Begeleiding: Emiel Van Loon

THE EFFECT OF LASER BEAM INCIDENCE ANGLE AND OF TERRAIN SLOPE ON THE VERTICAL ACCURACY OF LASER ALTIMETRY

CRISTI GHERASE

BACHELOR THESIS

AARDWETENSCHAPPEN

UNIVERSITEIT VAN AMSTERDAM

BEGELEIDING: EMIEL VAN LOON

Abstract

This study assesses the possible effects of the terrain slope and of the laser beam incidence angle on the accuracy of the lidar measurements. Ground survey using a total station was performed in 2 small areas in the Amsterdamse Waterleidingduinen. The surveyed points were interpolated using ordinary kriging and the lidar vertical error was calculated as the difference between the elevation of the obtained surface and the lidar points. Using the height of flight and the azimuth of the laser beams, together with the slope angle and the slope aspect of the points where the beams reach the ground, the incidence angle was obtained for each lidar point. The incidence and slope angles corresponding to the lidar points were then used in a linear regression analysis as predictors for the vertical error of the lidar measurements. The slope angle was found to be positively correlated with the absolute lidar vertical error due to the horizontal displacement of the lidar points. The same holds true for the incidence angle, since its value is partially dependent on the slope angle. A very weak positive correlation between the incidence angle and the under estimation of the true surface by the lidar measurements was also found, but the result is not conclusive and further research is needed for its confirmation.

1 TABLE OF CONTENTS

Table of Contents...... 2

1 Introduction...... 3

1.1 Basic principles and the AHN 2000...... 3

1.2 The influence of the slope angle on the accuracy of Lidar measurements...... 4

1.2.1 Horizontal displacement...... 5

1.2.2 Laser footprint distortion...... 5

1.2.3 Laser impulse distortion...... 5

1.3 The effect of the incidence angle on the accuracy of Lidar measurements...... 6

1.4 Research goal...... 6

2 Methods...... 7

2.1 Locations...... 7

2.2 Total station specifications and acquisition of the ground survey data...... 8

2.3 Calculation of the incidence angle...... 9

2.4 Interpolation details...... 12

2.4.1 Interpolation Parameters...... 12

2.4.2 Standard error map...... 14

2.4.3 The slope and aspect rasters...... 15

2.5 Elevation Values...... 16

2.6 Lidar flight strip misalignment...... 17

3 Lidar strip adjustment...... 18

4 Results...... 20

5 Conclusions and discussion...... 27

References...... 29

1 INTRODUCTION

2 Lidar (acronym for Light Detection and Ranging) technology has quickly become the main method for the elevation mapping of terrain surfaces. The accuracy of the of the measurements, the independence on light and largely on weather conditions, the short time it takes to collect and process data for large areas and the fact that the data is immediately available in digital format as xyz coordinates are among the reasons for the method’s increasing popularity in recent years. Application of the technology include topographic mapping, coastal management, environment monitoring, hazard areas mapping, vegetation height measurement and urban planning.

Although the principle of laser detection and ranging has been known since the 1960’s, it was only much later, in the 1990’s, that it began to be used for airborne terrain mapping. The main cause for the delay is that before the introduction of the GPS for civilian use in 1993 it was impossible to obtain the precise geographic coordinates of the scanned ground points (Shan 2008).

1.1 BASIC PRINCIPLES AND THE AHN 2000

The central element of a Lidar system is the laser scanner, which generates high frequency, short wave light pulses that travel in a certain direction. The emitted light bounces off any reflective surfaces it may encounter and returns to the scanner. A precise clock measures the time interval between the emission of the pulse and its return. The distance between the scanner and the reflective object is given by: distance= speed of light* travel time/2.

The laser scanner is mounted on an aircraft (usually a small airplane or a helicopter) together with a differential GPS and an INS (Inertial Navigation System) unit. While the GPS provides the absolute location coordinates once per second, the INS unit refines this information with high frequency measurements of the rotational movements of the Figure 1: discrete return scanning pattern aircraft (yaw, pitch and roll). (from Lewis 2007)

This study uses data from the “Algemeen Hoogte Model van Nederland 2000” (AHN 2000).

The altimetric data of the AHN 2000 was acquired using a discrete return scanning system, in which light beams were sent under a varying angle (called scanning range) across the direction of flight, creating a zigzag pattern of points on the ground (fig. 1). A very large scanning range coupled with a high flying altitude covers a wider area during one flight at the expense of decreased accuracy, while a very narrow range (pulses sent perpendicular to the ground) is more accurate but impractical because it requires a large number of flights to cover a certain area.

The laser scanners used by discrete return systems emit conically shaped laser beams, whose diameter increases with distance from the aperture (divergence). The projection of the beam on the ground is called the footprint. For a flying altitude of 1000 meters and a divergence angle of 0.25 miliradians – values characteristic of the Optech ALTM 1020 scanner with which part of the AHN data was obtained – the resulting footprint diameter is approximately 25 cm.

3 Different heights may be detected within the footprint area of a single beam (Fig. 2) If the beam falls on the edge of an object above the ground, such as a construction or a tree, that part of the beam (called the first return) will be recorded earlier by the sensor because the distance it travels is shorter.

Discrete laser sensors are able to distinguish between different returns of a single received pulse. If the purpose of the measurement is the height of the ground (as it is for the AHN), the last return - which corresponds to the lowest height - is usually selected for the data set because it has the highest probability of having reached the ground.

The “Productspecificatie AHN 2000” document which accompanies the lidar dataset states that the lidar measurement have a random error (noise) with a standard deviation of 15 cm. However, areas with a standard deviation of up to 35 cm have been discovered during tests (Kuijlaars, 2002).

This error is partly due to the technical limitations and the calibration of the Figure 2: the principle of multiple return different instruments (laser sensor, GPS, INS) and partly due to external factors, scanning (from Evans, 2006) such as the presence of aerosols in the atmosphere (which scatter the laser beams and weaken the signal), the nature of the reflective surface (buildings and dense vegetation covers prevent the pulses from reaching the ground) and the terrain topography.

1.2 THE INFLUENCE OF THE SLOPE ANGLE ON THE ACCURACY OF LIDAR MEASUREMENTS

A multitude of recent studies have tried to investigate and quantify the various factors that limit the laser scanning accuracy. The large majority of these studies have focused on the effects of the vegetation on the measurements, because the presence of vegetation is the most significant cause of error and because the mapping of the vegetation itself is an important application of the Lidar technology .

By comparison, the topography of the ground as a possible factor affecting lidar accuracy has received little attention. A few studies (Adams 2002, Su & Bork 2006) have tried to address this issue alongside with the effects of vegetation but produced inconclusive or mixed results, as the measurements were performed in heavily vegetated areas with limited relief.

The reasons for assuming that slope angle has an impact on the Lidar accuracy are the possible vertical errors caused by horizontal displacement, by laser footprint distortion and by laser impulse distortion.

1.2.1 HORIZONTAL DISPLACEMENT

4 Unlike on flat terrain, inaccuracies in the horizontal position of a scanned point on a slope can readily lead to errors in elevation (Fig. 3). This dependency is expressed by the formula: elevation error = horizontal displacement * tan (slope angle).

This is the maximum vertical error which occurs when the horizontal displacement happens along the direction of the slope. A generalized formula is:

Vertical error= horizontal displacement *cos (β) * tan (slope angle), where β is the angle between the horizontal Figure 3: The effect of horizontal displacement on elevation displacement line and the horizontal component of the accuracy (from Hodgson, 2004) direction of the slope. The [horizontal displacement *cos β] is the projection of the horizontal displacement on the horizontal component of the direction of the slope.

1.2.2 LASER FOOTPRINT DISTORTION

The other reason for assuming slope influences Lidar accuracy is related to the distorted laser footprint on a sloping surface. At a right angle of incidence, the laser footprint has the shape of a circle and all the points on its circumference are at the same distance from the laser sensor. On a slope, the angle of incidence is less than 90 degrees and the laser beam projection on the ground approximates an ellipse (the intersection of a cone with an inclined plane). In case the system still distinguishes between a first and last return for such small height differences (comparing to the height of a tree or building), then the last return will be given by the extremity of the ellipse that is lower on the slope, while the horizontal coordinates will be those of the center of the beam. The resulting elevation error is expressed by the formula: Z error = footprint diameter/2 x tan (slope angle). For a 25 cm footprint and a 35 degree slope the resulting vertical error is nearly 6 cm.

The height of a point on a slope is in this case consistently underestimated, and the difference from the real height is directly proportional to the slope angle.

1.2.3 LASER IMPULSE DISTORTION

5 Even when the sensor records the entire received pulse as a single return, the angle of the slope may still affect the ranging accuracy. The distribution of the energy in a laser pulse is Gaussian in both the beam’s length and width (Lohani 2008). If, ideally, the surface the pulses encounters is perfectly flat and perfectly reflective and the angle of incidence is exactly 90 degrees, then the shape of the reflected pulse will be identical to that of the emitted pulse (the black line in Fig. 4) In reality the reflective surfaces, and this is especially true of the soil, never meet these conditions, and the shape of the returned pulse (represented as coloured lines in Fig. 4) becomes distorted in proportion to the surface Figure 4: Ideal (black) and distorted reflected laser pulses (in inclination and roughness. The distortion of the received pulse colour) (from Lohani 2008). negatively affects the accuracy of the ranging.

1.3 THE EFFECT OF THE INCIDENCE ANGLE ON THE ACCURACY OF LIDAR MEASUREMENTS

The slope angle is an appropriate variable to study the effect of laser footprint distortion or laser impulse distortion only when the vertical angle under which the beam is emitted is zero. This is only the case for the middle line of the flight strip. For the rest of the strip, a more suitable variable is the incidence angle of the beam, which is a combination of the beam’s vertical angle and direction, and the slope angle and orientation (aspect). The distortion of the laser’s footprint and impulse are amplified on slopes inclined in the direction of the beam, and reduced on slopes facing opposite to the beam’s direction.

As far as known, there are no studies that deal with the effect of the incidence angle on the accuracy of the lidar measurements.

1.4 RESEARCH GOAL

The purpose of this study is to investigate the influence of the lidar beam incidence angle and of the slope angle on the accuracy of the lidar elevation measurements.

Having a clear understanding of these factors is especially important for the application of the lidar measurements to the study of geomorphologic processes and to coastal monitoring, as well as for the elevation mapping of terrain with pronounced relief.

6 2 METHODS

Ground survey with a total station was performed in 2 small areas in the Amsterdamse Waterleidingduinen. The method of determining the influence of slope and incidence angle on the lidar accuracy consists of interpolating surfaces for both sites from the ground survey data points and obtaining the elevation differences between the surfaces and the lidar points. Using the height of flight and the azimuth of the laser beams, together with the slope angle and the slope aspect of the points where the beams reach the ground, the incidence angle was obtained for each AHN data point. The incidence and slope angles corresponding to the AHN points were then used in a multi- linear regression analysis as predictors for the over or underestimation of the ground surface by the lidar measurements.

2.1 LOCATIONS

The locations chosen for ground survey had to meet a set of conditions:

- The vegetation cover had to be very low at the time when the AHN data was acquired (winter 1997-1998) in order to minimize the influence of vegetation on the lidar data. Ideal areas were deemed those covered with moss and/or short grass. Ammophila arenaria (Helm gras) was avoided, as this is a pioneer species which usually indicates a previously unvegetated area. Blowout sand areas (either in the past or present) with no vegetation were unsuitable because of their erodability. A vegetation map from 1997 was used to identify suitable areas.

- Various slope classes had to be represented at the locations, ranging from flat to the maximum angle of repose for sand (up to 35 degrees).

- The orientation of the slopes (or aspect) had to be diverse as well in order to obtain a wider range of incidence angles.

- The surface had to be relatively uniform, as altitude differences on a small scale (such as rabbit holes or erosion caused by large grazers on soft ground) would not be reflected in the interpolated surface due to the relatively large (1m to 1.5 m) sampling spacing (the distance between 2 measurements).

The task of finding locations in the AWD that met these criteria proved difficult, and allowances had to be made for each condition. The Google maps aerial photo in fig. 5 shows the location of the 2 areas in which ground survey was conducted. Fig. 5: Google Maps aerial photo of the AWD showing the 2 fieldwork locations

7 Site A is largely covered with moss. The Wood Small-reed (duinriet) that grows here is short and infrequent and is unlikely to affect the lidar measurements.

Site B is less than ideal for this type of investigation, as it is predominantly covered with low (10-15 cm) grass. Young small bushes are also growing in patches, and scattered dry branches suggest that bushes have grown here in the past as well.

The 2 sites have altitudes between 13,3 – 22.2 m and slope angles ranging from 0 to 36 degrees, with a mean angle of 16 degrees. The slopes of site A face mostly towards the East, while the largest part of site B is inclined towards the North.

The illustration below show the surfaces of the 2 sites interpolated from the ground survey points (represented as the blue dots). Their total area is 870 m². The surface produced from the lidar points serves as the background. For a better impression of the area’s relief a 1.5 vertical exaggeration has been applied. The white dots represent the ground survey locations.

Fig. 6: Surfaces for site A (left) and site B (right) produced by interpolating the total station points represented by the white dots. The surface interpolated from the AHN points serves as background. x1.5 vertical exaggeration has been applied.

2.2 TOTAL STATION SPECIFICATIONS AND ACQUISITION OF THE GROUND SURVEY DATA

The ground survey was carried out with a Nikon DTM 520 total station. A total station calculates the relative positions of surveyed points by measuring the angles (horizontal and vertical) and distances between the instrument itself and the reflective prism placed at the target points . In order to determine absolute locations, the total station needs a minimum of 2 points of known coordinates (usually the point above which the instrument is set and a 2nd point called backsight), which it uses to construct a x/y/z Cartesian coordinate system axes corresponding to the northing/easting/altitude (see fig.7).

8 Fig. 7: Basic total station coordinates setup. From

The total station used for the ground survey is Nikon DTM 520 model. It has an angle measurement precision of 3", a range of up to 2000 m and a ranging accuracy of ± (2mm + 2ppm * distance). This means that the instrument is capable of measuring the angle between 2 points situated 2000 m away from the total station and less than 3 cm away from each other. The maximum error of the measured distance between one of the points and the instrument would be 6 mm.

In practice, however, this accuracy is often decreased. It has been my experience that at distances above 100 m, less than ideal atmospheric conditions (such as heat haze or dust) combined with an optical phenomenon called parallax (by which the targeting cross-hair shifts according to the position of the eye relative to the telescope) make it difficult to precisely locate the target center of the reflective prism. Other sources of inaccuracy are related to the placement of the total station and of the reflective prism above reference points or points to be surveyed, and to the measurement of their height above the ground.

In order to assess the accuracy of the instrument, the altitude of several points and the distances between them were measured from different positions. The total error of the measurements is estimated to have a maximum of 3 cm and a standard deviation of under 1 cm, both vertically and horizontally. By comparison, the data points obtained with the AHN lidar system have a random error (noise) with a standard deviation of 15 cm (Rijkswaterstaat Adviesdienst Geo-informatie en ICT 2000). Therefore, the ground survey data can be safely used to test the accuracy of the lidar data.

2.3 CALCULATION OF THE INCIDENCE ANGLE

As mentioned in the introduction, the lidar points form a zigzag pattern on the ground, across the direction of the flight. The line of flight projected on the ground coincides with the middle of the strip formed by the points on the ground. Knowing the flight altitude and the shortest distance between the lidar points and the middle of the strip, it is possible to calculate for each point the vertical angle under which the laser beam was emitted (the angle the beam makes with the zenith), and the azimuth of the beam (the horizontal angle which the beam’s projection on the ground makes with the North axis of the northing/easting coordinate system).

9 The azimuth of the line between two points P1 and P2, with the direction of the line being P1 to P2, depends on the quadrant in which point P2 lies relative to P1, and can be calculated as follows:

- For the NE quadrant, azimuth = arctan[(P2x-P1x)/(P2y-P1y)]

- For the SE quadrant, azimuth = 180° - arctan[(P2x-P1x)/(P2y-P1y)]

- For the SW quadrant, azimuth = 180° + arctan[(P2x-P1x)/(P2y-P1y)]

- For the NW quadrant, azimuth = 360° - arctan[(P2x-P1x)/(P2y-P1y)]

(P2x-P1x) and (P2y-P1y) are the projection of the horizontal distance between the points on the northing and easting axes, respectively.

The value of the incidence angle of a laser beam is defined in this study as the angle between the beam and the normal to the terrain plane through the point where the beam intersects the terrain. Thus, a beam that falls perpendicularly on the ground will have an angle of incidence of 0 degrees (rather than 90 degrees).

Fig. 8: Incidence angle (source: Wikipedia)

Fig. 9 was used in the calculation of the incidence angle Fig. 9: Geometrical determination of the incidence angle for a single lidar point (O). The red triangle represents the terrain surface, whose slope angle is given by ∡OSR. The plane on which the points R, S and L lie is the sea level, serving as a horizontal base. The line OQ is the laser beam that falls on the surface. The line RL is the beam’s projection on the horizontal plane, and point P lies in the middle of the strip, at height LP above sea level. Line ON is the normal to the slope, and the triangle MNQ is parallel to the horizontal plane RLS. The line MN gives the aspect of the slope (the direction in which the slope faces), while the line QN gives the horizontal direction of the laser beam, from which its azimuth can be calculated.

The known parameters are the coordinates (xyz) of the lidar point and of the nearest point P on the middle line of the strip (from which the azimuth of the line OP can be calculated), the length of the line QP ( the altitude of flight), and the angle and aspect of the slope. From these values, the length of the sides of the green triangle QON can be calculated, from which the angle ∡QON can be obtained. The angle ∡QON, between the beam and the normal to the slope, is the actual incidence angle of the beam OQ with the slope.

10 A hillshade map produced in ArcMap for the entire site A offers a way to visualize the values of the incidence angle. The hillshade is an illumination map, with the azimuth and the vertical angle (scan angle) of the beam as parameters for the source of light. The values of a hillshade map lie between 0 (the darkest areas, where no light falls directly) and 255 (the brightest areas, where the light falls perpendicular to the surface).

The values of the map only approximate the true values of the incidence angles, because each location of this map receives light under the same angle, while the vertical angle (the scan angle) of the lidar beams varies with each point. However, the 2 values are comparable, and the analysis using hillshade values produced very similar results to the Fig. 10: Hillshade map of site A one using the incidence angle values.

Because manually measuring the distance between each lidar point and its corresponding nearest point on the middle of the strip is time consuming, the simplification has been made that the position of the laser scanner remains fixed (determined according to the center of the site) for all the points belonging to one site, while in reality the aircraft carrying the scanner is in continuous flight. The error introduce in this way in the value of the incidence angle is negligible (less than 1/10 of a degree). The scan angle does vary within the 2 sites in proportion with the distance from the scanner, as in reality.

The exact flight altitude is not known, but is specified by the AHN administration to be between 500 m and 750 m. Incidence angles have been calculated for an average flight altitude of 625 m.

The incidence angles obtained range between 6.28° and 44.45° for site A and between 0.49° and 48.38° for site B, with an average of 20.36° and 28.55°, respectively.

2.4 INTERPOLATION DETAILS

Contrary to the findings reported by Su & Bork (2006), the ordinary kriging interpolation method produced a lower cross-validation RMSE – with an average for both sites of 0. 046 m – and a lower maximum absolute error compared to the inverse distance weighted (IDW) method, which had an RMSE of 0.09 m for both sites.

11 A possible explanation for this may be the difference in topography: the two sites examined here have much higher slope angles (an average of 17 degrees, with a maximum value of 36 degrees) than the terrain subject to the study by Su & Bork (which reports slope angles between 2 and 4 degrees). At higher slope angles, the weights based on the degree of spatial autocorrelation that the ordinary kriging uses make it a better predictor than the IDW method, which uses weights based on distance alone. Fig. 11: Artifacts on a surface Moreover, the surface produced with the IDW method exhibited the so called produced using inverse distance “bull’s-eyes” – small bumps and hollows around the data points, as can be seen in weighted interpolation figure 11.

All surfaces used in this study were produced in ArcMap.

2.4.1 INTERPOLATION PARAMETERS

Below is a summary of the chosen ordinary kriging interpolation method. The interpretation of the various statistical parameters and plots was made with the help of the “Using ArcGIS Geostatistical Analyst” book by Johnston et al. (2001).

Table 1: Interpolation parameters

Site Model Type Range Nugget Partial sill Lag size Number of lags Neighbours

A Gaussian 9.36m 0.0012m 1.22m 0.79m 12 5

B Gaussian 8.18m 0.0012m 1.28m 0.69m 12 5

Figure 12 shows the fitted semi-variograms:

g g 2.7 3.27

2.16 2.62 1.62 1.96

1.08 1.31 0.54 0.65

0 1.19 2.37 3.56 4.74 5.93 7.11 8.29 9.48 0 1.03 2.07 3.1 4.14 5.17 6.21 7.24 8.28 Distance, h Distance, h 12 Fig. 12: Fitted semi-variograms for site A (left) and site B (right)

Cross validation was used to evaluate the different models and select the most appropriate one. Obtaining mean prediction errors close to 0 and low RMSE values is essential, but choosing a model that also gives an average standard error close to the RMSE value and a RMS Standardized close to 1 is equally important. The last two values are used to assess the uncertainty of the predictions.

While the RMSE value mostly stayed under 0.08 m, changing the various parameters of the semi-variogram and the search neighbourhood drastically altered the Root-Mean-Square Standardized value.

The results of the cross validation are given below:

Table 2: Cross-validation results

Site Mean Root-Mean-Square Average Standard Error Root-Mean-Square Standardized

A 0.0004 0.04293 m 0.04316 m 1.001 m

B 0.0014 0.05048 m 0.05249 m 1 m

The scatter plot of the predicted vs. measured values show the points on a straight line (the regression functions are y=1.001*x + 0.017 for site A and y= 1.000 * x + -0.002 for site B), which means that the interpolation method does not under- or overestimate values for certain ranges (such as high values tending to be underestimated and low values overestimated).

Fig. 13: Scatter plot of the predicted vs. measured values for site A (left) and site B (right). The Q- Q plots below (fig. 14) show the quantiles of the standardized error (the difference between the predictions and the measured values divided by the square root of the kriging variance) against the corresponding quantiles of a standard normal distribution. The fact that the scatter points for site A approximate the straight dashed line indicates that the prediction errors are normally distributed. For site B, the distribution of prediction errors shows positive skewness. The lower accuracy of the predictions is also reflected by the higher RMSE value compared to that of site A. This is perhaps caused by the shape of the sampling area, which is more elongated and irregular than site A.

13 Fig. 13: Q-Q plots of the interpolation standardized error for site A (left) and site B (right)

2.4.2 STANDARD ERROR MAP

Standard error maps of the kriging predictions were produced for both sites. These show the degree of uncertainty in the z values of the predicted surface and were used to filter out AHN points at less reliable locations of the map. The prediction standard error maps for the 2 sites are shown below (fig. 15).

Fig. 15: Kriging standard error maps for site A (left) and site B (right). The white dots represent the AHN points selected for analysis. The values of this map allow the estimation of the intervals into which the true values of the surface are likely to fall. Because the prediction errors are normally distributed (as seen in the QQPlot), around 95 % of the time the true surface will lie between the interval given by the predicted value ± 2 times the standard error value.

The values of the maps range from 0.047 m to 0.086 m. AHN points in areas with a standard error larger than 0.062 m were excluded from the analysis.

By comparison, the standard error values of the surface interpolated from the AHN points lie between 0.185 and 0.225 m in the areas where the point density is complete (where no points have been filtered out as vegetation). This is due to the different point densities of the two datasets – approximately 1 point every 4 m for the AHN and 1 point every meter for the ground survey data.

14 2.4.3 THE SLOPE AND ASPECT RASTERS

A grid cell size of 20 cm was used for the slope and aspect rasters. From these rasters the slope angles and aspect values were extracted at the location of the lidar points used in the analysis. Figure 16 shows the slope and aspect maps for both sites.

AsFig. the 16: illustrationsSlope (left) and show, aspect the(right) slope maps angle for site values A (above) and and distributions site B (below) are similar for the two sites, but the orientation of the slopes is very different: site A has mostly an east aspect, while site B a north and south aspect.

Although within the established ranges for the interpolation standard error values, a total of three lidar points were excluded from the analysis because they were situated right on the edge of the 2 areas, where the slope angles – and therefore the slope aspect – cannot be well determined. The reason for this is that on the edge of the map the slope values are calculated only from the neighbouring raster cells within the map, while the elevation from all neighbouring cells (in all directions) should be used for a reliable determination of the slope.

2.5 ELEVATION VALUES

15 The results of this study are based on 67 points from one lidar flight strip. Initially, 31 more points belonging to a second flight strip were included, but due to displaced horizontal coordinates they were left out from the results – see chapter 2.6, “Strip misalignment”.

The elevation values were extracted directly from the geostatistical layer at the location of the lidar points. In this way, the added inaccuracy due to cell size when using an elevation raster was avoided.

The differences in value between the interpolated surface elevations and the elevations given by the lidar points will be called “lidar errors”, because the assumption has been made that the total station measurements reflect the true value, while the lidar is the method whose accuracy is tested. In order to avoid confusion, the term “residuals” will be used exclusively in connection to the regression analysis.

For site A, the lidar error mean is 0.195 m, indicating an underestimation of the surface, and for site B the mean is -0.128 m, indicating an overestimation of the surface. Taken together, the 2 sites have an error mean of 0.0615 m.

The Root Mean Squared error has also been calculated according to the formula:

n RMSE=√(∑ i=1(Zlidari-Ztsi)²/n) where Zlidar is the altitude measured with the lidar system, Zts the altitude measured with the total station, and n is the number of points for which the difference has been calculated.

The RMSE for site A is 0.2398 m and for site B is 0.2475 m.

2.6 LIDAR FLIGHT STRIP MISALIGNMENT

Site B is situated in an area where 2 flight strips of lidar points overlap. The surface produced by interpolating the points from both strips showed a strange pattern of artifacts in the overlap zone (fig. 17 ).

It soon became apparent that the bumps and hollows in this surface were the result of horizontal misalignment of one, or possibly both flight strips. By juxtaposing in quick succession the surface obtained only from one strip with the surface obtained from the other, a small animation was produced in which the terrain features would jump back and forth all in the same direction. Simply on a visual basis it is estimated that the 2 strips are shifted horizontally relative to each other by a distance of 1.5 to 2 meters.

31 points from the northern strip are within the extent of the ground surveyed area. Their RMS error obtained by comparison

Fig. 17: Surface artifacts as a result of flight strip misalignment with the total station measurements is exceedingly high, 0.46 m as opposed to 0.24 m for the southern strip, and 12 points underestimate the surface by more than 0.5 m. Because most of the 31 points are situated on north facing slopes, it was impossible to determine with precision the magnitude and direction of the shift in relation to the real coordinates. Therefore, the points belonging to northern strip were not used in the analysis. The results of this study are based on the 67 remaining points from the other flight strip.

The Rijkswaterstaat “Kwaliteitsdocument Laser Altimetrie Noord Holland ” that accompanies the AHN data mentions GPS initialization errors and multipad effects – the reception of indirect, reflected signals – as sources of strip misalignment, together with calibration errors of the inertial navigation system unit. It also states that the strips in the dataset have been corrected, and, although systematic elevation errors are sometimes still present, the lidar points “hardly differ planimetrically from the reality” (Kuijlaars, 2001). This perhaps appears to be true on flat terrain, but on terrain with pronounced relief the planimetric shift becomes easily identifiable.

Because of the learned lesson from the case of the northern strip, the southern lidar strip also had to be checked for horizontal displacement. Using the lidar points from both site A and site B, the average lidar error per aspect class was therefore calculated. Although some aspect classes are vastly underrepresented (only 2 lidar points are found on the SW slopes, and 3 on the S slopes), the results (table 3) largely confirm the fact that horizontal displacement is also present in the case of the southern lidar strip.

Table 3: Average lidar error per aspect class

Aspect class N NE E SE S SW W NW

Average error (m) 0.073 0.184 0.188 0.144 -0.094 -0.388 -0.228 -0.232

The average value of the lidar points overestimates the surface elevation on the west, south west and north west facing slopes, and underestimates it on the slopes facing the opposite directions.

Using the lidar error obtained as elevation differences between the total station points and the surface interpolated from the lidar points offers a more detailed picture of the displacement, due to the larger number of differences. The circle of the aspect angles was divided into 18 intervals of equal size (20 degrees), and the average lidar error was calculated per interval. The polarization of the errors in 2 diametrically opposed directions largely demonstrates the presence of horizontal shift (fig. 18). However, the direction of the shift is not accurately reflected in the figure, because the extent of the shift depends on the slope angle. Fig. 18: Bar plot of the average lidar error per aspect class.

17 3 LIDAR STRIP ADJUSTMENT

The strip’s misalignment is probably the most important factor for the errors measured, so it is likely that the large vertical error resulting from misalignment will obscure or distort the effects of other factors, such as the incidence angle. In order to obtain more reliable results, a correction of the lidar error values that should compensate for the displacement had to be attempted.

The following method for correction has been devised and applied, using all the lidar points from both site A and site B. a. The lidar errors were treated as if they were solely the result of horizontal misalignment. A potential shift was calculated for each lidar point according to the formula:

potential shift = lidar error/tan(slope angle). b. Each potential shift together with its direction (the slope’s aspect for that point) form a vector. The total (potential) horizontal shift was obtained by adding up all the vectors. The length of the horizontal displacement is the magnitude of the resultant vector averaged for the number of lidar points.

However, the magnitude of the resultant vector cannot simply be divided by the total number of points. The reason for this is that, since a horizontal displacement is unidirectional, not all potential shift values should contribute equally to the length of the displacement: horizontal displacement causes the largest vertical error for points on slopes inclined in the direction of the displacement or its opposite, and zero vertical error for those on slopes perpendicular to the displacement. If a horizontal shift exists, but most of the points on which its calculation is based are on slopes perpendicular to the shift, dividing the magnitude of the resultant vector by the total number of points will underestimate the length of the shift. Estimating the direction of the shift first will allow us to derive the weights necessary for the calculation of its length.

In this correction method, all potential shift values must, unfortunately, contribute equally to the determination of the direction of the resultant vector, which will be taken as an estimate of the direction of the horizontal displacement.

The direction of the resultant vector is obtained by a similar method to that used in the calculation of the azimuth of a line. The formula depends on whether the sums of the vectors’ components on the east axis and on the north axis are positive or negative:

- If sum(east components) >0 AND sum (north components)>0,

Direction = arctan(sum (east component)/sum(north component));

- If sum(east components) >0 AND sum (north components)<0,

Direction = 180-arctan(sum (east component)/sum(north component));

- If sum(east components) <0 AND sum (north components)<0,

18 Direction = 180+arctan(sum (east component)/sum(north component));

- If sum(east components) <0 AND sum (north components)>0,

Direction = 360-arctan(sum (east component)/sum(north component)).

The value obtained for the direction of the displacement is 88.83 degrees. However, because the lidar errors were calculated by subtracting the lidar point values from the ground survey surface, the value of 88.83° actually represents the direction the ground survey surface shifted relative to the lidar points. The real direction of the lidar strip’s displacement is 88.83°+180°=268.83°, towards the west. In the rest of the study, for practical purposes, the direction of the horizontal displacement will continue to mean the angle of 88.83°.

Now it is possible to assign weights the lidar points according to their slope’s aspect divergence from this angle, using the formula: weights = absolute(cos(aspect angle-88.83°)).

Lidar points on slopes facing roughly in the direction of the shift (88.83° and 88.83° + 180°) will receive values close to 1, while those on slopes perpendicular to the shift will have values close to zero. The sum of all the weights is smaller than the number of points (67). It is only equal to that number in the extreme case when all lidar points are situated on slopes facing exactly in the direction of the shift, or the opposite direction. The sum cannot be zero, because in this case it would not be possible to determine the direction of the shift.

The length of the displacement was obtained by dividing the magnitude of the resultant vector (calculated by adding up all potential shift vectors) by the sum of the weights described above, and the value obtained is 0.642m.

Based on this value and the displacement direction, the adjusted horizontal coordinates of the lidar points were calculated. The slope and aspect values were extracted from the interpolated surface at the new locations, and the incidence angles were again determined.

The root mean square value for the corrected lidar error is 0.112 m, a dramatic improvement over the original RMSE of 0.242 m. This suggests that the correction has been highly effective.

The main drawback of this correction method (which possibly applies to all correction methods in general) is that, as mentioned above, it relies on the assumption that the lidar errors are caused only by horizontal misalignment. As a result, the elevation error of a lidar point caused for example by vegetation or simply by measurement noise will influence negatively the determination of the horizontal shift, especially if the respective lidar point is situated on a slope with a low angle. This is further amplified by the small number of lidar points (67) available for the correction, and their uneven distribution on the different slope aspect classes. It is therefore very likely that a degree of horizontal displacement still remains after the correction, or that it was introduced by overestimating the length of the displacement.

19 4 RESULTS

For the regression analysis, an extra variable was created in order to be used as a predictor which accounts for the (possible) remaining horizontal displacement. Manually increasing or decreasing the direction angle of the horizontal displacement only increased the RMSE value, which suggests that the angle of 88.83° may be accurate. Under this assumption, the extra variable for the analysis was calculated by transforming the circular aspect variable into a linear one. The new variable, which is given the name “transformed aspect”, has its highest value of 0, corresponding to the aspect angle of 88.83°, then, following the aspect circle, decreases down to 88.83°+180°=268.83° where it reaches its lowest value of -180, and then increases again to 0.

Multiple linear regression analysis was performed with all the data from both site A and B, with the corrected lidar error as the dependent variable, and the transformed aspect together with the slope and incidence angle based on the new coordinates as predictors. Table 4 below summarizes the model.

Table 4: Regression model summary. Using corrected lidar error as dependent and transformed aspect, incidence angle and slope angle as predictors.

R R Square Adjusted R Square Durbin-Watson D-W test sig. F statistic sig. Lilliefors test sig.

0.522 0.273 0.238 2.128 0.8318 <0.005 0.3098

The R square value indicates that the model accounts for around 30% of the variation found in the corrected lidar error, and the low p-value of the F statistic (<0.005) shows that the model’s ability to explain this variation is not due to chance. The significance values of the Durbin Watson test and of the Lilliefors test mean that the regression residuals do not exhibit autocorrelation and that they are normally distributed, which strengthens the model’s validity.

Table 5: Regression coefficients. Using corrected lidar error as dependent and transformed aspect, incidence angle and slope angle as predictors.

Model Unstandardized Standardized t Sig. 95,0% Confidence Interval Collinearity Coefficients Coefficients for B Statistics

B Std. Error Beta Lower Bound Upper Bound Tolerance VIF

(Constant) ,036 ,031 1,156 ,252 -,026 ,097

Slope angle -,003 ,002 -,235 -1,445 ,154 -,008 ,001 ,442 2,262

Incidence Angle ,003 ,002 ,327 1,895 ,063 ,000 ,006 ,393 2,542

20 Model Unstandardized Standardized t Sig. 95,0% Confidence Interval Collinearity Coefficients Coefficients for B Statistics

Transformed ,001 ,000 ,563 4,769 ,000 ,001 ,001 ,840 1,190 Aspect

The coefficients table shows only the aspect angle to be significant in explaining the variation in the corrected lidar error. The positive standardized coefficient of the transformed aspect suggests that the length of the displacement has been underestimated, because the lidar error increases from the west (268.83°, corresponding to a transformed aspect value of -180) towards the east (88.83° - value zero for the transformed aspect).

The p-value of the incidence angle (0.063) is only slightly larger than the 0.05 confidence level, which means that it may still have an effect as predictor. However, its relatively low tolerance value of 0.393 means that more than 60% of the variation in the corrected lidar error explained by this variable can also be explained by other predictors. Also, the variance inflation factor (VIF) indicates that the values of the regression coefficient for both incidence and slope angle have been “inflated” due to collinearity.

The collinearity is caused by the fact that the incidence angle can only take values within the range of the slope angle ± the vertical angle of the laser beam (which is between 10.7° and 12.7° for site A and 15.5° and 18° for site B).

In order to gain a better picture of the importance of the incidence angle, the analysis was performed again with only the transformed aspect and incidence angle as predictors, since the slope angle proved not to be significant.

Table 6: Regression model summary. Using corrected lidar error as dependent and transformed aspect and incidence angle as predictors.

0.498 0.248 0.224 2.108 0.8661 <0.005 0.062

The values of the new model are very similar to the first one, except for the decreased p-value of the Lilliefors test, which suggests that the residuals are not entirely normally distributed. As the scatter plot of the lidar error vs. the transformed aspect variable shows, there is higher variability in the vertical error on the east facing slopes (in the direction of the displacement). This is probably due to the much higher number of points on the eastern slopes rather than to decreased accuracy of the lidar measurements on those slopes.

Fig. 19: Corrected lidar error plotted against the transformed aspect variable

21 Table 7: Regression coefficients. Using corrected lidar error as dependent and transformed aspect and incidence angle as predictors.

Unstandardized Standardized 95,0% Confidence Interval Collinearity

Model Coefficients Coefficients t Sig. for B Statistics

(Constant) ,014 ,027 ,517 ,607 -,040 ,068

Incidence Angle ,001 ,001 ,142 1,217 ,228 ,000 ,004 ,871 1,148

Without the slope angle, the incidence angle shows no effect, which suggests that its near significance was only the outcome of the inflated variance due to collinearity. The slope angle showed no effect either (p-value = 0.945) when the incidence angle was excluded from the analysis.

Both slope and incidence angle showed only a very slight correlation with the absolute corrected lidar error, with the incidence angle being slightly more important and significant. The regression results are summarized in table 8. Due to collinearity, the analysis was performed separately for each predictor.

Table 8: Regression models summary. Using absolute corrected lidar error as dependent and slope and incidence angle as predictors.

Absolute corrected lidar error vs. R R Square Adjusted R Square D-W test sig. F statistic sig. Lilliefors test sig.

Slope angle 0.222 0.049 0.034 0.2049 0.074 0.0882

Incidence angle 0.245 0.06 0.045 0.2429 0.048 0.0772

Unstandardized Coefficients Standardized Coefficients 95,0% Confidence Interval for B

Model Sig.

B Std. Error Beta Lower Bound Upper Bound

Slope angle ,002 ,001 ,222 ,074 ,000 ,004

Incidence Angle ,001 ,001 ,245 ,048 ,000 ,003

22 Fig. 20: Scatter plots of the absolute corrected lidar error vs. slope (left) and incidence angle (right)

By contrast, both slope and incidence angle proved to be stronger predictors for the absolute uncorrected lidar error. Table 9 gives the results of the 2 regressions models (again, the analysis was performed separately for each predictor).

Tabel 9: Regression model summary. Using absolute uncorrected lidar error as dependent and slope and incidence angle as predictors.

Absolute corrected lidar error vs. R R Square Adjusted R Square D-W test sig. F statistic sig. Lilliefors test sig.

Slope angle 0.496 0.246 0.234 0.8753 <0.005 0.1610

Incidence angle 0.366 0.134 0.121 0.7196 0.002 0.0745

Unstandardized Coefficients Standardized Coefficients 95,0% Confidence Interval for B

Model Sig.

B Std. Error Beta Lower Bound Upper Bound

Slope angle ,008 ,002 ,496 ,000 0.004 ,011

Incidence Angle ,004 ,001 ,366 ,002 ,001 ,006

Since its significance as a predictor for the absolute corrected lidar error was very low, slope angle’s association with the absolute uncorrected lidar error is probably a consequence of the large horizontal displacement. Because the incidence angle is partially dependent on the values of the slope angle, the effect it appears to have on the absolute lidar error is likely the indirect result of horizontal displacement as well. Fig. 20 shows the absolute uncorrected lidar error plotted against the slope and the incidence angle.

23 An interesting result is that the importance of the incidence angle in accounting for variability in the absolute lidar error is increased relative to that of the slope angle when using the corrected lidar data. This could suggest that when other major sources of error, such as horizontal displacement, are reduced, the effect of the incidence angle, although extremely weak (R²=0.06), may become apparent.

An analysis was also performed with the original lidar error values as the dependent variable and the slope angle, incidence angle and transformed aspect as the independent variables. Table 10 and 11 summarize the results of the regression.

Tabel 10: Regression model summary. Using uncorrected lidar error as dependent and slope angle, incidence angle and transformed aspect as predictors

0.860 0.740 0.268 2.036 0.8568 <0.005 >0.500

The model is highly significant and accounts for a large amount of variability in the lidar error. Moreover, the regression residuals show no autocorrelation and are normally distributed.

Tabel 11: Regression coefficients. Using uncorrected lidar error as dependent and slope angle, incidence angle and transformed aspect as predictors

Unstandardized Standardized 95,0% Confidence Interval Collinearity Coefficients Coefficients for B Statistics Model t Sig.

(Constant) ,107 ,041 2,643 ,010 ,026 ,188

Slope angle ,007 ,003 ,236 2,353 ,022 ,001 ,013 ,412 2,429

Incidence Angle ,002 ,002 ,113 1,095 ,278 -,002 ,006 ,392 2,554

24 When using the uncorrected lidar error, the aspect variable is a very strong predictor, very probably the result of the large horizontal displacement. When used as the only predictor, the transformed aspect accounts for more than 60% for the variation found in the uncorrected lidar error (R square=0.636), with the model being significant (p<0.005) and valid (Lilliefors test p>0.5, Durbin-Watson test p=0.5166).

The incidence angle, unlike the slope angle, is not significant in this case. If it has any effect, it is probably obscured by the strong influence of aspect on the lidar error. Again, there is collinearity between the incidence and the slope angle.

In turn including the one in the analysis next to the aspect variable while excluding the other sees both slope and incidence angle as significant predictors, but with the model using the incidence angle having a slightly lower R square value than the one using the slope angle – 0.717 compared to 0.735.

This can be again seen as an indication that the incidence angle has an effect only as a consequence of its association with the slope angle. The reason why the slope angle is now significant is probably due to the horizontal displacement, since the largest number of lidar points are situated on east facing slopes, which is the direction of the displacement. When each used alone as predictors for the uncorrected lidar error, neither slope angle nor incidence angle showed any significance.

Fig. 21: Scatter plots of the uncorrected lidar error against the slope angle (left) and against the incidence angle (right)

In the lidar error vs. slope angle plot (fig. 21, left) it is visible how the values of the lidar error actually diverge symmetrically in two directions (above zero and below zero) with increasing slope angle – a fact that can be attributed to the horizontal displacement, which leads to negative errors on slopes facing in the direction of the displacement and positive errors on slopes facing the opposite direction. The lidar error vs. incidence angle plot (fig. 21, right) shows a similar pattern, although less pronounced.

25 5 CONCLUSIONS AND DISCUSSION

The analysis suggests that the only discernable effect the slope angle has on the lidar measurement accuracy is related to the horizontal misalignment. Using the uncorrected lidar data, the slope angle showed a very slight correlation with the vertical error only when used as a predictor next to the aspect variable. When used alone, the correlation was not significant. The variable was also not a significant predictor for the corrected vertical error.

By contrast, the slope angle showed a significant and relatively strong correlation (R square = 0.246) with the absolute uncorrected vertical error. However, it is not possible to conclude that the random error of the lidar measurement increases with the slope angle, since the errors were much more strongly associated with the slope aspect. The correlation between the slope angle and the absolute vertical error is thus likely to be the result of horizontal misalignment. This conclusion is further reinforced by the fact that the slope angle shows almost no association with the absolute corrected vertical error.

Adams and Chandler (2002) found a very weak (R²=0.02, significance not reported) correlation between slope angle and lidar measurements, with the lidar values underestimating the true surface with increasing slope angle. Hodgson and Bresnahan (2004) and Hodgson et al. (2005) also report a slight under prediction of the surface associated with higher slope angles. Both Hodgson and Bresnahan (2004) and Su and Bork (2006) found the absolute vertical error to be greater on steeper slopes. However, none of these studies investigated the relationship between the lidar vertical error and the slope aspect classes in order to rule out the possibility of horizontal misalignment.

If there is any significant correlation between lidar vertical accuracy and terrain topography other than that induced by horizontal displacement, no study has yet been able to unequivocally demonstrate it.

The results obtained from the uncorrected lidar data generally suggest that the incidence angle is a predictor for the lidar vertical error only by the strength of its partial dependence on the slope angle values (the incidence angle cannot take values outside the range of the slope angle ± the vertical angle of the laser beam). Its correlation with the lidar vertical error is, as in the case of the slope angle, largely attributable to the lidar strip’s horizontal misalignment. Using the corrected lidar data, the analysis revealed the existence of a very weak (R²=0.06) positive correlation between the laser beam incidence angle and the under estimation of the true surface by the lidar measurements, which may be independent from the slope and aspect angle. This is suggested by the fact that, for the corrected data, the effect of the incidence angle increased relative to that of the slope angle, compared to the

26 proportion found for the uncorrected data. This should not be the case if both effects were entirely dependent on the horizontal displacement. Nevertheless, a much larger dataset is needed to establish whether this effect is genuine.

As far as known, there are no published studies that confirm or infirm this interpretation.

A number of lidar points belonging to a flight strip for which it was not possible to correct the horizontal misalignment had to be discarded. In consequence, the results of this study are based on a small dataset, and therefore should not be regarded as conclusive. Moreover, the points’ distribution on the different slope and aspect classes is uneven. Another factor that negatively affects the validity of the findings is the uncertainty regarding the vegetation of the area at the time when the lidar data was acquired. All these factors also combine to diminish the precision with which the dataset was corrected for horizontal misalignment. Another problem with a correction based on such a small area is that it inevitably incorporates other possible sources of systematic errors characteristic for that area, and may therefore obscure their effect.

Future investigations using data from the upcoming Algemeen Hoogte Model van Nederland 2 should be able to overcome some of these issues more successfully, since the AHN2 will have a much higher sampling density (7 to 10 points per square meter) than the AHN 1 which this study uses. With this extreme sampling density it will be much more efficient to interpolate the surface from the lidar points and to test it for vertical error at the location of the ground survey points, although in this case the value of the incidence angle will have to be estimated, since it is bound to the location of the lidar points.

However, as long as the strip misalignment remains a characteristic of the lidar data, studying the relationship between terrain morphology and the accuracy of the lidar data is a problematic enterprise.

As far as the general applications of lidar data are concerned – especially those for which a high accuracy of the elevation is required, this study recommends that extensive tests to eliminate the possibility of horizontal displacement should be carried out before the lidar data is relied on. This recommendation is particularly strong when using lidar data in the study of geomorphologic processes and in coastal change monitoring, as any misalignment of the lidar strips which becomes readily apparent on slopes carries the risk of being misinterpreted as landscape change.

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