Abstract

In the geosciences, fine-scale detail of geomorphic surfaces, commonly parameterized as roughness, is growing in importance as a source of information for modeling natural phenomena and classifying features of interest. Terrestrial light detection and ranging (LiDAR) scanning (TLS), now well known to geologists, is a natural choice for collecting geospatial data. While many recent studies have investigated methodologies for estimating surface roughness from point clouds, research on the influence of instrumental bias on those point clouds and the resulting roughness estimates is scant. A scale-dependent bias in TLS range measurements could affect the outcome of studies relying on high-resolution surface morphology. Growing numbers of research applications in geomorphology, neotectonics, and other disciplines seek to measure the roughness of surfaces with local topographic variations (referred to as asperities) on the order of a few centimeters or less in size. These asperities may manifest as bed forms or pebbles in a streambed, or wavy textures on fault-slip surfaces. In order to assess the feasibility of applying TLS point cloud data sets to the task of measuring centimeter-scale surface roughness, we evaluated the relationship between roughness values of dimensionally controlled test targets measured with TLS scans and numerical simulations. We measured and simulated instrument rangefinder noise to estimate its influence on surface roughness measurements, which was found to decrease with increasing real surface roughness. The size of the area sampled by a single point measurement (effective radius) was also estimated. The ratio of the effective radius to the radius of surface asperities was found to correlate with the disparity between measured and expected roughness. Rangefinder noise was found to overestimate expected roughness by up to ∼5%, and the smoothing effect of the measurement size disparity was found to underestimate expected roughness by up to 20%. Based on these results, it is evident that TLS point cloud geometry is correlated with instrument parameters, scan range, and the morphology of the real surface. As different geological applications of TLS may call for relative or absolute measurements of roughness at widely different scales, the presence of these biases imposes constraints on choice of instrument and scan network design. A general solution for such measurement biases lies in the development of calibration processes for TLS roughness measurement strategies, for which the results of this study establish a theoretical basis.

INTRODUCTION

Quantitative descriptions of the surface texture of natural materials are increasingly sought in modern geological research. Statistical descriptors of surface morphology, or roughness, are found in many applications, including the modeling of fluvial processes or mass movement, detection and classification of features of interest in cluttered scenes, and the modeling of processes at work on or within a body of rock. Surface roughness may be defined as a measure that describes the distribution of heights of asperities, or protrusions of a surface, over a datum approximating the surface morphology at a lower resolution (Pollyea and Fairley, 2011). In this paper, we assume hemispherical asperities for describing scale (asperity radius). Different applications in the earth sciences involve various physical scales of asperities, which in turn inform the extent or resolution of the surface datum used. Approximate scales of asperities measured in several studies and application-relevant clast size ranges are collected in Figure 1 (see also Brown and Scholz, 1985; McKean and Roering, 2004; Sagy et al., 2007; Heritage and Milan, 2009).

Near-field observations have long been used in the study of surface texture (Limerinos, 1970; Brown and Scholz, 1985). These measurement practices are laborious, requiring extended time in the field and careful preservation of samples or casts for later study in the laboratory (Buffington et al., 1992). In practice, a grid is laid out on the sample surface, and heights are measured manually or with a profiling instrument.

In the past two decades, the geosciences have embraced the use of remote measurement techniques for capturing natural surface geometry (Buckley et al., 2008; Butler et al., 1998; Rowlands et al., 2003; Sagy et al., 2007; Renard et al., 2013). Photogrammetry offers a low-cost entry into rapid three-dimensional (3D) data capture, and the rapid proliferation of terrestrial light detection and ranging (LiDAR) scanning (TLS) technology has made scanners increasingly available to researchers. These measurement techniques have additionally been pursued as objective, in situ methods for measuring surface roughness. TLS point clouds have been demonstrated to satisfactorily replicate and even improve upon the results of manual, near-field sampling techniques when measuring streambed roughness (Heritage and Milan, 2009). Airborne and terrestrial LiDAR roughness measures have been employed to detect, characterize, and monitor landslides in a number of studies (Glenn et al., 2006; Hsiao et al., 2004; McKean and Roering, 2004). Roughness from LiDAR data has been shown to differentiate geological features and structural trends as well, and it has applications in automated surface classification procedures (Wallace et al., 2006; Wolf et al., 2005).

Some methodological investigations suggest interesting consequences of using LiDAR in this capacity. In Pollyea and Fairley (2012), a manufactured test target was used to identify a bias in roughness measurements corresponding to observation angle, and this was used to recommend fitting the measurement datum plane with general least squares and scanning close to the surface normal whenever possible to minimize this bias. A significant difference still remains between the best estimated value and the expected value of roughness. A possible explanation for this difference is a range measurement smoothing effect resulting from the size disparity between surface features and the instrument’s measurement beam, as suggested in other LiDAR research (Huang and Wang, 2012; Morris et al., 2008). Because applications of surface roughness in geology often seek to estimate morphological power-law relationships (Power et al., 1987; Fardin et al., 2001, 2004; Renard et al., 2013), scale-dependent systematic errors in surface height statistics are potentially disruptive to the application of LiDAR methodology to geological applications.

The main objective of this paper is to investigate the disparity between roughness measured for manufactured targets and expected values obtained through numerical simulation in order to estimate the influence, if any, of the aforementioned scale-related smoothing effect on measurements of surface geometry. As the hypothesized smoothing effect is controlled by instrument parameters, range to the measured surface, and the morphology of the surface itself, the specific results developed for the instrument and measurement conditions in this study may not apply in all circumstances. Therefore, a detailed explanation of the developed procedure is provided for the benefit of TLS users interested in understanding how the smoothing effect will influence their work. Likewise, though the smoothing effect is studied here for the purpose of surface roughness, it directly influences measured range values and therefore has implications for TLS measurements of small-scale geometry of natural surfaces.

Two test targets with different sizes of uniform hemispherical asperities were constructed and scanned at ranges from ∼3 m to 50 m. Using the known target dimensions, a mathematical model of each target was generated in order to calculate expected roughness values. The procedure for estimating rangefinder noise is demonstrated here, and we used the results to improve the accuracy of the point cloud simulation. We introduce the concept of effective radius, describing the area on the target surface sampled by the instrument with a single measurement pulse under ideal conditions, along with a technique to estimate it.

The point cloud data were collected with a Riegl VZ-400 terrestrial laser scanner. The instrument operates on the pulsed time-of-flight measurement principle, measuring range from the traveltime of a discrete pulse of 1550 nm coherent light. The energy of the return pulse is sampled at a very high frequency, and the digital record is used to interpret the exact arrival time of the pulse. The manufacturer specifications indicate a single point accuracy and repeatability of 5 and 3 mm, respectively. Beam divergence is specified as 0.3 milliradians (mrad). Distances from 0.5 m to several hundred meters may be recorded (Riegl Laser Measurement Systems, 2011). Test targets were scanned at ranges from 3 to 45 m, at center-to-center sample spacing not greater than 4.4 mm. A scan frequency of ∼80,000 points per second was reached. All scans were conducted from a view angle as close to the surface normal as possible, typically less than 5° away. All simulations sampled the theoretical target surface at 1 mm intervals.

METHODOLOGY

Constructing and Scanning Targets

A simple test target design (Fig. 2A) similar to that used by Pollyea and Fairley (2012) was chosen to simulate a natural surface with uniform textural scale. A 6 × 13 square grid of 85 mm edges was centered on a 0.609 × 1.244 m plane (target plane). Hemispherical asperities were placed at 36 randomly selected nodes. Two targets were built with identical node patterns, with 19 mm (target A) and 32.5 mm (target B) asperities. These roughness element sizes were chosen to represent surface texture in the size range corresponding to pebble-sized clasts or small asperities on a fault scarp, as seen in a common TLS campaign (Sagy et al., 2007; Heritage and Milan, 2009). Sheets of 19 mm medium-density fiberboard equal in dimensions to the target plane were used to construct the targets. Using a power drill with spade bit, 38 mm holes were cut in a board for target A, and 19 mm radius table tennis balls were set to the bottoms of the holes and secured with adhesive caulk. To construct target B, 32.5 mm radius plastic toy balls were carefully cut in half and fixed to the surface of a board with adhesive caulk. To avoid the influence of range walk, or the systematic bias of range measurements correlated with return energy (Pfennigbauer and Ullrich, 2010), it was necessary to paint the plastic hemispheres and metal hardware on the targets flat white, ensuring consistent return energy values. The test targets were scanned at high resolution at ranges up to 45 m. Point clouds were cropped manually to remove points influenced by edge effects (Fig. 2A) (Boehler and Marbs, 2003).

Calculating Roughness

Roughness for point clouds of targets A and B, both scanned and simulated, was calculated by fitting a plane to the cropped point cloud via a least-squares adjustment. The standard deviation σ (in mm), 
graphic
of the set (of size N) of all orthogonal residual distances d (in mm) of points to the fitted plane was chosen to numerically represent the roughness of the surface.

Measuring Beam Effective Radius

An empirical method was designed to estimate the extent of the target surface sampled by the scanner in a single pulse, obviating the complications of simulating the rangefinder hardware (Pesci et al., 2011). Although this measure yields only a relative description of the quantity being measured, by definition it accounts for the effects of other aspects of the range calculation process (including beam power distribution and receiver response characteristics), which may be impossible or impractical to measure and simulate (Wagner et al., 2006). Since aspects beyond the strict diameter assigned to the laser beam are incorporated into the measurement, it will be referred to here as effective radius (ER). The scanner ER was estimated by a process adapted from Lichti (2004).

A 0.6-mm-diameter metallic wire was stretched taut at an angle ∼45° from the scanner vertical axis in a wooden frame and painted flat white to ensure that consistent return energy values and accurate range information were collected. This wire was scanned at high resolution, at ranges from 3 to 38 m. A line was fit to the cropped wire point cloud via the method of least squares, and orthogonal point residual distances to the plane defined by the fit line and scanner origin were measured. The mean of the top 2% greatest-magnitude residual distances was assumed to represent the effective radius of the measurement beam. This selection threshold, halfway between two and three standard deviations on the normal distribution, was chosen to define the largest possible radius while mitigating the influence of any outliers. Examination of wire profiles captured by other instruments may require a higher or lower threshold to obtain an accurate measurement of the effective radius, due to presence or absence of outliers.

Measuring Rangefinder Noise

Because any random error in the scanner’s rangefinder apparatus would be indistinguishable from actual surface roughness, it was necessary to characterize the rangefinder noise under experimental conditions. A 19-mm-thick sheet of medium-density fiberboard of identical dimensions to the target plane was selected for its visual smoothness and flatness. The board was scanned at high resolution at ranges from 3 to 45 m. In total, 180,000 points in the cropped point cloud were selected at random, and the point cloud was separated into eight equal segments (Fig. 2B) to rule out the possibility of slight, long-radius curvature of the board contributing to measured roughness. A plane was fit to each segment via a least-squares adjustment, and the standard deviation of the vector composed of the residual distances of each point to its respective plane was calculated in order to describe rangefinder noise.

Point Population Normalization

Due to the correlation of the ratio of points representing asperities and backing surface with both range (due to loss of points to edge effects in physical target scans; Fig. 2A) and asperity radius (in both simulation and physical target scans), it was necessary to normalize the point population splits of both real and simulated point clouds. LiDAR edge effects, or “mixed pixel” returns, are an artifact of the scanner’s method of operation caused by the measurement beam spanning an edge with a significant offset along the beam propagation direction (Boehler and Marbs, 2003). In order to modify the relative point populations, points were classified into asperity and background populations. Points on the background were removed at random from the data set until a target ratio of some number of background points to one asperity point was reached. For simulated point clouds, class membership was known a priori, and in scanned point clouds, membership was decided on the basis of orthogonal residual distance to a plane fit to the data set. The roughness of the target point cloud was then calculated by fitting a new plane to the population-normalized data set and calculating the standard deviation of orthogonal point residuals.

Simulated Target Roughness

Two-dimensional (2-D; x,y) rasters with 1-mm-square cells and side lengths corresponding to measured dimensions of the targets were used to represent the target surfaces in a numerical simulation. Height values (z, in mm) for each of ∼757,000 cells were calculated based upon proximity to known asperity coordinates (xn,yn) with the following formula: 
graphic
where r describes the node radius of 19 or 32.5 mm. Points greater than r from any node location were assigned a z value of 0 mm. Random numbers on the Gaussian distribution, of zero mean and standard deviation equal to that calculated for rangefinder noise, were subsequently added to the z values. Point populations were normalized using a priori class membership information before roughness was calculated.

RESULTS AND DISCUSSION

Scanner Parameters: Effective Radius and Rangefinder Noise

A piecewise linear function was found to describe the change in effective radius over range (Fig. 3A). In the near field, the effective radius increases slowly, and in the farfield, it increases comparatively rapidly. Two linear trends were fit to the data via least-squares method, separated by a break point at 12 m. The precise location of the break point was picked manually, as its placement within a few decimeters is expected to have minimal influence on the results. Distinct trends are present in the curvature-reduced rangefinder noise data (Fig. 3A), but their fluctuations are small and can only account for a very small change in roughness values, as demonstrated in simulation (Fig. 3B). The mean of all measurements of rangefinder noise, σ = 1.6 mm, was chosen as the descriptor of rangefinder noise to be used in simulated target point clouds.

Point Population Normalization and Simulated Roughness

To normalize scanned point cloud roughness for range and asperity size, two methods for determining class (background board or hemispherical node) membership for each point were evaluated on simulated point cloud data using known class membership information. Targets A and B were simulated in 30 trials with 1 mm point spacing and 1.6 mm range noise. The residual distance of points to the least-squares plane was the sole criterion available for classification. The K-means method (MacQueen, 1967; Bui and Moran, 2001) was used to solve the one-dimensional clustering problem with two clusters, corresponding to background and node points. The K-means method operates by iteratively moving data points between clusters to reach the lowest possible sum of all Euclidian distances from all points to their respective cluster centroids.

Executing the K-means method on the data with additional clusters did not improve classification performance. An inspection of clustering results revealed that the K-means algorithm misclassified a significant proportion of points, so an additional logical step was employed to reclassify points from the background cluster based upon within-cluster residual distances. All points with residuals greater than three rangefinder noise standard deviations toward the scanner were shifted to the node point cluster. Note that here a priori refers to known class membership carried over from simulated point cloud generation, and a posteriori refers to the assignments made by one of the clustering methods. Using a priori class membership information, two tests evaluated classification performance: X scores (Table 1) represent the percentage of points belonging to the a priori node class correctly assigned to that a posteriori class, and Y scores represent the percentage of points assigned to the a posteriori node class belonging to the a priori background class.

Native population ratios were calculated for simulated point clouds using known class membership counts. Target A’s population ratio was found to be approximately 17.75, and target B’s ratio was approximately 5.5, so a target population ratio of five board points to one node point was chosen for both targets. Roughness values were calculated for simulated point clouds using all three normalization methods. Both clustering methods yielded overestimates of normalized roughness values calculated from data sets normalized with known class membership (Table 2). Roughness values of population-normalized simulated point clouds for targets A and B were calculated for data sets generated with and without simulated rangefinder noise of σ = 1.6 mm in 30 trials each. For target A, the addition of rangefinder noise increased measured roughness 4.7% from a mean of 5.07–5.31 mm. For target B, roughness increased 1.7%, from 8.64 to 8.79 mm.

Scanned Point Cloud Roughness

Measured roughness values arrived at postnormalization are displayed in Figure 4. Note that the dependent axis shows roughness as a percentage of the expected value from simulation. At a break point near 10 m (Fig. 4, left column), K-means 3σ normalized scanned roughness values peak, overestimating expected roughness values by up to 2%. Values overestimating roughness are expected to be a product of the normalization process, as the two classification methods tested lead to an overestimation of roughness in simulated test targets. Irregularities in test stand construction, such as improperly affixed nodes or long-wavelength curvature of the backing board may also contribute.

In Figure 4 (left column), roughness underestimation values depart from the trend expected from the point residual smoothing effect, which predicts that roughness will have a negative correlation with range. From Figure 4 (right column), two distinct phenomena appear to affect the data: one correlated with the effective radius:asperity radius ratio (the hypothesized smoothing effect), and one correlated with range. Overlap of the primary trends in Figure 4 (right column) confirms the hypothesis that the surface smoothing effect is controlled by the ratio of scanner effective radius to asperity radius. The positive correlation of roughness with range may be due to the fact that the emitter and detector in TLS systems are often separated by a short baseline, which causes their respective fields of view to overlap incompletely at close range and the instrument to record anomalous energy levels from the return pulse (Pfeifer et al., 2008). The variable separation of up to 3 percentage points between the trends in Figure 4 (right column) is expected to be caused by differences in the performance of normalization methods used with scanned and simulated data sets. Agreement in normalization performance X scores between the two target designs is higher with the K-means method, which shows the smaller gap of the two methods in Figure 4 (right column). Again, differences in construction methods between the two targets may also contribute to the gap. Due to differences in dimensional control on materials sourced for the test stands, roughness elements on target B show greater visual irregularity than the elements on target A. For ranges beyond the break point range observed for the instrument being used, roughness underestimation data presented as in Figure 4 (right column) may be used as an approximate estimator of experimental error for studies investigating roughness on known, limited scales of surface features. The data may also be modeled with a linear regression as a calibration function for TLS roughness measurements, though the presence of surface asperity scale in the function parameters makes the calibration nontrivial. Calibrating roughness data will require a methodology for accurately calculating roughness values limited to features within a given scale range in the target scene. Signal processing techniques such as decomposition in the Fourier or wavelet domain may provide a practical means for separating morphological features on the basis of scale. Ultimately, the developed procedure should be conducted on real rock surfaces with complex morphology. The results demonstrate that the apparent morphology of a given surface depends upon both the real morphology of that surface and the distance between it and the scanner. Merging two or more point clouds of the same surface captured at different ranges will not necessarily yield a more accurate depiction of the surface (and hence roughness) than the scan taken from the closest range.

Without prior knowledge of a particular instrument’s characteristics, it is therefore recommended to capture point clouds from a short range, i.e., ∼10 m if practical, and to keep the acquisition range consistent throughout the study. It is worthwhile to note that for the scanner used in this research, the manufacturer designates a minimum acquisition range of 0.5 m, while roughness measurements taken from that short of a range would significantly underestimate surface roughness. While the benefit or risk of merging point clouds from different view directions (but similar ranges) cannot be predicted from these results, it is expected that any disagreement between the morphology of two merged point clouds regardless of source will increase the estimated roughness.

Developed Methodology for Scanner Characterization

The flow chart in Figure 5 summarizes the experimental process. The physical and numerically modeled targets are shown in boxes, and the quantities and empirical parameters are given in ellipses. All test equipment used in these experiments was built with inexpensive tools and materials, following the methodology outlined. By scanning a flat target (such as a sheet of fiberboard or plywood with minimal curvature) at desired ranges, the rangefinder noise can be inspected through the residuals after a first-order plane fit on the scanned point cloud. For the scanner used in this experiment, a Gaussian distribution was found to adequately characterize the range noise, but other distributions may be more appropriate for different instruments. In order to measure the scanner effective radius, a very fine, reflective wire must be stretched tight between two supports and scanned at the desired ranges. The residuals to a line fit to the wire, projected orthogonal to both the line and the view direction, yield the data set used to characterize ER. Designing, building, and simulating a rough target are the most time consuming and sensitive parts of the process. The backing board used must be measured and prepared carefully so that its dimensions coincide with the numerically simulated point cloud. To realistically simulate the scanning process of typical TLS instruments, it is suggested to simulate the point cloud as a set of heights over a plane, on a regularly spaced rectangular grid. Random range noise (parameterized by the results of the rangefinder noise test) should be added to the point heights before roughness is calculated.

CONCLUSIONS

As geologists turn to modern remote sensing instruments such as terrestrial laser scanners for geospatial data collection, knowledge of the characteristics of the instruments used and their impact on data quality becomes important. Surface roughness statistics are used to parameterize models of natural phenomena such as stream flow and movements along faults, and the accuracy of these models depends upon the accuracy of the portrayal of fine-scale surface morphology in point cloud data. Complicating the use of TLS in a scientific capacity, systematic biases in its range measurements correlate with measurement range, instrument characteristics, and the morphology of the real surface. Point cloud surface morphology captured with a 3D terrestrial laser scanner was demonstrated to underestimate expected roughness statistic values derived from simulation. The disparity between estimated and expected roughness was found to vary with two parameters characterizing the scanner measurement apparatus. For a given asperity scale, an increase in beam effective radius will result in a decrease in measured surface roughness, and an increase in rangefinder noise will result in an increase in measured roughness.

The test instrument’s effective radius was shown to increase in a piecewise linear fashion as the distance between the instrument and the target surface increased. Rangefinder noise was also shown to vary with range; however, its influence on estimated roughness decreases with increasing real surface roughness. Scans collected at very close range showed an unexpected correlation of estimated roughness with range, but not with effective radius:asperity scale ratio. Although the results shown herein represent the particular instrument and scanning conditions tested, we present the developed procedure for further studies under various conditions. It is important to plan network geometry to maintain consistent, short (∼10 m is suggested) measurement ranges to target surfaces throughout a scanning campaign and resist the temptation to merge point clouds of the same surface from different ranges before calculating roughness, as the combined effect of the investigated biases makes roughness measurements highly dependent on scan range. With the development of techniques for isolating surface asperities in real scenes on the basis of morphological scale, the test procedure demonstrated in this paper could serve as methodology for generating calibration curves for TLS roughness studies, and it could be adapted to a general measurement range correction process. Future work will focus on the application of signal processing methodology to this end, and the development of more accurate calibration techniques utilizing real rock surfaces. The influence of registration errors on merged point cloud roughness is also an interesting case study for future research.