”.nkV .M'._"-"‘.’"°'m-—1a—>— -——.. t..~—- _ A GRAVITY STUDY OF BIG HAND AND , I COLUMBUS-NIAGA-RAN REEF FIELD-S IN ST. CLAIR COUNTY, MICHIGAN -‘ Thesis 'for the Degree oIIIII. SI. MICHIGAN STATE UNIVERSITY WARREN E. KEITH - ' _ 1967 ;' ABSTRACT A GRAVITY STUDY OF BIG HAND AND COLUMBUS-NIAGARAN REEF FIELDS IN ST. CLAIR COUNTY, MICHIGAN By Warren E. Keith The gravity method is one of the principal exploration techniques utilized in the search for petroleum reservoirs associated with Niagaran reef structures in southeastern Michigan. However, not all reef structures have obvious gravity anomalies. The Big Hand oil and Columbus gas fields, which are separated by only two miles are an excellent illustration of two nearby reef structures which appear to give different gravitational results. A gravitational study of these two reefs shows that the Columbus reef has an associated +0.27 mgal anomaly, while no anomaly is correlated with the Big Hand reef. Theoretical gravity anomalies computed from geological information indicate the gravity anomaly associated with the Big Hand reef is cancelled out by a negative anomaly originating from a structure in the overlying F-salt. These calculations also .show that structural relief on the A—2 carbonate and the A-l carbonate and reef body contrasting with the A-2 salt accounts for the major portion of the gravity anomalies associated with the reefs. A GRAVITY STUDY OF BIG HAND AND COLUMBUS-NIAGARAN REEF FIELDS IN ST. CLAIR COUNTY, MICHIGAN By Warren E. Keith A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Geology 1967 ACKNOWLEDGMENTS The author wishes to sincerely thank Dr. William J. Hinze for his invaluable guidance and genuine interest during the preparation of this study. Acknowledgment is also made to Dr. H. F. Bennett and Chairman C. E. Prouty for their suggestions and helpful criticism pertaining to this study. Thanks is also expressed to the Department of Geology, -Michigan State University, for the use of their World Wide gravimeter and to Michigan State University for the use of the CDC 3600 computer. 11 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi LIST OF APPENDICES viii Chapter I. INTRODUCTION 1 Purpose of Study 1 Location of Area . . . . 2 Physiography of Area. . . . 2 II. GEOLOGY A Regional Geology of Michigan Basin A Stratigraphy . 6 General Stratigraphy of Michigan Basin . . . . 6 Silurian Stratigraphy . . . . . 9 Geology of Reef Structures. . . . . ll Niagaran Reef Structures . . . . ll Columbus and Big Hand Reefs . . . 12 III. GRAVITY STUDY 31 History of Reef Studies by the Gravity. 31 Gravity Maps . . . 35 Bouguer Gravity Map . . . . . 36 Cross- -profile Residual Map. . . . 39 Least Squares Residual Map. . . . 39 Upward and Downward Continuation Maps. . . . . . A3 Second Derivative Maps . . . . . 50 Interpretation. 50 Theoretical Gravity Study of Reef Structures. . 53 Theoretical Study of. the Bedrock Surface. . 63 Interpretation of Observed Anomalies . 65 iii Chapter Page IV. CONCLUSION . . . . . . . . . . . 68 BIBLIOGRAPHY . . . . . . . . . . . . . 7O APPENDICES. . . . . . . . . . . . . . 7A iv Table 1. LIST OF TABLES Salina and Niagaran Strata in the Subsurface of St. Clair County. . . . . Strata Densities. Gravity Data and Location Coordinates. Results Obtained from Lease Squares Polynomial Equation . . . . . Upward Continuation Coefficients Downward Continuation Coefficients. Second Derivative Coefficients Example of Upward and Downward Continuation and Second Derivative Calculations Page 10 63 95 97 100 101 103 105 LIST OF FIGURES Area of Investigation. . . . . . . . . Regional Tectonic Features of Michigan Basin . Stratigraphic Sucession in Michigan . Structure Structure Structure Structure Structure Structure Structure Structure Contour Contour Contour Contour Contour Contour Contour Contour Map Map Map Map Map Map Map Map of Niagaran Formation of A—l Carbonate Unit of A—2 Carbonate Unit of B-Salt Unit. of C-Unit of F-Salt Unit. of Dundee Lime. of Traverse Lime Isopach Map of A-l Evaporite Unit: Isopach Map of A-2 Evaporite Unit. Cross-Section of Columbus Reef. Geologic Cross-Section of Big Hand Reef. Relationship of F-Salt Structure to Big Hand Reef Bouguer Gravity Map Portion of Regional Gravity Map of Michigan . Cross-Profile Least Squares Second Degree Least Squares Fourth Degree , Residual Map . . . . . . . . Residual Map-Observed Minus the Residual Map-Observed Minus the vi Page 1A 16 17 18 19 21 22 23 25 26 27 28 3O 37 38 A0 A1 A2 Figure Page 22. Interpolated Bouguer Gravity Anomaly Map. . . . AA 23. Bouguer Gravity Anomaly Upward Continued 2000 feet . . . . . . . . . . . . A6 2A. Bouguer Gravity Anomaly Upward Continued 2000 feet and Downward Continued 3000 feet- . . . . A8 25. Bouguer Gravity Anomaly Upward Continued 2000 feet and Downward Continued A000 feet. . . . . A9 26. Second Derivative of Bouguer Gravity Surface . . 51 27. Second Derivative of Upward Continued 2000 foot Surface. . . . . . . . . . . . . . 52 28. Bedrock Elevation Map . . . . . . . . . . 5A 29. Approximation Polygons for Columbus Reef Structure . . . . . . . . . . . . . . 55 30. Theoretical Gravity Anomalies Resulting From Columbus Polygons . . . . . . . . . . . 56 31. Approximation Polygons for Big Hand Reef Structure . . . . . . . . . . . . . . 57 32. Theoretical Gravity Anomolies Resulting From Big Hand Polygons . . . . . . . . . . . 58 33. Approximation Polygons for F-Salt Structures and Resulting Theoretical Gravity Anomoly. . . . . 60 3A. Theoretical Gravity Anomoly Resulting from Big Hand Reef and F-Salt Structures and Summation of these anomolies . . . . . . . . . . . . 61 35. Portion of Gravity Map and Six Ring Template . . 10A vii LIST OF APPENDICES Appendix Page A. Field Methods. . . . . . . . . . 75 B. Reduction of Data . . . . . . . . 78 C. Isolation Techniques . . . . . . . 8A viii CHAPTER I INTRODUCTION Purpose of Study The purpose Of this investigation is to determine the applicability and geological significance of the gravity method in the study of the Columbus gas and the Big Hand oil pool. The Columbus and Big Hand fields produce from Niagaran reef structures. Many similar reef structures in south- eastern Michigan have been found by the gravity method. How- ever, the Big Hand reef does not exhibit an obvious gravity anomaly and therefore was not found by the gravity method. The primary objective was to determine if the gravity method could have been used to detect the presence of this reef. The Columbus gas field has a well-defined gravity anomaly associated with it. The main purpose of studying and defining this anomaly is to compare it with the nearby Big Hand reef. A geological study of the reef structures was made and calculations were performed to determine the gravity effect of these structures. The objective of this portion of the study was to delineate the geological factors which produce or negate an observable gravity anomaly. Location of Area The area of investigation lies entirely in T5N, R15 and 16E. This places it in the political townships of Columbus and St. Clair, St. Clair County, Michigan. The location of the study area is indicated in Figure 1. Physiography_of Area The area of investigation is extremely flat lying. Only in the vicinity of the Belle River and Rattle Run stream channels is there appreciable topographic relief and there it is limited to approximately 50 feet. There is, however, a gradual increase in elevation from 635 feet above sea level in the southeast to 7A0 feet in the northwest. The area lies in the Erie-Huron Lowland and is covered with glacial drift which varies in thickness from 120 to 210 feet. The drift has been described as composed of glacial lake clay and-lake bed sand. Rich topsoil has developed in the glacial drift. This topsoil and level topography make the area very suitable for farming. As a result, predominately agricultural communities have developed in this region. ST. CLAIR COUNTY WA RISE I RISE \ T5N AREA OF INVESTIGATION FIGURE I CHAPTER II GEOLOGY Regional Geology of Michigan Basin The Michigan Basin underlies the entire southern Peninsula and the eastern part of the northern Peninsula of Michigan. It also extends into the portions of Ontario bordering on Lake Huron, Lake St. Clair, and Lake Erie, as well as the northwestern part of Ohio, northern Indiana, northeastern Illinois, and eastern Wisconsin. The Michigan Basin has been structurally described as an intracratonic, sedimentary basin. The rocks subcrop beneath the Pleistocene glacial drift in a roughly circular pattern, centered in the southern Peninsula of Michigan. Proceeding from the rim to the center of the Basin, the rocks become younger and generally thicker. The Basin is rimmed by tectonically positive features. Figure 2 illustrates the location of these features. On the north, northeast and northwest, the crystalline Precambrian rocks of the craton form the outer limits of the Basin. The southern boundary is formed by the two prongs of the bifurcated Cincinnati Arch, the prong to the northeast is the Findlay Arch and the northwest prong is the Kankakee Arch. The Algonquin Arch, which is an extension of the Findlay Arch, forms the eastern border of the Michigan Basin and the Wisconsin Arch is the western limit. In Cambrian time the Michigan Basin was low relative to the Canadian Shield to the north and the Wisconsin Arch to the west. An isopach map (Cohee in U.S.G.S. Prelim., Chart 33) of the Cambrian andwthe Canadian series (early Ordovician) shows a general thickening in the southern Peninsula of Michigan, but the greatest thickness occurs in northeastern Illinois. Thus, it appears that Michigan was part of a much larger basin. Thinning 0f the Paleozoic sediments in Ontario indicates that the Algbnquin Arch also was present at this time. Wide spread structural changes, evidenced in a major erosional unconformity, occurred at the end of Canadian time. The Findlay and Kankakee Arches may have developed as a result of these changes. However, an isopach map of the upper Ordovician rocks does not indicate the exist of these Arches. During the Silurian period the Basin-continued to develop. However, the rocks of the early-and middle Silurian period are thickest around the edge of the Basin and become thinner toward the center of the Basin. Some geologists have interpreted this as evidence of a positive feature in the center of the Basin. Other feel it is simply a case of nondeposition because of a lack of sediments. During Salina (late Silurian) time the Michigan Basin began to subside rapidly. The area of greatest subsidence coinciding with the present center of the Basin. By this time, the Basin had become very restricted. As a result, great thicknesses of evaporites were deposited, as much as 2500 feet in the center of the Basin. During the Devonian, the Basin continued to subside. The center of greatest subsidence shifted eastward from where it had been in the late Silurian. Throughout the Mississippian andPensylvannian- periods, the Michigan Basin retained the general shape-that it had acquired during the late Silurian and Devonian. However, after the Devonian, there was very little subsidence and the Basin filled in with Mississippian and Pennsylvannian sediments. The previous summary is derived from detailed discussions by Pirtle (1932), Cohee (19u8), Ver Wiebe (1952), Landes (1956), and Ehlers and Kesling (1962). Stratigraphy General Stratigraphy of Michigan Basin The generalized stratigraphy of the Michigan Basin is given in Figure 3. This geological column has been pre- pared by the Michigan Department of Conservation, Geological Survey Division. CANADIAN SHIELD 4‘)” BA 3 I N o o ‘G s I / V . 4. 3 44. 0 4’4 5 '4' § AXIS :3 OF ARCH 52-, REGIONAL TECTONIC FEATURES FIGURE 2 STRATIGRAPHIC SUCCESSION IN MICHIGAN Pmocmumm RA SYSTEH SERIES STAGE 2 m V mom ”ARI”! a MAM 8 QUATRHARY moon Gui-i- n“ " M Z I“ huh and I. a... u- o...— m h M L: “ ‘ ‘ ——v—L ‘- ‘- a. u I a. U I I Wm.~'nmfi-w;~#‘ *3 d d u I ‘ h suasumcs NOMENCLATURE e... wrflwl“ stat:- IOCK-SYIAW Iouummfluuum] an”? ..u.~.&uu._. hlwl~*~70— wwm-U,d~d- WWW .hm ”m INFORMALTEMS Mddflmdun-fl-nfin WadWDmJMprbh 35—ho- mam-cw mm “B Madman-d bps-l- III—uh— nub-gnu! It‘: our-w- ‘- I... -c at. Wit l-IC tau—u- alt-duh Ila—l «- Inn—dd Inn—pad “fl h‘hmu—uoh his h—dI—w-flbh halal- Io—In alu- ”ml lav-uh— lrmGwn— lu-Ill- GOI- wow-ma..— h“'— “‘5- hfiICqu a... had-cu ouc- Murmu— mdun-l-m Hou— out- In.‘ I... hlfl‘h -— COI- but.“ u— a.“ W'- ul- Mikh- '— Gflllfl bud C ”it? Cut-.- Iflh— .— w I lath MGM Anu— an... wmd III—O“- mh— "~— t—Glfin uh.- uh Md Nap- h-__. I‘m—u Ion-Gnu. GO.- mun-swam “ah—hib— -tbdfl—xh-Huhw. “I‘m, "*dhb CHART I nu FIGURE 3 Silurian Stratigraphy I The stratigraphy of the Silurian system is of primary importance to this study. It was in this period that the reef structures began, flourished, and died. During this time their influence was most dramatically demonstrated. The lower Silurian in southeastern Michigan is represented by the Cataract group, which is composed-of the Manitoulin dolomite and the Cabot Head shale. The middle Silurian is represented by the Niagaran group, which consists of the Clinton formation and the Lookport dolomite. Upper Silurian rocks are divided into two groups, the Salina and Bass Island. Cataract Group.--The Manitoulin dolomite is the oldest Silurian formation in the Basin. It consists of buff-gray to gray, thin bedded, and cherty dolomite. The Cabot Head shale overlies the Manitoulin. The lower part of the Cabot Head is an argillaceous dolomite. The upper half is made up of interbedded green shale and thin layers of gray argillaceous dolomite, as well as layers of red shale and gypsum. Niagaran Group.—-The Clinton shale overlies the Cabot Head. It has been described as a gray to greenish-gray, dolomitic shale. The overlying Lockport and Salina have been exhaustively studied by Landes (l9A5) and Evans (1950). Their classification is shown in Table l. 10 Table l.—-Salina-and Niagaran Strata in the Subsurface of St. Clair County. Formation Unit Description ULTJ Salina mcd “F I\) A-2 A-l A-l DOLOMITE: brown, finely crystalline; shaly dolomite; some anhydrite. SALT: in thick beds separated by beds of shale, shaly dolomite, gray and buff, and brown, crystalline dolomite; anhydrite nearly always present. SHALE: with argillaceous, gray and buff dolomite. SALT: nearly pure; thin partings of buff dolomite. SHALE: gray, dolomitic. SALT: thick salt beds with thin dolomite layers- DOLOMITE: brown, brown gray, gray and dark gray, finely crystalline; some dark bituminous shale. SALT: where salt is absentuthe base of A-2 is marked by anhydrite. DOLOMITE: buff, brown, brown gray and dark gray, dense to medium crystalline; some dark bituminous shale. ANHYDRITE: at base. Guelph- Lockport or Niagaran Niagaran Group DOLOMITE: tan, gray brown and brown, very finely to coarsely crystalline and vugular; often finel laminated near top. DOLOMITE: light and.dark gray mottled, finely crystalline. . DOLOMITE: light to blue gray, finely to coarsely crystallin.e ‘1 -xv Bass Island Group.—-The youngest strata in the Silurian is the Bass Island group. It-represents the closing phase of the evaporite cycle. It is characterized by a clean dolomite. 11 This summary of Silurian stratigraphy is a resume of literature published by Newcombe (1928), Landes (l9A5), Evans (1950), Ehlers and.Kesling (1962), and Alguire (1962). Geology of Reef Structures Niagaran Reef Structures During the Niagaran epoch (middle Silurian), reef structures flourished and developed into what Lowenstam (1957) has described as a reef archipelago. The Michigan Basin occupied the southwestern part of the archipelago. The reefs within the archipelago were isolated structures with varying densities of distribution through space and time. The structures range in areal extent from a few feet to several miles, and in height from 10 feet to nearly 1000 feet. Although they existed as individual units, the reefs greatly influenced the overall environment of the Silurian sea. They acted as a large-scale sediment trap. It has been estimated that the inter—reef beds may be as much as twice the thickness of equivalent beds in reef-free areas. The reef archipelago probably acted as a barrier to. circulation, which resulted in the saline conditions occurring in the Cayugan epoch. The two reefs studied in this investigation, the Columbus and Big Hand, are part of the reef archipelago which presumably developed along the Findlay-Algonquin Arch. They were positioned in that part of the Basin which was l2 subsiding rapidly, as a consequence, the reefs had to grow rapidly upwards in order to remain near sea level. Thus, they developed into pinnacle reefs. The Columbus and Big Hand reefs are typical of fully develOped reef structures. A typical reef consists of two structural components-~the reef core and reef flank.' The reef core is a massive build up of high—purity carbonates. Genetically, the core is the growth center of the reef. It is composed of the skeletal framework of the reef builders, with the interstitial spaces of the framework filled with bioclastic detritus of the niche dwellers and some terrigenous clastics (Lowenstam, 1957). I The reef flank borders the reef core and dips away from it at varying angles. The reef flank is genetically made up of reef-building organisms, coarse to clay-size and intertonguing inter—reef deposits. The reef flank, in contrast to the reef core, is well stratified. This, then, is the setting and development of the reefs structures which were studied in this investigation. Geology of Columbus and Big Hand Reef Structure contour and iSOpach maps of selected formations have been prepared in this study. The formations which were structure contoured are the Niagaran formation, the A-l carbonate unit, the A—2 carbonate unit, the B— salt unit, the C unit, the F- salt unit, the Dundee formation l3 and the Traverse lime formation. The structure contour maps are_presented in Figures A through 11 in ascending order according to age and stratigraphic position. They are presented in this order to show the decreasing influence of the reefs structure on the subsequent formations. Isopach maps have been prepared for the A-l anhydrite and the A-2 evaporite to show the relationship between the reef structures and the lateral variations in thickness of these units. A structure contour map of.the Niagaran formation, Figure A, shows the location and configuration of the. Columbus and Big Hand reefs. The Columbus reef is centered between sections 21 and 22 of T5N, Rl5E, and trends approximately north—south. The base of the reef is at an elevation of —2A00 feet and the highest control contour places the top of the reef at -2100 feet. Thus, the reef has a relief of at least 300 feet. The reef is approximately 0.75-miles Wide and 1.5 miles in length. The Big Hand reef, located entirely in the eastern half of section 2A of T5N, Rl5E, lies two miles east of Columbus. This reef is positioned 100 feet up-dip from the Columbus reef. Its lateral dimensions are only about half as large as those of the Columbus structure-~the-width is 0.5 miles and the length is 0.75 miles. However, it has a relief of nearly 275 feet. bd—I Imp-air. 20:53.0“. 55652 “no no... “23 machzco wmahoamhm 3.30.! . PF: :3 22.8 .5 moi—w?— DZ>>>>>>>>>> < 4w>u4 (mm H13hh8:00 2.140 .bm man—E... 025.. 65 024 mam—23.50 mam—m. znh 17 uuuuu I\\§\\ \§\X\ \\ \ WNW \\\\\\§: \ Ca . K 20 Figure 9 shows the structure on top of the F-salt unit. This formation is approximately 1300 feet above the Niagaran surface. As illustrated-in this figure, there is no structural reflection associated with the Columbus reef. However, there is a large structural high, with A0 feet of. closure, located to the south southwest of the Big Hand reef. This structure is too large, both in areal extent and relief, to be directly related to the reef structure.. Salt flowage may be the explanation of this structure, but in order to develop such a large structure due to salt flowage, it seems necessary to postulate a fracture zone in this area. The fracture zone could have provided an avenue into which the salt migrated. There are other possible explanations for the develOpment of this positive structural feature. Uniform deposition of the salt and subsequent removal of the salt from adjacent areas by solution may be one explanation. Another posibility is that local conditions near the reef_ structure were such that more salt was deposited originally. This structure could be instrumental in explaining the gravity picture associated with the Big Hand reef. Figure 10 and 11 are structure contour maps on the Dundee lime and the Traverse lime.- There are no apparent structures, either on the Dundee or the Traverse maps which are associated with the Columbus reef. The structure maps on the Traverse and Dundee above the Big Hand reef show 10 feet of negative closure to the northwest of the reef 21 ...u>u.. (um ”13h40 Huo. HJ<>¢mhz_ KDOhZOU t2: ._...hzaoo ¢.<...o Fm moqmi 024... gm oz< mam—2:400 mnE ono_.I IU 20h 22 O. uCDOE UJ.‘ PEP—Ch. wwwww Jw>u4 (mm H 13...: F... O. ”41):; Sable”. m2... mmozao H “.0 n.0h n22 map—.200 washoamhm 3.6! .>h8:8 5140 .ha mOJwE 025.. 05 024 9.523400 \IIIIII\4\ \WI ON 9 0 § 0 o o o o o o o x o o 6 o o \ o o. ._ . c \ v. _ . . o. t . o o o o \ \ ~ ~ I NQE 23 20h mm;— \ . . ucao... no upau¢un . v..- 3. v: o EIIII ILL Uddun Aw>wd (mm ”33h10 Ft 0. UJ(>¢W._.: 505.200 m2... wmmw>u4 (um . 13.30 H... n “44:38. "50.5.00 mtmom<>m .l< n22 zo...8:oo 51.5 Hm man—Mr... 024... 0.0 oz< mam—2:400 uni \ \x 53 N03. \\ \ m2: . /II 20... 26 n. urac... I...‘ UhabS—C . .3 a: 0‘. O In mm u4w N04 «.28 10(109 216.30.! .>h2:00 8240 F» won—ME 024... 9m 024 £523.50 wnE Zn... bu qu 9" h. a... Zn... 27 -l500 -I600 "ITOO "I800 -I900 “2000 "2I00 — 2200 " 2300 - 240 254I5 4.1.14 aesALT 4.4.14 4.4.14.4 4.1 4.14.1 4.14.4 4 4.14.1 4.1 .14 .14.14 44444444 44444 .I.J_J_I_I.I.I_I_I_I / /JJ’ / ] XJJ J—l—JJ,% J/ r7f’1//~ ZJ / / 7 Z 1 1 , / 7 / T J//J/J/-/T/1 /,f,]/ 4/74 4 V ,J / / A-g glassware J, / 9‘ f! / .T //-[JT-7 / 7- i - — — - '.A-l — ‘5”-14 _I4 4.1 Wk 4 4.14.14.1 4.14.14 , “ 2:;4 4.4.14.14 4.14.4.I4_14 .i=L 4.1.14.14 4444444444 / /*/r Y ,444444 A-2 EVAPORITE _J l / f / \V _J_I.J.I.J 44444 \" 444 E 0““ V ,0“ / / // /r x / j’ /5 /’ \_- €"w°«\/ El 475 uugnaAJfi /f \‘gfissn““\- r.‘ 1 L /f/ 1 , / / / 1 mod DATUM I SEA LEVEL GEOLOGIC CROSS - SECTION COLUM BUS REEF FNRMEI4 28 23594 24799 241499 246I5 245I9 JJ _]_| 44 44 4444444: _._,J_,-' J—‘-‘:'- __-__:5-: 32‘: 33:- 3112233.: _ _ _ __ : -_—_-:_:: —_—_—__-_: :' :: :2." : C-UNIT _-__— — "5°°::::_: 1:::: ::.:_-_':_ 1‘: -1: :_—::::1_;_ 321-552: 33.: E-_—’ -'-J 33 ----- :-.Z-:_—.: M4 444 44 44 444444444 44444 444-4444 4 -l600 -l700 B-SALT 44444 44 44 4 -I800 444444444 4 44 444 44444 44444444 44 44444444 44444 4444444 , 7S- 4444 _ / J / / I ”“579/‘21, / / /,/,/,134 if _A-z cAaaom'rz - ’ / F’ / / / / / I I L/7L7f - [I'll-7 \ gist: - 111 11111111 44444 4 ‘ 44 44444444 2'0044444 444444 44 A-ZEVAPORITE 44444 444444 44 44444444 44444 4444 44444444 44444 4 E444444 zzooll’ll 4 1’F111 9? § :777/ 7f 7A-ll onion";- 7JJ / f / / f DATUM I SEA LEVEL //// ]/,/7/,f/j NIAGARAN GEOLOGI C CROSS - SECTION BIG HAND REEF FIGURE l5 29 Figure 16 demonstrates the relationship of the F-salt structure to the Big Hand reef structure.. The location of the profiles are shown on the structure contour of the Niagaran formation, Figure A. 3O m. was... uuwm 02¢: 0.0 O... umnhoamhm ._...000 mum". “.0 20:50.. 4\\ \ . . . °° ° . ., a: ‘ \\ \R \ - ° " 003“.» . _ \\ \\ \‘\ . . 0 ~ 0 WW§§ . o a "l' \mmm \\\\\ ° mmmww \\\\\\ . °° . ' \\\\ WW “\wa ° / . >\\\\‘-\\ \\\‘\ ° COLUMBUS AND BIG HAND FIELDS s'r. cum couvmr. uncmm INTERPOLATED BOUGUER GRAVITY ANOMALY OJ MG CONTOUR NTERVAL 3 nmzz 45 anomolies. Therefore, a grid interval of 1000 feet was used. This-interval gave much better results. Because of the limited regional data it was necessary to use only 6 rings instead of the full lOrings which are suggested by Henderson (1960). The radius of the 6th ring is 5000 feet.> Therefore, when the Bouguer values are upward continued, 5000 feet Of the survey area is lost from the edges of the original map. An additional 5000 feet is lost when the upward continued values are downward contined (Servos, 1965). The original Bouguer map and the following upward and downward continued and second derivative maps show the same areal coverage. Regional data covering a strip 10,000 feet wide around the edges of the Bouguer gravity map, was-used but is not shown on the maps because it was necessary only for the calculation of values around the edge of the upward and downward continued and second derivative maps. The Bouguer gravity values were first upward continued- in an attempt to distinguish between anomalies caused by near surface features and anomalies caused by features at greater depth. Figure 23 is the upward continued gravity surface. This surface has been upward continued 2000 feet. The gravity values have been greatly smoothed by this process. The large anomalous area in the vicinity of the Columbus reef is indicated on this map only by a widening of the contour lines. 46 2 ID .— RI6E RISE R|5E COLUMBUS AND BIG HAND FIELDS ° <§\\\\\\\\\\\\\\\§ + IICHIGAN ST. CLAIR COUNTY, BOUGUER GRAVITY ANOMALY UPWARD CONTINUED ZOOOF'IZ NTOUR NTERVALT SCALE mas 0 ‘ IN . IR __ $5 — 0 I000' _ zooo _‘soao"‘ M}Q___ I 55553 «1» «no "it . ,1: \ \\l \\ 0 \\\\‘§\‘\\\ a 00° T5N Fl“ 23 47 After the Bouguer surface was upward continued, it was then_downward continued to bring out the anomalous features. Figure 24 is a map of the 3000 feet downward continued surface. The anomaly associated with.the Columbus reef was} isolated. The closure on the anomaly is +0.5 mgals. The positive anomaly in sections 20 and 21 is present on this map. The negative anomalies on the east and west side of the Columbus reef have diminished somewhat in relation to the positive anomalies.- However, they are still present.» This can mean one of two things. The source of the anomalies is deeply buried, or the source of the anomalies is near surface, but the character of the anomalies is such that it resembles a deep source and therefore this method cannot distinguish between the two. Directly to the northwest, and partly overlapping the Big Hand reef, there is a slight increase in the gravity, values. However, there is no positive closure on this map that is associated with this-reef. The same upward continued surface was then downward continued an additional 1000 feet. The resulting map is_ shown in Figure 25. The same anomalous features that were present in Figure 20 have been isolated on this map. The isolated gravity low, which occurs in sections 25 and 26, is correlativity with the structural high on the F—salt (Figure 9). 48 2m... VN gt pun. : .- - 008 o mIm 1.1% ... . Ir..r-.ll..~ll...nn.rk|.nqw _ "fl N\_ '3 O mud! UJdum 01 94.0 . 4(>km.rz_ KDOFZOU .tooon 8:2:on 053238 oz< .t 88 332.58 923.5 >4t>._.23oo C340 Hm m04m_...._ 023.. 0.0 024 m3m§3400 was... 20... 49 znh wads O. and h 4433; C30h200 .tooov 952.58 053238 92 H... 83 832.58 9.5:: >4<202< >t>_._.<>_mwo ozooww 3.30.8 .rhzaou 5440 P» moi—mi 023... 0.0 024 m3m23400 zm... 52 20... hug » an: 88 300 89 002 68. I? P - .u ...I A .. .v. i.|‘ "I' o . . A qu I» Q. I: o mud! 04100 ”:83 5.0 . 4<>¢ut§ 130.200 005.030 00322200 0043...: Ha OOON no 02.5.2000 020000 23.20... $538 2.4.8 .5 0040.“. 024.... 0.0 oz< 03023400 00.0 20... 53 Theoretical gravity studies of the Columbus and Big Hand reef structures and bedrock river channels map help to answer these questions. Theoretical Gravity Study of Beef Structures A theoretical gravity study was carried out on the. Columbus and Big Hand reef structures. This was done in an- attempt to determine if these reef structures produce a gravity anomaly. And, if an anomaly is produced, what is its magnitude and what parts of the reef structure contribute most to the anomaly. The two-dimensional-gravity computation method was used to calculate the gravity effect of these reefs (Talwani, Worzel, Lamar, and Landisman, 1959). The two-dimensional geologic sections used in these calculations are shown in Figure 14 and 15. Gravity anomalies are, of course, caused by lateral- variations in density. In this computation method, bodies. of.the same density contrast are approximated by polygons. The polygons used in this study and their density contrasts are shown in Figures 29 and 31. The resultant anomaly due to these structures is shown at the top of the figures. The gravity anomalies resulting from each individual body are shown in Figures 30, 32 and the anomaly resulting from combining these individual anomalies is shown at the top of the figure. 54 »»»»»»»»»»» 00000 n.<2 29.53040 xoomomm 23.222 {:58 2.2.6 .5 0040.... 024... 0.0 024 03023400 _,_ . 2.. .1. 55 UGALS '°°° THEORETICAL GRAVITY ANOMALY RESULTING ERou POLYGONS 'ISOO ‘Z/m “IGOO ' -0.35 ’I7OO -l800 -I900 0.5 D -2000 ‘ZIOO “2200 B 0.5 -2300 % -0J 4‘00 / -o.3 DATUM I SEA LEVEL ‘ APPROXIIIATIOII Raucous FOR cowusus REEF sTRUCTURE FIGREZS 56 r03 MGALS we? ' SUMMATION OF ANOMALIES MGALS THEORETICAL GRAVITY ANOMALIES RESULTING FROM COLUMBUS POLYGONS . FIGJRE 30 Approximation Polygons for Columbus Reef Structure Pol on Strata Constrasts with A A—l carbonate A-2 salt A' A—l carbonate A—2 salt B reef body A-2 salt C A-2 anhydrite A—2 carbonate D A-2 carbonate B—salt 57 MGALS [0.3 ~02 -O.I ' mod 1 THEORETICAL GRAVITY ANOMALY RESULTING FROM POLYGONS /‘ -— 0.35 -ISOO -IGOO -ITOO -ISOO O 0.5 -l900 -2000 400 -2200 -OI 02 \-02 /j -O.3 j -2300 W “TU": SEA LEVEL ‘ APPROXIMATION POLYGONS FOR BIG HAND REEF STRUCTURE FIGURE 3| Iooo' 58 mm 0030.... 0200>40n. 024... 0.0 200... 02....43000 00.442024 >h.>400 440....00001... 0m .4420 24 “.0 20:42 230 . .000. . NOL SWVON 59 The magnitude of the anomaly resulting from the Columbus reef is +0.27 mgals, Figures 29 and 30. As is shown in Figures 31 and 32, the magnitude of the anomaly resulting from the Big Hand reef is +0.15 mgals. Figure 33 shows the approximation polygons for the Fesalt structure and the theoretical gravity and anomaly resulting from it.- The anomaly has a magnitude of -0.16-mga1s. This, of course,' is superimposed upon a regional anOmaly caused by the sloping F-salt and G-unit interface. In Figure 34 the anomalies due to the reef structure and the F-salt structure are shown at the bottom of the diagram and the summation of these two anomalies is shown at-the top. The resultant anomaly exhibits a positive 0.04 mgals on the west end of the profile. and a negative 0.07 mgals on the eastern portion of the profile. An error was introduced in the gravity calculations because the reef structures were approximated by two dimensional features. The error is about 25 per cent of the calculated value. This would reduce the magnitude of the Columbus reef anomaly to about +0.02 mg. and the Big Hand anomaly to about +0.12 mg. Because of the size of the structure on the F-salt, it is safe to approximate it as a two-dimensional feature. Therefore, the magnitude of this anomaly does not have to be reduced. The polygons in Figure 29 which are designated by the letters A, A', B, C, and D are of primary interest. Comparing 6O mm 0030.... 003h030h0 F440I... 00“. 0200>40n. 20....42.x00an.4 0200>40n. 200.... 02....43000 >442024 >..._>4m0 440:0...001... 4w>uJ 400 H23h40 0.0.! 000.! N01 . boo. .6. 61 V0 0030.... 00¢3h030h0 P440I... 024 “.000 0242 0.0 2000 02343000 >442024 >....>400 440F080...» 54.824 Emma 92: a... 00.442024 ...440I.... 024 0000 024... 0.0 .00 20.042230 O‘IVOI 62 Figure 29 with Figure 14 shows that polygons A and A' are those parts of the A-l carbonate which constrast laterally with the A—2 salt. Polygon B is that part of the reef body which also constrasts with the A-2 salt. Polygon C approximates the portion of the A-2 anhydrite which constrasts with the A-2 carbonate. Polygon D is that portion of the A-2 carbonate which constrasts laterally with the B— salt. The curves labelled A, A', B, C, and D in Figure 30 are the theoretical gravity effects resulting from the polygons A, A', B, C, and D of Figure 29.‘ It can be-seen that the majority of the theoretical anomaly of the entire Columbus reef structure comes from that part of the reef body and the A-l carbonate (polygons A, A', and B) which constrasts with the A-2 salt and from the structure on top of the A-2 carbonate (polygon D). As iS illustrated in Figures 31 and 32, the same results were obtained for the Big Hand reef. The results of this study does not invalidate the conclusions of Servos (1965). The theoretical anomaly obtained from this study slightly underfits the observed gravity anomaly. Consideration of the effect of peripheral dolomitization could posibly account for the difference between the theoretical and observed gravity anomalies. Also, the geologic conditions in this study area and the area of Servos' study area may not be the same. The densities used in these calculations were taken from density determinations of well samples and from formation 63 density logs.' The formation density logs are from wells which are both in the reef structure and in the off-reef facies. These density values, shown in Table 2, agree with typical average densities for these rock types. Table 2.--Strata densities. ngmggign Density-gm/cc Niagaran 2.70 A-l Evaporite 3.00 A-l Carbonate 2.80 A-2 Salt 2.20 A-2 Anhydrite 3.00 A-2 Carbonate 2.70 B- Salt 2.20 C- Unit .2.55 D- Salt 2.20 F- Salt 2.20 Theoretical Study of the Bedrock Surface Klasner (196”), Servos (1965), and Roth (1965) have investigated the gravitational effects of bedrock river channels. They found that river channels in the bedrock which had been filled in with low density drift material can cause negative anomalies with magnitudes as high as '0.3 mgals. Figure 28 is a map of the bedrock topography in this study area. There are bedrock valleys present on this map. One channel trends northeast across sections 26 and 25 and then swings northwest across section 23, where it widens 6A and-appears to divide. One branch continues northward through section 1" and the other branch appears to continue northwestward, where it connects with a depression in section 15. There-is another depression in the eastern portion of section 21. Lack of control makes it impossible to determine if this depression is only an isolated low or if it continues to the north and south. Comparing this bedrock topographic map with the Bouguer minus the Ath degree, the cross-profile, the 3000 foot and A000 foot downward continued and the second. derivative residual gravity maps shows the correlation between the gravity lows and bedrock depressions.- There is not a one to one correlation, but there definitely is.a general trend.‘ The gravity values-tend to decrease over- the bedrock channel which occurs in sections-26, 25, 23, and 1A. The northern end of the large negative anomaly. east of the Columbus reef coincides with the bedrock low in section 15. The southern end of this negative anomaly occurs along the section line between sections 22 and 23. This. is where the bedrock channel appears to widen and divide. There is a noticeable lack of geologic control in the area and thus, the magnitude of this channel is not known. The negative gravity anomaly to.the west of Columbus is considerably smaller in magnitude and narrower in areal extent than the negative anomaly to the east of Columbus. This anomaly appears to coincide in section 21 with the depression 65 in the bedrock surface. The gravity anomaly extends both to the northwest and southwest. As has been stated previously, the extent of the bedrock depression cannot be determined because of the lack of data. Thus, it appears that there is a general correla- tion between bedrock channels and negative gravity anomalies. With increased control data, the buried bed- rock channels possibly would be.shifted-in actual position so that even better correlation would exist. Interpretation of Observed Anomalies A basic understanding of the types of anomalies which can be eXpected from various geological features has now been established. From this, it is possible to make a reasonable interpretation of the anomalies which are observable on the gravity maps. The positive anomaly located over the Columbus reef is no doubt associated with the reef structure. The profile of the geologic cross-section.along which the theoretical gravity calculations were made is located on the original minus the Ath.degree~least squares map, Figure 21. The observed residual gravity anomaly along this profile reaches a magnitude of about 0.22 mgals. This very closely matches the theoretical value of 0.20 mgals. For this reason, it is felt that the observed anomaly is a product of the reef structure. 66 A comparison of the width of the residual anomaly and width of the calculated anomaly suggests no definite conclusions. This is because the width of the residual anomaly varies with the isolation technique and the edges of the positive anomaly are masked by the negative anomalies which are believed to be the result of bedrock channels.- The negative anomolies, which flank the Columbus reef on the east and west, were first thought to be of the "sombrero" type. Yungul (1961) considered the negative "sombrero" ring to be an anomaly which is related: to the reef structure. After further consideration, it is now believed that these negatives are due to bedrock river channels. There are several reasons for drawing this_ conclusion. First, when conducting the theoretical gravity study of the Columbus reef, no density contrasts were found. in the geological sections which could cause a resultant negative anomaly-of this magnitude. Secondly, gravity data to the north and south of this area indicate that these negative features do not end at the boundaries of the map. Instead, they extend to the north and south. Their shape and linear extent also are characteristic of river channels. Finally, it has been shown, through theoretical studies of river channels in the bedrock, that they can produce negative anomalies of this magnitude and there are river channels in this area which general correlate with the negative gravity anomalies. 67 The gravity picture in the vicinity of the Big Hand reef is dominated by the presence of the structure on the F-salt. The negative anomaly due to this feature almost completely cancels out the positive anomaly caused by the reef structure, Figure 34. Although there is no negative closure associated with this feature, the gravity values. do decrease in value. The Slight increase in positive values above the northwest corner of the reef is probably due to the reef structure where it has not-been completely nullified. The positive anomaly to the west of Columbus compares favorably in magnitude with that of the anomaly caused by the Columbus reef structure. This would tend to indicate the presence of a structure which is comparably with that of the Columbus reef structure. Present drilling indicates the presence of a structure in this area. This is shown on the geological maps in Figure A and 12. Although a substantial number of wells have been drilled in this area, the structure probably has not been fully explored. This should prove to be an interesting area. CHAPTER IV CONCLUSION In conclusion, it can be stated: 1. The Columbus reef structure produces a recog- nizable gravity anomaly with a magnitude of about +0.27. mgals. The width of the residual anomaly does not correspond with the width of the reef structure for the width of the residual anomaly varies with the isolation technique and, the boundary of the anomaly is complicated by the flanking negative anomalies. 2. The Big Hand reef structure is estimated to produce a gravity anomaly of about +0.12 mgals, however, this positive anomaly is almost completely negated by a negative anomaly of 0.16 mgals which originates from a structure on the F-salt. This positive feature has a closure of approximately “0 feet which is not directly associated with the Big Hand reef. If this structure was not present the anomaly due to the reef could probably be isolated and recognized as resulting from a reef structure. From this, it may be concluded that in areas of known structures on the F—salt, the possibility of a reef structure occurring should not be ruled out simply because of no gravity anomaly. 68 69 3. Bedrock river channels can produce negative anomalies with a magnitude of 0.3 mgals. Bedrock river channels do exist in.this study area and the negative anomalies which flank the Columbus anomaly are probably the result of such features. A. The major source of the gravity anomaly, resulting from the Columbus and Big Hand reef structures, is probably due to that portion of the reef body and the A-l carbonate which contrasts with the A-2 salt and the part of the A-2 carbonate which contrast with the B-salt. 5.- All of the interpretation techniques successfully isolated the Columbus anomaly. It was, however, necessary to use good geological Judgment in applying these techniques‘ in order to obtain the best results. The original Bouguer surface minus the Ath degree better isolates the anomalies and for this reason it is felt that the Ath degree polynomial surface best approximates the regional gravity.’ In the upward and downward continuation.and second derivative methods, a mesh interval of 500 feet, which is about 1/6th the depth to the reef structure, was found to be too small. A mesh interval of 1000 feet, l/3rd the depth, gave better results.- Second derivatives on the upward continued surface gave more easily interpretable results than second derivatives on the original Bouguer surface. BIBLIOGRAPHY 7O BIBLIOGRAPHY Alguire, S. L. (1962). Some geologic and economic aspects of Niagaran-reefs in Eastern Michigan: Michigan Basin Ggological Society Annual Field Conference, Cohee, G. V. (19MB). Thickness and.lithology of Upper Ordovician and Lower and Middle Silurian rocks in-the- Michigan Basin: U. S. Geol. Survey, Oil and Gas. Invest., Preliminary Chart 33. Cumings, E. R., and Shorck, R. 8.: (1928). Niagaran coral reefs of Indiana and adjacent states and their stratigraphic relationships: Geol. Soc. Amer. Bull. V.-39. p- 579-620. Ehlers, G. M., and Keling, R. V. (1962). Silurian rocks of Michigan and their correlations: Michigaanasin Geological Society Annual Field Conference, p. 1-20. Elkins, Thomas A. (1950). The second derivative method of gravity interpretation: Geophysics, V. XVI, Jan.,' Ells, G. D. (196A). Personal Communication. Evans, C. s.. (1950). Underground hunting in the Silurian of Ontario: Geol. Assoc. Canada Proc., V. 3, pp. 55—85.v Henderson, R. G. (1960). A comprehensive system of automatic computation in magnetic and gravity interpre-- tation:; Geophysics, Vol. 25, no. 3, p. 569-585. Henderson, R. G., and Zietz, Isadore. (19A9). The computa- tion of the second vertical derivatives of geomagnetic fields: Geophysics, Vol. 14, p. 508-53A. Hinze, W. J.- (1963). Regional gravity and magnetic maps of the Southern Peninsula of-Michigan: Michigan Dept.‘ of Conservation, Geological Survey Division, Report of Investigation 1. Klasner, J. S. (1964). A study of buried bedrock valleys near South Haven, Michigan by the gravity method: Masters Thesis, Michigan State University. 71 72 Krumbein, W. C., and Graybill, F..A.A (1965). An Intro— duction to Statistical Models in Geology: McGraw- Hill, New York.- Landes, K. K. (l9A5). The Salina and Bass Island rocks in the Michigan Basin: U. S. Geol. Surv. Oil and Gas Inv. Preliminary Map No. 40. Landes, K. K.' (1956). Petroleum Geology: John Wiley and Sons, Inc., New York, pp. 298-301. Lowenstam, H. A. (1950). Niagaran reef of the Great Lakes area: Jour. Geology, Vol. 58, p. A30-A87. Lowenstam, H. A.. (1957). Niagaran reefs in the Great Lakes area: Geol. Soc. Amer., Mem. 67, Treatise on marine ecology and paleoecology, V. 2, Paleoecology chapt. 10, p. 215-2A8. LI “—1—“.I1? ‘ Nettleton, L. L.. (1940). Geophysical prospecting for oil: McGraw-Hill, New York. Nettleton, L. L. (195“). Regional, residual, and structures: Geophysics, Vol. 19, p. 1-22. Newcombe, R. B. (1920). Oil and gas deveIOpment in Michigan: Mich. Geol. Survey Pub. 37, Series 31. Peters, L. J. (1949). The direct approach to magnetic interpretation and its practical application: Geophysics, Vol. 1A, No. 3, p. 290-319. Pirtle, G. W. (1932). Michigan structural basin and its relationship to surrounding areas: Bull., A.A.P.G., Vol. 16, pp. 1A5-l52. Pohly, R. A. (195A). Gravity case history: Dawn No. 156 pool, Ontario: Geophysics, Vol. 19, p. 95-103. Rosenbach, Otto. (1953). A contribution to the computation of the "second derivative" from gravity data: Geo— physics, Vol. 18, No. A, p. 89A—912. Roth, J. N. (1965). A gravitational investigation of fracture zones in Devonian rocks in portions of Arenac and Bay County, Michigan: Ms. Thesis, Michigan State University. ' Serves, Gary Gordon. (1965). A gravitational investigation of Niagaran reefs in southeastern Michigan: Ph.D.- Thesis, Michigan State University. 73 Talwani, M., Worzel, J., Lamar, and Landisman, M. (1959). Rapid gravity computations for two-dimensional bodies with application to the Mendocino submarine Eracture zone: Jour. Geophys, Res., Vol. 6“, p. 9-59. Ver Wiebe, W. A. (1952). North American Petroleum: Edwards Brothers, Inc., Ann Arbor, Michigan pp. 58—72. Yungul,.S. H. (1961). Gravity prospecting for reefs: Effects of sedimentation and differential compaction: GeOphysics, Vol. 26, No. l, p. A5-56. AP PENDI CBS 714 APPENDIX A FIELD METHODS 75 FIELD METHODS Regional gravity coverage of the study area was obtained from interested parties. The regional coverage consists of~ gravity stations spaced 1/A mile apart along the existing road network. It, therefore, was necessary only to add de- tailed station coverage in the near vicinity of the Columbus and Big Hand reefs. The detail coverage consists of cross- country traverses with a station spacing of 660 feet. These traverses were spaced at l/A mile intervals wherever it was possible.~ Elevation control was obtained from U. 8. Geological Survey and St. Clair County bench marks. The station eleva-. tions were established with a Zeiss self-leveling level and the distance between stations was determined by stadia interval. The World Wide gravity meter number AS-was used to take the gravity readings. This meter has a calibration constant< of 0.10093 mgal/div and a reading accuracy of 0.01 mgals. The meter is subject to time variations. These time variations, which are called drift, are due to tidal variations,' temperature changes, material fatigue and instrument handling. In order to correct for drift, it was necessary to take hourly readings at preestablished base stations. These base 76 77 stations were strategically positioned so that a minimum of time was required to reach them from all points in the survey area. If the meter drift was more than a 0.1 mgal/hr, the stations read during that hour were repeated. Each station was read until values were obtained which agreed within 0.2 of the scale division (0.02 mgal). APPENDIX B REDUCTION OF DATA 78 REDUCTION OF DATA It is necessary to remove from the observed gravity readings those factors which are not directly related to geological conditions.» This must be done for each gravity station. The value obtained after correcting for these factors is called the.Bouguer gravity anomaly. The Bouguer gravity anomaly was calculated according to the formula: GBA = 8o + 8e + m _ $1 + gt where GBA = Bouguer gravity anomaly $6 = observed gravity gafln = elevation and mass correction gl = latitude correction gt = terrain correction Observed Gravity The observed gravity values are obtained by multiplying the meter readings by the meter calibration constant and subtracting out the drift. The observed gravity values are then in milligals. 79 80. Elevation and Mass Correction The elevation and mass correction is a combined correc- tion involving the free-air correction and the mass correction. The free—air correction accounts for the change in the acceleration of gravity with elevation. The correction is made by first assuming an elevation datum which is common to all stations in the survey area. The difference between the station elevation and the elevation datum is multiplied by the factor 0.09406 mgals/ft. This gives the free-air correction. The elevation of the lowest station was taken to be datum. Therefore, the correction was always positive. The material between the datum and the station obviously increases the acceleration of gravity. Therefore, a mass correction must be made to account for this increase.. This is done by multiplying the difference in elevation between the datum and the station by the factor (0.01276 x material density). Combining these two effects gives the elevation and mass correction formula: 8e + m =-(0.09406 - 0.01276p)h where h = elevation difference between the datum and the station 0 = density of the material. 81 Servos (1965) determined that the density of the glacial in this area is 2.1 gm/cc. This is the value that was used in the elevation and mass correction. Latitude Correction The acceleration of gravity increases from the equator to the poles. This, of course, is due to the fact that the radius of the earth is less at the poles than at the equator and the centrifugal force decreases toward the poles. A correction is therefore necessary to account for this north- south variation in gravity. The correction is made by first establishing a common- latitude datum for the survey area and measuring the perpen- dicular distance from this datum to the station. This distance is then multiplied by the correction coefficient, K. The constant, K, according to Nettleton (1940) is: K = 1.307 sin 20 mgals/mile where e is the mean latitude of the survey-area. The mean latitude for this survey is 42° 52' 30". The constant, K,. is therefore, equal to 0.0002474 mgal/ft. Terrain Correction The relief in the survey area is low enough that effects due to terrain were negligiable and therefore could be ignored. The survey stations were judiciously located, so that local elevation variations such as stream channels and ditches, did not affect the accuracy of the gravity readings. 82 Accuracy_of Bouguer Reductions Inaccuracies in Bouguer values result from four possible causes. These are (1) errors in observed readings, (2) errors in station elevations, (3) errors in distance measure-. ments, and (4) errors in the density used in the mass correction. Errors in observed readings are the results of meter drift and human error and can only be determined by repeating meter readings. A number of stations chosen at random were reread. The maximum variation in repeatability was 0.04 mgals. The maximum allowable closure error in elevations was 0.1 of a foot per mile. The longest closed traverse loop was 4 miles. Therefore, the maximum allowable elevation error. was 0.4 of a foot. But in actuality the closure error was never greater than 0.2 of a foot. This would cause an error in the combined elevation and mass corrections of 0.013 mgals, for a density of 2.1 gm/cc. Errors in distance, of course, will cause errors in the latitude corrections, if the error is in the north-south direction. A maximum error of 50 feet was allowed in the field measurements. Another 50 feet of error was allowed in plotting the stations on the base map for a total error of 100 feet. For a latitude correction factor of 0.0002474 mgals per foot, this amounts to an error of .025 mgals. Slight errors in the density used in the elevation and mass corrections can cause large inaccuracies in areas of 83 rugged topographic relief. This can be seen from the formula used in the calculation of this error. This formula is: Error - 0.001280h where 0.00128 magnitude of error in-mgals per foot for each 0.1 gm/cc error ianensity p = error in density in units of 0.1 gm/cc h = maximum relief in feet In this area the maximum relief between stations is only-10v feet. Assumming an error of 0.1 gm/cc was made in the density determination, this would lead to an error of 0.013 mgals. Combining the effects of the errors in observed readings, station elevations, distance measurements and near surface density, would cause a maximum relative error between adjacent stations of 10.091 mgals in the Bouguer gravity.‘ Assuming a normal'distribution; the standard deviation is essentially 1/3 of the maximum error. Therefore, the standard deviation. is :0.030 mgals. APPENDIX C ISOLATION TECHNIQUES 84 ISOLATION TECHNIQUES The gravity method has been used as an exploration tool for a number of years. However, there has been a hesitancy to accept the method because of the ambiguity of the results. In recent years there has been a change in attitude toward the application and interpretation of the gravity method. This change has come about largely because the geOphysicists who are now applying it have a better under- standing of the geological significance of the method. The digital computer has greatly increased the use— fulness of the gravity method for it has made possible the application of interpretation techniques which prior to the advent of the computer required a tremendous amount of time. In the following sections the theory and application of various interpretation techniques which are used in this study will be discussed. Cross-Profile Method One of the most common methods uded for the isolation of local anomalies is the grid method known as cross- profiling. The first step in using this method is to cover the area with a square network of profiles. The grid is positioned so that the profiles in one direction are parallel to the regional gravity trend. The profiles in 85 86 the other direction are at right angles to the regional trend. The Bouguer gravity values-are plotted along these profiles. The regional gravity gradient is then estimated with a smooth curve. If the regional gradient has been estimated correctly, the points of intersection between the cross-- profiles should have the same value. The regional gradient is adjusted until these intersection points agree., The final step is to subtract the regional gravity gradient from the Bouguer gravity values. This gives the residual gravity anomaly. This method is subject to the personal bias of the interpreter. This can be an advantage if the interpreter has a good understanding of the regional geology. If he does not, it can lead to erroneous results. Least Squares Method The purpose of the least squares technique is to isolate residual gravity anomalies from the Bouguer gravity map. A Bouguer gravity map describes a three—dimensional surface. The least squares method consists of approximating this three-dimensional surface with a polynomial equation. The polynomial equation is increased in degree until it roughly approximates the Bouguer surface. This rough approximation is considered to be the regional gravity trend. The residual gravity is then the difference between this approximation and the original Bouguer surface. 87 The degree of the polynomial, which is used to approximate the regional gravity trend, depends on the complexity of the Bouguer gravity and also depends on the personal bias of the interpretor. The following Mathematical Treatment is taken from Krumbein and Graybill (1965). The Bouguer value, G(x,y), at any point on a Bouguer map is a function of the x and y coordinates of that point° Thus, the Bouguer value at any point can be approximated methematically by a polynomial equation. The basic polynomial equation is: (1) G(x,y)* =.8 + s X + a Y + . . . epq xp Yq 00' 10 01 where G(x,y)* is the approximation of the Bouguer value G(x,y), the p's and q's are the degree to which the term is raised and the 8's are coefficients. This is the general equation for an nth degree polynomial. For instance, an equation of first degree would be: * = G(x,y) 800 + 810 X + 801 Y a second degree equation would be: Y+B X2+B'XY+8 Y2 r x = GKX’V) 800 + 810 X + 8 20 . 11 02 01 a third degree equation would be: . 2 2 * = . G(x,y) 800 + 310 X + 301 Y + 820 X + 811 XY + 802 Y X2Y + 8 XY2 + 3 3 + 830 X + 821 12 803 Y and so on. 88 As was explained above, the approximation G(x,y)* is the value of the regional gravity trend at the point (x,y). It is, therefore, obvious that the regional gravity of any point on a Bouguer map can be obtained from the basic polynomial equation, simply by inserting the x and y coordinates of the point. First, however, the values of the coefficients in the equation must be determined. One method used for determining these coefficients is the least squares method. According to the least squares method, the estimation of the coefficients will be such that the sum of the squares of the error is at a minimum. The error is the difference between the values- obtained from the polynomial equation and the actual observed values. The least squares method states that (D) must be a minimum where n n 2 (2) D = i e = 2 (Gi(x,y) - Gi(X,y)*) (Gi(x,y) - e - a, x13)2 K 2 . 0. J: 1 n = number of observations and k = number of terms in the G(x,y)* equation. The values of B which make (D) a minimum, are the 3 least squares estimators of Bj. For (D) to be minimum, the partial derivative of (D) with respect to each 81 must be equal to zero. Thus, 89 E%% = -2 X [Gi(x,y) - BO - X 8 X1 1': O ( ‘0. 1=1 ' . i=1 3 J ( ( n k I§% = -2 z [Gi(x.y> - so - 2 8 X13] X11 = 0 ( 1 i=1 J=l J ( 3 ( (2D n k = -2 2 [G (x,y) - B — z B X 1 X = 0 Ezfig i=1 i 0 3:1 j 13 12 ( ( k (39 = —2 2 [G (x,y) - so - z 8 X 1 X k = 0 (25k i=1 i j=l J 13 i Dividing by 2 and transposing the Gi(x,y) terms, equations (3) become: n n n n n 80 + 81 Z X + 8 z X + . . B Z X - 2 G (x,y) i=1 il 2 i=1 12 k i=1 1k i=1 i n n 2 n n so 2 X + 81 z X + s 2 X X + . . . B 2 X X 1=1 11 1=1 11 2 1=1 11 12 k 1=1 11 ik n = z X G (x,y) 1:1 11 1 n n n n 80 2 X + 81 Z X X + B Z X + B Z X X 1=1 12 1=1 11 12 2 i=1 12 k 1=1 12 ik n = z X G (x,y) 1=1 12 1 n n n~ n 2 BO 2 X. + 81 Z X X + 82 Z X X + . . . B X X i=1 1k 1=1 1k 11 1=1 ik 12 k i=1 ik n = z X G.(x,y) i=1 ik 1 These are the normal equations and they may be written in matrix form as follows: 90 L 2.14 gm mu an an HaH 2H2 H N C HIfi xfix mg m C. HIH ‘xH HH 5 H.H , H m2 2H2 H HHx 2H2 a ma x m 0 HH NH H x .H C C HuH ,HuH x H HHx me a C C HIH w H CHI! :1 v4 91 It is easily seen by carrying out the multiplication that this is indeed the matrix form of the normal equations. If the previous matrixes are denoted as: 1’1 1’1 n n 2 X 2 X 2 X 1=1 11 1=1 12 1=1 ik n 1’1 n 2 X11 2 Xil 2X11X12 2X11X1k 1=1 1=1 1=1 1=1 T = n nX X n nX X E X 2 12 11 Z X 2 12 ik 1=1 12 1=1~ 1= 12 =1 11 1’1 n 1'1 2 X1k ZXikXil ZXikXi2 2 Xik 1=1 1=1 1=1 1=1 ,_j F- -- I-vn - Bo 2 G1(X'y) 1=1 n, 81 2 X G (x,y) 1.1 11 1 B = G = n 82 2 X G (x,y) 1=1 12 1 n 3k 2 X G (X Y) .__ __ 1=1 ik 1 ’ —J then T8 = G s = T‘1 G’ where T"1 is the inverse of T. 92 This is demonstrated for the second degree polynomial equation: G(x,y)* = s + e X + B Y + e X2 + B X Y + 8 Y2 00 10 01 20 ll 02 where 800 = so; 810 X = 8112(1; 801 Y = 82 X2; 820 X2 = 83 X3; 811 XY = 84 X4 and 802 Y» = 85 X5 The normal equations for this 2nd degree equation are: n n n 2 n n 800 + 810 1:1 Xi + 801 1:1 Yi + 820 1:1 Xi + 811 1:1 Xi Yi n 2 n + 802 iilYi - 1:1 Gi(x,y) n n n n 3 800 1:1 X1 + 810 1:1 X1 + 801 1:1 X1 Y1 + 820 1:1 X1 n n 2 n + 811 1:1 X Y1 + 802 1:1 Xi Y = 1:1 Xi Gi(x,y) n n n 2 n 2 800 1:1 Y1 + 810 1:1 X1 Y1 + 801 1:1 Y1 + 820 1:1 X1 Y1 n 2 n 3 n +8ll iilxi Yi + 802 1:1 Y1 = 1:1 Yi Gi(x,y) n 2 n n 2 n 4 800 1:1 Y1 + 810 1:1 X1 + 801 1:1 X1 Y1 + 820 1:1 X1 + 811 2 Xi Yi + 802 2 Xi vi = 2 X ‘G1(X’y) i=1 i=1 i=1 93 n n n X2 Y + a X Y2 + a X Y + 810 1E1 1 1 01 151 1 1 n 800 1§1 1 1 n 2 2 n 3 _ n + 811 1§1 X1 Y1 + 802 iii X1 Y1 " 1E1 X1 Y1 G1 (x,y) n 2 n 2 n 3 n 2 2 B00 1§1 Y1 + 810 1§1 X1 Y1 + 801 1§1 Y1 + 820 iii X1 Y1 n Y3 n Yu _ n 2 +811 1§1 X1 1 + 802 1§1 1 ‘ 1§1 Y1 G1 (x,y) The matrix form of the normal equations are shown on the following The matrix for the system of linear (first degree) equa- tions is those elements in the matrix for the second degree equation which are enclosed in the dashed lines. This is easily seen by setting up the matrix for the linear equations in the same manner in which the second degree was set up. As a further example consider the data in Table 3. When this data is substituted into the matrix for the system of linear equations, it becomes: 12.0 2H0.0 “50.0 80 “.06 2U0.0 6300.0 9000.0 81 103.15 “50.0 9000.0 21875.0 82 160.77 -1 T8 = G; a = T’ G >> A A I A 0 A >a >; I >3 N >: A a q I a \J - a. .'>: N >4 I >4 H >4 0. v v | v (5 V >4 H HI H H H v (5 (D : (5 >4 (D H H HI (\1 H H (\J H (5 N >4 : N X >1 H H HI H H H QWII SIN" CNII I CINII CNII CNN H H HI H H H II O 0 HI 0 H (\J O H CI (\1 H O m (n m | (n. ca «1 (\1 0H (\I H m H >4 >4 >4 (\I H H m H (\l H H :r H >-I X >4 N >< >4 H H H H H H :1 N" :1 WI! CNN C Wu Cw" C Wu H H H H H . H H (\l H H (\l H m H >4 >4 >4 >4 >« >4 H (\I H H m H (\J H H ><1 N N >4 X N H H H H H H C N” C. W" CHI! C. N" SW" C NII H H - H H H H H N H >4 >4 >4 (\I H m H (\J H :1- H m H (\I H >< N H H H H H H :1 N" C. N" CNII C.‘ Wll CNN 5: NH H H H H H - ____________ , H l H (\J H >4 II >4 >-¢ H H (\l HI N H H m H {>4 N >4 : N N >4 H H HI H H H Cw" QNII CNII I Cw" Cw" cm" H “I". 'I—I: H 9H . I HI H (\l H >-I : >-| >4 .r-I (\J H HI (‘0 H (\J H H >4 >4 >< : >< >< >4 H H HI H H H CWII SW" SW" I QNII CW" CNN H H HI H H H I l I I H : {>4 H .r.” (\J H -r-I (\l H >< >4 : N N {>4 H HI H H H C EN" CW" I CNII CW" SSW" H HI H H H 95 Table 3.-—Gravity Data and Location Coordinates after Krumbein and Gray— bill. Sample Number X Y G(x,y) 1 5.0 12.5 .23“ 2 15.0 12.5 .205 3 25.0 12.5 .220 “ 35.0 12.5 .510 5 5.0 37.5 .225 6 15.0 37.5 .212 7 25.0. 37.5 .21“ 8 35.0. 37.5 .730 9 5.0 62. .20“ 10 15.0 62.5 .202 11 25.0 62.5 .23“ 12 35.0 62. .870 ___ —l ——_ 80 12.0 2“0.0 “50.0 F_-“.O6 81. = 2“0.0 6300.0 9000.0 103.15 82 “50.0 9000.0 21875.0 160.77 The matrix T must first be inverted. One method of doing' this is to divide each element of the matrix of cofactors by the determinant of the matrix and then transpose the results. For instance, the first element of the matrix of cofactors is 56,812,500. The determinant of T is 90,000,000. The first element of inverse matrix is 0.631250. Proceeding in a 1 like manner, the remaining elements of the T- matrix are determined. The inverse matrix is then: 96 —1 0.631250 -0.013333 —0.007500 T =< -0.013333 0.000667 0.00000 -0.007500 0.00000 0.000200 The matrix of coefficients is then: 80 0.631250 -0-013333 —0.007500 “.06 Bl~ = -0.013333 0.000667 0.00000 103.15‘ B 2 -0.007500 0.00000 0.000200 160.77 Multiplying the two matrixes on the right, the coefficient matrix becomes: '_ '7 —0.0183 0.01“6 0.0017 comm NI—‘O ll Substituting the coefficients into the polynomial equation, the predicting equation becomes: G(x,y)* = -0.0183 + 0.01“6 X + 0.0017 Y The approximation of the observed values is obtained by substituting the x and y coordinates into the predicting equation. For the data used in this example the results are as listed in Table “. The equations of higher degree are obtained in the same manner. The techniques for calculation of.these mathematical operations have been programed for the digital computer. 97 Table “.--Resu1ts Obtained from Least Squares Polynomial Equation after Krumbein and Graybill. Sample Observed Regional Values Residual Number Values-G(x,y) G(x,y)* G(x,y)-G(x,y)* l .23“0 .0762 .1578 2 .2050 .2225 -.Ol75 3 .2200 .3689 -.1“89 “ .5100 .5152 -.0052 5 .2250 .1188 .1062 6 .2120 * .2652 —.0532 7 .2l“O .“115 -.l975 8 .7300 .5578 .1722 9 .20“O .1615 .O“25 10 .2020 . .3078 -.1058 ll .23“0 .“5“1 —.2201 12 .8700 .6005 .2695 Analytical Techniques In gravity investigations, it is often desirable to know the gravitational field above or below the plane of observation (ground surface). Many investigators have dealt with this problem including Evjen (1936), Peters (19“9), Elkins (1951), Rosenbach (1953), Henderson (1960), and others. The basic problem is the same, but the manner in which it is approached has led to many different solutions. Henderson's method was used in this study for the determination of upward and downward continuation and 2nd derivatives. This method will be outlined in the following paragraphs and an example worked out to demonstrate the use of the method. 98 The horizontal change in gravity on the plane of observation is represented as A¢(x,y,z), where x and y are the horizontal axes and z is the vertical axis which- is positive vertically downwards. A¢(x,y,z) satisfies the Laplacian V2[A¢(x,y,z]=0 and therefore, can be treated by the conventional potential theory methods. The so—called "upward continuation integral" solves the problem of computing the gravational field above the plane of observation. The integral in polar coordinates is: (1) Wm.) J ma .3; (r) r dr 0 (r2 + m2 a2)3/2 m = 1,2,3, . . .~. n where 2'" A37 = 1/21r 5 M (r, 0) d0 0 is the average value of A0 (x,y,z) on circles of radius r about the point at which A0 (-ma) is to be determined. The term "a" is the distance between stations which” course, must be in a square grid pattern. Using a mean value theorem over the interval ri i r i ri.+ 1 for each interval, equation (1) can be approximated numerically. Equation (1) then becomes: 99 n-l r‘14'1 (2) A¢(-ma) = Z [(r1+1 - ri)-1 S A¢(r) dr] ma 1=0 r1 [rf + (ma)21‘1/2 — [ri + 1 + (ma)2]‘1/2 + 0 (1/rn) (a) is set equal to l and equation (2) is successively evaluated for m = l, 2, . . .~5. In otherwords, the gravitational field is determined for a distance a, 2a, 3a, “a, and 5a above the observation plane. The 5 sets of coefficients are thus obtained for the upward-continuation formula. These coefficients are shown in Table 5. The working equation then is: (3) A¢(-m) 2 Z A¢(r1) K (r m) is where K (r1, m) are the upward continuation coefficients. Let P(x,y,0) be the point at which the upward continued values of gravity have been determined. According to equation (3), these values are at distinct points. They are at a distance of a, 2a, 3a, “a, and 5a above P(x,y, 0). A polynomial can then be fitted to these points. Henderson uses the Lagrangian interpolation method for determining this polynomial. This polynomial is: (u) A¢(Z) =2 (-1)m z(z+a) (z + 2a) . . . . (Z + na)(A¢(_ma)) an (Z + ma) (n-m) I m! where A¢(-ma) is equation (3). Now, to determine the gravitational field below the surface the polynomial is simply extrapolated into this region. The working formula for down- ward continuation is obtained by substituting equation (3) I‘ll lit-l... I39! Table 5.--Upward Continuation Coefficients after Henderson. 100 i A¢(ri) K(r1,1) K(ri,2) K(ri,3) K(ri“) K(ri,5) 0 a§(0) 0.11193 0.0u03u 0.01961 0.011u1 0.007u2 1 13(1) 0.32193 0.12988 0.06592 0.03908 0.02566. 2 agfi/E) 0.06062 0.07588 0.05260 0.03566 0.02509 3 agK/5) 0.15206 0.1u559 0.10563 0.07u50 0.05377 u 13(f8) 0.05335 0.07651 0.071u6 0.058u1 0.0u611 5 Ag(/T3) 0.06586 0.09902 0.10226 0.09173 0.0778u 6 13(/25) 0.06650 0.11100 0.12921 0.12915 0.11986 7 13(f50) 0.05635 0.10351 0.13635 0.15u7u 0.16159 8 Ag(/T36)0.03855 0.07379 0.10322 0.12565 0.1u106 9 13(V27E)0.02273 0.0uu6u 0.06500 0.08323 0.09897 10 A¢(/625)0.03015 0.05998 0.08917 0.117uu 0.1uu58 1 _ ‘ A¢(-m)=2 A0 (r1) K (r1, m). 1=0 into equation (“) and successively setting Z = a, 2a, 3a, “a, and 5a. becomes: 0 After some algebraic manipulating the equation IIMIH A0 (k) = 1 (5) M (r1) D (ri,k) 0 where D(ri,k) are the downward continuation coefficients. They are given in Table 6. The second derivative formula can be obtained by taking First equation (“) the second derivative of equation (“). must be put in equivalent determinant form: 101 00 .0000000000 ommnvAxvea oH 0000.0 I 0000.0 I 0000.0 I 0000.0 I 0000.0I 0mmmxvm0 00 0000.0 I 0000.0 I 0000.0 I 0000.0 I 0000 0I 00000000 0 0000.0 I 0000.0 I 0000.0 I 0000.0 I 0000.0I Ammmxvm0 0 0000.0 0000.0 0000.0 I 0000.0 I 0000.0I Abm\000 0 0000.0 0000.0 0000.0 0000.0 I 0000.0I 0mmwia 0 0000.0 0000.0 0000.0 0000.0 I 0000.0I Am00000 0 0000.0 0000.0 0000.0 0000.0 0000.0I 000000 0 0000.00 I 0000.0 I 0000.0 I 0000.0 I 0000.0I Am\vm0 0 0000.00 0000.0 0000.0 0000.0 0000.0 000000 0 0000.000I 0000.00I 0000.00I 0000.00I 0000.0I A0000 0 0000.000 0000.00 0000.00 0000.00 0000.0 00000 0 00.0000 00.0000 00.0000 00.0000 00.0000 000000 0 .comhmucmm mmumw memHOHMMmoo GOHpmschcoo pmmzczomll.0 00909 102 0 1 -Z/a (Z/a)2 . (~Z/a)n A¢(0) 1 0 0 . . . 0 A¢(-a) 1 l 1 . . . 1 A¢(-2a) 1 2 . 22 . . . 2n (Ha) AI<2>= — |v|:l ° ° ° A0(-na) 1 A2 n2 . . . nn I where IVIis the Vandermonde determinant. It is obtained be deleting the first two rows and columns of (“a). Putting n=5 in (“a) and differentiating once with respect to z, the first derivative is obtained. I I 2 3 u 0 O -l 2(Z/a) -3(Z/a) “(Z/a) -5(Z/a) A¢(0) 1 0 0 0 0 0 A¢(-a) 1 1 12 13 In 15 A¢(-2a) 1 2 2 2 2 2 <6)§fli§L-- 2 -1 A¢(-3a)1 3 32 33 3“ 35 522 ’(a 'Vl) A¢(—“a)1 u u; u3 ufi u5 A¢(-5a)1 5 5 53 5 55 Putting z=a and substituting (3) into (6), the working equation for second derivatives is obtained: 2 10 [2.1%21] Z“E“X$ (01) D"(ri, k) 62 z=k i=0 103 where D"(ri,k) are the second derivative coefficients. The coefficients are given in Table 7. Table 7.-—Second Derivative Coefficients after Henderson. i A¢(ri) D"(;1,0) D"(ri,l) D"(ri,2) D"(r1,3) 0 A§(0) 2.8299u 7.08u08 14.15751 2u.7u755 1 13(1) —2.u9489 -6.93715 -1u.51327 —26.02351 2 13(/2) 0.05173 0.36265 0.96018 1.92719 3 13(/5) -0.39uu6 —0.8076u — 1.u2970 — 2.30269 u ag(/8) 0.00932 0.13050 0.35907 0.72u7u 5 13(/T3) 00.00732 0.07231 0.22256 0.u6253 6 agg/25) 0.0030“ 0.06502 0.17330 0.33920 7 agj/50) 0.00219 0.02312 0.05501 0.09985. 8 13(/T36) 0.000u0 0.00565 0.01239 0.02070 9 13(/27F) 0.0000“ 0.00103 0.00210 0.00322 10 A¢(/625) 0.00000 0.000u3 0.00085 0.00122 A simple example will be done here to demonstrate the use of the upward continuation equation (3), the downward continuation equation (5), and the second derivative equation (8). Figure 35 is a portion of a gravity map with the stations in the necessary square grid pattern. In practice, it is usually impossible to obtain field stations in such a pattern. So, it is necessary to interpolate between the actual stations to procure a grid pattern. In this example, the distance between stations is 500 feet. A template of circles is placed over the point at which it is desired to obtain upward continuation, downward continuation, and second derivative values. Henderson 1! III; 5.. . 10“ . 04'ch. 9937 39.,» ,gm ”9'14 ”£7.39 ”9.51. 956.” 399.59 339.7; ”9.17 336.97 ”15.94 389.09 359.99 367.31 369.42 334.903.39.9{039970 \39917 \93194 ”1'37 3,"; o . 0”’——_-—.“\\\ 389.19 3091’ 399-39 989, 95’ 99953 997.&3 399,99 o7\«p\a9. 79 309/.“ 399.97 399/33 739.97 ”Vb/38:66 Y7o\ 30977\399” 999“ >199; 339.95 5893’ 3093/9 [89.47 )89‘7 99765 989-7 369.76 3L3; 333.9! 3899} \. 393.25 39.9.” 399.40 ’89." 73.956 )09. 13.9.88 ’9'933 399.2 ,3‘933 999.90 339.49 mitt 0569690 39971 589,77 359.09 7396! 53999 999.25 99-91} 9094'" ”59-47 Jab-fl ”9.69 ”07; 399.78 p98} ngflé 739.90 59924 ”.93, faflmxaafiwfi} 137.87 3339/ PORTION OF GRAVITY MAP AND SIX RING TEMPLATE FIGURE 35 105 has found that ten circles with radii or r=a, a/2, a/5, a/8,‘ a/T—, a5, a/50, a/136, a/27“, and a 25, adequately sample the field. This is why there are 11 rows of coefficients in the tables of coefficients--one for each ring and one for the center point. It can be seen from Figure 35, where 8he ring template has-been placed over the point P, that the number of grid points falling on the circles having the above radii, are respectively 1,“,“,8,“,8,12,12,8,8,l2. The average value of the points-falling on the first 5 rings was determined and the average value was then multiplied by the appropriate coefficients. Thus, the upward continuation, downward continuation, and second derivative values were calculated for the point P. The calculations are shown in Table 8. The calculations involving the remaining 5 rings are carried out in the same manner. Table 8.--Examp1e of Upward and Downward Continuation and Second Derivative Calculations. W Center 1st 2nd 3rd “th 5th 389.66 389.65 389.73 389.72 389.77 389.77 . 389.7“ 389.70 389.78 389.76 389.82 389.66 389.56 389.77 389.“5 389.82 389.57 389.56 389.58 389.“8 389.75 389.53 389.“2 389.“? 389.3“ 389.“8 389.“0 389.55 389.“7 389.66 1558.62 1558.55- 3116.88 1558.“6 3116.79 Average Values 389.6“ 389.61 389.66 389.66 389.62 389.60 106 1 unit or 500 ft. above original surface 389.66 (.11193) + 389.66 (.32193) + 389.6“ (.0602) + 389.61 (.15206) + 389.62 (.05335) + 389.60 (.06588) - “3.61 + 125.““ + 23.62 + 59.2“ + 20.79 + 25.66 - 298.36 1 unit or 500 ft. below original surface 389.66 (n.89u8) - 389.66 (-3.0113) + 389.6u-(.0081) - 389.61 (.560u) - 389.62 (.0376) - 389.60 (.0689) - 1907.31 - 1173.38 + 3.16 - 218.3“ - 1“.65 - 26.8“ I “77.26 2500 ft. above original surface 2.89 + 10.00 + 9.78 + 20.95 4.17.96 + 30.33 + 10.70 - second on original surface 389.66 (2.8299“) - 389.66 (2.u9u89) + 389.6“ (.05173) -389.61_(+.39““6) + 389.62 (.00932) - 389.60 (.00732) = 1102.71 - 972.16 + 20.16 - 153.69 + 3.63 - 2.85 8 l2.20 HICHIGQN STRTE UNIV. LIBRQRIES 31293102776063