Journal of Geodesy (2003) 77: 279–291 DOI 10.1007/s00190-003-0325-6 Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan C. Hwang, Y.-S. Hsiao Department of Civil Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan e-mail: [email protected]; Tel.: +886-3-5724739; Fax: +886-3-5716257 Received: 5 August 2002 / Accepted: 21 February 2003 Abstract. A new orthometric correction (OC) formula is discuss various height systems and their advantages and presented and tested with various mean gravity reduc- disadvantages in practical applications. Rigorous OC tion methods using leveling, gravity, elevation, and computation is expensive because it requires observed density data. For mean gravity computations, the gravity values at benchmarks along the leveling route. Helmert method, a modified Helmert method with Also, the conventional thought is that the OC is small, variable density and gravity anomaly gradient, and a especially in areas of low elevation, so it can be neglected modified Mader method were used. An improved in most cases. As already mentioned in Heiskanen and method of terrain correction computation based on Moritz (1967), different methods for the OC may yield Gaussian quadrature is used in the modified Mader different OHs and the differences can reach several method. These methods produce different results and centimeters. This implies that OHs from leveling may yield OCs that are greater than 10 cm between adjacent mismatch the true orthometric heights by several centi- benchmarks (separated by 2 km) at elevations over meters if the OC computation is not sufficiently accurate. 3000 m. Applying OC reduces misclosures at closed It is common practice to evaluate the precision of a leveling circuits and improves the results of leveling gravimetric geoid model by comparing modeled geoidal network adjustments. Variable density yields variation height with ‘geoidal’ heights computed from the differ- of OC at millimeter level everywhere, while gravity ence between GPS-determined ellipsoidal heights and anomaly gradient introduces variation of OC of greater the ‘orthometric heights’ obtained from leveling. As just than 10 cm at higher elevations, suggesting that these stated, OHs from leveling could be in error by several quantities must be considered in OC. The modified centimeters if the OC is not applied or is not properly Mader method is recommended for computing OC. applied. Recent progress in both theory and numerical technique has greatly improved the precision of geoid modeling, and a sub-centimeter accuracy appears within Keywords: Gravity anomaly gradient – Geoid – reach (see e.g. Sanso` and Rummel 1997). Thus, incorrect Orthometric correction – Mean gravity – Terrain OHs from leveling will make such geoid model evalua- correction tion unreliable. Furthermore, use of incorrect OHs for geoid model evaluation will be very likely to occur over a region with a rugged terrain and high mountains, for example the Rocky Mountains in North America, the Alps in Europe, and the Central Range in Taiwan (see 1 Introduction below). In view of the popular use of GPS and geoid modeling for OH determination today, the subject of The orthometric height (OH) can be obtained by spirit OC deserves more attention. leveling (see e.g. Heiskanen and Moritz 1967; Moffit and One approximation of the OH is used in Taiwan, and Bossler 1998). However, height differences from leveling in earlier vertical networks normal gravity was used to must be corrected for non-parallel equipotential surfaces convert leveling heights to OHs. In a recent effort by the using the orthometric correction (OC) in order to obtain Ministry of the Interior of Taiwan to completely revise OHs (Heiskanen and Moritz 1967, Chap. 4). Recent Taiwan’s vertical datum, leveling and gravity data have work on the OC can be found in, for example, Strang van been collected at Taiwan’s first-order benchmarks. Ele- Hees (1992), Kao et al. (2000) and Allister and vation, density, and gravity data are also available from Featherstone (2001). Dennis and Featherstone (2003) other sources, for example the Institute of Agricultural and Forestry Aerial Survey, and the Institute of Earth Correspondence to: C. Hwang Sciences, Academia Sinica (Yen et al. 1990). In this 280 paper, these data will be used to investigate the theories where DnAB is the sum of all geometric height differences and methods of OC, for which Taiwan is an ideal testing between A and B obtained directly from leveling, gi is area because of its rugged terrain and complex geolog- surface gravity value at leveling set i, dni is the geometric ical structure. In the following sections, a new OC for- height difference at leveling set i, and k is the number of mula will be derived and results of OC computations sets. A leveling set may span up to 100 m horizontal using various gravity reduction methods over Taiwan distance between backsight and foresight, depending on will be presented. the terrain and the level used (see e.g. Moffit and Bossler 1998). In Eq. (4), the two integrals have been approx- imated by finite, discrete sums. Substituting Eqs. (3) and 2 Theories (4) into Eq. (2) yields 2.1 Orthometric height and orthometric correction 1 Xk g DH ¼ Dn þ ðÞg À g dn þ H A À 1 ð5Þ AB AB g i B i A g The OH is the height above the geoid measured along B i¼1 B the curved plumb line. Leveling alone will yield a geometric height difference between two consecutive Thus an OC formula is obtained as benchmarks, which in turn yields an OH difference by Xk applying the OC (Heiskanen and Moritz 1967, Chap. 4). 1 gA OC ¼ ðÞg À g dn þ H À 1 : ð6Þ Thus the OC plays a critical role in obtaining OHs from AB g i B i A g B i 1 B leveling. A new OC formula is derived below. By ¼ definition, the OH (H) at a benchmark is the ratio This formula is to be compared with the formula given between its geopotential number (C) and its mean by Heiskanen and Moritz (1967, p. 168) gravity along the plumb line (g) between the surface and Xk the geoid. Thus, for two benchmarks A and B gi À c gA À c gB À c OCHM ¼ 0dn þ 0 H À 0 H ð7Þ AB c i c A c B CA i¼1 0 0 0 HA ¼ gA ð1Þ where c0 is normal gravity at some latitude (usually CB 45°Nor45°S). If A is sufficiently close to B in horizontal HB ¼ gB distance (below 2 km), Eq. (6) can be simplified as The difference between H and H is follows. Within a short distance, gravity can be assumed A B to be a linear function of height, as well as horizontal DHAB ¼ HB À HA distance. Then, Eq. (4) gives C C ¼ B À A ZA gB gA 1 1 gA þ gB ðÞg À gB dn ¼ À gB DnAB ð8Þ 1 CA CA gB gB 2 ¼ ðÞþCB À CA À B gB gB gA ZB Thus, a simplified formula for the OC is 1 CA CA ¼ g dn þ À ð2Þ gB gB gA 1 gA þ gB gA A OCAB ¼ À gB DnAB þ HA À 1 ð9Þ gB 2 gB where dn is the differential geometric increment of height and g dn ¼ dC,withg being surface gravity. Furthermore where gA and gB are the surface gravity values at A and B. From Eq. (9) and the assumptions used to derive it, CA CA gA À ¼ HA À 1 ð3Þ gravity values need only be measured at the two adjacent gB gA gB benchmarks (A and B in this case), without knowing the and gravity values at the k leveling sets between them. As an example, Fig. 1 shows the relationship between gravity ZB ZB and elevation collected at 32 leveling sets along a 2-km- 1 1 g dn ¼ ðÞg À gB þ gB dn long leveling line in central Taiwan. The mean elevation gB gB here is approximately 1800 m and the height difference A A between the start and end benchmarks is approximately ZB ZB 1 100 m. In this case, the gravity values are almost linearly ¼ dn þ ðÞg À gB dn correlated with elevation. For this particular leveling gB A A line, the OC was computed using Eq. (9) and the more rigorous Eq. (6). The difference in OC is less than Xk 1 Xk dn þ ðÞg À g dn 0.1 mm. Thus, the small perturbation from linearity i g i B i i¼1 B i¼1 should have little effect on the accuracy of the approx- imation in Eq. (9). 1 Xk ¼ Dn þ ðÞg À g dn ð4Þ In Eq. (9) it remains to determine the mean gravity AB g i B i B i¼1 values along the plumb lines at A and B. A simple 281 The use of mean gravity computed by Eq. (12) in Eq. (1) yields the Helmert OH. 2.2 Error due to density variation and gravity anomaly gradient As will be shown below, the gravity anomaly gradient over Taiwan can be very large in mountainous areas, so it cannot be neglected. Vanı´cˇ ek et al. (2000) reported that use of the gravity anomaly gradient will change OHs by up to a few decimeters over the Canadian Rocky Mountains. Also, rock density over Taiwan is variable and cannot be assumed to be constant. Therefore, in this paper, it will be investigated how density variations and gravity anomaly gradients affect the result of OC. From Eq.