R. W. R. RUTLAND, PH.D. University College London, England Graphic Determination of Slope and of Dip and Strike This stereographic method is applicable not only to vertical aerial photographs but also to tilted and oblique views. INTRODUCTION applied to slope determination. The attitude HE METHOD OF DETERMINING dip and of only one line is then required and con­ T strike from aerial photographs de­ siderably less work is involved than for strike scribed by Wallace (1950) is widely quoted in and dip determination. No corrections for text-books of photogeology and structural relief displacement or vertical exaggeration geology (e.g., Ray 1960, p. 64, Badgley 1959. are necessary. This method of determining p. 202-206). The method is probably not slopes may be valuable to geomorphologists, widely used, and this may account for its geologists and others as an alternative to having escaped criticism for so long. methods using specialized equipment, such as However, it has advantages over other the stereo-slope com parator (Heckman 1956). methods of determining dip and strike from It may also be used for determining true aerial photographs. No specialised equipment slopes as a basis for determining the vertical is required other than a WuIA' net. The exaggeration factor in stereoscopic viewing. ABSTRACT: This stereographic method for determining strike and dip requires only that the two geologic outcrop lines of the bedding be observed. They need not be the same level nor from the same bedding plane. The method can also be applied to the determinatioll of slope. The attitude of only one line is required and considerably less work is involved than for the strike and dip problem. No corrections for relief displacement or vertical exaggeration are required. The method can also be applied to oblique photographs. method can be used when stereometer and Finally, unlike the other methods, the mirror stereoscope are not available and it stereographic method can be applied to can be applied to more limited data than oblique photographs. required by other methods. Familiarity wi th stereographic projection The stereometer method requires that the methods (e.g.. Phillips 1954) is required, but strike and direction of true dip be estimated the principles are simple and most geology in stereoscopic viewing before the parallax students recei\'e training in these methods. measurements can be utilised. (e.g., Hemphill, This paper therefore attempts to eliminate 1958). Tn many terrains the evidence is the en'ors in the method as described by generally inadequate for this purpose. The vVallace, and to present a procedure which is stereographic method described here requires applicable to both vertical and oblique photo­ only that two outcrop lines of the bedding be graphs. observed. They need not be at the same level nor from the same bedding plane, although DISCUSSION OF THE METHOD they must be close enough so that it can be The principle of the method is simple (cr. assumed that they belong to the same bedding Wallace, op. cit. p. 270). Figure 1 illustrates attitude. Both dip and strike are derived the fact that for any short feature Be on the from the determination. ground surface (which may be an outcrop The stereographic method can also be line or a line between two points at top and 178 GRAPHIC DETERMINATION OF SLOPE .\\"D OF DIP ,\\"D STHIKE 179 A tions made in the field because then the Camtro angles to the rake lines can be measured in the POlitlon plane normal to the line of sight whate"er the orientation of the line of sight (\\'allace op. cit. problem 1, pp. 270-272). Jn aerial photo­ graphs, howeYer, the plane in which angles are measured is the plane normal to the PLAN VIEW ON PHOTOGRAPH optical axis of the camera system, and not the plane at right angles to the line of sight. For example, in vertical aerial photographs the sight line is normal to this plane only at the principal point of the photograph. Else­ where on the photographs the sight line will usually have a plunge between 90° and 60°. The measured angles to the rake lines should Plumb therefore be plotted in the horizontal plane Point in stereographic projection and not in the Flight lin t P plane normal to the sigh t Ii ne. This is the procedure adopted in the revised method FIG. 1. Relationship between an outcrop line on given below. the ground and its image on the aerial photograph. In the method given here the fundamental BC, the outcrop line on the ground surface, will plane of the stereographic projection is not be seen as the rake line BD on the photograph owing to relief displacement". For further explana­ necessarily horizon tal. Tt is taken to be the tion see text. plane of the photograph, which is normal to the optical axis. The optical axis may suffer bottom of a slope), a plane ABD in which the slight tilt from the vertical in vertical air feature is observed can be defined by the photographs, or it may be strongly tilted in sight line AB and the rake line BD (which oblique air photographs. In ei ther case, cor­ represents the projection along the sight line rection for the til t can be made at the end of of the feature B C on the photograph). This the procedure, which is therefore applicable plane must con tai n the true posi tion of the to both vertical and oblique photographs. required linear feature. Thus, if the same The geographical orientation of the pro­ feature is observed along the different sight jection is found in different ways for vertical lines of two photographs of a stereographic and oblique photographs. In the case of pair, two planes can be obtai ned, each of "ertical photographs. t he north line through which contains the true position of the re­ the principal point can be plotted in the quired feature. If these two planes are plotted fundamental plane as the zero meridian. For in stereographic projection, then their inter­ oblique photographs the line normal to the section provides the true orientation of the axis of tilt in the plane of the photograph is required outcrop line or slope line. This is all chosen as zero meridian. Its azimuth, the that is necessary for determination of slopes. direction of tilt, must be known. In the case of dip and strike determina­ In order to plot the attitudes of any rake tions, the process must be carried out for two line and its sight line in the stereographic outcrop lines in a locality where the bedding projection, it is necessary to choose a suitable plane orientation appears to he constant. reference line from which the angles to the e.g., outcrops on either side of a ,·alley. Thus required sight and rake lines can be easily two apparent dips can be obtained. The plane measured in the plane of the photograph. containing these two apparent dips can be This may he the zero meridian or the projec­ obtained. The plane containing these two tion of the sight line itself on the photograph apparent dips in the stereographic projection (BP in Figure 1). The latter is the line joining is the plane of bte bedding and gi"es the true the principal point to the observed feature dip and strike. and is here pI·eferred. vValJace (1950) used this principle for the For vertical photographs, the projection of determination of dip and strike both from the sight line can now be plotted on the indirect observations in the field and from pri mi ti ve ci rcle of the stereographic projec­ aerial photographs. Jn all cases he plots the tion at the appropriate horizontal angle, measured angles to the rake lines in the plane measured on the aerial photograph, from the normal to the sight line on the stereographic north Ii ne (the zero meridian); and the trends projection. This is correct for direct observa- of the rake lines can also be plotted on the 180 PHOTOGRAMMETRIC ENGINEERING HORIZONTAL PL-AN-E-A-T ~5~T-0-S-IGH-T-L-IN-E------·--1 ~ " ../ -:. II ~ ~ C> 0 I-' ~-~j~lf ~~ :I: 0 ~ • " (,•.-t 0 -I 2 - • •• ~--_~'" -In: -~-- ~ ~ ~ II ~ ~~ '___ z ~ • V k. "'-, rT1 Z (,I) .. ti'~ '" C) ~ ! ........ So. ~ \ ~ I o \. 0 I 1----------1<1 ....0 \2 I .0. IL- '---L._~ ~ \ \ ! I I !--__...cF...o"'ld"-,,i~",---,L...i-,,-n-,,-. ~ ,I ! / / / / / / 45° PLANE NORMAL TO SIGHT LINE FIG. 2. Diagram showing the angular relations between lines in the plane normal to the sight line and between the same lines projected along the sight line onto a horizontal plane. primitive circle at the measured angles from sight line and angles between the same lines the sight line trend. For oblique photographs, projected by orthographic projection along the projection of the sigh t line is plotted, the sight line onto a horizontal plane. The with reference to the zero meridian (whose construction is made by the standard method azimuth is the direction of tilt) in just the of preparing right sections in structural geo­ same way as for vertical photographs. The logy (see Badgley 1959, pp. 51-55) and is angle between the projection of the sight line self-explanatory. I t is easily seen that hori­ and the zero meridian is measured in the zon tal lines and their normals in the plane plane of the photograph and this angle is normal to the sight line retain their ortho­ plotted on the primitive circle of the projec­ gonal relationship when projected onto a tion.