The Forward Propagation of Integrated System Component Errors within Airborne Lidar Data Tristan Goulden and Chris Hopkinson
Abstract Error estimates of lidar observations are obtained by applying the General Law of Propagation of Variances (GLOPOV) to the direct georeferencing equation. Within the formulation of variance propagation, the most important consideration is the values used to describe the error of the hardware component observations including the global positioning system, inertial measurement unit, laser ranger, and laser scanner (angular encoder noise and beam divergence). Data tested yielded in general, pessimistic predictions as 85 percent of residuals were within the predicted error level. Simulated errors for varying scan angles and altitudes produced horizontal errors largely influenced by IMU subsystem error as well as angular encoder noise and beam divergence. GPS subsystem errors contribute the largest proportion of vertical error only at shallow scan angles and low altitudes. The transformation of the domination of GPS related error sources to total vertical error occurs at scan angles of 23°, 13°, and 8° at flying heights of 1,200 m, 2,000 m, and 3,000 m AGL, respectively.
Introduction Precision is a quantification of all of the random error or noise associated with observations. The observed random error of a measuring instrument results from the combination of many internal and external parameters pertaining to its operation. Considering the mechanics of the principle components and operational scenarios in lidar systems, the precision depends significantly on the height of the aircraft above the target surface, the vector of the laser pulse, external factors including terrain attributes such as slope and reflectance, atmospheric conditions, as well as the abilities of the on-board ranging, scanning, attitude, and positioning systems mentioned above. This paper details the prediction of precision, or random error, through well-known error propagation routines applied to the unique characteristics of lidar systems. The methods applied here can be considered a “forward” propagation of error indicating that predicted errors are independent of external influences. Once lidar observations make contact
Tristan Goulden is at Dalhousie University 1, Faculty of Engineering, Sexton Campus, 1360 Barrington Street, P.O. Box 1000, Halifax, N.S B3J 2X4, and formerly with Applied Geomatics Research Group 2, NSCC Annapolis Valley Campus, Lawrencetown, N.S. B0S 1M0e (
[email protected]). Chris Hopkinson is with the Applied Geomatics Research Group 2, NSCC Annapolis Valley Campus, 50 Elliott Rd, RR1 Lawrencetown, N.S. B0S 1M0. PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
with the physical terrain, the laser energy interacts with surface features that contribute error to the returned energy. Quantifying this interaction would be considered the “reverse” propagation of error. The neglect of terrain features in the “forward” propagation creates a highly theoretical prediction. However, this intentional limitation of scope enables a quantification of how individual sub-systems in the complete lidar system contribute to individual point coordinate errors. A thorough understanding of contributing hardware error sources provides several benefits to a data analyst. Namely, post-mission observed errors of final coordinates can be compared to error predictions to judge whether the equipment is performing as expected. Such a prediction is often referred to as “uncertainty” as it quantifies how a number of observations will distribute if an infinite number of samples (population) existed (Mikhail and Gracie, 1981). A deviation from this statistical postulate gives reason to believe that unaccounted for systematic influences, human error, or poor data collection procedures have contaminated the observations with errors that can be reduced or eliminated on subsequent surveys. Knowledge of the predicted system error based on simulated acquisition criteria will enable more rigorous mission planning techniques and allow users to efficiently meet client specifications. The development of lidar precision prediction capabilities will also aid in creating universal lidar standards as the statistical distribution due to unavoidable random error in a variety of survey settings will be known. Within the lidar industry, the distinction between the precision and accuracy of commercial systems has become confused leading to poor representations of system capabilities. Major manufacturers of lidar systems, such as Optech, Inc., who sells the majority of the world’s lidar systems (TMSI, 2005), quotes an accuracy value for their ALTM (Airborne Laser Terrain Mapper) models in their advertising brochures and system literature. The quoted accuracy values are 15 cm, 25 cm, and 35 cm for flying altitudes of 1,200 m, 2,000 m, and 3,000 m AGL, respectively. The equivalent horizontal accuracy is reported to be 1/2000 the flying height above ground (Optech, Inc., 2004). The accuracy values are indicated to be at the 1-sigma (68 percent) confidence level. It is common to quantify the
Photogrammetric Engineering & Remote Sensing Vol. 76, No. 5, May 2010, pp. 589–601. 0099-1112/10/7605–0589/$3.00/0 © 2010 American Society for Photogrammetry and Remote Sensing May 2010
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precision of an instrument with a confidence level, and not the accuracy. Accuracy is a quantity that cannot be solely represented by a postulated probability distribution function as a confidence interval implies. Accuracy is more commonly found by the root mean square error (RMSE). Therefore, it must be assumed that the numbers quoted above represent the predicted precision of the ALTM 3100 system and not the accuracy. However, a similar level of accuracy can be achieved by block adjustment (positional translation) of the lidar data point cloud using strip matching or ground control techniques: a practice that is common in the lidar survey industry. Details of block or strip adjustments is detailed in Filin and Vosselaman (2003) and Vosselman and Maas (2001). Several papers have been presented that attempt to quantify or correct the systematic error of lidar systems such as Baltsavias (1999), Latypov (2002), Latypov (2005), Krabill (2002), Schenk (2001), Filin (2001), Filin and Vosselman (2003), Filin (2003), and Vosselman and Maas (2001). A quantification of systematic error can be performed by purposely introducing systematic biases into error models and analyzing the effect on final point position. A mathematical description of error sources was given in an error recovery model developed in Schenk (2001) and Filin (2001) from the direct georeferencing equation formulated in Vaughn et al. (1996): x1 X0 dx 0 ex 1 ey y1 ! Y0 1 RwRGRINS dy 1 RmRs 0 Jz K JZ K J "r K Q J e K PJd K 1 0 z z
(1)
where x1, y1, and z1 are the location of the laser point, X0, Y0, and Z0 are the location of the phase centre of the GPS antenna, Rw is the rotation matrix from the WGS84 datum to a local ellipsoidal reference frame, RG is the rotation from the local gravity frame to the ellipsoidal frame, RINS is the rotation from the body frame to the local gravity frame, dx, dy, and dz are the offsets from the laser transmission point, and the phase center of the GPS antenna in the body frame, Rm is the boresight angular values which rotate between the body frame and laser scanning frame, Rs is the rotation by the observed scan angle, r is the observed range observation and ex, ey, ez are the random error components of the observation in the same reference frame as the laser point (Vaughn et al., 1996), (Schenk, 2001), (Filin, 2003). The current focus on accuracy has led to the quantification and elimination of systematic errors. Often quantification of random errors (ex, ey, ez) came as a product of adjustments seeking the quantification of systematic errors as in Filin (2003). These random errors are associated with the adjustment of a block of data and are not modeled on an individual point basis. Such accuracy studies provide valuable postprocessed information for the performance of the system; however, they do not provide a predictive model or a priori information from which the quality of post-processed data can be compared. In addition, work has been done by Hodgson and Bresnahan (2004), Hodgson et al. (2005), Bowen and Waltermire (2002), Reutebuch et al. (2003), Töyrä et al (2003), and Hopkinson et al. (2005) to attempt to quantify the error resulting from external system contributions such as terrain slope and vegetation cover. Further studies (e.g., Chasmer et al. 2006; Hopkinson, 2007) have attempted to isolate and quantify the influence of individual lidar sensor acquisition parameters on the point cloud for a variety of land-cover types. These studies relied on post-mission statistical analysis and did not attempt predictions of error. The information these studies provide is valuable for 590
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specific surveys being analyzed however does not guarantee meaningful predictions in subsequent surveys. Existing research has not provided a rigorous study on the prediction of random error in lidar observations using error propagation techniques. Recently, the British Columbia Base Mapping and Geomatics Services became interested in this subject and a report was provided by Habib et al. (unpublished data, 2006) while a similar study was undertaken by Glennie (2007). Although several papers use functional covariance propagation as a step in their procedure for solving other problems, such as those found in Filin (2003), Friess et al. (2006) and Schaer et al. (2007), these studies, in general, do not provide detailed descriptions of procedures, testing or assumptions of the error propagation, especially those relating to the quantification of the performance of individual system components. Without this information, it is impossible to analyze or confidently repeat results independently. No study has isolated results to only random errors in hardware components and properly validated the models through real world observations and rigorous statistical testing. The error in a coordinate can be represented in onedimension as a single line, in two dimensions as an ellipse, and in three dimensions as an ellipsoid. Since lidar data points are represented in three dimensions as x, y, and z coordinates, each lidar observation will have an associated three dimensional error ellipsoid. Properly scaled, the volume of the ellipsoid will represent the space in which a lidar observation has a 68 percent probability of falling. This is only the case when all systematic influences have been accounted for and eliminated, and the random errors follow a normal distribution. The results of this study produce such an ellipsoid based solely on propagating the error through the geometric relationship of the hardware components. This is a crucial first step that must be considered prior to producing a “reverse” propagation which includes error due to surface conditions. This paper is organized into the following sections: 1. Sources of error within lidar system components, 2. Deriving the predictive error model from the direct georeferencing equation, 3. Testing and validating the predicted uncertainty, and 4. A sensitivity analysis of the magnitude of predicted error for a variety of survey configurations.
A detailed review of lidar sensor systems and concepts is outside the scope of this paper, as it is an introduction to the relevant mathematical and statistical techniques. For review of the fundamentals of lidar systems the reader is referred to Wehr and Lohr (1999). For a review of normal distribution statistics and the law of propagation of errors the reader is referred to Wolf and Ghilani (1997) and Mikhail and Gracie (1981) for focus on point positioning problems. It should be mentioned that for the purpose of this paper, analysis primarily focused on the error associated with the use of the Airborne Laser Terrain Mapping (ALTM) lidar systems developed and manufactured by Optech, Inc., a company based in Toronto, Canada. ALTM systems were chosen for two reasons: first, the Applied Geomatics Research Group (AGRG) owns and operates an ALTM 3100 giving the authors access to data, hardware, and manuals for this system; and second, Optech, Inc. manufactures and sells more lidar units to private industry than any other manufacturer of lidar systems (TMSI, 2005), making this analysis of potentially high relevance to a broad crosssection of the community. Details of the ALTM 3100 system may be found on the Optech, Inc. website (http://www.Optech.ca/). PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Sources of Uncertainty within Lidar Components When dealing with an integrated lidar system, there are five significant hardware sub-system error sources (Schenk, 2001). These have to be understood to quantitatively determine their performance prior to inclusion in the error propagation model. They are as follows: 1. 2. 3. 4. 5.
Global Positioning System Unit, Inertial Measurement Unit, Scanning Mirror Unit, Laser Ranging Unit, and Integration of components.
To better understand the connection between the components, a flow chart that illustrates their relationship has been given in Figure 1, followed by an explanation. The main components listed above can be seen in the center column of the flow chart. They represent the observational components of the system. Also included with the components are their respective observation rates. The final block culminates in the combination of all the system components to produce the observed point data. The integration of the components is included on the right and left side of the flowchart, representing calibrations and time synchronizations, respectively. Literature relating to bore-sight misalignments between the IMUs and imaging devices is readily available, as it has been an issue for the direct georeferencing of aerial imagery. Information on the misalignments can be found in Cramer (1999), and Shwarz et al. (1993), and for specific lidar applications in Morin and El-Shiemy, (2002), Skaloud and Lichti (2006), Schenk (2001), Vaughn et al. (1996), and Filin (2003). Figure 2 displays each of these physical integration corrections connected to the components in which they affect. Namely, eccentricity distances for the GPS and IMU systems and the bore-sight misalignment for the IMU and laser scanner systems. On the left side of Figure 1 is the connection between the timing components. Timing is the most critical part of the integrated lidar system, as laser range observations can be collected at rates exceeding 100 KHZ and must be registered
Figure 2. Calibration considerations: scalar eccentricity distances between GPS and IMU frame as well as angular boresight angles between laser scanner frame (LSF) and IMU frame.
to timing systems within the GPS and IMU sub-systems, which in the case of the ALTM, operate at lower frequencies of 2 Hz and 200 Hz, respectively. Any error in the cross-registration of this timing scheme as well as the interpolation of low frequency data (GPS and IMU) to that of high frequency (laser ranges) will seriously impact results (Schenk, 2001; Vaughn et al., 1996). The separate observation rates of each piece of equipment are an important consideration when propagating and testing random errors. Random errors associated with pieces of equipment with low observation rates (GPS, IMU) will exhibit themselves similarly within sequential samples of laser points taken at high observation rates. This similarity will cause errors in final point positions to be correlated. Correlated observations can adversely impact testing and interpretation of quality control data as the effects can be difficult to predict and separate from systematic error sources. Table 1 presents the documented known, random error sources in the total lidar solution and identifies which factors are considered in the analysis presented here. Several
TABLE 1. ERROR SOURCES CONSIDERED IN THE ANALYSIS OF HARDWARE COMPONENTS
Error Source
Figure 1. Flow chart diagram of lidar system components. The left column illustrates timing integration considerations, the right column any calibration considerations, and the center column represents how each of the main components in the airborne laser scanning system relate.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Effect
Considered in Analysis
Global Positioning System IMU
Number and Location of Satellites, Atmospheric Effects etc. Initialization Parameters and drift
Yes
Laser Ranger
Time interval Meter Atmospheric Effects Beam incidence angle Refraction Local Terrain Effects
Yes No No No No
Scanning Unit
Angular Encoder Refraction Beam Divergence Boresight Angles Eccentricity Distances Timing Considerations
Yes No Yes Yes Yes Yes
Measurement System integration
Yes
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factors have been excluded to isolate sources of random error directly related to the hardware components and their integration. The following section details the logic for choosing appropriate error magnitudes for the considered error sources. This selection is the most critical part of such an analysis, because it is the only interpretive information required as input by a data analyst and will determine the success of the error predictions. Derivation of an Error Prediction Formula The error prediction formula can be derived from the direct georeferencing equation by applying the General Law of Propagation of Variances (GLOPOV). The law states that the sum of the squares of the partial derivative of each observable quantity, multiplied by the observation’s individual predicted error, will result in the total propagated error when observations are statistically independent (Wolf and Ghilani, 1997) as follows: CovSolution ! ACovObservation AT
(2)
where 0F1 0x1 0F2 0x1 A! o 0Fn 0x1
£
Cov ! ≥
sx22 sx1x2
0F1 0x2 0F2 0x2 o 0Fn 0xZ
Á Á ∞ Á
sx1x2 sx22
o
o
sxnx1
sxnx2
0F 0xn 0F2 0xn o 0Fn 0xn
§
(3)
and
Á Á
sx1xn sx2xn
∞ Á
o s2xn
¥
where F is the function, x is the observables, and s represents the uncertainty. The function (F) being analyzed is Equation 1, the solution is the final x, y, and z coordinates, and the observations of interest are: roll, pitch, yaw, x mirror coordinate, y mirror coordinate, z mirror coordinate, scan angle, and range. The terms related to the eccentricities and boresight angles have not been explicitly included, and the following section explains the reasoning. The partial differentiation of the function F with respect to each observable is taken in A to linearize the non-linear equation allowing it to be solved. The covariance matrix of the solution, Covsolution, will be fully populated including covariances (off diagonal elements) between each of the final coordinate errors resulting from the dependence of each coordinate solution to the common observations. These covariances must be appropriately taken into account to determine correct estimates of the error in each dimension. The covariance matrix of the observations can be simplified to: CovObservations ! diag C s2x1
s2x2 Á s2xn D
(4)
as the observations from each hardware component are statistically independent being generated from individual sources. One exception is the error associated with the position and orientation system, which is discussed with the aircraft coordinates and attitude components. The difficult terms of the equation are the hardware component errors used to populate CovObservations, since each will have to be investigated independently. Since the direct georeferencing equation is well defined, the partial 592
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derivatives are definite mathematical derivations and not open to interpretation. For this reason, they will not be explicitly included. The following sections will outline the suggestions for the system component errors and the reasoning behind them. Aircraft Coordinates and Attitude Components The error of the coordinates and attitude information is determined from the combined observations of the GPS and the IMU. They are integrated through a tightly coupled solution using positioning software developed by Applanix Corporation (Toronto, Canada), the company that manufactures the airborne vehicle positioning and orientation system (POS AV) embedded within the ALTM 3100. From the postprocessed positioning data, a combined RMS error (RMSE) can be reported using two determinations of the solution. The RMSE is then separated into two data sets: one outlining the positional accuracy and the other for attitude accuracy. The combined RMSE is determined at 1 HZ along the entire trajectory and is used as the prediction of the position and orientation hardware components. RMSE for the position and orientation data is only the best available interpretation of positional and attitude error and is not entirely correct. It ignores the co-variances in the position errors and could be contaminated by systematic influences, which will result in estimations that are generally scaled from their true size and not in the proper orientation. Quantitatively validating the use of RMSE for error prediction is a costly and difficult procedure since flight data is extremely variable and difficult to reproduce independently (Schwarz et al., 1993). Methods for comparison using multiple ground base GPS stations (Cannon, 1992) and with photogrammetric methods (Schwarz et al., 1993) have been suggested. Despite these drawbacks the RMSE estimation is considered the optimal solution for estimating the error of combined GPS/IMU position observations (Hare, 2001; Grejner-Brzezinska and Wang, 1998). Scan Angle and Beam Divergence The error in the scan angle of the ALTM 3100 system is given as approximately "0.003° ("10.6#; Personnel Communication, 2006). Leica, another manufacturer of scanning mirror lidar systems similar to the ALTM 3100 claim an accuracy of "0.001° ("3.6#; Morin, 2002) for their system, although neither claim was supported with evidence. A further investigation was performed on two leading producers of angular encoder systems, Renishaw, Gloucestershire, United Kingdom and Heidenhain, Illinois, USA. Each company provided specification sheets with error values and detailed explanations of how they had been determined. Pessimistic estimations made about an assumed model of the encoder system used in lidar systems yielded results of "6.3" ("0.0018°) and "9.1" ("0.0025°) for Renishaw and Heidenhain, respectively (RESR, ("0.0025°) 2006; Heidenhain, 2005). In house tests of shot-to-shot scan angle difference records during AGRG calibration surveys suggests that for the ALTM used in this study, the one sigma precision in scan angle lies in the range of "0.001° and "0.002°. These reported and observed scan angle errors were slightly better than the Optech, Inc. reported value of "0.003°, therefore the Optech quote was adopted for simulation purposes as a pessimistic estimation. In addition to the error caused by the encoder system, an additional error results from the beam divergence of the laser pulse. As a lidar pulse propagates through the atmosphere it expands. The pulse area upon surface contact (beam footprint) is dependent on flying height and beam divergence. It is well known that lidar observations are subject to significant horizontal errors caused by the edge of the beam footprint making first contact with the surface features. PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Breaklines in the terrain are especially prone to positional error because after the beam footprint edge makes contact, the remaining energy experiences a time delay prior to surface contact. Although the returned energy from the edge of a beam footprint can be low, it is frequently sufficient for detection by the receiving optics. In such cases, an erroneous range results, which when applied to the beam centreline vector has the effect of a potentially large horizontal and a slight vertical error. The consequence of beam divergence with respect to the error in airborne laser scanners was discussed in Glennie (2007) and Schaer et al. (2007) and for terrestrial laser scanners in Lichti and Gordon (2004). Lichti and Gordon (2004) suggested an error level of one-quarter the beamwidth, although an even distribution of energy across the pulse width was assumed. However, as explained in Glennie (2007), the pulse shape is known to form an approximate biaxial Gaussian distribution with peak pulse power, represented by the peak of the Gaussian curve, along the beam centerline. This infers reduced error estimates as pulse returns would likely be returned from the high energy center. Following this, Glennie (2007) chose to retain the one-quarter estimate as a pessimistic estimation since surface conditions such as slope were not considered. It is prudent to identify here that beam divergence is described in mRad as this is the angular spread of a circular cross-section of the Gaussian beam as it propagates. Theoretically, the tails of a Gaussian curve extend to infinity, and therefore it is practical to describe the beam footprint extents as a percentage of the peak power. Commonly, the factor of 1/e2 is used to describe the beam power (irradiance) at approximately 14 percent of peak power (Marshall, 1985). However, within the lidar industry, beam diameter is often described at the 1/e level representing approximately 37 percent of peak power and creating a reduced footprint. Based on the properties of Gaussian curves, the 1/e2 and 1/e definitions will contain approximately 95 percent and 84 percent of the total emitted energy of the laser pulse respectively. A selection of one-quarter of the 1/e beam divergence to represent the error level as was adopted in Glennie (2007) results in an extent that represents 94 percent of peak power and containing 27 percent of the total energy of the pulse. Figure 3 displays the locations of the different beam divergence definitions on a Gaussian curve and the percentage of total pulse power they contain.
Figure 3. Gaussian energy distribution of a laser pulse. Several definitions of the pulse footprint are shown represented as a percentage of peak power as dots along the Gaussian curve. Associated with each of these definitions and illustrated in separate shades of grey is the percentage of total returned power within the area bounded by each of the defined footprints.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
For the error analysis presented here, the error due to beam divergence was chosen to be equal to the beam divergence definition provided by the manufacturer. It is felt this is appropriate to illustrate a pessimistic case where a breakline could return significant energy from the edge of the beam footprint and initiate a return. In addition, using the manufacturer’s definition offers a convenient basis for comparing system performance with stated beam divergence definitions. Range Range is calculated by dividing the travel time of the laser pulse in half and multiplying by the speed of light as follows: Range "
t c. 2
(5)
Using the SLOPOV (Special Law of Propagation of Variances) the error in the range can be calculated as follows (Wolf and Ghilani, 1997): s2Range " c
0Range 0t
2
d s2t .
(6)
According to Baltsavias (1999) the error in the time counter in typical lidar systems is approximately !0.1 nanoseconds. Therefore, the above equation can be written as: c 2 2 s2Range " c d [0.1]2 " [1.5cm] . 2
(7)
Integration of Components The errors due to the distance eccentricity determinations between the on-board GPS antenna, the IMU sensor, and the scanning mirror require special attention. These errors can be obtained directly from an algorithm in the Applanix processing software obtained from in-flight observations or a terrestrial survey. To simplify the derivation of the error prediction model, the error of the eccentricities has not been propagated through the mathematical model. The eccentricity error values have merely been rotated from the body frame to the local ellipsoidal frame. The rotated values can be directly added to the position errors to achieve a total error of the scanning mirror coordinate. Quantifying angular boresight errors proves challenging, because lidar observations are used in their calculation and therefore will be correlated with several other sources of error. Additionally, there is no established means for independently checking whether the determined error is correct (Skaloud and Lichti, 2006). Morin (2002) and Skaloud and Lichti (2006) have suggested methods for calculating boresight misalignments through a least squares adjustment from which sample standard deviations can be obtained. However, this method requires strict flight plans or use of targets which were not feasible for this analysis. Even within these methods, there is no guarantee that other sources of error have not contaminated the results. In addition, predicted errors result from minimum convergence criteria of relative datasets and may not represent true absolute error. In addition, Glennie (2007) and Schaer et al. (2007) report that error magnitudes for these parameters when utilizing the least squares approach are well within the error tolerance of the IMU sub-system. Using manual adjustment methods can result in errors that approach the error in the IMU subsystem and through careful corrections, they can be reduced further. Therefore, it is assumed that the boresight angular errors are contained within the predicted RMSE IMU May 2010
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errors for roll, pitch, and yaw. This allows the mathematical model to be further simplified and eases the derivation of the propagated error formula. If users wish to explicitly include these error parameters, a careful error analysis of the chosen method for the determination of the boresight misalignments must be done to determine the error levels which can then be introduced into error equation developed through the GLOPOV. All necessary terms in Equation 1 have now been considered. Equation 2 may now be used to solve for the three-dimensional error ellipsoid associated with each observation. To calculate the error for numerous lidar observation points, a script was developed in MATLAB®. The following section details the testing of the script.
Testing and Verification The testing and verification of the error prediction algorithm is not a straightforward procedure. It is difficult to isolate sources of random error within a test data set that inevitably contain multiple sources of additional systematic and random errors. Ideally, a statistical analysis of this nature would be done by repeating observations of a target with known coordinates. The observed standard deviation could then be compared with the predicted error. This approach is impossible with lidar observations since they are captured in variable flight conditions with high speeds and observation rates. Data are spread across large areas, with different predicted and actual error values at every point which creates difficulty in obtaining a sample data set of sufficient quantity for statistical analysis. To overcome these problems, a validation site must be observed and control points isolated in close proximity to several observed points. If the predicted error in small local areas has negligible change, all observations within this area are assumed to be statistically similar and are treated as a sample set. Lidar observations from calibration flights were used for test data because calibration data are used to adjust the boresight misalignments of the lidar unit. These parameters describe the mis-orientation of the IMU reference frame and the scanning mirror frame, and tend to drift over time. Using data that have been corrected for flight line specific boresight misalignments mitigates any systematic influence of this component within the predicted error model simulation. Calibration control sites were surveyed through postprocessed kinematic GPS procedures. The testing was split into two sections, one for horizontal data and one for vertical data. To perform the horizontal test, wall edges of a tall building were surveyed. For the vertical test, an airport runway was surveyed. Buildings are used for the testing of horizontal error because their vertical walls allow horizontal error to be readily quantified. Runways are ideal for vertical
Figure 4. Map of Nova Scotia showing locations of building and runway calibration sites. The building was used for testing the horizontal error while the runway was used for the vertical error. The GPS base stations for the Nova Scotia surveys were located either on the roof of the building site or mid-way between the building and runway at Kingston.
data because if the flight line is flown perpendicular to the runway, an entire scan angle can be contained within it. Runways are also relatively flat which reduces errors caused by terrain effects. The description of each validation procedure will be hereafter split into sections describing the horizontal and vertical procedures. Figure 4 shows a map outlining the locations of the calibration sites, Middleton and Waterville, Nova Scotia, representing the building and runway site, respectively. A secondary calibration site in Aurora and Oshawa, Ontario representing the building and runway sites, respectively, was used for data on Julian Day 139; the configuration is similar to that of the Middleton and Waterville sites. Table 2 shows the lidar system settings that were chosen for the calibration surveys. Three statistical tests were used to determine whether predicted errors were acceptable. The first test considered whether observed data distributed normally which was done to ensure the testing methods were appropriate. If results were not normally distributed, then test data might not be the result of a random process. An Anderson-Darling test for
TABLE 2 DETAILS OF FLIGHT SPECIFICATIONS FOR ALL CALIBRATION FLIGHTS USED FOR MODEL TESTING Julian Day
Building versus Runway
GPS Base Location
PRF (kHz)
Building Runway Runway Building Runway Building Runway Building Runway Runway
Middleton Kingston Kingston Middleton Middleton Buttonville Buttonville Middleton Middleton Middleton
70 50 70 50 50 33 50 70 50 70
86 88a 88b 129 129 139 139 146 146 146
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Scan Frequency (Hz) 4 18 39 15 20 17 35 15 18 8
Scan Angle (degrees)
Flying Height (m, AGL)
Flying Speed (knots)
15 20 20 4 18 4 25 4 20 19
1200 1000 1000 1000 1450 1000 1100 1000 1000 1000
144 150 150 110 124 123 121 117 120 120
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
normality was done to ensure the data was normally distributed (Anderson and Darling, 1952). The test was coded in MATLAB® and made available on the World Wide Web by Trujillo-Ortiz et al. (2007) and was performed at the 95 percent confidence level. The second test calculated the number of observed samples that were less than the predicted error. If the observed data were distributed as predicted by the error simulation results, then 68 percent of the observations should be within the predicted range. This test must be performed only after any systematic biases have been removed, which was done by subtracting the mean of the sample data from each individual observation. The error values with the mean removed are termed residuals. The third test compares the sample standard deviation with the predicted error. The test follows a description from Chapter 13 of Vanícˇek and Krakiwsky (1982). The test determines whether the observed and predicted errors are statistically compatible at the 95 percent confidence interval. The test is given as follows: 1N " 12s2 2 jxN"1 ,1 " a2
6 s2 6
1N " 12s2 jx2N"1, a2
,
(8)
where N is the number of samples, s is the sample standard deviation, s2 is the predicted error, a indicates the confidence 2 interval, and jxN"1 , 1 " a2 is obtained from the chi-squared distribution. Failure of the test because the predicted error is higher than the statistic on the right-hand side indicates that the predicted error is pessimistic, and actual errors are performing better than expected. Failure because the predicted error is less than the left-hand side indicated the prediction was optimistic, and actual errors are greater than expected. Horizontal Validation The horizontal validation observations were taken on each corner of the building, plus several positions along the edges were observed to create a building footprint. Calculation of the error values was automated by ACalibPro (Optech, Inc., Toronto, Ontario), a program developed by Optech, Inc. for calibration of lidar systems. As observations are flown over the building, the edge is detected by a sudden upward change in observed elevation values. Figure 5 displays the lidar scan lines as they intersect the rooftop. The distance each pulse was away from the building edge when the vertical change was first detected was compiled as the sample data for testing purposes. The predicted error changed negligibly (!1 mm) along the rooftop edge allowing all observations to be treated as a single sample set. The distance at which a pulse first makes contact with the roof edge has an inherent semi-random pattern based on how the flight configuration settings result in ground point spacing. For testing purposes, the interest is in the random variation in the distance of the pulses to the building edge due to the random errors in the lidar system. In order to separate these phenomena, each pulse that made contact with the building edge was analyzed to determine how many subsequent pulses occurred until pulse footprints were located completely on the building roof and not enveloping both the ground and roof simultaneously. If a similar number of pulses were required, then the random effect due to point spacing was considered removed. Only the horizontal distance extracted from pulses that made first contact with the roof and required a similar number of subsequent pulses to be located completely on the roof were considered for analysis. The selection procedure is diagrammed in Figure 5. PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Figure 5. Pulses that make contact with the building rooftop and ground due to overlapping beam footprint represented by black dots. Along each scan line only the first pulses that made contact with the rooftop and then had a similar number of subsequent pulses before being located soley on the rooftop were chosen for horizontal error analysis. Under this testing scheme the beam divergence will cause an evident systematic trend in the error calcualtions represented by the dashed line.
Under these testing conditions, the error that results from the beam divergence must be given special consideration because it will be represented as a systematic bias which is understood by analyzing Figure 5. The first pulse that intercepts the building edge is isolated for error analysis and contains a predictable systematic bias equivalent to the extent of the beam divergence. This effect will be evident in the observed RMSE calculation of error and not in the standard deviation. For this reason, two predicted error values have been produced, one including the effect of beam divergence and one without. The success for the error predictions including beam divergence must be made only through direct comparison with the observed RMSE, as the testing schemes presented are designed for comparison of random errors only. Every data set except for Line 3 on Day 129 was normally distributed at the 95 percent confidence level. Table 3 outlines the results for several different days of flights including the predicted error represented by the semi-major axis of the horizontal error ellipse with and without beam divergence, the observed standard deviation (random component) of the distances, the observed mean (systematic bias), the observed RMSE the percentage of residual (systematic bias removed) distances that were within the error prediction without beam divergence, and whether the sample standard deviation is compatible with the predicted error at the 95 percent confidence level. If properly predicted, 68 percent of the residual distances should land within the predicted error. All data are presented in millimeters, and the day identification number corresponds to the Julian day of the year 2006. The observed horizontal errors produced residuals which were consistently lower than the predicted errors. This situation is ideal for ensuring observations will meet May 2010
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TABLE 3. Predicted Error (m)
Day 86 Line 1 Line 2 Line 3 Line 4 Day 129 Line 1 Line 2 Line 3 Line 4 Day 139 Line 1 Line 2 Line 3 Line 4 Line 5 Line 6 Day 146 Line 1 Line 2 Line 3 Line 4 Line 5 Line 6
MODELED AND OBSERVED HORIZONTAL ERROR STATISTICS
Sample Statistics (m)
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w/o BD
BD
Std
Mean
RMSE
0.094 0.100 0.094 0.092
0.182 0.172 0.173 0.166
0.072 0.074 0.088 0.058
0.230 0.236 0.146 0.310
0.240 0.247 0.169 0.316
72 73 67 88
Compatible Compatible Compatible Pessimistic
0.088 0.078 0.085 0.091
0.154 0.151 0.154 0.147
0.054 0.055 0.121 0.109
0.257 0.238 0.120 0.300
0.262 0.240 0.168 0.319
91 94 56 59
Pessimistic Compatible Optimistic Compatible
0.089 0.093 0.091 0.089 0.091 0.091
0.163 0.161 0.163 0.162 0.164 0.160
0.046 0.084 0.055 0.049 0.046 0.060
0.188 0.420 0.235 0.210 0.213 0.196
0.194 0.428 0.242 0.216 0.218 0.205
97 71 91 97 98 89
Pessimistic Compatible Pessimistic Pessimistic Pessimistic Pessimistic
0.077 0.087 0.092 0.083 0.084 0.081
0.127 0.145 0.149 0.156 0.146 0.146
0.086 0.056 0.086 0.054 0.058 0.062
0.250 0.280 0.251 0.288 0.240 0.285
0.263 0.286 0.265 0.293 0.247 0.292
63 86 56 90 87 85
Compatible Pessimistic Compatible Pessimistic Pessimistic Pessimistic
targeted error requirements, however may not be ideal for efficient planning procedures, as it will tend towards oversampled datasets. Vanícˇek and Krakiwsky (1986) indicate that there are several reasons for an incompatibility between sampled standard deviations and predicted errors. The most obvious is that the mathematical model is incorrect. Since the direct geo-referencing equation is well defined, problems may be the estimations of individual component errors. These values were purposely chosen to be pessimistic, as it is often wise to consider “worst-case-scenario” estimations. In addition, the semi-major axis of the error ellipse was used as the error prediction. This value is the largest possible magnitude within the horizontal error ellipse and residuals will not consistently distribute at this upper limit. Vanícˇek and Krakiwsky (1986) also indicate that if the residuals are correlated, then the sample standard deviations could be comparatively small compared to predicted error. This scenario is likely with lidar observations since each piece of observation equipment operates at a different rate as outlined above. For example, if the Pulse Repetition Frequency (PFR) is set to 50,000 KHZ, then 50,000 individual laser pulses will be correlated with the random errors of only two GPS observations which are taken at 1 to 2 Hz. The random GPS errors will exhibit themselves systematically within the set of laser pulses with which they are associated, and the error will be represented within the RMSE value and not the standard deviation. The data taken along the rooftop would suffer from this effect, as it takes less than two seconds (or four GPS observations) to traverse the length of the calibration building. The resulting standard deviation of these points will be lower than predicted since it is influenced by only the laser range, scanning mirror, and inertial measurements. Technically, systematic errors should not be propagated using the GLOPOV, however these errors are still random when considering an entire survey, which indicates that when performing quality control procedures, care must be taken to acquire validation data throughout an entire survey area. Control data taken in small clusters will 596
Residuals within predicted error (%)
generate optimistic standard deviations and contain a systematic bias partially derived from random position and orientation errors. As previously mentioned, the effects of beam divergence would manifest systematically under the presented testing scheme. It is apparent from the observed RMSE values that the predicted error including beam divergence is optimistic, as observed RMSE values are in general greater than the prediction. Given that a pessimistic estimate of scan angle error was adopted, this likely indicates that the definition of 1/e for the limits of the beam footprint is overly optimistic. The area of the beam footprint containing enough energy to initiate a return signal is larger than initially hypothesized. The more common 1/e2 definition of the beam footprint is likely more appropriate. Using this definition expands the 2 original footprint (1/e definition) radius by a factor of 12 (Marshall, 1985). The resulting radius of the expanded beam footprint at a flying height of 1,200 meters at nadir is approximately 0.21 meters compared to 0.15 meters for the 1/e definition. The 1/e2 definition conforms better with the RMSE values presented in Table 3 indicating it is more appropriate for error analysis on breaklines. Other error sources which could have contributed to the total systematic error could be slight residual errors in boresight calibration, atmospheric effects, or ranging errors. Vertical Validation The Berwick vertical validation data consisted of over 600 post-processed kinematic GPS points with a standard deviation of less than one centimeter, while the Oshawa data was provided by Optech, Inc. and is the validation data used for their airborne sensor tests and calibrations. Statistics were calculated by subtracting the elevation of all lidar from the elevation of a GPS control point within a two meter sample area surrounding the GPS control point. The standard deviation of the elevation differences could be compared with the predicted vertical error. Within the two meter PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
radius it is assumed that the runway was sufficiently flat and will not introduce error due to ground slope. On a sufficiently flat runway the vertical error due to beam divergence would also be insignificant as returns would likely be along the assumed beam centerline and therefore were not included. In some test flights it was evident that a systematic bias had affected the results. The source could be related to atmospheric conditions or intensity-based range biases and also the possibility of small GPS base station or aircraft GPS antenna height measurement blunders. To remove any systematic bias in the data, the mean offset was subtracted from all vertical distances to obtain a set of residual values. Figure 6 shows a typical plot (Day 88, Line 1) of the absolute value of residuals and their modeled errors. In general, data were found to be normally distributed within areas with similar scan angles where predicted uncertainty was negligibly different. Figure 7 displays a typical histogram along the Middleton runway from each of port, nadir, and starboard sections. Sample sets that failed this test for normality displayed the typical Gaussian curve, however, minor deviations in skew or kurtosis caused the failure.
Table 4 summarizes the test statistics representing a separate day and split into three sections showing the port, nadir, and starboard results. Shown in Table 4 from left to right are the predicted error, RMSE, sample mean, sample standard deviation, percent of residuals below predicted error, and whether the predicted uncertainty and sample standard deviations are compatible. The results of the vertical validation are encouraging. Although error predictions were in general pessimistic, there was consistency among all the days with the number of residuals that fell underneath the predicted errors. Similar to the horizontal data, there is evidence of a correlation between the residuals owing to the dependence of localized observations to low frequency positioning data which accounts for the pessimistic error predictions and partially for the observed RMSE. Considering these factors, and that surface and atmospheric characteristics of the reverse propagation are ignored, it is not surprising to observe data that does not strictly conform to the hypothesis that 68 percent of data will be underneath the predicted error values and that results show variation in the conformity of sample standard deviations with predicted errors.
Simulated Predicted Error in Varied Flight Configurations
Figure 6. Residuals in vertical data. Grey points indicate residual magnitudes of observations along test runway within 2 m radius of GPS validation position; the black line shows the calculated uncertainty of those same points.
(a)
In order to place the presented results into a more practical context, an analysis showing the total error magnitude and percentage contribution for separate flight configurations was performed. Altitudes and scan angles were varied for the analysis because they typically experience a controlled change between lidar missions. Altitudes were selected to be 1,200 m, 2,000 m, and 3,000 m AGL following the flying heights presented in the Optech, Inc. accuracy statement. The scan angle was varied from 0° to 25° following the operation range on the ALTM 3100. The attitude values were set to 0° in roll, pitch, and heading throughout to simulate optimal flying conditions. The errors for the GPS and attitude component were set following achievable RMSE values routinely seen in the Applanix post-processing positioning software with good satellite geometry throughout the survey, i.e., 3 centimeters for the horizontal (X and Y) components of the GPS, and 5 centimeters vertically (Z), 0.005° for roll and pitch, and 0.01° for heading. The scan angle error was set to 0.003°, with an additional 0.0025 mRad (0.0143°) to account for beam divergence. The beam divergence was applied parallel and perpendicular to the scan line direction to account for the circular nature of the beam propagation. The error in the laser range was set to 1.5 cm.
(b)
(c)
Figure 7. Typical histogram plots of data from three separate sections of the swath: (a) Port, (b) Center, and (c) Starboard. The three graphs divide the scan angle into sections of similar predicted error for display purposes. It can be seen that results typically showed distributions resembling a Gaussian shape.
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TABLE 4. VERTICAL ERROR PREDICTION AND TESTING RESULTS: THE LEFT THE DISPLAYS PREDICTED VERTICAL UNCERTAINTY, OBSERVED ROOT MEAN SQUARE ERROR, SAMPLE MEAN, SAMPLE STANDARD DEVIATION, PERCENTAGE OF OBSERVED VERTICAL RESIDUALS LESS THAN UNCERTAINTY, AND COMPATIBILITY OF SAMPLE STANDARD DEVIATION WITH PREDICTED UNCERTAINTY DIVIDED INTO PORT, NADIR, AND STARBOARD SECTIONS Line Port 1 2 3 4 Nadir 1 2 3 4 Starboard 1 2 3 4 Line Port 1 2 3 4 Nadir 1 2 3 4 Starboard 1 2 3 4 Line Port 1 2 3 4 Nadir 1 2 3 4 Starboard 1 2 3 4 Line Port 1 2 3 4 5 Nadir 1 2 3 4 5 Starboard 1 2 3 4 5
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Predicted Error (m) 0.051 0.051 0.051 0.050
RMSE (m) 0.065 0.096 0.084 0.100
Day 88 Mean (m) !0.047 !0.087 !0.070 !0.092
Std Dev (m) 0.045 0.040 0.047 0.038
Percent 71 76 68 82
Pessimistic Pessimistic Pessimistic Pessimistic
0.046 0.047 0.046 0.046
0.068 0.093 0.099 0.102
!0.056 !0.085 !0.092 !0.096
0.039 0.038 0.037 0.035
73 75 78 82
Pessimistic Pessimistic Pessimistic Pessimistic
0.048 0.050 0.052 0.049
0.068 0.089 0.104 0.099
!0.058 !0.080 !0.095 !0.093 Day 129
0.039 0.047 0.041 0.035
78 78 79 86
Pessimistic Pessimistic Pessimistic Pessimistic
0.054 0.057 0.055 0.060
0.128 0.143 0.079 0.128
!0.120 !0.138 !0.067 !0.122
0.046 0.038 0.043 0.041
75 90 79 90
Compatible Pessimistic Pessimistic Pessimistic
0.053 0.055 0.054 0.052
0.123 0.139 0.070 0.118
!0.117 !0.134 !0.060 !0.111
0.036 0.035 0.036 0.039
87 91 88 87
Pessimistic Pessimistic Pessimistic Pessimistic
0.057 0.054 0.055 0.055
0.130 0.125 0.142 0.099
!0.121 !0.116 !0.134 !0.086 Day 139
0.050 0.046 0.047 0.048
75 72 78 71
Compatible Pessimistic Pessimistic Pessimistic
0.059 0.056 0.057 0.058
0.041 0.051 0.040 0.046
!0.006 !0.008 !0.015 !0.024
0.041 0.050 0.037 0.038
84 76 87 87
Pessimistic Pessimistic Pessimistic Pessimistic
0.051 0.051 0.051 0.051
0.034 0.036 0.029 0.034
0.003 0.022 !0.001 !0.017
0.034 0.036 0.029 0.029
87 83 94 94
Pessimistic Pessimistic Pessimistic Pessimistic
0.058 0.059 0.059 0.060
0.055 0.047 0.058 0.055
0.021 0.001 0.014 !0.001 Day 146
0.051 0.042 0.057 0.055
83 85 79 81
Pessimistic Pessimistic Pessimistic Pessimistic
0.048 0.046 0.044 0.044 0.043
0.036 0.139 0.076 0.160 0.112
!0.020 !0.136 !0.066 !0.156 !0.106
0.030 0.031 0.036 0.035 0.035
93 88 76 75 75
Pessimistic Pessimistic Pessimistic Pessimistic Pessimistic
0.045 0.045 0.042 0.041 0.042
0.032 0.134 0.089 0.164 0.114
!0.015 !0.131 !0.082 !0.161 !0.109
0.028 0.032 0.034 0.034 0.033
92 85 76 74 75
Pessimistic Pessimistic Pessimistic Pessimistic Pessimistic
0.046 0.046 0.043 0.042 0.043
0.030 0.137 0.092 0.167 0.116
!0.018 !0.134 !0.088 !0.165 !0.113
0.024 0.029 0.029 0.030 0.029
99 90 85 81 85
Pessimistic Pessimistic Pessimistic Pessimistic Pessimistic
Compatibility
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Figure 8a and 8b display the horizontal and vertical error magnitudes at altitudes of 1,200 m, 2,000 m, and 3,000 m AGL. The graphs separate the error magnitudes for the GPS subsystem and the remaining error sources. This separation was performed because the GPS error is constant under varied conditions making it a convenient benchmark for comparing error magnitudes of the remaining sub-systems. As shown in Figure 8a, the constant error of the GPS sub-system contributes only a fraction of the total horizontal
(a)
(b)
error at the displayed scan angles and altitudes. The horizontal errors are more heavily affected by angular dependant errors such as beam divergence, angle encoder error, and attitude errors. It is interesting to note that the along-track error is consistently lower at small scan angles because the angular encoder error has no effect in this direction. As the scan angle increases, errors propagate more heavily into the along-track coordinates and surpass the across-track error at approximately 13° which is why the error plots take a distinctive turn at this point. Inspecting Figure 8b shows that the GPS sub-system error contributes the majority of the total vertical error for low survey altitudes and small scan angles. As scan angle increases, GPS error remains constant at 5 cm and other factors begin to contribute a heavily higher proportion of the total error. This is especially apparent at higher altitudes where the change is more rapid. With the chosen altitudes of 1,200 m, 2,000 m, and 3,000 m AGL, the majority of the error ceases to be due to the GPS sub-system at approximately 23°, 14°, and 9°, respectively. This change is more apparent in Figure 8c which shows the percentage contributions of the vertical GPS error and the remaining errors. The error in GPS begins by contributing approximately 91 percent of total error, but then quickly degrades especially at high altitudes and large scan angles. The change between the majority contributing source of error is represented in Figure 8c by the crossing of the curves at each altitude. Therefore, users should be aware that although high altitudes and large scan angles decrease total survey time, these configurations will have an adverse effect on the error in final coordinates, in particular on the vertical error, which increases more rapidly at high altitudes than horizontal error. This fact should remind users of the importance of flying surveys in clear conditions to avoid turbulence that cause sharp changes in roll and pitch as these conditions can increase the error in determinations of these values. The vertical error will also be further increased by the existence of ground slope or vegetation which will increase the percentage contribution of the vertical GPS error.
Conclusions
(c)
Figure 8. Simulated error magnitudes percentage contributions under a variety of survey configurations: (a) Horizontal error magnitude at increasing altitude and scan angles: all error sources excluding GPS error shown as solid lines, the GPS error is shown as a dashed line; (b) Vertical error magnitude at increasing altitude and scan angles: all error sources excluding GPS error shown as solid lines, and GPS error is shown as a dashed line; (c) Percentage contributions of GPS vertical error and remaining sources of vertical error shown at 1,200 m, 2,000 m, and 3,000 m AGL represented by the solid, dotted, and dashed lines, respectively.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
This paper has summarized the development and testing of a predictive error model for lidar observations resulting from major components within the system, including the GPS unit, the IMU, operational laser scanning, and ranging units. This type of analysis is useful for predicting the anticipated random error component in lidar data prior to the actual acquisition, i.e., identifying areas that may be problematic within surveys and assist with decision making regarding areas needing re-flights. It can also help to quantify the success of a survey mission in terms of meeting contracted data accuracy requirements. Although it does not represent a complete model of the predicted error, it will guide users to predict the unavoidable, random error in their surveys due to instrument sub-system precision and allow for justification of observed errors beyond those predicted from (a) systematic calibration factors that can be identified, modeled, and eliminated from the dataset, or (b) a result of external factors such as terrain and landcover that are difficult to corrected. The model validation analysis presented has demonstrated that the error prediction routine developed here provides estimates that are as expected considering the assumptions and simplifications that were necessary to perform this analysis. The horizontal validation showed results that were pessimistic, which is due in part to being unable to account for sample position correlation from the May 2010
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interpolations necessary to accommodate variable equipment observation rates. It may also partly be the result of adopting a pessimistic estimate of scan angle measurement precision due to the inability to establish reliable and consistent quantifications either in the lab or in the published literature. Beam divergence effects are most evident on horizontal break lines such as building roof edges similar to those used in the horizontal validation procedure. This was evidenced by observed RMSE values being higher than predicted horizontal errors when the 1/e beam divergence component was considered. With the expanded beam divergence definition, the pulse radius upon contact for the range of tested flying altitudes is approximately 180 cm to 250 cm at nadir. Recalling the RMSE values from Table 3, it appears this would be a more appropriate definition for error analysis of breaklines. The vertical validation showed that under the ideal conditions of calibration surveys, and without any systematic bias the model simulated errors that were close to expected and observed. Again, considering that external terrain and land-cover effects were not accounted for, it is not surprising to see deviations between predicted and observed errors. Ongoing research is addressing the feedback influences of terrain and land-cover on lidar point position uncertainty, and a more robust and comprehensive model is under development. Although vertical error in GPS observations is frequently perceived as the largest source of vertical error, this is not always the case. From a sensitivity analysis of the error magnitudes under various survey configuration scenarios, it was demonstrated that increasing survey altitude and scan angle results in increasing horizontal and vertical random errors. Up to a flying altitude of 3,000 m AGL and a scan angle of 9°, GPS is predicted to be the largest component of vertical error. However, for horizontal error, the pitch, roll, yaw, scan angle, and beam divergence components contribute most significantly for all survey configurations. It is clear that choices made during mission planning and data collection do not only impact the cost of a survey but will also systematically, and to a large extent predictably, influence the accuracy of the collected data.
Acknowledgments The authors would like to acknowledge the financial and technical support of CARIS, Optech, Inc., the Natural Sciences and Research Council (NSERC) of Canada, the Canadian Innovation Fund (CFI), and the Applied Geomatics Research Group (AGRG). In addition, we would like to thank Laura Chasmer for support during airborne operations and the anonymous reviewers for their insightful comments.
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