In the following, the results of the experiments are presented and discussed. In

Section 6.1, the relationship between the intensity of the backscattered laser light and the precision of the distance measurements is examined leading to three intensity-based stochastic models for the three scan rates of the Z + F Profiler 9012A (i.e., 254 kHz, 508 kHz and 1016 kHz).

Section 6.2 analyzes the performance of the close range optimization and connects this to the theory from

Section 3.3.

#### 6.1. Determination of Intensity-Based Stochastic Models

As described in

Section 4.1, scans with all six settings of the Z + F Profiler 9012A were performed. These six settings comprise scan rates of 254 kHz, 508 kHz and 1016 kHz (

Table 1). In order to derive an intensity-based stochastic model for each scan rate, all pairs of standard deviations

${\widehat{\sigma}}_{r}$ and mean intensity

$\widehat{I}$ (Equations (

4) and (

5)) from all experiments were pooled for the respective scan rate, whereas, in experiments 1–3, planar targets were scanned on the comparator track, in experiment 4, normal environments were scanned. For the latter, appropriate surfaces were manually extracted (e.g., walls, ground, ceiling). These surfaces were analyzed in the same way as the planar targets on the comparator track. This leads to an extensive data basis (

Table 3).

Figure 8a–c show the standard deviations

${\widehat{\sigma}}_{r}$ of the distance measurements as a function of the mean intensities

$\widehat{I}$ for all three scan rates. Please note that the horizontal axis is plotted logarithmically. Clearly, all three curves show a similar behavior that applies to an exponential function. For lower intensities, higher standard deviations are obtained. Furthermore, a dependency on the scan rate is apparent: the higher the scan rate, the higher the standard deviation. However, all curves rapidly decrease in the interval between 0 Inc and 500,000 Inc. For higher intensities, the precision only slightly improves and it seems reasonable to assume that the estimated curves continue in the same way.

In the course of experiments 1–3, measurements with a scan rate of 1016 kHz were carried out on both white and colored targets as well as targets with varying incidence angles. In addition to it, the measurements covered a range between 0.4 m and 22 m. Yet, all of these different measurements with respect to object properties and scanning geometry basically follow the exponential behavior of the curve. The black and green targets produced the lowest intensities, whereas the white, blue and red targets produced higher intensities. The measurements under different incidence angles between 0

${}^{\circ}$ and 80

${}^{\circ}$ lead to very different intensities decreasing with higher incidence angles. The dependency on the distance is discussed in more detail in

Section 6.2.

Please note that the measurements in the normal indoor and outdoor environments cannot be separated from the measurements on the comparator track. This verifies that our approach can be applied in normal use. It is only necessary that the scans cover the spectral range of the intensities and that some constraints regarding the scanning geometry are taken into account (

Section 4.2).

Our results confirm the findings published in [

24,

25,

26]. According to this, the intensity is a suitable measure to quantify the precision of the distance measurements of a laser scanner. Due to the fact that the pairs between standard deviation

${\widehat{\sigma}}_{r}$ and mean intensity

$\widehat{I}$ show a characteristic exponential behavior, it is reasonable to approximate the data with the functional model [

25]:

For all three scan rates, a model according to Equation (

6) was fitted to the data points in

Figure 8. This was performed by using a Least-Squares adjustment within the Gauß–Markov model [

47,

48], where the standard deviations

${\widehat{\sigma}}_{r}$ were utilized as observations and the mean intensities

$\widehat{I}$ as constants. By applying a variance component estimation [

47], the stochastic model of the adjustment was adapted to the data allowing for a reasonable estimation of the parameter’s covariance matrix as well as a reasonable filtering of outliers by classical data snooping techniques [

47]. For better comparison of the differences between the three scan rates,

Figure 8d shows the three estimated models without data points in one plot.

Table 4 lists the estimated parameters

$\widehat{a}$,

$\widehat{b}$ and

$\widehat{c}$ of the three models with their estimated standard deviations

${\widehat{\sigma}}_{a}$,

${\widehat{\sigma}}_{b}$ and

${\widehat{\sigma}}_{c}$. Please note that all parameter sets were significant with respect to their estimated covariance matrix. The good approximation is also indicated by the coefficients of determination [

49], which were 0.99 for all three models.

As described in

Section 3, the noise specifications of the Z + F Profiler 9012A are provided for a standard scan rate of 127 kHz (

Table 2). In order to obtain values for the scan rates of 254 kHz, 508 kHz and 1016 kHz, conversion factors have to be used (

Table 1) that can be deduced from the law of error propagation for uncorrelated observations with equal precision [

47]. In this regard, the ratio between the estimated curves is of interest because it quantifies the noise reduction by selecting a lower scan rate. This ratio can be compared to the conversion factors from theory. The ratio between the models for 1016 kHz and 508 kHz is 1.41 on average, which is exactly

$\sqrt{2}$. This is consistent with the ratio of

$2.8/2.0$ between the related conversion factors and meets the expectation that, due to the halved scan rate, two measurements are averaged. The ratio between the models for 1016 kHz and 254 kHz is 1.81 on average. However, due to the quartered scan rate, a factor of

$2.8/1.4=2$ was expected.

The lower precision at a scan rate of 254 kHz is most probably caused by the discretization of the distance measurements. The measurements are provided with a resolution of 0.1 mm; however, the representation in the digital signal processing is about 0.17 mm per bit. As a result, the precision at a scan rate of 254 kHz is limited by the discretization. An alternative explanation for the smaller ratio is positive correlations between the measurements that reduce the increase in precision [

50]. The improvement of the precision by a factor of 2 only applies to uncorrelated observations.

#### 6.2. Investigation of the Close Range Optimization

Regarding the special hardware optimization of the Z + F Profiler 9012A for measurements in the close range (

Section 3.3), the data on the comparator track were analyzed in more detail. During the experiments, both white and colored targets were scanned in a range between 0.4 m and 22 m in increments of 0.2 m and 1 m, respectively. These data can be utilized to investigate the measurement noise and the mean intensity as a function of the distance between sensor and object.

As described in

Section 6.1, each target scan comprises

i angular steps of the 2D laser scanner and, thus,

i pairs of standard deviation

${\widehat{\sigma}}_{r}$ and mean intensity

$\widehat{I}$. These pairs belong to the individual lines in the related point cloud (

Figure 7a). For better interpretation, one of these pairs was determined to be representative for the specific target scan. This was realized by choosing the angular step of each target scan containing the median of all standard deviations along with its mean intensity.

Figure 9a shows the representative standard deviations and mean intensities as a function of the distance between laser scanner and object for the scans on the white target from experiment 1. The individual curves correspond to the different settings of the Z + F Profiler 9012A (

Table 1). Clearly, all curves follow a similar pattern. The precision of the distance measurements is increased directly in front of the 2D laser scanner, but quickly drops to a local minimum at about 2 m. Following this, the precision worsens in the range between 2 m and 5 m with a maximum value at approximately 3.5 m. For distances larger than 5 m, the curves sink to the global minimum around 8 m before increasing steadily up to the end of the investigated range. It is likely that this behavior continues beyond 22 m due to the signal attenuation with the squared distance [

1,

2,

44].

According to

Figure 9a, the hardware-based close range optimization of the Z + F Profiler 9012A has its optimum at about 2 m, producing a noise that is of the same magnitude as the noise at about 8 m. Especially at 3.5 m, one has to be aware of increased noise, which cannot be completely compensated by the close range optimization. The special behavior of the curves in the range below 5 m is caused by the intensity of the backscattered laser light. This is proved by

Figure 9b, which shows the related mean intensities of

Figure 9a as a function of distance. The noise and intensity curves behave in exactly the opposite way. Furthermore, the intensity curves in

Figure 9b are equivalent to the simulated blue curve in

Figure 2a. This demonstrates that theory and reality agree well. Moreover, these results verify that the intensity-based stochastic models are also valid in the range below 5 m since the data from the close range also contribute to the determination of the stochastic models in

Figure 8.

As can be seen from

Figure 9a, curves belonging to the same scan rate overlap in good accordance. In addition, the factors of 1.4 and 1.8 between the scan rates (

Section 6.1) can be identified again. This emphasizes that both the intensity and the scan rate are the decisive factors for the distance precision. According to this, an intensity-based stochastic model only applies to a certain scan rate. Please note that only the scan rate matters and not how it is generated. In the case of Z + F Profiler 9012A, for instance, a scan rate of 1016 kHz can be generated by three different combinations of mirror speed and point resolution (

Table 1). However, the black curves in

Figure 9 (mirror speed: 200 rps, scan rate: 1016 kHz) are slightly different in this respect since they do not completely overlap with the blue and cyan curves. In

Figure 9a, the black curves are a bit higher, in

Figure 9b a bit deeper in the middle part. However, this is consistent for the relationship between precision and intensity. Thus, the stochastic model from

Figure 8a is not affected. An explanation for this special behavior could not be identified so far. Probably, there is a connection to the high rotational speed of 200 rps of the deflecting mirror.

The distance-related investigations were also carried out for the colored targets. The results are shown in

Figure 10. All curves refer to a mirror speed of 50 rps and a scan rate of 1016 kHz. For better interpretation, the corresponding curves from

Figure 9 are added to

Figure 10. The characteristic shape of the curves as observed for the white target can also be found for the colored targets. However, the level of precision is very different. Whereas the precision of the white, blue and red curves is almost the same, the precision of the green and black curve is considerably worse (

Figure 10a) due to the lower intensity of the laser light (

Figure 10b). The characteristics of the curves are more distinctive with decreasing reflectivity of the surface. For example, the noise of the black curve increases by 0.5 mm between 2 m and 3.5 m, whereas the difference for the white target is only 0.15 mm. The precision of

${\widehat{\sigma}}_{r}=1.6\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$ for the black target at 3.5 m seems to be small, but this means that 99.7 % (i.e.,

$\pm \phantom{\rule{3.33333pt}{0ex}}3\phantom{\rule{3.33333pt}{0ex}}{\widehat{\sigma}}_{r}$) of the measurements are scattered over an interval of almost 1 cm. For precise measurements of dark asphalt surfaces, this is highly relevant. In practice, therefore, it is advisable to mount the Z + F Profiler 9012A not in a distance of 3.5 m to the road surface, but rather 2 m or 5 m.