Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
GPS and Digital Terrain Elevation Data (DTED)
                 Integration
                                    M. Phatak, ...
values which provides basic quantitative data for systems        EARTH GRAVITY MODEL (EGM) DESCRIPTION
and applications th...
φ , λ and h as well as a           4) Fit a 2-D polynomial of degree 4 in the variables
Thirdly, the grid of 45 points of
...
(φ u − φ 3 )3600
                                                                                                         ...
[9]                                      approximation using Taylor series gives the following
                           ...
m(n) = m(n − 1) + (n + 1) with m(0) = 1.
Equation [11] is solved for ∆p x , ∆p y , ∆p z and ∆b .                          ...
The value of I is chosen to be 4, and J to be 2 giving                                                 [24]
total number o...
[27]
                                                                                                          List of loc...
12                 104.83               294.87
Table 4 Horizontal errors
                                                 ...
intentions for this study rather it was simply convenient
and of high value for where it was collected. Similarly,
for San...
Table 6 Sierra Nevada Horizontal Statistics

                     Integrated LSQ       Non-Integrated LSQ

Average (m)    ...
Navigation System. The Journal of Navigation, Royal
Table 7 SF Horizontal Statistics
                                     ...
Upcoming SlideShare
Loading in …5
×

GPS and Digital Terrain Elevation Data (DTED) Integration

3,484 views

Published on

Published in: Technology, Business
  • Be the first to comment

GPS and Digital Terrain Elevation Data (DTED) Integration

  1. 1. GPS and Digital Terrain Elevation Data (DTED) Integration M. Phatak, G. Cox, L. Garin, SiRF Technology Inc. (DTM). These models are available from various BIOGRAPHIES government agencies and depending on the application, DTMs have relatively accurate vertical elevations Makarand Phatak is currently Staff Systems Engineer at sufficient enough to be used for aiding GPS constrained SiRF Technology Inc. where he has been for over three heights for E-911 or LBS. For the purpose of this study years. Prior to joining SiRF he worked at Siemens for two the National Imagery and Mapping Agency (NIMA) years on pattern recognition applied primarily to speaker Digital Terrain Elevation Data (DTED®) Level 0 was pattern recognition identification and Aerospace Systems used. To integrate DTM data in LSQ constrained height for five years on design of GPS navigation algorithms conditions a fourth order polynomial equation is formed. such as GPS and GPS/INS integrated Kalman filters and Results from integrating DTM data into the LSQ fault detection and identification. He holds a Ph. D. navigation algorithm is very promising. degree in control systems from the Indian Institute of Science, Bangalore, India and M. Tech. in electrical engineering from the Indian Institute of Technology, Kharagpur, India. INTRODUCTION Geoffrey F. Cox is currently Staff Engineer at SiRF GPS weak signal acquisition and tracking with the use of Technology, Inc. where he as been since 2000. Before high sensitivity autonomous or network centric GPS joining, SiRF Mr. Cox worked in various areas of GPS receivers in urban and rural indoor or outdoor application development for Satloc, Inc., Nikon, Inc., and environments will allow for navigation in these NovAtel, Inc. from 1996 to 2000. He holds M. Eng. in environments, MacGougan, G. et. al. (2002) and Garin, L. Geomatics Engineering from the University of Calgary. et. al. (1999). However, strong signal shading and fading effects compounds a GPS receiver’s ability to acquire and Lionel J. Garin, Director of Systems Architecture and track SVs and if the effects are strong enough the Technology, SiRF Technology Inc., has over 20 years of resulting navigation using a minimum number of 3 SVs experience in GPS and communications fields. Prior to for 2D or 4 SVs for 3D navigation can have negative that, he worked at Ashtech, SAGEM and Dassault impacts of desired horizontal accuracy. Accurate 2D Electronique. He is the inventor of the quot;Enhanced Strobe navigation is further impacted if the GPS receiver handset Correlatorquot; code and carrier multipath mitigation device is used in a high elevation urban/suburban technology. He holds an MSEE equivalent degree in environment for example, Denver, CO, USA or Calgary, digital communications sciences and systems control Alberta, CAN. Typical application scenarios for 2D GPS theory from Ecole Nationale Superieure des navigation is to use some type of externally provided Telecommunications, France and BS in physics from height and its uncertainty. Such methods are: last known Paris VI University. calculated height (stored in receiver memory from previous 3D navigation), network provided height through cell location or Base Station (BS) as well as ABSTRACT through the use DTM models, Moeglein, M. and N. Krasner (1998) and external sensors, Stephen, J., and G. The use of GPS for personal location using cellular Lachapelle (2001). telephones or personal handheld devices requires signal measurements in both outdoor and indoor environments for E-911 and Location Based Services (LBS). Therefore, DIGITAL TERRIAN ELEVATION DATA (DTED®) minimal satellite (SV) measurements such as a 3 SV or 4 DESCRIPTION SV acquisition, tracking and navigation fix in many situations will be encountered. To overcome 2D In 1999, the NIMA released a standard digital dataset navigation side effects, such as increased horizontal DTED® Level 0 for the purpose of commercial and uncertainty and error, because of constraining an public use. This DTED® product provides a world wide inaccurately obtained height in Least Squares (LSQ) can coverage and is a uniform matrix of terrain elevation be minimized using an integrated Digital Terrain Model
  2. 2. values which provides basic quantitative data for systems EARTH GRAVITY MODEL (EGM) DESCRIPTION and applications that require terrain elevation, slope, and/or surface roughness information. DTED® Level 0 GPS is referenced to the WGS-84 ellipsoid and the elevation post spacing is 30 arc second (nominally one computed navigation heights are either, above/below kilometer). In addition to this discrete elevation file, ellipsoid ( h ). It is desired to aid the constrained height separate binary files provide the minimum, maximum, through the means of Orthometric ( H ) to h conversion. and mean elevation values computed in 30 arc second To do this, the EGM is employed to obtain h from the square areas (organized by one degree cell). DTED® H . For the purpose of this paper, the EGM (1984) was used. Currently there is an EGM (1996) The DTED® Level 0 contains the NIMA Digital Mean available for use from NIMA and arguably would not Elevation Data (DMED) providing minimum, maximum, improve on reducing absolute LE errors associated with and mean elevation values and standard deviation for each using the DTED® Level 0 unless a more accurate 15 minute by 15 minute area in a one degree cell. This DTED® Level is used or other sourced DTM. initial prototype release is a quot;thinnedquot; data file extracted from the NIMA DTED® Level 1 holdings where available and from the elevation layer of NIMA VMAP Table 2 EGM - 84 Level 0 to complete near world wide coverage. The Horizontal DATUM WGS-84 specifications for DTED® Level 0 are: Coverage World Wide 10o by 10o Grid Spacing Table 1 DTED® Performance Specification (1996) Relative Vert. Accuracy 3 (m) LE Horizontal DATUM WGS-84 ALGORITHM OUTLINE Coverage World Wide 1o by 1o First, the idea is to form a fourth equation from the Tile (Individual File) DTED®. This equation is derived from a polynomial (in Coverage 2 variables of northing φ and easting λ ) surface fit to Grid Spacing ~1 Km (30’) the appropriate terrain. To select this appropriate terrain Absolute Hor. Accuracy 90% Circular Error (CE) the 3 SV measurements are solved first for a fixed h . ≤ 50 (m) The fixed h is the average value of the h in the neighborhood of the BS (Base Station). Typically, the Absolute Vert. Accuracy 90% Linear Error (LE) boundary of this neighborhood is a few tens of kilometers ≤ 30 (m) away from the BS (as the center). Error in the fixed h is 90% CE WGS ≤ 30 (m) Relative Hor. Accuracy taken as the standard deviation of h in the neighborhood. over a 1o cell (point to point) With this information the 3 SV position solution with 90% CE WGS ≤ 20 (m) Relative Vert. Accuracy fixed altitude comes with an estimated error ellipse. over a 1o cell (point to point) Secondly, it is required to construct grid points along the directions of the major and minor axes of the error ellipse. The step sizes are made proportional to the magnitudes of One technical issue to point out is the NIMA still reserved the major and minor axes respectively. The center of the the right to not include sensitive military installations here ellipse is the 3 SV position as obtained before. Along the in the US and abroad. Those areas deemed sensitive do semi-major axis 9 points are selected (4 in the positive not include elevation data but rather horizontal positions direction, 4 in the negative and one at the center) and and will appear delineated as an empty space if using a along the semi-major axis 5 points are selected (2 in the mapping package to visualize the terrain. Therefore, if an positive direction, 2 in the negative and one at the center) E-911 or LBS system requires seamless DTM coverage to cover a rectangular grid of 4 sigma along each axis. In there are other models available with similar performance this process, 45 points are chosen in the rectangular grid. specifications, namely GTOPO30 Global Elevation Altitude values above the mean sea level ( H ) at these Model, GTOPO30 Documentation (1996). points are obtained from the DTED® by indexing the four corner points in which the grid point resides and then GTOPO30 is a global digital elevation model (DEM) with using bilinear interpolation between these corner points. a horizontal grid spacing of 30 arc seconds The obtained H values are converted to the WGS 84 h (approximately 1 kilometer). The DEM was derived from several raster and vector sources of topographic by adding the Geoid N separation at the 3 SV position information. The coverage and accuracy specification are point. similar to that of the DTED® Level 0.
  3. 3. φ , λ and h as well as a 4) Fit a 2-D polynomial of degree 4 in the variables Thirdly, the grid of 45 points of of φ and λ with a total of 15 coefficients to the φ and λ is found using LSQ 4-th order polynomial in 45 points obtained in step 3. Find the maximum method. There are 15 coefficients to determine. To residual error for the polynomial fit. If this error handle ill conditioning the polynomial is found in new exceeds a threshold of 100 m stop processing variables that represent a scaled deviation from the center with appropriate error message, else proceed point (the 3 SV position solution). Also a robust along to step five. numerical method of Q-R decomposition is used; with Q computed using modified Gram-Schmidt procedure (to 5) Solve GPS equations with 3 SV pseudorange make Q only orthogonal rather than orthonormal); this is measurements and the equation of the to avoid square root operations. The equation of the polynomial along with the maximum residual polynomial with so determined coefficients is the 4-th error of step 4 to find position and horizontal equation. The maximum deviation of the grid point error ellipse parameters. altitude from the surface fit is the error associated with this 4-th equation. If this error exceeds a given threshold 6) For the φ and λ as obtained in step 5 check (empirically derived and set to 100 meters) then the polynomial fit is declared poor and unusable. Then more whether the corresponding point belongs to the than one polynomial surface fits are required. rectangular grid of step 3 and if yes accept the solution of step 5 as a valid solution else reject it Lastly, the 3 GPS equations and the 4-th polynomial as invalid. equation are solved in coordinates of φ , λ , h and clock bias rather than using Earth Center Earth Fixed (ECEF). COMPLETE EQUATIONS The ECEF coordinate formulation is retained and change from ECEF to chosen coordinates is achieved by working DTED® Level 0 Indexing with the corresponding Jacobian. The Jacobian corresponding to the 4-th equation comes from the φu λu , the nearest Given a user latitude, and longitude, derivative of the polynomial. If there is a convergence then it is checked whether the converged solution is South-West corner of an available DTED® data file is within the rectangular grid of polynomial fit. If it is not found and used as a reference to find an index in that data then the method is repeated for the next surface fit if file. This index is used to retrieve the H . The equations available. are found on the next page. y = λu − λr [1] Algorithm In Steps where, 1) With the reference location at the center retrieve Orthometric heights at points 1 km apart in the λr Reference Longitude for the South West Corner Easting and Northing directions. A total of of an available DTED data file. (2 ⋅ N + 1) 2 points are considered on a grid of λu Apriori/User Longitude. (2 ⋅ N + 1) × (2 ⋅ N + 1) points. Convert the y Difference in degrees Orthometric H to WGS 84 h . Determine average h and set h error equal to the standard x = φu − φ r [2] deviation over the grid of points.. 2) Solve GPS equations with 3 SV pseudorange where, measurements and average h and the h error in φr step 1 to find the position and corresponding Reference Latitude for the South West Corner of horizontal error ellipse parameters. an available DTED data file. φu User Latitude. 3) With the position of step 2 at the center, retrieve H at points on a rectangular grid constructed x Difference in degrees . along the major and minor axes of the ellipse. A total of 45 points are considered on a grid of  y ∗ 3600  9 × 5 points. [3] brow =    ∆ λ spacing   
  4. 4. (φ u − φ 3 )3600 [5] x' = (∆ φ ) where, − bcol spacing where, ∆λspacing DTED Level 0 Grid Spacing of 30” Arc Seconds x' Weighted Ratio from user/reference Latitude brow Integer Row value within DTED data grid spacing and DTED column location. file in the range [0, 129]. (λ u − λ 3 )3600 y' = [6] (∆ λ − brow )  x * 3600  spacing [4] bcol =   where,  ∆ φ spacing    y ' Weighted Ratio from user/reference Longitude where, grid spacing and DTED column location. ∆φspacing DTED Level 0 Grid Spacing of [7] 30” Arc Seconds H i = H1 + (H 2 − H1 )x'+(H 4 − H1 ) y'+(H1 + H 3 − H 2 − H 4 )x' y' bcol Integer Column value within DTED data file in the range [0, 129]. where, The values of brow and bcol are used to find the index in H 1 , L , H 4 , represent 4 Orthometric heights in a given the data file and then this index is used to access the searched row and column result. altitude value. H i is the interpolated Orthometric height and 45 of these points are determined. 1× 1Deg. y' Height above Ellipsoid Estimation H3 H4 To estimate the 45 points of h requires the estimation of H 3, 4 x' the Geoid N from the EGM-84 as a function of φu and Hi λu . H1, 2 H1 Row H2 Once N is estimated a linear calculation is used to determine h , Schwarz, K, and Krynski, J. (1994). Figure (φ r , λr ) 2 below illustrates the relationship for computing between Col the DTED® model and EGM-84 model. SW Corner Reference Figure 1: Indexing and Bilinear Interpolation Scheme h=N+H [8] Bilinear Interpolation pt25 The H is obtained as above and interpolated to the given pt1 user latitude, φu and longitude, λu as follows. The brow Topography H 25 H1 h25 h1 φ3 Geoid Model and bcol correspond to the altitude H 3 , latitude, and True Geoid N 25 N1 λ3 ; longitude, see figure 1. Three more altitudes H 1 , Ellipsoid (WGS-84) and H 4 are obtained from (brow+1) and bcol, H2 (brow+1) and (bcol+1) and brow and (bcol+1) Figure 2: Vertical Relationships respectively. Then, H , at φu and λu is obtained as Constrained LSQ Solution from 3 SV Pseudoranges follows: and Average h The equations to be solved are:
  5. 5. [9] approximation using Taylor series gives the following equations. ( s1x − px ) 2 + ( s1 y − p y ) 2 + ( s1z − pz ) 2 ⋅ (1 − m1 ) + b = ρ1  − l1x − l1 y − l1z 1  ∆px   ∆ρ1  ( s2 x − px ) 2 + ( s2 y − p y ) 2 + ( s2 z − pz ) 2 ⋅ (1 − m2 ) + b = ρ 2 − l 1 ∆p y  ∆ρ 2  − l2 y − l2 z  2x ⋅ ( s3 x − px ) 2 + ( s3 y − p y ) 2 + ( s3 z − pz ) 2 ⋅ (1 − m3 ) + b = ρ3 = , [11]  − l3 x 1  ∆pz   ∆ρ3  − l3 y − l3 z ( p 'x − px ) 2 + ( p ' y − p y ) 2 + ( p'z − pz ) 2 ⋅ sgn(h) = h     − d x − dy − dz 0  ∆b   ∆h    where ( s ix , s iy , s iz ) are the ECEF coordinates of antenna phase center of SV i at the receive time, ( p x , p y , p z ) where, l i is line of sight unit vector pointing from are the ECEF coordinates of the GPS receiver antenna receiver to SV i and d is down direction unit vector phase center, b is common offset in pseudorange pointing along the downward normal to the WGS-84 measurements, ρ i is i -th pseudorange measurement, ellipsoid, ∆p x , ∆p y , and ∆p z are differential position coordinates, ∆b is differential pseudorange offset, ∆ρ1 , mi satellite mean motion correction term (given below), ∆ρ 2 , and ∆ρ 3 are differential pseudoranges, and ∆h is ( p' x , p' y , p' z ) are ECEF coordinates of projection of ( p x , p y , p z ) on the WGS-84 ellipsoid, ρ i is the differential h . The line of sight unit vector is given by measured pseudorange for SV i , h is the height above WGS-84 ellipsoid and sgn( h) = 1 if h > 0 , [12] sgn(h) = −1 if h < 0 , and it is undefined when h = 0 (here the equation itself reduces to an identity but the , 1 differential version of the equation is still defined; see li = ⋅ (si − p * ) ( s ix − p * ) 2 + ( s iy − p * ) 2 + ( s iz − p * ) 2 below). The h is given by the average h as obtained x y z from the step 1 of the above algorithm. The receive time is assumed to be have error less than about 10 ms so that the satellite positions as computed from the ephemeris the down direction unit vector is given by have good accuracy. The mean motion correction term, mi is given as − cos λ* ⋅ cos φ *    d =  − sin λ* ⋅ cos φ *  , [13] 1 mi = ⋅ (vi + ω × s i ) o (si − p ),   − sin φ * [10]   c where × denotes vector cross product, o denotes vector (φ * , λ* , h * ) are the WGS-84 geodetic where dot product, vi is the velocity vector of SV i , s i is the * coordinates of p (the equations for change of position vector of SV i , ω is the Earth rotation vector coordinates are not included in this document, Kaplan, D. and p is GPS receiver position vector, the x , y , and (1996)), z coordinates of all vectors are in ECEF and all except ω correspond to the antenna phase centers. ∆ρ i = ρ i − ρ i* , [14] * * * Let ( p , p , p ) be the ECEF coordinates of the x y z ρ i* where are obtained from the left hand side of [9] at reference or approximate position which serves as the the initial guess point, and * initial guess for ( p x , p y , p z ) and let b be the initial guess for the pseudo-range offset. Expanding the left hand side of [10] around the initial guess to a first order ∆h = h − h * . [15]
  6. 6. m(n) = m(n − 1) + (n + 1) with m(0) = 1. Equation [11] is solved for ∆p x , ∆p y , ∆p z and ∆b . For degree, n = 4 , m = 15 . The coefficients, ci , Then the estimates of position and clock bias are updated as i = 1,L, m are obtained by solving the following linear equation [20] using least squares method.  p x   p *   ∆p x  ˆ [20] x  p   *  ∆p  ˆy  c0  py   y   = + [16] . c   p z   p *   ∆p z  ˆ  1 z  ˆ  *     c2  ξ1  φ1 λ1 φ12 φ1 ⋅ λ1 λ12 L λ1n 1 b   b   ∆b       1  c3  ξ 2  φ2 λ2 φ2 φ2 ⋅ λ2 λ2 L λ2n 2 2 =  ⋅ M   c4   M  M M M M M OM    This is the Newton-Raphson update. Next, the initial 1  M  ξ r  φr λr φr φr ⋅ λr λr L λrn 2 2 guess is replaced by the new estimate as cm−2     cm−1   p*   px  ˆ x  *  ˆ  where the subscript i (except on the coefficient)  py  =  py  , [17] represents i -th point of the terrain. The points are chosen  p*   pz  ˆ as follows. The center point (as given by φ c , λc , and z  *  ˆ  b   b   ξc ) is the point which corresponds to the solution obtained in step 2. This solution also comes with horizontal error ellipse parameters of the semi-major axis, and the iterations are continued until ∆p x , ∆p y , ∆p z a e , the semi-minor axis, be , and the angle the semi- and ∆b become less than respective thresholds. θ e , measured major axis subtends with the east direction, anti-clockwise positive. This information is used to Polynomial Surface Fit create a grid of points as With the grid of 45 points a 2-D polynomial is set up in  ∆n  cos θ e − sin θ e   i ⋅ ∆a e  φ λ  ∆e  =  sin θ ⋅ the auxiliary variables and which are given in [21] , cos θ e   j ⋅ ∆be     φ λ as terms of and, e i = − I ,L , I , j = − J , L , J , φ = q ⋅ (φ − φ c ) , λ = q ⋅ (λ − λ c ) , [18] and φc where q is a scale factor (chosen as 100), and and  φ i  φ c  ∆n /(( N c + hc ) cos φ c ) λ  =   +  λc respectively [22]  are the northing and easting of the  j  λ c   ∆e /( M c + hc )  solution obtained in the step 1 of the algorithm given above. The polynomial equation is given by where, [19] [23] a(1 − e 2 ) a ξ = p(φ,λ) = c0 ⋅φ +c1 ⋅ λ + c2 ⋅φ +c3 ⋅φ ⋅ λ +c4 ⋅ λ +L+cm−2 ⋅ λ +cm−1. 2 2 n Mc = and N c = , (1 − e sin φc ) 2 2 3/ 2 1− e 2 sin2 φc The polynomial fit is not performed on h but it is rather performed on the down component, ξ . The expression where a is the semi-major axis of the WGS-84 ellipsoid relating ( p x , p y , p z ) and (φ , λ , ξ ) is given later in and e is its eccentricity. [25]. The total number of coefficients, m for degree, n are given by the recursive formula,
  7. 7. The value of I is chosen to be 4, and J to be 2 giving [24] total number of points, r = 45. As seen earlier, degree (s1x − px (φ,λ,ξ))2 +(s1y − py (φ,λ,ξ))2 +(s1z − pz (φ,λ,ξ))2 ⋅ (1− m ) +b = ρ1 of the polynomial n , is chosen as 4th order giving number 1 (s2x − px(φ,λ,ξ))2 +(s2y − py (φ,λ,ξ))2 +(s2z − pz (φ,λ,ξ))2 ⋅ (1−m2) +b = ρ2 of coefficients, m = 15 . The system of equations in [20] (s3x − px(φ,λ,ξ))2 +(s3y − py (φ,λ,ξ))2 +(s3z − pz (φ,λ,ξ))2 ⋅ (1−m3) +b = ρ3 therefore has 45 equations and 15 unknowns and is solved c0 ⋅φ(φ) + c1 ⋅ λ(λ) + c2 ⋅φ(φ)2 + c3 ⋅φ(φ) ⋅ λ(λ) + c4 ⋅ λ(λ)2 +L+ cm−2 ⋅ λ(λ)n + cm−1 =ξ with the help of modified Gram-Schmidt procedure as follows. Equation [20] in the usual matrix notation is A ⋅ C = H and the objective in least squares solution is to minimize where, ( A ⋅ C − H ) T ⋅ W ⋅ ( A ⋅ C − H ) , where W is positive [25] definite weighting matrix. The optimum solution is obtained by solving the set  px  − cosλ1 sinφ1 − sinλ1 − cosλ1 cosφ1  φ  AT ⋅ W ⋅ A ⋅ C = AT ⋅ W ⋅ H . This set can be written  p  =  − sinλ sinφ cosλ − sinλ cosφ  ⋅ λ  y  1   as B ⋅ B ⋅ C = B ⋅ Γ ⋅ H , by using the decomposition T T 1 1 1 1  pz   cosφ1 − sinφ1  ξ  0   W = Γ T ⋅ Γ and using B = A ⋅ Γ . This new set can −1 further be written as R ⋅ C = D ⋅ Q ⋅ H , where B T or is decomposed as B = Q ⋅ R , with R unit upper φ   − cos λ1 sin φ1 − sin λ1 sin φ1 cos φ1   p x  λ  =  − sin λ 0  ⋅  py  triangular (diagonal elements of R are all ones and lower cos λ1     1 diagonals are all zeros) and such that Q ⋅ Q = D , D T ξ   − cos λ1 cos φ1 − sin λ1 cos φ1 − sin φ1   p z      being a diagonal matrix. The upper triangular set of equations can be solved easily using back-substitution φ1 and The transformation in [25] depends on the latitude method. In the above, two decompositions are used. The longitude λ1 as corresponding to the position obtained in first is: W = Γ ⋅ Γ . This can be done using Cholesky’s T step 2, Kaplan, D. (1996). The set of equations [24] is method. Usually, W is diagonal and then so is Γ and it solved using usual Newton-Raphson method. This time can be obtained simply taking square roots of the diagonal the initial guess is given by (φ c , λc , hc ) which are the elements of W . Even simpler case is when W = I , transformed coordinates (second equation in [25]) of the where I is identity matrix and then Γ = I as well. This position obtained in the step 2. Further, let bc be the simple equal weighting is used in the solution of [20]. The B = Q⋅R. second decomposition is This initial guess for the pseudorange offset, taken again from the solution of step 2. Expanding the left hand side of decomposition can be obtained by modified Gram- [24] around the initial guess to a first order approximation Schmidt method which gives Q , R and D by avoiding using Taylor series gives the following equations. square root operations since Q is only orthogonal (not orthonormal); Golub, G. and Van Loan, F., (1983). [26] LSQ Solution from three SV pseudoranges and polynomial surface equation 1 ∂px / ∂φ ∂px / ∂λ ∂px / ∂ξ 0 ∆φ ∆ρ1  − l1x − l1y − l1z − l − l − l 1 ∂py / ∂φ ∂py / ∂λ ∂py / ∂ξ 0 ∆λ ∆ρ2 ,  2x ⋅  ⋅   =   2y 2z 1 ∂pz / ∂φ ∂pz / ∂λ ∂pz / ∂ξ 0 ∆h ∆ρ3 The equations to be solved are same as in [9] with the two − l3x − l3y − l3z       exceptions. The last (forth) equation is replaced by α β −1 1 ∆b  ∆ξ  0  0 0 0 altitude equation as a polynomial in φ and λ . With this where, the expressions for various derivatives are given change it is convenient to consider the first three below. Equation [26] is solved for ∆φ , ∆λ , ∆ξ and equations in the unknowns of φ , λ and ξ rather than in ∆b and then the procedure is same as that used in the the in ECEF frame. So, the equations are written as: solution of [9].
  8. 8. [27] List of locations Continued α = q ⋅ (c0 + 2 ⋅ c2 ⋅ φ + c3 ⋅ λ + 3 ⋅ c5 ⋅ φ 2 + 2 ⋅ c6 ⋅ φ ⋅ λ + c7 ⋅ λ 2 + L + cm −3 ⋅ λ n −1 ) Pt. Place Latitude Longitude Altitude β = q ⋅ (c1 + c3 ⋅ φ + 2 ⋅ c4 ⋅ λ + c6 ⋅ φ 2 + 2 ⋅ c7 ⋅ φ ⋅ λ + 3 ⋅ c8 ⋅ λ 2 + L + n ⋅ cm −2 ⋅ λ n −1 ) 7 West 49.33303 -123.1170 66.918 ∂px / ∂φ = − cosλ1 sinφ1 Vancouver, ∂px / ∂λ = − sinλ1 Canada ∂px / ∂h = − cosλ1 sinφ1 8 Calgary, 51.05502 -114.0831 1068.79 ∂p y / ∂φ = − sinλ1 sinφ1 Canada ∂p y / ∂λ = cosλ1 9 Toronto, 43.6570 -79.38959 76.71 ∂p y / ∂h = − sin λ1 cosφ1 Canada ∂pz / ∂φ = cosφ1 10 Montreal, 45.51374 -73.56042 52.545 ∂pz / ∂λ = 0 Canada ∂pz / ∂h = − sinφ1 11 Innsbruck, 47.25732 11.39317 594.554 Austria 12 Zermatt, 46.02137 7.74862 1668.853 The value of ∆ξ is the difference between the right hand Switzerland 13 Berchtesgaden, 47.63058 13.00635 599.634 side and the left hand of the last equation in [24] for the Germany chosen initial condition. 14 Klappen, 63.45426 14.11699 374.498 Sweden Simulation testing 15 Penderyn, UK 51.76133 -3.52172 264.045 Measurements were simulated with Gaussian 16 Fort William, 56.82175 -5.09585 39.96 measurement errors with standard deviation of 10 meters. UK One set of measurements is simulated for the visible 17 Chamonix- 45.92519 6.87376 1040.082 satellites for each of the 30 locations given below. The all Mont-Blanc, possible 3 satellite combinations have been created to get France all possible 3 satellite geometries for each location. These 18 Vatican City, 41.90207 12.45701 -3.82 combinations give on the average 100 measurement sets Italy per location. 19 Brussels, 50.84837 4.34968 36.671 Belgium Results and analysis 20 Barcelona, 41.36245 2.15568 84.365 Spain 21 New Delhi, 28.65974 77.22777 198.838 Table 3 List of locations India 22 Mumbai, India 18.95001 72.82963 23.045 Pt. Place Latitude Longitude Altitude 23 Tsun-Wan, 22.36929 114.11651 -13.099 1 Stockton, 37.94404 -121.3446 -8.493 Hong Kong California 24 Taipei, Taiwan 25.03879 121.50934 -9.155 2 San 37.75296 -122.4464 146.904 25 Singapore 1.30031 103.84861 2.299 Francisco, 26 Seol, South 37.56332 126.99138 22.404 California Korea 3 Camden, 44.28846 -69.06771 65.24 27 Tokyo, Japan 35.67640 139.76929 -22.467 Maine 28 Honshu, Japan 35.43063 138.71154 1375.919 4 Knoxville, 35.96004 -83.92057 241.391 29 Sapporo, Japan 43.05465 141.34358 2.922 Tennessee 30 Shanghai, 31.23445 121.48123 -7.625 5 Golden, 39.72184 -105.2100 1847.155 China Colorado 6 West 34.09083 -118.3830 87.583 Hollywood, California
  9. 9. 12 104.83 294.87 Table 4 Horizontal errors Surface fit errors continued Pts. 67% 95% Num. Num. % Pt. 67% Surface fit 95% Surface fit Hor. Hor. of of Yield err. (m) err. (m) Error Error solns. combi. 13 79.56 128.90 (m) (m) 14 26.75 29.45 1 39.85 75.41 45 56 80.36 15 48.98 53.22 2 40.08 80.45 71 84 84.52 16 51.64 56.23 3 45.75 130.41 141 165 85.45 17 94.02 442.49 4 45.57 152.02 44 56 78.57 18 73.24 74.51 5 25.83 86.22 45 56 80.36 19 42.41 43.54 6 31.50 104.84 45 56 80.36 20 82.17 90.69 7 51.98 92.72 47 56 83.93 21 19.97 20.18 8 33.55 54.24 46 56 82.14 22 48.94 49.47 9 35.21 74.04 68 84 80.95 23 15.66 18.87 10 48.56 162.14 101 120 84.17 24 31.67 31.92 11 47.11 100.01 67 84 79.76 25 15.63 16.56 12 60.02 194.07 27 84 32.14 26 56.08 59.85 13 57.33 126.71 64 84 76.19 27 66.67 66.72 14 36.10 68.77 143 165 86.67 28 82.14 106.32 15 43.79 136.38 145 165 87.88 29 53.19 53.73 16 65.32 135.25 108 120 90.00 30 24.59 24.73 17 66.61 124.77 33 84 39.29 18 72.56 174.15 70 84 83.33 Field testing 19 29.59 92.65 103 120 85.83 20 87.89 195.27 70 84 83.33 The aim of the field tests is to exercise the integrated 21 18.38 42.77 48 56 85.71 DTED in two major areas of interest, namely a mountainous region and an Urban Environment with 22 54.45 119.44 49 56 87.50 significant changes in elevation. 23 35.29 100.53 181 220 82.27 24 40.63 74.91 103 120 85.83 The Sierra Nevada Mountains Interstate 80 Corridor was 25 23.72 58.34 73 84 86.90 chosen for its extremely undulating terrain, significant 26 52.69 134.42 129 165 78.18 elevation changes (steep grades), and a peak elevation of 27 76.91 190.28 95 120 79.17 Donor Pass which is found along the freeway at an 28 62.23 141.95 93 120 77.50 elevation of ~2200 m (~7219 ft) HAE. Furthermore, the 29 44.61 116.95 45 56 80.36 terrain along the Truckee River I-80 corridor leading to 30 39.39 103.05 95 120 79.17 Reno, NV with its narrow steep canyon walls, ~500 m wide river valley, to the NW and SW provides a difficult Table 5 Surface fit errors region to model. The Truckee – Reno river gorge will exercise how well the polynomial surface fit is executed. Pt. 67% Surface fit 95% Surface fit The city of San Francisco, CA also provides an excellent err. (m) err. (m) Urban Canyon environment and undulating terrain to test 1 9.02 9.02 the algorithms. 2 3.82 16.00 3 15.14 17.92 DATA COLLECTION AND PROCESSING 4 4.95 7.21 5 14.38 24.81 Dynamic GPS data for the Sierra Nevada was collected 6 21.77 33.14 August 18, 2000 as part of a four day cross-country trip to 7 33.67 37.27 Northern New England as part of SiRF’s BETA receiver 8 28.47 29.69 software release testing. Raw GPS code and carrier, 9 20.41 21.11 ephemeris, and almanac data was logged during this trip 10 37.45 38.37 at typical freeway speeds of approximately 105 to 110 11 48.54 66.96 Km (65 to 70 mph). The data was not collected with the
  10. 10. intentions for this study rather it was simply convenient and of high value for where it was collected. Similarly, for San Francisco Urban Canyon testing archived raw GPS data was used from SiRFLoc release testing and it was collected on July 7, 2002 at typical city streets speeds of stop and go type traffic conditions, 0 to 30 km (0 to 25 mph). Raw GPS data processing using the integrated LSQ and DTED required an exclusively developed Win2K PC post processing software to simulate server based Network Centric MS-Assisted navigation. To access the DTED files for post processing, at run-time, the applicable files for California and Nevada where stored on the PC’s hard drive. For the purpose of this study only three SV measurements Figure 3: Sierra Nev. Mtns. I-80 Corridor 20 Km2 were used to compute both integrated DTED and HAE Search nonintegrated aiding, just the average h derived from the simulated network BS. The three SV measurements used Figure 4 is a horizontal plot zoomed in to the location of in the integrated LSQ where not constrained to any of the I-80 corridor between Truckee, CA and Reno, NV particular criteria such as signal strength (C/No) level, along the Truckee River gorge. There are three dynamics elevation angle or azimuth; just any three SV plots overlayed on I-80. The black dots are the Kalman measurements stored in the matrix of observations were filter trajectory, the red dots are the integrated LSQ and used. DTED, and the yellow dots are the non-integrated LSQ using average h . To provide quantitative statistics the real time Kalman filtered navigation results logged along with the raw GPS measurements for the Sierra Nevada (WAAS corrections) cross-country trip and San Francisco (no corrections) results was used as the truth trajectory. Results and analysis Sierra Nevada Mtns - Interstate 80 Corridor Dynamic Test Figure 3 is a 20 Km2 area representing height above ellipsoid for the Sierra Nevada mountain range along the I-80 corridor. The trajectory, shown in red, is the 3SV only integrated LSQ and DTED. The figure is produced to give the reader an idea of the terrain that is required to be modeled under dynamic conditions. Figure 4: Sierra Nev. Mtns. I-80 Corridor Cross Track Errors The next sets of figures are the statistical plots showing horizontal errors. The top plot in Figure 5 is the HDOP for the 3SV only integrated LSQ and DTED. The HDOP for the non-integrated LSQ using average h is identical and is not shown. The middle plot is the corresponding Horizontal Error in meters. The bottom last plot in Figure 5 is the non-integrated LSQ using average h . In Table 6 the horizontal statistics are compiled and the results show a significant improvement in integrated LSQ horizontal accuracy than the non- integrated LSQ.
  11. 11. Table 6 Sierra Nevada Horizontal Statistics Integrated LSQ Non-Integrated LSQ Average (m) 13.4 75.9 Std. (m) 44.7 101.2 RMS (m) 46.6 126.5 Figure 7: SF Urban Canyon - 20 Km2 HAE Search Figure 5: Sierra Nev. Mtns. I-80 Corridor HDOP and Horizontal Error San Francisco Urban Canyon – Financial District Figure 7 is a 20 Km2 area representing height above ellipsoid for the City of San Francisco. The trajectory, shown in red dots, is the 3SV only integrated LSQ and DTED. The figure is produced to give the reader an idea of the terrain that is required to be modeled under Figure 8: SF Urban Canyon Cross Track Error dynamic conditions. The next sets of figures are the statistical plots showing Figure 8 is a horizontal plot zoomed in to the location of horizontal errors. The top plot in Figure 8 is the HDOP the San Francisco’s Financial District that consists of an for the 3SV only integrated LSQ and DTED. The extremely multipath prone area that can also significantly HDOP for the non-integrated LSQ using average h is degrade receiver signal tracking capability. There are identical and is not shown. The middle plot is the three dynamic overlay plots on the city test streets. The corresponding Horizontal Error in meters. The Bottom black dots are the Kalman filter trajectory, the red dots are last plot in Figure 9 is the non-integrated LSQ using the integrated LSQ and DTED, and the yellow dots are average h . In Table 7 the horizontal statistics are the non-integrated LSQ using average h . compiled and the results show an improvement in integrated LSQ horizontal accuracy than the non- integrated LSQ. Note: the impact of mulitpath is clearly shown in these results and SV visibility is also impacted as well, as shown in HDOP.
  12. 12. Navigation System. The Journal of Navigation, Royal Table 7 SF Horizontal Statistics Institute of Navigation, 54, 297-319. Integrated LSQ Non-Integrated LSQ National Imagery And Mapping Agency (NIMA) (1996) Performance Specification Digital Terrain Elevation Average (m) 128.8 262.1 Data (DTED), MIL-PRF-89020A 19 April 1996. Std. (m) 227.4 587.1 Superseding MIL-D-89020. Defense Mapping Agency, 8613 Lee Highway, Fairfax VA 22031-2137 RMS (m) 261.3 643.0 National Imagery and Mapping Agency (NIMA) (1996, 1997) World Geodetic System 1984/96 Earth Gravity Model Office of Corporate Relations Public Affairs Division, MS D-54 4600 Sangamore Road, Bethesda, MD 20816-5003 The Land Processes (LP) Distributed Active Archive Center (DAAC) (1996), Online GTOPO30 Documentation, U.S. Geological Survey, EROS Data Center, 47914 252nd Street, Sioux Falls, SD 57198- 0001 Web: http://edcdaac.usgs.gov Golub, G. H. and Van Loan, C. F., (1983) Matrix Computations, The John Hopkins University Press, Baltimore, 1983. ( problem p6.2-4). Kaplan, E. D. (Ed.) (1996) Understanding GPS Principles Figure 9: SF Urban Canyon HDOP and Horizontal and Applications, Artech House, Boston, 1996. (section Error 2.2.3.1). CONCLUSIONS Schwarz, K. P. and Krynski, J. (1994) Fundamentals of The use of an integrated LSQ and DTED or other DTM Geodesy, Lecture Notes ENSU 421, Dept. of can have a positive impact on improving LSQ navigation Geomatics Engineering, University of Calgary, horizontal accuracy. The case where 3 SV only Calgary, AB, CAN. (Section 3.2 p 26) Presented by G. navigation is encountered the improvement can be highly MacGougan at ION National Technical Meeting, San beneficial to E-911 or LBS where horizontal accuracy is Diego, 28-30 January 2002. critical for dispatching emergency services or providing premium navigation for personal services. REFERENCES MacGougan, G., Lachapelle, G., Klukas, R. Siu, K. Department of Geomatics Engineering Garin, L., Shewfelt, J., Cox G., (2002) SiRF Technology Inc.Degraded GPS Signal Measurements With A Stand- Alone High Sensitivity Receiver, ION National Technical Meeting, San Diego, 28-30 January 2002. Garin, L. J., M. Chansarkar, S. Miocinovic, C. Norman, and D. Hilgenberg (1999) Wireless Assisted GPS-SiRF Architecture and Field Test Results. Proceedings of the Institute of Navigation ION GPS-99 (September 14-17, 1999, Nashville, Tennessee), 489–497. Moeglein, M. and N. Krasner (1998) An Introduction to SnapTrack Server-Aided GPS Technology. Proceedings of the Institute of Navigation ION GPS-98 (September 15-18, 1998, Nashville, Tennessee), 333–342. Stephen, J., and G. Lachapelle (2001). Development and Testing of a GPS-Augmented Multi-Sensor Vehicle

×