Sayısal arazi modellerinden hacim hesaplarında en uygun enterpolasyon yönteminin araştırılması

Abstract

ÖZET Gelişen teknolojinin mesleğimize yansımasıyla Jeodezi ve Fotogrametri Mühendisliği alanında hızlı bir otomasyon süreci yaşanmaktadır. Bilgisayar donanımlarının ve yazılımlarının inanılmaz bir hızla gelişmesi, elle yapılan uygulamaları bilgisayar ortamında daha hızlı ve kolay yapılır hale getirmiştir. Kısaca, arazinin sayısal gösterimi diye tanımlayabileceğimiz Sayısal Arazi Modeli (SAM) yaygın bir uygulama alanına kavuşmuştur. Bir sayısal arazi modeli uygulaması, çeşitli enterpolasyon yöntemlerinden birini veya bir kaçım içeren uygun bir bilgisayar yazılımı gerektirir. SAM geniş anlamıyla, sayısal yükseklik modelini ve sayısal durum modelini birlikte içerir. Dar kapsamda düşünüldüğünde, sadece sayısal yükseklik modeli olarak algılanır. Bu çalışmada, sayısal yükseklik modellemesinde kullanılan enterpolasyon yöntemleri toplu halde verilmiştir. Yer alan yöntemler şu şekilde sıralanabilir: 1. Ağırlıklı aritmetik ortalamayla enterpolasyon 2. Polinomlarla enterpolasyon 3. Multikuadrik enterpolasyon 4. Kayan yüzey yardımıyla enterpolasyon 5. Yüzey toplamlarryla enterpolasyon (Lineer prediksiyon) 6. Sürekli parça parça polinomlarla enterpolasyon 7. Dikdörtgen gridde enterpolasyon 8. Üçgenler ağında enterpolasyon Enterpolasyonda önemli bir yeri olan üçgenleme işlemine ayrı bir yer verilmiştir. Üçgenleme algoritmaları içinde özel bir yeri olan Delaunay Üçgenlemesi ayrıntılı olarak anlatılmıştır. Sayısal Arazi Modellerinde kullanılan hacim hesapları bir bölüm içerisinde sunulmuş ve seçilen 7 farklı arazi grubu için hacim uygulamaları yapılmıştır. Uygulanan hacim hesaplan şu 3 ana başlık altında toplanabilir: 1 Üçgen prizmalarla hacim hesabı 2 Dikdörtgen prizmalarla hacim hesabı 3 Kesitlerle hacim hesabı Elde edilen sonuçlar ışığında, hacim hesaplan için kullanılmakta olan enterpolasyon yöntemleri, grid büyüklüğü, kesit aralığı ve kritik daire yarıçapı için değerlendirmeler ve öneriler yapılmıştır.

SUMMARY RESEARCH OF THE MOST SUITABLE INTERPOLATION METHOD FOR VOLUME DETERMINATION IN DIGITAL TERRAIN MODELS Digital terrain model, which is the subject of this study, simply, can be defined as digital representation of the terrain. The representatin is based on measurements on some points which are called control points, reference points or sample points. A digital terrain model involves an interpolation method and a computer program. D.T.M. is a digital representation of the terrain, based on measurements on the reference points by means of detailed computer program which contain an interpolation method. Reference points are known with their x,y planimetric coordinates and z coordinates. Reference points for digital terrain models must be chosen in such locations that they can serve for a satisfactory analytical representation of the terrain They can be chosen in three different patterns : 1. in more or less random positions 2. on characteristic terrain features such as welldefined linear features, ridges, contour lines, fall lines, highs and lows, or on parallel profiles 3. at the nodes of a regular grid In addition, in each case points will be chosen on such important features as breaklines in the terrain. Control points for digital terrain models can be obtained from,. terrastrial measurements. photogrammetric measurements. topographic maps. stereo satallite images An other classification can be done according to the way of selection of points as follows,. selective sampling. adaptive sampling. composit sampling This classification is usually used for photogrammetric methods. The control points are important for the accuracy of D.T.M. The accuracy that can be obtained from the interpolation methods, of course, upon both density and distribution of the reference points and the choise of method. The accuracy of a digital terrain model depends on,. density and distribution of reference points. accuracy of reference points. used interpolation method, and. terrain features( terrain structure) Interpolation, generally means prediction of measurement values on interpolation points, based on measurement values on the reference points. In digital elevation models, it means, prediction of z value for an interpolation point known with x and y coordinates. In this study, only interpolations in digital elevation models have been dealt with. Interpolation of curves may be subject of another study. The methods studied on, can be summarized as follows,. Interpolation with weighted average. Interpolation with polynoms. MuMquadric interpolation. Interpolation with moving surfaces. Interpolation with summation of surfaces (Linear prediction). Interpolation with simultaneous patchwise polynomials. Interpolation in a rectangular grid. Interpolation in a net of triangles An interpolation in the D.E.M. can be done in three ways:. pointwise. patchwise, or. simultaneously with a function The second and third methods interpolate the surface simultaneously with a function. The first and fourth methods interpolate the surface with pointwise approach. The others use patchwise approach. The mathematics of the methods as follows,. Interpolation with weighted average The height of an interpolation point is found by weighted arithmetic average of the heights of selected reference points. Z0=(Pl*Zl+P2*Z2+ +Pm*Zm)KP1+Pz+ +Pm)- The weight (p) is a function of the distance between interpolation point and reference points. Interpolation with polynoms In this method, all interpolation surface is expressed with a second or third degree polynoms. ~(*,v) = I? nZk n. t.a.JyJ v '' k = 0 I =0,J = K-l İJ J XIVParameters of the polynom are calculated by using the condition that all reference points must satisfy the polynom. If the number of reference points are equal to number of the parameters, the unknown parameters can be calculated without adjustment. Otherwise, they are calculated by adjustment, which is based on least squares method.. Multiquadric interpolation The surfaces can be assumed to be multiquadric surface, which can be represented by the series, l.%Cj[Q(XjtyJtxty)] = &z in which Az is a function of x and y resulting from summation of a single class of quadric surfaces Q. The vertical axis of symmetry of each quadric term is located at a discrete position xt,yr The associated coefficient C determines the algebraic sign and flatness of the quadric term. The equation, generally used is, 1/2 J^CjliXj-xf+iyj-yf+k] -As with constant k which equals zero or positive value. The unknown constants C are calculated by solution of equation group.. Interpolation with moving surfaces The method requires a surface calculation for any interpolation point. Generally this surface will change its orientation from an interpolation point to another one. For this reason it is called a moving surface. Usually, a polynom is used as expressed in the section related with interpolation with polynoms. For the interpolation of a point, the coordinates and height of each of the surrounding reference points are substituted in the equation. Each reference point is given a weight that is decreasing function of the distance to the interpolation point and unknown parameters in the equation are solved by the least squares method.. Interpolation with summation of surfaces (Linear prediction) The interpolation is also called linear least squares interpolation or prediction. To qualify as a stationary random function, the systematic trend in the data must be eliminated by reducing the heights to a reference surface. The surface is called a trend surface. The values at the interpolation points are assumed to be linear function of Az values on the reference points. A distance function, called correlation or covariance function is either defined or calculated. By using least squares method problem is solved.. Interpolation with simultaneous patchwise polynomials The interpolation method starts by dividing the region of interest into square or rectangular elements. A surface is defined for each grid element by low degree xvpolynom in such a way that the total surface is continuous and possibly smooth. Each local polynom is either a full 16 term bicubic polynom or a part of it. The first and second derives at the grid nodes and the heights of grid nodes are calculated using height of interpolation points in the grid. A 16 term bicubic polynom is expressed as, ^ = ûoo +o01y+any2 +<w3 +ûrıo* +anxy +a12xy2 + auxy3 + a20x2 + a21x2 y + a^x2 y2 +a23x2y3 +a30x3 + a3lx3 y + a32x3 y2 +a33x3y3 The coefficients of the polynom are determined with the use of derive and height values at the nodes of grid by least squares adjustment.. Interpolation in a rectangular grid Heights at the nodes of a rectangular grid may be obtained either by one of the interpolation method or by measuring directly. Every grid part is expressed as 16 term bicubic polynom or a part of it. The local polynoms can be an 12 term bicubic polynom, an 8 term biquadratic polynom, an 4 term bilinear polynom or two linear triangles. The parameters of each local polynom must be computed from the given heights at the nodes of the its grid and any other provided values at the boundaries of the element. The other values can be heights of the middle of the grid lines or derive values at the corner of grids calculated by using heights of grid nodes The Voronoi Diagram and Delaunay Triangulation Of a Point Set. Interpolation in a net of triangles This method uses the reference points as the vertices of a net triangles that cover the interpolation area without overlapping. If the reference points have been measured in an irregular pattern, the triangles will have irregular shapes. There are various algorithms that can be used for triangulation. Some of these algorithms are Optimal, Greedy, and Delaunay triangulation. The optimal triangulation is defined to have a minimal sum of edge lengths. An apparently alternative approach to the optimal criterion is obtained by triangulating with the requirement that no edge should be included if there is a shorter edge that would properly intersect it. The Delaunay xvitriangulation assigns triangles in the data set by the criterion that no data lies inside the circumcircle of any other triangle. Delaunay triangles define nearest natural neighbors in the sense that the data points at the vertices are closer to their mutual circumcenter than is any other data point. Various methods can be applied for interpolation. Some of them are linear interpolation, curvilinear interpolation, interpolation with natural neighbor coordinates, least squares interpolation with normal vectors and least squares interpolation with use of minimum area. The aim of the study are,. To present interpolation methods used in D.T.M. together for digital terrain model users,. To research accuracy of volume calculations used in D.T.M. and. To determine effect of the grid size, number of cross section and diameter of critical circle on volume values The applied methods for volume calculations can be classified in three groups, 1. Volume calculations with triangular prisms, 2. Volume calculations with rectangular prisms and 3. Volume calculations with cross sections The applied interpolation methods as follows, 1. Volume calculation according to a reference plane 2. Volume calculations with linear interpolation in triangles 3. Volume calculations with multiquadric method 4. Volume calculations with weighted average 5. Volume calculations with least squares method (Gauss function) 6. Volume calculations with least squares method (Hirvonen function) 7. Volume calculations with moving surfaces 8. Volume calculations with 2. degree polynom 9. Volume calculations with weighted average using Delaunay neighbors 10. Volume calculations with cross sections xvu