ANALYSIS AND RECOVERY OF SYSTEMATIC ERRORS IN AIRBORNE LASER SYSTEM

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1 ANALYSIS AND RECOVERY OF SYSTEMATIC ERRORS IN AIRBORNE LASER SYSTEM Zhihe Wang*, Rong Shu,Weiing Xu, Hongyi Pu, Bo Yao Shanghai Institute of Technical Physics, CAS, 500 Yutian Road, Shanghai , P. R. of China Coission VI, ICWG V/I KEY WORDS: Airborne laser syste, Syste errors, Error recovery, adjustent odel, Surface extraction ABSTRACT: Although soe ature anufactures of airborne laser syste (ALS) have been published for soe years, however, in china, the developent of ALS just is on the starting step. Shanghai Institute of Technical Physics (SITP), CAS is developing a new airborne laser instruent. It is the best difficult task to deterining the systeatic biases of ALS. The ultiate goal is to deterine the aster systeatic errors and to correct the raw laser points. By analyzing the systeatic error source firstly, the adjustent odel presented in paper enable odelling and reoving the actual errors in laser point sets. Interesting surfaces and regions can be deterined by a least-squares plane fit through a subset of laser points. The proposal odel of solution is based on integrating the observations and adequate control planes and the redundancy in the overlapping areas of laser data sets. It has been deonstrated that oderate slopes are sufficient to generate reliable solutions. In addition, the precision of ranging and scan angle aiing to the sae target point is tested by laboratory experient at the end of paper. 1. INTRODUCTION Although soe ature anufactures of airborne laser scanning (ALS) have been published for soe years, however, in china, the developent of ALS just is on the starting step. Shanghai Institute of Technical Physics (SITP), CAS is developing a new airborne laser instruent. Its laser pulse rates have achieved 50k Hz and its scan rates have 40 Hz. It is capable of recording ulti return signal instead of either the first or the last return. It provides a 3D point cloud as a priary product. The achieveents in developing appropriate ALS data processing software, however, have been rather arginal. It is the best difficult task to deterining the systeatic biases of ALS. The ultiate goal is to deterine the aster systeatic errors and to correct the raw laser points such that only rando errors are left. The factors affecting laser-target position accuracy are nuerous. Huising and Goes Pereira (1998) report about systeatic errors of 20 c in elevation and of several eters in position between overlapping laser strips, Crobaghs et al. (2000) identify systeatic trends between overlapping strips. Apart fro the target reflectivity properties and laser-bea incidence angle, the ain liiting factors are the accuracy of the platfor position and orientation derived fro the carrier-phase differential GPS/INS data and uncopensated effects in syste calibration. The calibration can be divided into that of calibration of individual sensors such as the laser range-finder and that concerning spatial (lever-ar) or orientation (bore-sight) offsets between the sensors due to a particular assebly. In ost syste installations, the lever-ars between LiDAR-IMU-GPS sensors can be deterined separately by independent eans, although this represents certain difficulties related to the realization of the IMU body frae. On the other hand, the deterination of the 289 bore-sight angles is only possible in-flight once the GPS/INS-derived orientation becoes sufficiently accurate. The existing calibration procedures, while functional, are recognized as being sub-optial since they are labor-intensive (i.e., they require anual procedures), non-rigorous and provide no statistical quality assurance easures. Furtherore, the existing ethods often cannot reliably recover all three of the angular ounting paraeters. The proble is worsened by the angular uncertainty due to the broad laser bea width (Lichti, 2004). On the other hand, the cross-section ethod sees to be popular in coercial systes and usually provides satisfactory results for the bore-sight estiate in the roll direction. However, its use for the recovery of pitch and yaw/heading direction is less appropriate. The use of the slope gradients in DTM/DSM for bore-sight estiation ade its way to a popular software package used for ALS data handling (Soininen and Buran, 2005). The principal weakness of this approach is the strong correlation of the bore-sight angles with unknown terrain shape. Also, the ipleented stochastic odel of the LiDAR trajectory assues tie-invariant behavior of the GPS/INS errors that is not realistic. θ α Z Figure 1. Illustration of scan angle errors θ X Y

The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 The organi zation of the paper is as follows.

2 The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 The organi zation of the paper is as follows. At the first, definition of syste errors and their influence with respect to apping results are described. The paper focuses on the proposed error odel, and its linearization is presented. The subsequent discussion concerns the recovery of the ain error paraeters such as range error and bore-sight angles and the analysis of control and tie inforation. Finally, paper provides a brief suary, and an outlook for further work. 2. ERROR SOURCES Generally speaking, there are three types of errors during a process of easureent: blunders, systeatic biases and rando errors. Blunders are significantly larger than the other two types. They can be easy detected and eliinated with use of epirical paraeters. Rando errors are always present and can never be eliinated, however, can be iniized by least-square solution and redundant observations. As to systeatic errors, they are caused by iperfect instruents. They can be represented through soe paraeters, which are estiated by a atheatic odel fro redundant observations. The type of errors including range error, scan angle errors, bore-sight angles and level ars fro the INS syste to the laser local coordinate syste are ephatically discussed in this section. 2.1 Range Error and Scan Angle Errors The ranging easureent is deterining the tie-of-flight of a light pulse, i.e., by easuring the travelling tie between the eitted and the received pulse. Various factors contribute to the range error. It has relation with not only optical and electronical designent but also target reflectivity because range accuracy is inversely proportional to the square root of the signal-to-noise ratio(s/n).the S/N lie on any factors, such as power of eitted and received signal, input bandwidth, background radiation, responsivity of the signal detector, etc. Furtherore, if airplane flies on rather a better altitude as 2000, range error is dependent on the atospheric disturbances because of variability of atospheric refractive index. For siplicity, herein, it is regarded as unknown sall constant. Scan angle errors are depicted in Fig.1 The ideal syste is systetric to the z-axis with a scan angle θ. The real scanning syste is rotated by the alignent error α and the scan angle is θ. It chiefly includes the following three error sources: Alignent error α the zero degree direction (broken line Z-axis) and the plub line (real line Z-axis) ay not coincide. The angle bias α is aounted to adding a constant angle k to scan angle. Scan angle error Δθ an inaccurate scan angle affect scan angle. It does not suffer fro non-linear effects and is therefore oitted. Scan plane error expresses that the scan plane and the X-axis are not perpendicular. The offset is described by the two exceeding sall angular errors. So it is oitted as well as. The alignent angle influences on the apping result ore than the latter two errors. Therefore, we contribute the alignent angle to error odel. These errors result to non-linear effect to the laser points position, in particular, at the end of the swath as the reported paper(crobaghs,2000). 2.2 Bore-sight Angles and Level-ar Vector Considering ounting errors of the laser scanning (LS) sensor with respect to the INS sensor, the LS and INS coordinate syste are not parallel. The alignent error is expressed by the sall rotation atrix. The bore-sight angles are the ain ai of the recovery syste errors because they are deterined well by other eans. Therefore, the bore-sight angles ust be estiated ore precisely during an in-flight calibration procedure. The practical influence of the bore-sight on the apping result is deonstrated by Fig. 2. This figure shows a cross-section of a building. It is illustrated with the profile that the discrepancies due to bore-sight angles are clearly visible on the inclined planes. Figure 2. Effect of bore-sight errors on a cross-section profile of a building roof fro 2 flight lines of different directions and heights The agnitude of the lever-ar vector between LS and INS origins can be easured quite accurately. It is usually neglected. As for the lever-ar vector between the center of GPS antenna and INS origins, the easuring accuracy can achieve to the level of c after installation on the ground. As ention above, if the bore-sight angles are considerably larger, the level-ar vector is thus not as accurate as its agnitude, so larger level-ar quantities ay be expected, i.e., they can not be oitted. There errors exhibit to soe extent linear effect to the laser points (Crobaghs,2000). 2.3 INS Errors and GPS Errors INS errors are derived fro shift or drift of INS sensor. GPS errors are caused by soe factors such as differential troposphere, ionosphere delay, ultipath and clock biases, etc.. Certainly, they are contributing to the apping accuracy and are deterined as estiated paraeters added to the adjusting odel (Filin and Vosselan, 2004). However, it is difficult to estiate these, and even the quality of the unknown paraeters estiates ay be degraded as additional correlation. At present, the POSPAC software of version 4.4 produced by APPLANIX copany can ipleent the difference of GPS based on ulti-base Kalan Filter and integrated inertial navigation and soothing optiization, which achieve a level that the residual effect in GPS/INS trajectory estiation are lower than 0.05 and in position and attitude respectively under the optiization of the calibration area and the flight conditions. Fig.3 depicts the position RMS error derived fro GPS/INS processing by POSPAC software. So the additional paraeters of INS and GPS biases are not added to the estiated odel. 290

The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 knowledge as planes, slopes.

The paraeters of a plane j are described as r [ S S S S ] T S j = 1 j 2 j 3 j 4 j (2) where S, S 1 2 and S3 are the direction cosines of the plane's noral vector and S 4 is the negative orthogonal

The observation equation for an point i expressed by its coordinates x, y, z lying on plane j has the for i i i s x s y + s z + s 0 (3) 1 j i + 2 j i 3 j i 4 j = Figure 3.

3 The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 knowledge as planes, slopes. Soe factors of there unknown planes are estiated together with the calibration paraeters. The paraeters of a plane j are described as r [ S S S S ] T S j = 1 j 2 j 3 j 4 j (2) where S, S 1 2 and S3 are the direction cosines of the plane's noral vector and S 4 is the negative orthogonal distance between the plane and the coordinate syste origin. The observation equation for an point i expressed by its coordinates x, y, z lying on plane j has the for i i i s x s y + s z + s 0 (3) 1 j i + 2 j i 3 j i 4 j = Figure 3. Navigator position RMS for X, Y, and Z direction Note that the direction cosines ust satisfy the following unit length constraint ERROR REVOERY MODEL Geo-Referencing of LiDAR Measureents s s + s 1 (4) 1 j + 2 j 3 j = As already entioned, it is necessary to copute the laser point geo-referencing that represent atheatical odels. The geo-referencing of the laser points is viewed as a function of the observation fro the above paraeters estiated. The LiDAR geo-referencing equation in a local reference frae can be given in eq.1: Cobining equation (1) with equation (2) consist of the for that constraint conditions and laser points position as a function of the systeatic errors: FOX (, ) = 0 (5) xl y l zl X = Y Z + R iu Δx Δy + R Δz 0 Rs 0 ρ where: x, y, z =the laser footprint position in the apping l l l frae X, Y, Z =position of the phase center of GPS antenna in the apping frae Δ x, Δy, Δz =ever ar vector fro the phase center of GPS antenna to laser scanner center R iu =the rotate atrix between the IMU frae and the apping frae described by roll, pitch and yaw observation R =a priori known rotation atrix fro the IMU frae to the LS coordinate frae that depends on the ounting situation R = 0 cosθ sinθ =laser scanner rotation s 0 sinθ cosθ θ =the LiDAR encoder angular value ρ =the LiDAR range at tie t 3.2 Recovery Function Model The following odel is based on constraining the target objects to the surface extracted fro the laser points and known (1) where, O is the observations, X is the systeatic errors. The geo-referencing of the laser points in the laser coordination syste is viewed as a function of the GPS, INS, range, scan-angle, and the systeatic biases. In section 2, the systeatic biases are selected fro bore-sight angles, ranging biases, and scanning angle biases. After adding the systeatic errors to the equation (1), the geo-referencing of the laser points is changed by the following for: x y z X = Y Z + R iu Δx Δy + ΔR Δ z 0 R Δ ρ + 0 Rs Rs Δρ with Δ ρ =the range bias Δ R s =rotation atrix with alignent errorα defined in section 2 1 κ ϕ ΔR = κ 1 ω =bore-sight rotation atrix ϕ ω 1 ω, ϕ, κ =bore-sight angles Since the equation (6) is non-linear and each laser point position is represented by ore than one observation, the adjustent odel ust be used. Substituting the equation (6) to (6) 291

The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 the equation (5), the linearized for of equation (5) is given in Eq.7.

residuals; w is the isclosure vector, i.e. the equation (5) evaluated at current estiate of the paraeters and observations The solution of equation (7) is adopted by the traditional approach of least-squares adjustent.

4 The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 the equation (5), the linearized for of equation (5) is given in Eq.7. xˆ + Av + w = 0 B (7) where B is partial atrix with respect to unknowns, naely the calibration paraeters; A is partial atrix with respect to observations; xˆ is vector of unknowns; v is the vector of residuals; w is the isclosure vector, i.e. the equation (5) evaluated at current estiate of the paraeters and observations The solution of equation (7) is adopted by the traditional approach of least-squares adjustent. Naely, the su of weighted squares of the residuals reaches iniization. Following standard procedures, the resulting final for of the noral equations used herein is: T T 1 1 T T 1 xˆ = ( B ( AC A ) B) B ( AC A ) w (8) vv vv with: ˆ 2 0 T T 1 ( Av) ( ACvv A ) ( Av) = n 2 T T 1 = ˆ ( B ( AC A ) B σ (9) Dˆ 1 xˆ xˆ σ 0 vv ) (10) Figure 6. Points of extracted slope surface for a building 2 where ˆ σ 0 the variance coponent; Cvv the covariance atrix of observation; n the nuber of target laser points; the nuber of unknowns. 3.3 Surface Extraction For the strip adjustent, surfaces are the natural candidates to be used. The selecting areas are suitable for the adjustent and iprove the estiation of the paraeters. For the effect of noise on the surface paraeters, artifacts are introduced into the observation. The surface odel is, therefore, used in the for a surface constraint in equation (3). Interesting surfaces and regions can be deterined by a least-squares plane fit through a subset of laser points. The extraction procedure that is used herein is based on iniizing the weighted quadratic su of the distances of the laser points to the plane (Lee, Schenk 2001). The standard deviation of unit weight σ 0 can, therefore, be interpreted as the standard deviation σ D of the shortest distance of a point to the plane. The plane is accepted if σ 0 is saller than, or equals, a threshold. Experience shows that ost of significant surface have a std. sall than 15 c. The threshold is the average standard deviation of the distances to the plane coputed by error propagation fro the standard deviations of the laser point positions tested for the plane fit. For a horizontal plane it is just a function of the z-coponents, and thus influenced only by the accuracy in z. The steeper the slope of the plane, however, the greater will be the effect of the x and y planietric coponents. If a plane is accepted, the neighboring points are tested statistically for the fit to the plane. If the fitting error reains saller than the given threshold, the points are used to update the plane paraeters using sequential least-squares. Figure 6 to 7 show exaples for extracting surface. In Fig.6, red points represent the each planes of the building. In Fig.7, the central region is extracted as well as, but don t split up the botto plane. Figure 7. Points of one flattop plane 4. LABORATORY EXPERIMENT AND DISCUSSION 4.1 Laboratory experient Range biases and scan angle biases as entioned in section 2 result to non-linear effect to the laser points position. The precision of range and scan angle for single laser point is analysed by laboratory experient will help in reducing the effect and iproving the perforance of the ALS. The proposal ethod here repeats range easureent aiing to the sae target through fixing the scan irror. Rate of the laser instruent and scan irror respectively are 35 khz and 25 Hz. We select nine targets to test point saples extracted fro whole laser point sets are analysed statistically for each target. The specific details about the result of range are listed in Table 1 with Rax axiu range value, Rin iniu range value, Rean ean range value, and STDR standard deviations of range. Range resolved easureent precision is the standard deviation in the easured range data about the ean easured value.the corresponding shape of range 292

The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 precision is shown as Fig.8. The ean of STDR is 0.187.

So the range biases considered as a syste error are introduced into the adjustent odel. Target R ax () R in () R ean () STD R () 1 165.965 164.763 165.465 0.187 2 161.701 160.491 161.197 0.196 3 154.

5 The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing 2008 precision is shown as Fig.8. The ean of STDR is As can be seen fro the table, the agnitude of range precision is outside the LiDAR precision specification (1 σ =2.5c) under the target distance is less than 250. So the range biases considered as a syste error are introduced into the adjustent odel. Target R ax () R in () R ean () STD R () Table 1. Range results Figure 8. Magnitude of range precision for nine targets The scan angle results are listed in Table 1 with Aax axiu scan angle value, Ain iniu scan angle value, Aean ean scan angle value, and STDA standard deviations of scan angle. The corresponding shape of scan angle precision is shown as Fig.9. The ean of STD A is The result deonstrate that scan angle easureent is so stable that scan angle errors as ention in section 2 can be oitted. Target A ax A in A ean STD A Figure 9. Magnitude of scan angle precision for nine targets Discussion The ention presented in paper enables the estiate the estiation of errors over general surfaces. No distinct landarks are needed to perfor the adjustent either as control or tie points. Consequently, there are only little restrictions on its application, as the adjustent odel is based on odeling the actual effect of the error sources on the geo-reference of the laser point on the ground. A syste based approach enables odeling and consequently reoving the actual effect of the error sources. Furtherore, inclusion or eliination of error sources as ore experience is gained becoes easier to ipleent. Error odeling concerns identifying the syste errors and odeling their effect on the geo-reference of the laser point. Least-squares offers a variety of possibilities for analyzing and testing the results. Tests can be perfored to check if the residuals are randoly distributed, thus, if all systeatic errors are reoved. The estiated standard deviation of unit weight allows for proofing the correctness of the a priori assuptions for the observation accuracies. Measures for the internal and external reliability can be used for blunder detection and for accessing the geoetry of the adjustent. They show how uch single observations contribute to the estiation of the unknown paraeters and how uch a single observation is controlled by the other observations of the network. Blunders in the individual observations are not expected to be present, as they are detected during the plane search. However, the blunder detection in the laser point adjustent would reveal if planes used as tie-planes didn t atch. Re-processing the laser points with the corrections deterined in the adjustent results in a geoetrically correct point cloud of which the accuracy can be described by the standard deviations derived by error propagation. At each of the tie-planes the laser point accuracy can be verified, by coputing the planes noral vectors through the individual laser points. This gives the residuals in all three coponents x, y, z together with the length of the noral vector, i.e. the distance of the laser point to the plane. Laboratory experient is perfored to analyse the range and scan angle perforance aiing to the sae target point. The agnitude of ranging precision and scan angle is coputed. The ranging precision chiefly depends on the tie easureent accuracy and the agnitude of S/N. The result can be considered as correction to the raw range-finder offset. Table 2. Scan angle results 293

6 The International Archives of the Photograetry, Reote Sensing and Spatial Inforation Sciences. Vol. XXXVII. Part B1. Beijing SUMMARY This research studied the calibration of a laser altieter syste. The adjustent odel presented enable odelling and reoving the actual errors in laser point sets. By analyzing the properties of the proposed ethod, it has been deonstrated that oderate slopes are sufficient to generate reliable solutions. The only requireent consists in having the surface eleents oriented in different directions. The copelling conclusion is that natural terrain will yield results that are accurate and reliable. In addition to the effort of developing a robust ethod based on tie-planes, the extraction of other tie-features (e.g. lines) will be investigated. Finally, it is noted that in the case where aerial photographs are taken during an ALS ission, the ALS and photograetric observations can be processed together in one siultaneous block adjustent. REFERENCES Crobaghs, M., E. De Min and R. Bruegelann (2000). On the Adjustent of Overlapping Strips of Laser Altieter Height Data. International Archives of Photograetry and Reote Sensing, 33(B3/1): Filin, S., Vosselan, G., Adjustent of airborne laser altietry strips. International Archives of Photograetry, Reote Sensing and Spatial Inforation Sciences 34 (Part B3), Huising, E. J. and L. M. Goes Pereira (1998). Errors and accuracy estiates of laser data acquired by various laser scanning systes for topographic applications. ISPRS J. of Photograetry and Reote Sensing, 53(5): Lee, I., Schenk, T., Autonoous extraction of planar surfaces fro airborne laser scanning data. In Proceedings of the Annual Conference of the Aerican Society of Photograetry and Reote Sensing (ASPRS), St. Louis, MO, USA. Lichti, D., A resolution easure for terrestrial laser scanners. International Archives of Photograetry, Reote Sensing and Spatial Inforation Sciences 34 (Part B5), Soininen,A., Buran,H., TerraMatch for MicroStation. Terrasolid Ltd, Finland. 294

The optimization design of microphone array layout for wideband noise sources

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