Sayısal arazi modellerinden hacim hesaplarında en uygun enterpolasyon yönteminin araştırılması
Research of the most suitable interpolation method for volume determination in digital terrain models
Yazar: MUSTAFA YANALAK
Danışman: PROF. DR. ORHAN BAYKAL
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Jeodezi ve Fotogrametri Mühendisliği Ana Bilim Dalı / Ölçme Tekniği Bilim Dalı
Derece: Doktora, 1997
Konu: Jeodezi ve Fotogrametri = Geodesy and Photogrammetry
Dizin: Sayısal arazi modelleri = Digital terrain models ; İnterpolasyon yöntemleri = Interpolation methods

Özet:
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.

Abstract:
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 +