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Computing the Deflection of the Vertical for Improving Aerial Surveys

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Computing the Deflection of the Vertical for Improving Aerial Surveys sensors Article Computing the Deflection of the Vertical for Improving Aerial Surveys: A Comparison between EGM2008 and ITALGEO05 Estimates Riccardo Barzaghi 1,*, Daniela Carrion 1, Massimiliano Pepe 2 and Giuseppina Prezioso 3 1 Department of Civil and Environmental Engineering (DICA), Politecnico di Milano, Milan 20133, Italy; [email protected] 2 Architecture, Built environment and Construction engineering (ABC), Politecnico di Milano, Mantova 46100, Italy; [email protected] 3 Department of Science and Technology (DIST), University of Naples “Parthenope”, Naples 80143, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-02-2399-6528 Academic Editor: Peter Teunissen Received: 22 April 2016; Accepted: 18 July 2016; Published: 26 July 2016 Abstract: Recent studies on the influence of the anomalous gravity field in GNSS/INS applications have shown that neglecting the impact of the deflection of vertical in aerial surveys induces horizontal and vertical errors in the measurement of an object that is part of the observed scene; these errors can vary from a few tens of centimetres to over one meter. The works reported in the literature refer to vertical deflection values based on global geopotential model estimates. In this paper we compared this approach with the one based on local gravity data and collocation methods. In particular, denoted by x and h, the two mutually-perpendicular components of the deflection of the vertical vector (in the north and east directions, respectively), their values were computed by collocation in the framework of the Remove-Compute-Restore technique, applied to the gravity database used for estimating the ITALGEO05 geoid. Following this approach, these values have been computed at different altitudes that are relevant in aerial surveys. The (x, h) values were then also estimated using the high degree EGM2008 global geopotential model and compared with those obtained in the previous computation. The analysis of the differences between the two estimates has shown that the (x, h) global geopotential model estimate can be reliably used in aerial navigation applications that require the use of sensors connected to a GNSS/INS system only above a given height (e.g., 3000 m in this paper) that must be defined by simulations. Keywords: deflection of the vertical; aerial surveys; EGM2008; Least Squares Collocation 1. Introduction In recent years, the increasing use of sensors connected to a GNSS/INS system has led to deep studies on the impact of the Earth’s actual gravity field [1–4]. In particular, the effect of the deflection of the vertical [5] has been investigated. Such analyses have shown the importance of considering this signal both in terms of degradation of the accuracy [6] and in terms of improvement of the observations [7–9]. Additionally, the most important companies specialized in GNSS/INS positioning (e.g., Applanix or Novatel) suggested the necessity of using a proper (x, h) model in order to achieve high performance results. In most of the works reported in literature, the deflection of the vertical components are computed at ground level, on the Digital Terrain Model (DTM). However, this approach is unsatisfactory when Sensors 2016, 16, 1168; doi:10.3390/s16081168 www.mdpi.com/journal/sensors Sensors 2016, 16, 1168 2 of 12 aerial applications are considered. As it is well known, the values of the deflection of the vertical vary Sensors 2016, 16, 1168 2 of 12 with altitude [10], decreasing while moving away from the ground level. Thus,Sensorsaerial in 2016applications order, 16, 1168 to evaluateare considered. the As impact it is well of known, the deflection the valuesof of the deflection vertical of in the aerial vertical surveys,2 varyof 12 it is importantwith to altitude define [10], a rigorous decreasing approach while moving to itsaway calculation from the ground that takeslevel. into account its variation with altitude.aerial Thus,Thus, applications proper in order are estimation to considered. evaluate formulasthe As impactit is well must of known, the be deflec consideredthe tionvalues of ofthe the that vertical deflection allow in computingaerial of the surveys, vertical ( x varyit, his) in any given pointwithimportant altitude in space, to [10],define i.e.: decreasing a rigorous while approach moving to away its calc fromulation the ground that takes level. into account its variation with altitude.Thus, Thus, in order proper to estimationevaluate the formulas impact mustof the be deflec consideredtion of thatthe verticalallow computing in aerial surveys, (ξ, η) in itany is x “ f pj, l, hq importantgiven point to in define space, a i.e.: rigorous approach to its calcx ulation that takes into account its variation with (1) altitude. Thus, proper estimation formulas# h must“ fh bepj considered, l, hq that allow computing (ξ, η) in any f , , h given point in space, i.e.: In this paper, two different methods for estimating (x, h) are presented, i.e., collocation(1) and f ,,h spherical harmonic representation. The collocation f approach,, h based on gravity data will be revised and (1) numerical resultsIn this willpaper, be two computed different formethods some for relevant estimating cases. (ξ, η) are presented, i.e., collocation and f ,,h Thesespherical results harmonic will berepresentation. then compared The collocation with the approach values based obtained on gravity using data thewill be other revised proposed estimationand methodnumericalIn this paper, based results two on will different one be comp of themethodsuted most for for recentsome estimating relevant and detailed (cases.ξ, η) are Global presented, Geopotential i.e., collocation Model and (GGM), sphericalThese harmonic results willrepresentation. be then compared The collocation with the approach values basedobtained on gravity using datathe other will beproposed revised the EGM2008andestimation numerical [11 ].method results based will onbe onecomp ofuted the mostfor some recent relevant and detailed cases. Global Geopotential Model (GGM), the EGM2008These results [11]. will be then compared with the values obtained using the other proposed 2. Predictingestimation the method Deflection based ofon theone Verticalof the most Using recent Gravity and detailed Data Global and Geopotential the Collocation Model Method(GGM), Thethe2. deflectionPredicting EGM2008 the[11]. of theDeflection vertical of is the the Vertical difference Using in Gravity direction Data between and the Collocation the direction Method of Earth’s gravity The deflection of the vertical is the difference in direction between the direction of Earth’s gravity vector and2. Predicting some reference the Deflection direction, of the such Vertical as the Using direction Gravity perpendicular Data and the Collocation to a given Method reference ellipsoid or the directionvector and of some some reference reference direction, gravity such field as the (the direction normal perpendicular gravity) [4 ].to Consideringa given reference the ellipsoid normal to the or theThe direction deflection of someof the referencevertical is gravitythe difference field (the in direction normal gravity)between [4]. the Consideringdirection of Earth’s the normal gravity to ellipsoid as the reference direction, at the geoid surface we have the situation sketched in Figure1, vectorthe ellipsoid and some as the reference reference direction, direction, such at asthe the geoid direction surface perpendicular we have the to situation a given referencesketched inellipsoid Figure where theor1, wherethe angle direction the is exaggeratedangle of issome exaggerated reference to explain to gravity explain more field more clearly (the clearly normal the thephysical gravity) physical [4]. aspect Considering of of the the problem. problem.the normal to the ellipsoid as the reference direction, at the geoid surface we have the situation sketched in Figure 1, where the angle is exaggerated to explain more clearly the physical aspect of the problem. FigureFigure 1. The 1. The deflection deflection ofof the vertical vertical at atgeoid geoid level. level. The deflection of theFigure vertical 1. The vector deflection can ofbe the decomposed vertical at geoid into level. two mutually perpendicular Thecomponents: deflection the of north-south the vertical or south vector (ξ), canpositive be decomposedtoward the north, into and two east-west mutually or first perpendicularvertical components:(η), positiveThe the deflection north-south in the east of direction.the orvertical south These vector (x ),two positive can compon be decomposed towardents are represented the into north, two in and mutually Figure east-west 2 withperpendicular respect or first tovertical (h), positivecomponents:the unit in sphere, the eastthe centred north-south direction. at point or These Psouth, having two(ξ), thepositive components z axis toward parallel are the to representedthenorth, Earth and rotation east-west in Figureaxis, or the first2 equatorialwith vertical respect to (circleη), positive parallel in tothe the east Earth’s direction. equatorial These twoplane compon and theents reference are represented meridian in in Figure a plane 2 with parallel respect to the to the unit sphere, centred at point P, having the z axis parallel to the Earth rotation axis, the equatorial theGreenwich unit sphere, meridian. centred at point P, having the z axis parallel to the Earth rotation axis, the equatorial circle parallelcircle parallel to the to Earth’s the Earth’s
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