( ) Eight geological shape curvature classes are defined in terms of the Gaussian and mean normal curvatures as follows: perfect saddle, plane
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1 Geological Shape Curvaure Principal normal curvaures, and, are defined a any poin on a coninuous surface such ha:, + and + () The Gaussian curvaure, G, and he mean normal curvaure, M, are defined in erms of he principal normal curvaures as: ( ) =, = + G M () Eigh geological shape curvaure classes are defined in erms of he Gaussian and mean normal curvaures as follows: < 0 = 0 > 0 synformal saddle, synform, basin > 0 > 0 > 0 < 0 = 0 perfec saddle, plane = 0 = 0 < 0 = 0 > 0 aniformal saddle, aniform, dome < 0 < 0 < 0 (3) Each class is associaed wih a name ha connoes an idealized shape and a color (Figure ). Geological curvaure classificaion color able KM > 0 synformal saddle synform basin KM = 0 perfec saddle plane KM < 0 aniformal saddle aniform dome KG < 0 KG = 0 KG > 0
2 Figure. Color able for geological shape curvaure. The narrow color bands labeled aniform and synform, and ha labeled perfec saddle, represen exacly zero values of M and G respecively. The small colored square a he cener labeled plane represens exacly zero values of boh M and G. The narrow whie band represens he mahemaically inadmissible class. The eigh disinc colors are used o decorae a geological surface, hereby idenifying he local shape of ha surface (Figure ). The daa for Figure are aken from Bellahsen e al., 006 who digiized conour lines on he srucure conour map of Forser e al., 996 for he base of he Jurassic Sundance Formaion a Sheep Mounain, Wyoming. There are likely o be subsanial errors in elevaion inroduced during consrucion of he srucure conour map and during gridding of he surface from he digiized conours. These errors have no been evaluaed, so we ake he surface as given and proceed wih he curvaure analysis of i as an illusraion of he conceps and mehods. We anicipae a fuure evaluaion based upon comparison o a gridded surface generaed by inerpolaion of Airborne Laser Swah Mapping (ALSM) daa. Geological curvaure classificaion of poins on base Sundance Fm z axis y axis x axis Figure. Geological shape curvaure classificaion for base Sundance a Sheep Mounain, WY. The MaLab code surf_curv_pure.m calculaes and plos he surface in 3D space (x,y,z), he principal normal curvaure magniudes, he principal normal curvaure direcions as vecors on angen planes, and he geologic shape curvaure classes. Also calculaed are he coefficiens of he firs and second fundamenal forms, and he shape operaor. Curves ha are concave upward (posiive z) have posiive curvaure. The inpu for he code is hree ab-delimied ex files conaining he gridded x, y, and z coordinaes, respecively, of poins on he surface. The grid of x- and y-coordinaes are organized in hese files as illusraed in (4). Each elevaion (zvalue) in he hird file is associaed wih he corresponding pair of (x, y) coordinaes in he firs wo files.
3 ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + min x min x dx max x min y min y min y min x min x + dx max x min y + dy min y + dy min y dy (4) min ( x) max ( x) max ( y) max ( y) The definiions given in (3) lead o cerain mahemaical and daa analysis issues such ha he values of Gaussian and mean normal curvaures, calculaed from coordinae daa on a geological surface, resul in a preponderance of poins falling ino he classes in he four corners of he color able (Figure ). This phenomenon is apparen for he Sundance surface from Sheep Mounain anicline when i is roaed so he view is direcly downward (Figure 3). The numbers of poins in each class are: 375 aniformal saddle; 0 aniform, 5 dome; 0 perfec saddle; plane; 83 synformal saddle; 0 synform; and 836 basin. Geological curvaure classificaion of poins on base Sundance Fm. Threshold k = y axis x axis Figure 3. Geological shape curvaure classificaion wih zero hreshold. The smaller black circle locaes of planar poins; he larger black circle locaes 3 of 0 synformal poins. Consider he Gaussian and mean normal curvaures for aniformal and synformal poins: = 0 and 0 requires = 0 or 0 G M Apparenly i is unlikely ha a calculaed value of principal normal curvaure is exacly zero for a naural surface. Furhermore, given he ineviable errors in measuremens of coordinae daa, i would be difficul o defend a value as being exacly zero. Consider he perfec saddle: < 0 and = 0 requires = = 3
4 I is unlikely ha calculaed values of principal normal curvaures would be exacly equal in magniude and opposie in sign. Again, errors in measuremen preclude a defense of his special case. Consider he plane: = 0 and = 0 requires = 0 = The argumens given above apply o his case. Finally, consider he unlabeled class shown as he narrow whie band on Figure : > 0 and = 0 requires = 0 G M This is mahemaically inadmissible because he produc of wo numbers, equal in magniude and opposie in sign, always is less han zero. I is possible o address he ineviable errors inroduced in measuremens of coordinae daa on geological surfaces, and o reconcile he problem ha naural surfaces have a preponderance of poins classified ino he corner posiions of he color able, by inroducing he concep of a normal curvaure hreshold. If he magniude of a principal normal curvaure is less han or equal o a posiive consan called he hreshold, k, hen ha curvaure is se o zero: if k hen = 0 if k hen = 0 for k 0 (5) This redefiniion resrics he ranges of he principal normal curvaures such ha: > k > k = 0, = 0 (6) k < < k The consequences for he Gaussian curvaure are found by wriing down a able of all possible values of using he ranges in (6): > k > k = 0 < k > k = 0 < k = 0 = 0 = 0 < k > k The cases in he lower lef of his able are ruled ou because. The resricions imposed by (6) on he Gaussian curvaure are relaed o he square of he hreshold. The consequences for he mean normal curvaure are found by wriing down all possible values of ( + )/ using he ranges in (6): 4
5 > k = 0 < k > > > + k k k, = 0 = 0 < k < k < k Again, he cases in he lower lef of his able are ruled ou because. Also, he unresriced range for he mean normal curvaure (upper righ case) is found if he principal curvaures have opposie signs and unequal magniudes. In summary, using a hreshold o resric he ranges of he principal normal curvaures as in (6), also resrics he range of he Gaussian curvaure, bu does no resric he range of he mean normal curvaure: > k = 0, M + < k (7) For he synformal saddle wih a hreshold imposed on he principal normal curvaures we have > k and < k. Because he range of he mean normal curvaure is no resriced by hese condiions, we mus impose he hreshold on i, such ha M > k. For he cylindrical synform wih a hreshold we have > k and = 0. The consequence for he mean normal curvaure is M k >. For he basin wih a hreshold we have > k and > kso he range of mean normal curvaures is resriced such ha M > k. Using similar argumens he ranges for he aniformal saddle, aniform, and dome are redefined. Togeher hese new ranges serve o redefine he ranges for he perfec saddle and plane. The eigh geological shape curvaure classes are: < k G = 0 > k synformal saddle, synform, basin > k > k > k < k = 0 perfec saddle, plane k M k < k G = 0 > k aniformal saddle, ani form, dome < k M < k < k (8) The color able corresponding o he ranges in (8) is provided as Figure 4. 5
6 Geological curvaure wih hreshold classificaion color able KM > k k synformal saddle basin k / synform KM = 0 plane undefined -k / -k perfec saddle KM < k aniformal saddle aniform dome KG < k KG = 0 KG > k Figure 4. Color able for geological shape curvaure wih a hreshold k operaing on boh principal normal curvaures according o (5). The ranges of principal normal curvaures for he base of he Sundance formaion wih no hreshold (Figure ) are: and (9) A meaningful hreshold for his surface mus be less han he greaes magniude of eiher principal curvaure, so k < 0.0. How is he value chosen? The crierion suggesed here relaes he hreshold o he sandard error σ associaed wih he elevaion daa. z d ρ A B C δz nσ δx x Figure 5. Geomeric relaions used o define he curvaure hreshold. 6
7 Referring o Figure 5 consider hree successive daa poins (A, B, and C) in he x-coordinae direcion wih consan spacing δx. A circular arc is drawn hrough hese poins wih radius ρ. The disance d is measured from he cener of he circular arc o he level line connecing poins A and C, and he elevaion change from ha line o he poin B is δz. From Pyhagorus we have: ( ) ρ = δx + d Subsiuing for d = ρ - δz and rearranging o solve for ρ, he radius of curvaure is: ρ = ( δx) + ( δz) To disinguish he arc ABC from he sraigh line AC we asser ha he elevaion change δz mus exceed some muliple of he sandard error of he elevaion measuremens, nσ. Noing ha he curvaure is he reciprocal of he radius of curvaure and subsiuing he muliple of he sandard error for he elevaion change, we define he hreshold: δz ( ) ( σ) k = nσ δx + n (0) Calculaed values of normal curvaure ha are less han or equal o he hreshold signal ha he elevaion daa are no precise enough o defend a non-zero curvaure. This mehodology may be exended o evaluae normal curvaures in any direcion on a surface, including he principal direcions. For example, suppose ha he sandard error for he gridded elevaion daa for he Sundance formaion a Sheep Mounain is σ = m. Using (0) wih n = we find he corresponding hreshold k = , which corresponds o a radius of curvaure of,50m. This hreshold is well wihin he range of magniudes of he principal curvaures (9). As in Figure he geological shape curvaure is compued, bu now i is consrained by he hreshold as implemened in he MaLab scrip surf_curv_hres.m (Figure 6). On he Sundance surface he numbers of poins in each class are: 44 aniformal saddle;,9 aniform, 68 dome; 94 perfec saddle;,48 plane; 36 synformal saddle; 988 synform; and 7 basin. The shape curvaure a 46% of he 4,64 grid poins on he surface is planar. Tha is, he Sundance formaion locally is indisinguishable from a plane a hese poins, given he precision of he elevaion daa. Cylindrical aniforms and synforms comprise 6% and % of he poins respecively, and are concenraed in he wo hinges of he folded Sundance surface. The shape curvaures are bes illusraed for he Sundance surface from Sheep Mounain when i is roaed so he view is direcly downward (Figure 7). Comparison o he shape curvaures for he surface wih zero hreshold (Figure 3) emphasizes he fac ha for mos of he more genly dipping back limb he shape canno be disinguished from planar given a m sandard error in elevaions. A similar conclusion may be reached for he much less exensive 7
8 forelimb. To address quesions relaed o he curvaure of exposed sraigraphic surfaces in he back- and forelimbs more precise elevaion daa would be essenial. The majoriy of poins in he cres of he fold are aniformal in shape, wih some domal and aniformal saddle poins. The majoriy of poins in he rough of he fold are synformal in shape, bu lile of he sraigraphy is exposed in his region so he elevaion errors are probably quie significan. Geological curvaure wih hreshold classificaion for base Sundance Fm z axis y axis x axis Figure 6. Geological shape curvaure classificaion wih a hreshold, k = , for he base Sundance a Sheep Mounain. See Figure 4 for color able. Geological curvaure wih hreshold classificaion for base Sundance Fm. 500 y axis x axis Figure 7. Geological shape curvaure classificaion for Sheep Mounain anicline wih a non-zero hreshold, k = , associaed wih a sandard error σ = m. 8
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