Assessment of surface deformation with simultaneous adjustment with several epochs of leveling networks by using nD relative pedaloid Ocena deformacij-s simultano izravnavo več terminskih izmer z nD relativnim pedaloidom Milivoj VuLic1, Ana Vehovec2 'Faculty of Natural Science and Engineering, University of Ljubljana, Aškerčeva cesta 12, SI-1000 Ljubljana, Slovenia; E-mail: milivoj.vulic@ntf.uni-lj.si 2 DDC svetovanje inženiring, Kotnikova 40, 1000 Ljubljana, Slovenia; E-mail: ana.vehovec@ddc.si Received: November 02, 2006 Accepted: November 14, 2006 Abstract: Relative error hyperellipsoid, 3D relative error pedaloid and 2D relative pedal curve are discussed. Izvleček: V članku govori o 3D relativnem pedaloidu pogreškov in 2D relativni pedali. Key words: Adjustment by parameter variation, nD relative error hyperellipsoid and hyperellipsoid, 3D relative error ellipsoid, 2D relative ellipse. Ključne besede: Posredna izravnava, nD relativni hiperelipsoid pogreškov in hiperelipsoid, 3D relativni elipsoid pogreškov, 2D relativni elipsoid. Introduction Consequence of underground extraction of coal is surface alteration. Negative consequences of mining are reflected above all as ground deformation, field landslides, formation of lakes, climate changes due to alteration of landscape, influence on subterranean waters and thermal springs, seismic effects of subterranean blasting. Ground subsidence is the most intensive above extraction fields but can also be observed on the edge fields. That is the reason for planning local observation networks, by which expanse of deformation can be deter- mined. Observation of networks is important because of closeness of outbuildings and other buildings. With simultaneous adjustment of several epochs of measurements, the field deformation can be determined. Theoretical basis of adjustment by parameter variation in geodetic leveling network Ultimate aim for adjustment of geodetic networks are point coordinates. Definitive or most probable point coordinates can not be obtained by direct mathematical processing of measured quantities (angles, lengths, height differences, etc.), they can be only determined by process of adjustment. This process is possible only if the number of measured data is greater then necessarily needed. In the leveling network adjustment one point with known absolute height should be given (or assumed). This holds for adjustment by parameter (height coordinate) variation. In the adjustment there are three types of quantities: • given quantities (constant values, which don't change by adjustment), • measured quantities, • unknown quantities. By adjustment unknown quantities are determined from given quantities through series of measured quantities on condition that the sum of squares of their residuals is minimal. With observation equation coefficients a., b .,...u., and absolute terms f. Coefficients are partial derivatives of functional relation between given, measured and unknown quantities. Their values depend on configuration and size of network. Absolute terms can be symbolically expressed as f = approximate - measured. Approximate values are computed from approximate coordinates. The adjustment is done considering: vrQ^1v=min or vrPv=min (a) v residuals, Qll correlation matrix of measured quantities, P weight matrix of measured quantities. Observation equations can be written in matrix form: V1 a, h • JC A = «2 b2 . * y + /2 at b, • t ft (b) Or shortly: v vector of residuals, A design matrix of observation equations, x unknowns vector, f vector of absolute terms. Coefficient matrix of normal equations N reads: v = A-x + f N = A PA Ipaa\ \pab\ \pha] \pbb] [pwa] \pub\ ■ P = diag[_pj p2 ••• p] (c) \pau\ \pbu\ [pMM]J(d) (e) When measurements are of the same accuracy then P = pI, where I is unit matrix. Vector of absolute terms of normal equations n: n = ATPf = wji \pbf] .[pw/1 = AJpIf = pATf (f) Vector of unknowns is: x = -N_1n = -YsXf QxcXc ^Q. XcXi xsyv + Q XvXy (p) 2d relative error pedaloid and ellipse Figure 3. 2D relative pedaloid and ellipsoid Slika 4. 2D relativni pedaloid in elipsoid By the analogy of relative ellipsoid, relative error ellipse is: Q ellipse SV Gayay Qayax Qayax Qaxax Qysys Qrsxs Qysyv Qrsxr ' 1 0 1 0 -1 0" QysXs Qxsxs Qxsyy Qxsxv 0 1 0 1 0 -1 Qysyv Qxsyy Qyvyv Qyvxv -1 0 QysXy Qxsxy Qyvxv O'xyxy 0 -1 Q&yay Qayax Qayax Qaxax . ' 1 0 Qysys Qysyv Qys*s QxSYy Qysy¥ Qyvyv Qysxv Qyvxy 0 1 Qysxs ~ Qysxy Qxsxs Qxsxy QxsYy ~ Qyvxy Qxsxv ^^xyxy -1 0 0 -1 '.SV Qayay Qayax Qayax Qaxax . Qysys ~ 2Qysyv + Qyvyv Qysxs Qyxxv QxsYy Qyvxv QySxS Qxsyf Qysxy Qyvxv öxfXP ^öXFXD ^^XyXy (q) (r) (s) For given point: Zaključek ' ellipse ST Qay&y Qayax Qay&x Qaxax . Summary ^^YyYy YyXy ^^YtrXtr ^^XrrXv (t) Multi epochs adjustment by parameter variation is simple enough, besides that the points subsidence is calculated directly. In this article nD hyperpedaloid and hyperel-lipsoid, error ellipse and error ellipsoid were formulated. Ocena deformacij s simultano izravnavo več terminskih izmer z nD relativnim pedaloidom Posredna izravnava več terminskih izmer je dokaj enostavna, poleg tega pa se ugrezke izračunava neposredno. V članku so predstavljeni nD hiperpedaloid in hiperelipsoid, elipsa pogreškov in elipsoid pogreškov. Prikazana je bila njihova izpeljava. Zanje je značilno, da so neodvisni od koordinatnega izhodišča. Their execution was shown. Their characteristic is that they do not depend on coordinate origin. Reference Mihajlovič, K., Vračarič, K., Geodezija III. Gradevinski fakultet Beograd, Beograd, 1985. Mihajlovič, K., Vračarič, K., Geodezija I. Gradevinski fakultet Beograd, Beograd, 1989. Mihajlovič, K., Geodezija II (2. deo), Naučna knjiga, Beograd, 1978. Mihajlovič, K., Geodezija: Izravnanje geodetskih mreža. Gradevinski fakultet Beograd, Beograd, 1992. Todorovič, R. T., Uvod v rudarsko škodo II. Naravoslovnotehniška fakulteta, Oddelek za geotehnologijo in rudarstvo, Ljubljana, 1996. Todorovič, R. T., O relativni elipsi pogreškov. Rudar-sko-metalurški zbornik. 1994: 41, 169-174.