This book provides a complete set of mathematical formulas needed to 1. Convert and store all types of spatial measurements, together with their accuracy estimates, as coordinates or coordinate differences, relative to a common set of three-dimensional 3D Cartesian coordinate axes x ,y ,z ; and 2. Transform this Cartesian information to ellipsoidal latitude, longitude, ellipsoidal height , State Plane x ,y , local ellipsoidal tangent plane north, east, up coordinates and orthometric height information needed in practical applications. This complete set of equations, combined in a single computer program, has been named the Global Spatial Data Model GSDM , by the writer. Observational data considered include surface mark-to-mark observations horizontal and vertical angles, distances, azimuths , orthometric height differences from leveling, and Cartesian coordinates and coordinate differences from GPS. The GSDM is an outstanding means for combining all types of measurements where the following conditions are met: • All observations have been referenced to the same coordinate system external to GSDM for example, using the NGS HTDP program ; and • Coordinates of points do not change over time relative to the coordinate system used by more than the accuracy required by the user. Given the time variability of NAD83 coordinates, one can expect to use GSDM directly in conjunction with NAD-83 related coordinates only with local urban areas or small county networks in the eastern United States. For larger-scale NAD83 networks, even in the eastern United States, to maintain 1 cm accuracy desired by many surveyors, over times of decades, would require reduction of observations to a common time epoch before entering them into a GSDM. Conversion to a common time epoch will always be required if ITRF or WGS84 coordinates are used and few-centimeter accuracy is desired. While conversion between coordinate systems using HTDP does not introduce error into observations, the error in conversion to a common time epoch must be included in the error budget. Another point that is discussed, but may be overlooked by the reader, is that observed angles using, for example, a total station are relative to a plane perpendicular to the local gravity vector at a station, not the ellipsoidal normal at that point. Deflection of the vertical corrections may be needed to produce the differential north, east and up ellipsoidal normal components discussed in Appendix A of the book. Chapter 1 is an essential guide to the rest of the book. It gives a brief summary of each of the 22 sets of equations that make up the GSDM and graphically illustrates how these equations fit together. The chapter also summarizes the covariance matrices used to provide the accuracy information for the various types of observations and coordinates. Chapter 2 briefly introduces the four types of coordinates to be considered: Earth-centered Cartesian, ellipsoidal, State Plane, and local tangent plane, as well as the observation types that will be considered. Chapter 3 is one of several chapters containing what might be called ancillary information. This chapter can be characterized as a short summary of basic mathematics, including arithmetic, algebra, plane and spherical geometry, trigonometry, calculus, the use of matrices, and probability and statistics. One cannot, of course, learn mathematics from these 42 pages, but they can serve as handy references to forgotten concepts. Chapters 4 and 6 go into detail about geometry relevant to development of the necessary GSDM equations. Chapter 4 discusses basic 2D and 3D Cartesian geometry, including circular and spiral curves. Chapter 6 covers ellipsoidal coordinates. It begins with the transformations in both directions between Cartesian and ellipsoidal coordinates. It then covers computations on an ellipsoid in going back and forth between latitudes and longitudes and angles, distances, and azimuths. Chapter 5, another ancillary chapter, is a summary of the field of geodesy, including an interesting short history of the development of geodesy. Chapter 7 is a nonmathematical discussion of datums, including currently relevant datums and coordinate systems, namely NAD83, WGS84, and the ITRF coordinate systems. The reader cannot depend on this chapter to understand coordinate systems and datums. I found it misleading in places. For example, the importance of time variability is underestimated, and also it is not made clear that the ITRF coordinate systems are fundamental and the NAD83 and WGS84 Cartesian coordinates are derived from ITRF coordinates. Chapter 8 covers physical geodesy, giving the relation between ellipsoidal, orthometric, and geoid heights and specifying how height systems fit into the GSDM. Chapter 9 is another ancillary chapter, giving a discussion of how GPS positioning works. Chapter 10 is a detailed development of the equations used in the GSDM to transform between ellipsoidal latitude and longitude and State Plane coordinates. Map projections considered in discussing State Plane coordinates are Mercator, Oblique Mercator, and Lambert Conformal. Chapter 11 begins with a discussion of why the GSDM was developed, but the bulk of the chapter is a numerical example of the covariance matrices for a small observation network. The final chapter, Chapter 12, covers issues involved in using the GSDM. There are three appendices. Appendix A gives the rotation matrices needed to convert from Cartesian coordinate differences between two points to local, north, east, and up coordinate differences. Appendix B gives the constants needed to derive the NAD83 State Plane coordinates. Appendix C gives a

Journal of Geophysical Research: Solid EarthVolume 92, Issue B10 p. 10711-10714 CommentariesFree Access Comments on “Saugus-Palmdale, California, field test for refraction error in historical leveling surveys” by Ross S. Stein, Charles T. Whalen, Sanford R. Holdahl, William E. Strange, and Wayne Thatcher Robert Reilinger, Robert ReilingerSearch for more papers by this author Robert Reilinger, Robert ReilingerSearch for more papers by this author First published: 10 September 1987 https://doi.org/10.1029/JB092iB10p10711Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL References Helm, D. C., Field verification of a one-dimensional model for transient compaction and extension of a confined aquifer systemSpecialty Conference on Verification of Mathematical and Physical Models in Hydraulic EngineeringAm. Soc. of Civ. Eng.College Park, MD, 1978. Holzer, T. L., Preconsolidation stress of aquifer systems in areas of induced land subsidence, Water Resour. Res., 17, 693– 704, 1981. Lofgren, B. E., Changes in aquifer-system properties with groundwater depletion, Evaluation and Prediction of Subsidence S. K. Saxena, 26– 46, Am. Soc. of Civ. Eng., New York, 1979. Reilinger, R. E., Elevation changes near the San G abrial fault, southern California, Geophys. Res. Let., 11, 1017– 1019, 1980. Reilinger, R. E., L. D. Brown, Neotectonic deformation, near-surface movements and systematic errors in U.S. releveling measurements: Implications for earthquake prediction, Earthquake Prediction: An International Review, Maurice Ewing Ser., 4 D. W. Simpson, P. G. Richards, 422– 440, AGU, Washington, D.C., 1981. Riley, F. S., Analysis of borehole extensometer data from central California, Land Subsidence, IASH Publ., 89, 423– 431, 1969. Robson, S. G., Water resources investigation using analog model techniques in the Saugus-Newhall area, Los Angeles County, California, U.S. Geol. Surv. Open File Rep.5021-04, 58, 1972. Stein, R., Discrimination of tectonic displacement from slope-dependent errors in geodetic leveling from southern California, 1953–1979, Earthquake Prediction: An International Review, Maurice Ewing Ser., 4 D. W. Simpson, P. G. Richards, 441– 455, AGU, Washington, D.C., 1981. Stein, R. S., C. T. Whalen, S. R. Holdahl, W. E. Strange, W. Thatcher, Saugus-Palmdale, California, field test for refraction error in historical leveling surveys, J. Geophys. Res., 91, 9031– 9044, 1986. Strange, W. E., The impact of refraction correction on leveling interpretations in southern California, J. Geophys. Res., 86, 2809– 2824, 1981. Terzaghi, K., R. B. Peck, Soil Mechanics in Engineering Practice, 729, John Wiley, New York, 1967. Citing Literature Volume92, IssueB1010 September 1987Pages 10711-10714 ReferencesRelatedInformation

Errors are introduced in orthometric height computations by the use of standard formulas to estimate mean gravity along the plumb line. Direct measurements of gravity between the Earth’s surface and sea level from bore hole gravimetry were used to determine the magnitude of these errors. For the seven cases studied, errors in orthometric height, due to the use of the Helmert method for computing mean gravity along the plumb line, were generally small (<2 cm). However, in one instance the error was substantial, being9.6 cm. The results verified the general validity of the Poincaré-Prey approach to estimation of gravity along the plumb line and demonstrated that the suggestion byVanicek (1980) that the air gradient is more appropriate is incorrect. With sufficient topographic information to compute terrain corrections, and density estimates from surface gravity, errors in mean gravity along the plumb line should contribute no more than 3cm to orthometric height computation.

Astronomic azimuths are used in classical geodesy, through the Laplace equation, to control the orientation of geodetic networks. The method most commonly used by the United States National Geodetic Survey for the determination of astronomic azimuth is often referred to as the “direction method”, and is based on observations of Polaris at any hour angle.

Aeromagnetic surveys of the Hawaiian Islands have revealed that the pr imary magnetic anomalies associated with the islands are dipole anomalies caused by the intrusive rocks of the volcanic centers and rift zones. Comparisons of the direction of magn etization indicated by the dipole anomalies with results of laboratory measurements on iavas show that in many cases the lavas possess reverse polarization while the intrusive rocks are normally polarized. These results must be taken into account when interpreting the magnetic field of subm erged marine volcanic features such as seamounts and when establishing periods of reversal in the earth's magnetic field. AN AEROMAGNETIC SURVEY covering the ma jor islands at the southern end of the Hawaiian chain has recently been completed with flight lines approximately 1 mile apart. The structural and geologic implications of this survey are discussed in derailby Malahoff and W oollard (in a forthcoming issue of Pacific Science ) , to which pap er the reader is referred for examination of the actual anomaly contour maps. The dominant magnetic anomalies observed over the islands were found to be positive-negative anomaly pairs-typical dipole anomalies. Such anomalies might be expected to result from bodies with near vertical sides magnetized parallel to the present earth's field, which, in this area, has an inclina tion of 30-40°. From a comparison of the location of the dip ole anomalies with the geologic and gravimetric data on the islands, it is apparent that they are caused by the intrusive rocks associated with the volcanic centers and major rift zones on the islands. Such a result is in agreement with laboratory determinations of susceptibility and intensity of remnant magnetization of some Hawaiian rocks made by the authors. These measurements show that the intensity of remnant magnetization is much great er in most intrusive rocks of the Hawaiian Islands than in the lavas. In both types of rocks the intensity of remnant magnetization greatly exceeded that of induced 1 H awaii Inst itute of Geophysics Contribution No. 100 . magnetization-by a factor of 1:10 in olivine poor samples. Several model computations showed that .it is possible to explain the observed aeromagnetic anomalies by assuming that the intrusive rocks were either normally or inversely polariz ed in a direction nearly parallel to the present earth's field. This is in agreement with measurements by Tarling (1963) of direction of remnant magnetization carried Out on surface samples, primarily lavas, which also indicated directions of remn ant magnetization nearly parallel to the present earth's field. Because the magnetic anomalies caused by the remnant magnetization of the intrusive rocks are dip ole anomalies, it is possible to determine by inspection whether the intrusive rocks are normally or inversely polarized. This gross direction of remnant magnetization ( normal or reverse ) is given in Table 1, along with the results obtained by McDougall and Tarling ( 1963) , and Doell and Cox ( 1963), and measurements made by the writ ers on surface samples of both extrusive and intrusive rocks. The paleomagnetic and age dating results obtained by McDougall and Tarling ( 1963 ) from Hawaiian lavas have been utilized by Cox, Doell, and Dalrymple ( 1964) in conjunction with data from other areas to establish alternating periods of norm ality and reversal in the earth's magnetic field during the last four million years. The reality of these reversals, their leng th (i f they exist ) , and the