified Danish robust estimator, while targets T3 and T15 were rejected iteratively by the Pope test. For the Riegl, targets R9 and R20 were rejected in the first iteration based on the modified Danish robust estimator, while targets R2, R3 and R5 were rejected iteratively by the Pope test.

The main results of the self-calibration process (Table 2
) are derived from the cofactor matrix of the unknowns, which is subdivided into two clearly differentiable blocks: the external parameters block that corresponds to geo-reference parameters (translation and angles) and the internal parameters block (calibration parameters). Note that *Std. Dev.* refers to the standard deviation of unit weight, while the *Value* refers to the value of the unknown parameter.Regarding external parameters (Table 2), one can see how the Trimble GX200 incorporates a vertical dual axis compensator because the angles in the XY plane are *α _{1}* = 0.0156° (56”) and

*α*= −0.0031° (11”), with a vertical deviation of 0.0157° (57”). However, that is not the case for the Riegl LMS-Z390i, which takes values of

_{2}*α*= −0.4333° (26’) and

_{1}*α*= −0.0302° (2’), with a vertical deviation of 26’.

_{2}In particular, for the Trimble GX200, the levelling effectiveness of the automatic dual axis compensator is analysed with (14) to compute the quadratic differences between the Z laser coordinates (Z_{L}) and the topographic coordinates (Z_{T}), divided by the number of control points minus 1:

*α*angle is 6 cm at 100 m. Note that this equation measures the effectiveness of the levelling compensation by the tilt angles and not the levelling accuracy because any accidental errors in the

_{1}*Z*coordinates due to the laser scanner being off level are accounted for by the

*α*and

_{1}*α*transformation angles.

_{2}Regarding calibration parameters (Table 2), the Trimble GX200 scale factor, *m*, exhibits a scale correction of 1mm/100m, while the Riegl LMS-Z390i exhibits 5mm/10m. It seems clear that the value shows by Riegl is not realistic and thus not useful. On the other hand, a special remark should be made to the offset vertical error, *θ _{0}*, which presents a standard deviation larger than its value for both laser scanners. Again, a possible correlation with this parameter could be a cause of its lack of significance.

With the aim of assessing the quality of the adjustment result, dependencies (correlations) between adjusted parameters are investigated and analyzed based on the covariance matrix of the unknowns. These correlations show the adequacy of the functional model and geometric configuration of observations. Higher correlation parameters indicate linear dependencies between parameters. They should be avoided particularly because the inversion of the normal equation matrix, and hence the adjustment solution, can become numerically unstable.

In particular, both laser scanners exhibit three important correlations (above 0.7):

- • A correlation between collimation axis error,
*c*, and horizontal axis error,*i*, which could affect the horizontal angle observations. It is important to remark that both errors take part in the correction of the horizontal angle. - • A correlation between range offset,
*k*and scale factor,_{0}*m*, which could confirm the possible unstable and opposite values obtained for both parameters in the case of Riegl. - • A correlation between offset vertical error,
*θ*and range offset,_{0}*k*, which could explain the lack of significance of vertical error,_{0}*θ*with a standard deviation larger than its value. Besides, any source of error in scanning mirror, such as its imperfection or its own performance, could be considered as a systematic trend in the offset vertical error._{0}

The TLS observation accuracies after the self-calibration are shown in Table 3 .As one can see in Table 3, the application of the self-calibration model improves the results in the case of Trimble GX200. The error interval between observables has been reduced as well as its dispersion. This is not the case for Riegl LMS-Z390i since the level of improvement is not significant. Although, this TLS provides a worse precision, an optimization of the accuracy cannot be achieved since the systematic trends are not eliminated completely. Likewise, the complexity of the deflection system itself together with the uncertainty in its design (closed to the final user) could be another important reason.

Note that minimum and maximum errors are computed for absolutes values. In addition, it should be remarked that the dispersion,*σ*, of laser observables is not proportionally related with the deviation of targets error.

As in the previous case, bivariant graphics between errors and observables are computed in order to see if any systematic trends remain in the data. The error plots are shown in Fig. 8 . It is evident that the behaviour of the errors in the vertical and horizontal angles for Trimble GX200 has changed when compared to the errors before calibration.

In particular, for Trimble GX200, the errors in vertical and horizontal angles have decreased by modelling these errors in the proposed self-calibration model (5), and no systematic trends are now present in the data. In fact, the correlation (C) between errors and observables is very low compared to that before the self-calibration. Analysis of Fig. 8(a) and 8(c) shows that the errors in the vertical and horizontal angles are approximately randomly distributed with zero means, and no systematic trends can be observed. Therefore, the set of calibration parameters for these scanners could be considered to be final.

On the other hand, it is obvious that the effectiveness of the self-calibration model is not the same for Riegl LMS-Z390i Fig. 8(b) and 8(d)), since the systematic trend is not completely rejected. More precisely, an offset can be observed in angular errors, especially in the case of the horizontal angular error (Fig. 8(b)). This is due, in part, to the randomly of data points which makes difficult the linear adjustment.

## 5. Conclusions and future perspectives

In this paper we have investigated and analysed the systematic instrumental errors that occur in time-of-flight measurements with TLS Trimble GX200 and Riegl LMS Z30-90i. These instrumental errors are attributed to the three main components of a TLS: the laser rangefinder, beam deflection unit and rotation platform. These errors are difficult to estimate and analyse: first, TLS are very complex and include several optical and mechanical components, each of which contributes its own error. A good understanding of these errors requires better knowledge of the TLS design system, which is usually not accessible to the user. All these factors make it extremely difficult to develop a standard procedure for TLS system calibration to identify and correct possible systematic measurement errors. Thus, TLS measurements could be more accurate and reliable and be applied to those cases where more precision is required, for example, monitoring structures and computing deformations.

Based on our investigations and results, the following concluding remarks should be mentioned:

- • The systematic instrumental errors and trends have been identified successfully for Trimble GX200, and thus, the precision and accuracy of the measurements have been improved.
- • The reported self-calibration approach indicates that within certain limits, results from TrimbleGX200 are of interest and could be useful for field work.
- • This is not the case for Riegl LMS-Z390i, since an optimization of the accuracy has not been achieved and several systematic trends remain.
- • Try to cope with correlation between parameters may produce better results. In particular, we could confirm that the controversial scale parameter,
*m*, it should not be included in the self-calibration model, especially when the test field dimensions are reduced (indoor test field). This parameter should be incorporated in outdoor test fields where larger distances can be considered. Nevertheless, in these cases the atmospheric influences must be considered as well. - • Accuracy, precision and reliability have been estimated for statistical analysis. This stochastic approach has been supported by several bivariant plots between TLS errors and TLS observations, which have lead to identification of possible systematic trends.
- • The self-calibration model proposed guarantees reliability because it has been reinforced with an iterative least squares process that incorporates robust estimators and statistical tests.
- • A software that diagnoses and corrects TLS field measurements before being processed,
*CalibTLS*, has been developed. - • The self-calibration model proposed is scalable since allows continuous improvement and feedback based on analysis and testing.
- • A step forward in TLS calibration protocols and their standardisation has been accomplished, so they can be applied easily and economically.

## Acknowledgments

Authors would like to give thanks to Consellería de Economía e Industria (Xunta de Galicia) and Ministerio de Ciencia e Innovación (Gobierno de España) for the financial support given: projects INCITE09 304 262 PR and BIA2009-08012.

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