Physical and Geometric Effects on the Classical Geodetic Observations in Small-Scale Control Networks

The deflection of the vertical at the geoid surface is usually defined as a spatial angle ($\epsilon$) between the normals to the geoid and ellipsoid surface (Fig. 1). The DOV thus also shows the slope of the geoid with respect to the Earth’s reference ellipsoid (Vanicek and Krakiwsky 1986; Sjöberg and Bagherbandi 2017). The DOV angle $\epsilon$ deviates from ${\epsilon }^{\prime }$ elsewhere due to the local features of the Earth’s gravity field and the curvature of the plumb line (cf. Vanicek and Krakiwsky 1986, pp. 491–506). In Fig. 1, ${\epsilon }^{\prime }$ denotes the DOV at the Earth’s surface.

The well-known equations of DOV components in the meridian and prime vertical directions at any point of interest on the geoid surface can be found in Heiskanen and Moritz (1967, p. 312). Corresponding equations can be written for the DOV components at the Earth’s surface using the quasigeoid and Molodenskij’s definition of the height anomaly ($\varsigma$). Molodenskji et al. (1962) discarded the geoid and defined a new surface, the quasigeoid, in which geoidal undulation is replaced by height anomaly. The north–south (${\xi }^{\prime }$) and east–west (${\eta }^{\prime }$) components of the DOV are obtained by Heiskanen and Moritz (1967, p. 313)where $R$ = mean radius of the Earth (the radius of a sphere whose volume equals that of the ellipsoid according to Moritz 2000); $\phi$ and $\lambda$ = latitude and longitude of the desired point, respectively; $\gamma$ = normal gravity on the earth’s surface; $H$ = topographic height (normal height in this case study because there is quasigeoid model in Sweden); and $\mathrm{\Delta }g$ = gravity anomaly (Heiskanen and Moritz 1967, p. 83). Therefore, Eq. (1) provides us with the fundamental links between the Stokes and Molodenskij approaches, which involve the topographical and curvature of plumb line effects (Heiskanen and Moritz 1967, p. 312; Vanicek and Krakiwsky 1986, p. 533). The formulae of determination of the DOV components using an Earth gravitational model (e.g., the EGM2008 model) are also presented in Supplemental Materials.

The curvature of the plumb line correction should be considered, especially in rough topography areas. This correction was presented by Vanicek and Krakiwsky (1986, p. 506) and Jekeli (1999). They showed that it mainly affects the north–south component, that is, $\xi$. In other words, the correction depends on the normal gravity field that does not change in the east–west direction. Therefore, the plumb line curvature does not affect the east–west component. See more details in Vanicek and Krakiwsky (1986, p. 506) and Jekeli (1999).

As already mentioned, conventional geodetic observations are affected by the physical effect, that is, by the DOV components. In other words, the gravity field corrections (i.e., corrections for the effects of the DOVs and geoidal undulations) should be considered before performing geodetic control network adjustment if vertical angles (zenith angles) are used to convert the slope distances to horizontal distances (cf. Shirazian et al. 2021).

After calculating ${\xi }^{\prime }$ and ${\eta }^{\prime }$, one can calculate the effect of DOV on zenith angles (${\delta }_{Z_P}$) at the Earth’s surface (Point $P$) using the following equation [Heiskanen and Moritz 1967, Eqs. (5)–(16)]:where ${\alpha }_{PQ}$ = geodetic azimuth from Point $P$ to Point $Q$. Eq. (2) shows the projection of the DOV onto the plane that goes through Points $P$ and $Q$ and the center of the Earth (for a spherical earth model but not for an ellipsoidal model) with geodetic azimuth ${\alpha }_{PQ}$. The slope distance (SD) is converted to the horizontal distance (HD) using the following equation:where $Z$ = observed vertical (zenith) angle. Finally, the effect of ${\delta }_{Z_P}$ on the horizontal distances can be determined by (i.e., considering with and without the DOV effect on the vertical angle)

The geometric effect in question causes curvature-skewness and curvature errors on the ellipsoid [Fig. 2(a)] and sphere [Fig. 2(b)], respectively. In order to derive the geometric effect on slope distances, the Earth is considered first as an ellipsoid of revolution and then as a sphere. This is illustrated in Figs. 2(c and d) with more details. For example, the normals at two distinct points on the ellipsoid do not generally lie in the same plane (they are skewed) and never coincide (except if the azimuth is zero or 90°). In other words, there is more than one definition of horizontal distance between Points $P$ and $Q$. This means that the horizontal distance from $P$ to $Q$ is not equal to the horizontal distance from $Q$ to $P$ due to the geometric effect (cf. Rollins and Meyer 2019). If we considered $P$ and $Q$ points on different meridians, then the normals pass through $n_P$ and $n_Q$, respectively ($n_P$ and $n_Q$ are the intersection of normals at $P$ and $Q$ with the $Z$-axis). Hence, assuming the ellipsoid model [Figs. 2(a and c)], the effect of the curvature of the reference surface is the difference between horizontal distance $PQ$ (computed in the horizontal plane of $P$ using the zenith angle) and the ellipsoidal distance $P_1{Q}_1$ (i.e., the geodesic length between $P_1$ and $Q_1$). The effect of skewness problem is the difference between ellipsoidal distances $P_1{Q}_1$ and $P_1{Q}_1^{\prime }$. The skewness is quantified by the angle $Q_1–Q–Q_1^{\prime }$ (shown by angle $\theta$) and the curvature-skewness by angle $\beta$ [assuming the spherical model, the angle between the normals is shown by ${\beta }^{\prime }$; see Fig. 2(d)]. This geometric property is called the curvature-skewness angle and affects the horizontal and vertical angles and distances indirectly (using the vertical angle to reduce slope to horizontal distances). It is important to emphasize that the angle $\beta$ shows the nonparallelism of normals at Points $P$ and $Q$ [because the normals at $P$ and $Q$ are not in the same plane; see Fig. 2(a)]. In other words, as can be seen in Figs. 2(a and c), one should not assume any angle between two normals at Points $P$ and $Q$ because they are not at the same normal plane. In practice (for simplicity), one can define an angle $\beta$ that is the angle between the normals at $P$ and $Q$ projected to the plane of the normal section between $P$ and $Q$ (see also Fig. 3).

In the following, we present the impact of the curvature-skewness problem on horizontal and vertical angles. The effect of the curvature-skewness error on the horizontal angles is explained in Rapp (1991). We call this effect ${\delta }_{{\alpha }_{PQ}}$ for the two Points $P$ and $Q$, and it iswhere $\mathrm{\Delta }h$ = ellipsoidal height difference between the two points; ${\alpha }_{PQ}$ = azimuth from $P_1$ to $Q_1$; $s_{PQ}$ is the length of the geodesic between $P_1$ to $Q_1$; ${\phi }_P$ is the latitude of $P$; and ${\hat{N}}_P$ (radius of curvature in the prime vertical at $P$) and ${\mathrm{\Psi }}_P^2$ are as follows:where $e^2=(a^2-b^2)/a^2$; ${(e^{\prime })}^2=(a^2-b^2)/b^2$; and $a$ and $b$ = semimajor and semiminor axes of the ellipsoid. The reader should be aware that Eq. (4a) is not useable if $P$ is not at one of the poles: ${\alpha }_{PQ}=0$, so $\sin {\alpha }_{PQ}=0$ and ${\delta }_{{\alpha }_{PQ}}=0$. For $s_{PQ}=\mathrm{2,000}\mathrm{m}$, ${\phi }_P={\alpha }_{PQ}=45°$, and $h=100\mathrm{m}$ on the GRS80 ellipsoid (Moritz 2000), ${\delta }_{{\alpha }_{PQ}}$ is about 0.005 (in arc seconds), which is negligible in small-scale networks. As mentioned before, the curvature-skewness problem has been presented only for horizontal angles in the classical geodesy literature (Krakiwsky and Thomson 1974; Rapp 1991). Hence, we present a method to quantify this problem for vertical angles, which is important for converting from slope to horizontal distances. We will show that the effect of the curvature-skewness error on vertical angles is significant and must be accounted for when dealing with precise geodetic networks. A schematic illustration of the geometric effect (corresponding to the curvature-skewness problem, assuming the ellipsoidal model) on vertical angles is shown in Fig. 3 (considering the horizontal plane passing Point $P$). Here we assume that the angle $\beta$ in Fig. 2(c) is the angle between the normals at $P$ and $Q$ projected to the plane of the normal section between $P$ and $Q$.

As in Fig. 3, the slope angle at Point $P$ isand this angle at Point $Q$ is

Thus, a unidirectional observing vertical angle will have an error of $\beta$ due to the curvature-skewness effect. If one measures the vertical angle reciprocally and takes the average of the measurements at the two endpoints of the line, then average reads

This means that even in the reciprocal measurement of the vertical angles, the effect of the curvature-skewness problem does not disappear, and half of that remains in the measurement. This imposes an error in the slope distance reduction, which is

A method is proposed in this paper to compute the curvature-skewness (C-S) error in Eq. (6) accurately. This method consists of the following steps:

In the first step, we establish a local geodetic coordinate system (east–north–up or enu system; see Fig. 4) at Point $P$ and compute the coordinate differences, that is, $\mathrm{\Delta }{e}_{PQ}$, $\mathrm{\Delta }{n}_{PQ}$, and $\mathrm{\Delta }{u}_{PQ}$. Then, the slope angle from Point $P$ to Point $Q$ will be

To determine the coordinates (at Points $P$ and $Q$), one can employ the Vincenty algorithm (Vincenty 1975) or the method proposed by Bowring (1983). Using the Vincenty formula and assuming a reference ellipsoid (GRS80), we computed the endpoint coordinates along with the backward azimuth with submillimeter accuracy for our purpose, starting from the geodesic length between two points, forward azimuth, and coordinates of the starting point (direct problem in Geodesy).

In the second step, similarly, we established another enu system at Point $Q$ and computed $\mathrm{\Delta }{e}_{QP}$, $\mathrm{\Delta }{n}_{QP}$, and $\mathrm{\Delta }{u}_{QP}$. Then, the slope angle from Point $Q$ to Point $P$ will be

Finally, one can compute the curvature-skewness effect on the horizontal distances (${\delta }_{\mathrm{HD}}^{\mathrm{C}-\mathrm{S}}$) according to Eqs. (5c) and (6).

Therefore, if we measure the vertical angles reciprocally, we have to add ${\delta }_{\mathrm{HD}}^{\mathrm{C}-\mathrm{S}}$ from Eq. (6) to $\mathrm{SD}\cos (V)$ to get the horizontal distance in the horizontal plane of $P$. In practice, we have to choose a common reference plane (or generally a surface) for all distances in the network.

Assuming a spherical approximation of the ellipsoid, the curvature effect on the slope distance reduction can be obtained by the following steps. First, the average of the latitudes of Points $P$ and $Q$ is considered to calculate the mean radius of curvature, that is, Gaussian curvature (cf. Heiskanen and Moritz 1967)

Then, the angle between two normals (at Points $P$ and $Q$) can be obtained by [Fig. 2(d)]

We denote the first point (i.e., $P$) as projected on the sphere surface ($h_P=0$); then, it will become $P_1$. Thus, the slope distance between Points $P_1$ and $Q$ is given by

The slope angle on the sphere at Point $P_1$ is obtained by [see the planar triangle $P_{1}Q{Q}_1$ in Fig. 2(d)]where $\omega =(\pi +{\beta }^{\prime })/2$. Finally, the curvature error on the slope distance reduction is obtained by

In this section, first, we present the determined DOV components at the Earth’s surface obtained from the SWEN17_RH2000 quasigeoid model and Eqs. (1a) and (1b), and then the DOV impact on the classical geodetic observations will be assessed. The north–south ($\xi$) and east–west ($\eta$) components in Sweden are shown in Fig. 6. We used the same abbreviation as the quasigeoid model (i.e., SWEN17) for the obtained DOV components. Generally, large values for the DOV components can be seen in Lappland (northwest of Sweden), Västerbotten, Ångermanland, Medelpad (northeast of Sweden), and Småland (south of Sweden).

The calculated DOV components can also be compared with one determined using spherical harmonic coefficients, such as the EGM2008 model (Pavlis et al. 2012), at the Earth’s surface (see Supplemental Materials). The statistics of the DOVs obtained from the EGM2008 and SWEN17 models in Sweden are presented in Table 1. The EGM2008 and SWEN17 models show that the DOVs are generally similar except for the rough topography areas (especially in the northwest of Sweden) because the EGM2008 model has a lower resolution than SWEN17_RH2000 (see also Fig. S1 in Supplemental Materials). Hence, we studied the impact of the DOV on geodetic observations using only SWEN17 in this paper.

Table 2 shows the magnitude of the north–south ($\xi$) and east–west ($\eta$) components at the selected points in Kebnekaise, Umeå, Mårtsbo, and Skövde. We selected these stations in different latitudes (from the north to the south) to scrutinize the influence of the DOV on geodetic observation in different regions. The Kebnekaise station is located in the northwest part of Sweden that has the highest mountains. The Umeå and Mårtsbo stations are close to the shorelines, with close to minimum height anomaly values (on the Swedish mainland). Finally, we selected the Skövde station in the south of Sweden, representing a medium value of height anomaly in the study area. We believe that this combination can be a good sample to examine the effect of the DOV.

To evaluate the obtained DOV components, one can compare them with those obtained from classic astronomical observations or advanced zenith camera systems [e.g., see (Hirt and Seeber 2002)]. At the time of writing the present paper, we had access only to the classic astronomical observations through Ekman and Ågren’s (2010) report (see also Wargentin 1759; Cronstrand 1811; Selander 1835). The comparison between the DOVs obtained by the SWEN17 model and astronomical observations can be seen in Table S1 in the Supplemental Materials.

We aim to show the effect of the DOV on the reduction of the slope distances to the horizontal ones and conclude whether its impact is significant (in the case of using vertical angles to convert slope to horizontal distances). Using Eq. (3b), one can calculate the impact of the DOV on the slope distance reduction by assuming a value for the zenith angle. According to the surveying guidelines and literature (e.g., USACE 2002, pp. 2–13; Krakiwsky and Thomson 1974, p. 37), it has been recommended to design the geodetic control points so that the stations’ elevations are as similar as possible in order to eliminate or reduce both physical and geometric effects (i.e., the DOV and curvature-skewness problem). Therefore, the zenith angles should usually be as close to 90° as possible and, in practice, vary within $90°\pm 5°$ to fulfil this criterion. However, this assumption causes another problem, atmospheric refraction error, because the observed zenith angles are very sensitive to the atmospheric refraction if the zenith angle varies between 90° and 85° (cf. Vanicek and Krakiwsky 1986, p. 366). Zenith angles close to 90° will, on the other hand, be an ideal case for the DOV and curvature-skewness effects. However, we assumed 85° for the zenith angle in our simulation. The results of the impact of the DOV on horizontal distances can be seen in Figs. 7–10 in the selected stations (i.e., Kebnekaise, Umeå, Mårtsbo, and Skövde). We plotted the absolute values of ${\delta }_{\mathrm{HD}}$ considering different baselines (varying between 0.4 and 5 km) and azimuths (varying between 0° and 360° with 10° intervals). As we show in Tables S2–S5 in the Supplemental Materials, the impact of the DOV on horizontal distances will be as large on the opposite side (i.e., 180° azimuth difference). This is the reason for illustrating the absolute values of ${\delta }_{\mathrm{HD}}$ in Figs. 7–10. In addition, similar plots are presented in Figs. S2–S5 (see Supplemental Materials) assuming 70° for zenith angle in the same stations. This extreme zenith angle value might occur when establishing a control network around high towers and landslide monitoring projects where target points are located above or under the horizontal plane with a large zenith angle value such as 70°.

The statistics of the impact of the DOV on horizontal distances (${\delta }_{\mathrm{HD}}$) can be seen in Tables S2–S5 in the Supplemental Materials assuming different baseline lengths (between 0.4 and 5 km), azimuths (between 0° and 360°), and two zenith angles (70° and 85°). For instance, considering 400 m baseline length, the absolute values of the DOV effect on the horizontal distance observation were 10.7 (for $Z=70°$) and 2.7 mm (for $Z=85°$) in Umeå. Similarly, these values were 6.6 (for $Z=70°$) and 1.7 mm (for $Z=85°$) in Mårtsbo. These values increase rapidly with the length of the slope distance. The values show that the influence of the DOV is significant and considerable. Moreover, studying the specifications of many brands of precise geodetic instruments suitable for control networks [e.g., Leica TM50 (Bern, Switzerland) total station and similar instruments] shows that the distance nominal accuracy is $0.6\mathrm{mm}+1\mathrm{ppm}$ (using a round prism). Therefore, any systematic error larger than instrument nominal accuracy should be taken into account before geodetic control network adjustment if the vertical angles are not measured reciprocally.

In addition, the results confirm that the effect of the DOV is also significant in specific directions (azimuths). In other words, the azimuth of the baseline is an influential parameter. With a closer look, the DOV effect will reach its maximum value (approximately) in azimuth equal to 150° and 140° in Kebnekaise and Umeå, respectively, for the applied baselines (Figs. 7 and 8). Similarly, the maximum effects can be seen (approximately) in azimuths equal to 100° and 40° in Mårtsbo and Skövde, respectively (Figs. 9 and 10). Moreover, the DOV impact depends on the azimuth of the baselines. The need for vertical angle measurements to establish geodetic networks, which suffer from the DOV effect and vertical refraction (or their simultaneous reciprocal measurements, which significantly increase the fieldwork and costs of the network campaign), is avoided by using the proposed solution by Shirazian et al. (2021) with the so-called network-aided reductions of slope distance observations. They presented an approach that considered a special geodetic observation strategy to significantly reduce the volume of operations of a precise geodetic network by omitting all measured vertical angles in the network. It changed the design concept for such geodetic networks because the network-aided method decreases data collection time and cost while keeping or even increasing the quality of control networks because it avoids the DOV problem in question (Shirazian et al. 2021). Otherwise, one should use the gravity field corrections (i.e., corrections for the effects of the DOV and geoidal undulations), which need a precise geoid model and gravity and elevation data. We emphasize again that DOV correction must be applied if one uses vertical angles for the reduction of slope distances.

Fig. 11 compares the forward and backward effects of the DOV, considering different distances and assuming a vertical angle equal to 85° in the selected stations. The figure shows the maximum absolute value of ${\delta }_{\mathrm{HD}}$ assuming azimuths 150°, 140°, 100°, and 30° for Kebneise, Umeå, Mårtsbo, and Skövde, respectively (in the directions that the maximum DOV effects occur). The impact of the DOV on the forward and backward measurements (reciprocal reading) is almost the same if the height difference between the start points and endpoints are small. It confirms that reciprocal reading can be a solution to eliminate the DOV effect when the height difference in the baseline is not large. However, simultaneously, reciprocal reading will increase the fieldwork and costs of the projects (as already mentioned), and it is not recommended. The results show that the differences are ignorable up to 3,000 m baseline length if both points are in the same elevation. For example, the differences are more significant in Kebnekaise stations due to the large height difference between the start points and endpoints; see more details in Tables S6–S9 in the Supplemental Materials. For example, Table S6 shows the positions of the Kebnekaise station and the endpoints. The position of the baselines’ endpoints (varying between 0.4 and 5 km) was calculated using the Vincenty algorithm. Using the obtained position of the endpoints, one can calculate the DOV effect on the assumed vertical angle from the endpoint toward the Kebnekaise station (backward) and finally ${\delta }_{\mathrm{HD}}$. The last column in Tables S6–S9 compares the forward and backward values.

The geometric effect on the zenith (vertical) angles and consequently on the slope distance reduction is assessed in this section. First, we present the results based on the ellipsoidal Earth model, and finally, the results will be compared with those derived from the spherical assumption. Using Eq. (6), one can calculate the impact of this error by determining forward and backward slope angles from Eqs. (7a) and (7b). We simulated the impact of this error by assuming a point at latitude 60° and longitude 15° and different baseline lengths (100 to 5,000 m) and height differences, varying between 0 and 500 m in Sweden. The geodetic latitude, longitude, and height are first converted into the Cartesian coordinates $X$, $Y$, and $Z$ and then into the enu system (see equations in Hofmann-Wellenhof et al. 2007, pp. 280–282, for the coordinate transformation). The GRS80 reference ellipsoid (Moritz 2000) was considered for this simulation. Using the obtained coordinates in the enu local geodetic coordinate system, one can calculate the reciprocal slope angles and finally ${\delta }_{\mathrm{HD}}^{\mathrm{C}-\mathrm{S}}$, as explained already. Fig. 12 shows the angles between normals for different baseline lengths. The results show that the curvature-skewness error is significant, and it should be considered for the baselines in the geodetic control networks (see the effect of this error on the slope distance reduction in Table 3 and Fig. S6). It varies between ${3.2}^{″}$ (for 100 m baseline length) and ${161.6}^{″}$ (for 5,000 m baseline length). According to guidelines (e.g., USACE 2002), geodetic network designers always try to design network stations at the same elevation. Therefore, the curvature-skewness effect on the slope distance reduction will be eliminated by designing a proper geodetic network following the guidelines. However, it is not easy to follow the recommendations because of the study areas’ situations (e.g., existing rough topography). The curvature-skewness problem does not depend on the azimuth, and its effect is ignorable ($<0.01\mathrm{mm}$) for short baselines (see Fig. S7 in Supplemental Materials). We mean to quantify the systematic effects affecting the slope distance reduction to the horizontal distance. For this purpose, we have to create enu coordinate systems at the endpoints of the baselines. Through a simulation, we created baselines for which all the station coordinates (geodetic $XYZ$ or $BLH$) were computed accurately with the exact (simulated) geodesic lengths and azimuths between the start points and endpoints. However, in real practice, we propose the method by Shirazian et al., which establishes a single enu system at one of the network points (the closest to the center of the network), measuring its accurate $BLH$ coordinates (by GNSS receiver, for instance).

Table 3 shows the curvature-skewness effect (${\delta }_{\mathrm{HD}}^{\mathrm{C}-\mathrm{S}}$) on the slope distance reduction in reciprocal measurements for different baseline lengths and height differences. It can be seen from the table that the curvature-skewness effect is below 0.4 mm (assuming 1 km baseline length) if the geodetic network stations are almost at the same level (for $\mathrm{\Delta }H\le 5\mathrm{m}$). This correction will be notable when one uses a very precise electronic distance meter (EDM), such as a Kern ME500 Mekometer with $0.2\mathrm{mm}\pm 0.2\mathrm{mm}/\mathrm{km}$ range accuracy (Bell 1992), and for large height differences. In other words, the effect of the skewness error is small for zero height differences. Thus, the use of the proposed method is recommended in the case of using highly accurate instruments and avoiding the spherical approximation in the computations.

Fig. S6 visualizes Table 3 and shows the correction for different lengths and height differences. From Fig. S6 and Table 3, one can infer that the correction ${\delta }_{\mathrm{HD}}^{\mathrm{C}-\mathrm{S}}$ increases as the height difference and baseline lengths increase. However, for specific baseline length, larger height differences amplify the effect of the curvature-skewness on the slope distance reduction. Hence, the curvature-skewness correction should be considered for the measured vertical angles to covert the slope to horizontal distances correctly (for large height differences), even if one measures the vertical angles reciprocally. Therefore, the obtained results show that both physical and geometric effects (i.e., the DOV and curvature-skewness problem) will be eliminated by designing the geodetic control points so that the stations are all nearly at the same heights. However, it will generate two further problems. First, as we mentioned already, the refraction error will increase by designing the network at the same elevation. The vertical angles will be very sensitive to atmospheric refraction for vertical angles equal to 90° (Vanicek and Krakiwsky 1986, p. 366). Therefore, there is a recommendation to observe the vertical angles under stable atmosphere conditions: at night or during the winter when there is a thermal inversion. Second, the normal matrix in a three-dimensional (3D) multilateration network becomes ill conditioned if the stations are all nearly at the same heights, which is possible for a geodetic control network. The ill-conditioned problem for the 3D multilateration network was emphasized by Meyer and Elaksher (2021). Therefore, both problems due to designing the geodetic network at the same elevation highlight the study done by Shirazian et al. (2021). Their proposed method is independent of vertical angles for reducing the slope to horizontal distances.

Furthermore, we compared the curvature-skewness error (using an ellipsoidal model) with the curvature error (spherical model) in this study. Similar results were obtained based on using the spherical assumption [Eq. (8e)]. For example, the geometric errors on the slope distance reduction were 38.8 (ellipsoidal model) and 39.5 mm (spherical model), assuming a 5-km baseline length and 100-m height difference (the difference between the two models was 0.7 mm). We presented both spherical and ellipsoidal models in this study. These can be useful, especially for applying rigorous formulas and avoiding using the spherical approximation. The results show that the spherical assumption is sufficient in all practical cases.

Remark: As can be seen in the results, the geometric effect in the slope distance reduction increases with baseline length (see Fig. S6 and Table 3), whereas the DOV effect does not change so much (Fig. 11). In other words, the forward and backward DOV effects on the horizontal distances (Fig. 11) are approximately equal, but there is a curvature-skewness error on normals. This means that the verticals (perpendicular lines to the geoid) must also be skewed (to keep an equal DOV effect in forward and backward stations), and their angles are close to the one on the reference ellipsoid, that is, the curvature-skewness angle. This issue will be important when one establishes a local astronomy coordinate system. Some nonexperts believe that considering a local coordinate system can overcome physical and geometric errors. This assumption is incorrect because the plumb lines at adjacent points are not parallel. To verify this issue, we conducted a test on real data, and the results and description are presented in the next section.

Quite like normals, verticals (to the geoid) might not coincide at a point and then be skewed. The angles between the verticals must be accounted for when using vertical angles to reduce the slope distances to horizontal ones. One needs to have an accurate geoid model to compute the vertical curvature-skewness angle and its corresponding correction on the vertical angles to attain accurate horizontal distances from slope distances. To show this, we selected four areas in Sweden (Kebnekaise, Umeå, Skövde, and Mårtsbo) and computed the geoid heights ($N$) for $0.01°\times 0.02°$ grids ($\mathrm{\Delta }\phi =0.01°$, $\mathrm{\Delta }\lambda =0.02°$) from the SWEN17 model at each grid point in each area. The reader can find the data of the test areas in the Supplemental Materials (see Fig. S8 and Table S10). Then, we fitted a surface to the 3D grid in each area. According to our results, the best-fitting surface was a triaxial ellipsoid for all areas. There were 35 grid points in each area, of which 30 points were used for surface fitting, and the rest were used to validate the fitted surface. A geoid height at a GNSS levelling point (at which both geodetic height $h=211.9217\mathrm{m}$ and orthometric height $H=180.2623\mathrm{m}$ were available) in the Skövde area ($\phi =57.97749308°$, $\lambda =14.43111579°$) was also used for this validation. The maximum discrepancy between the fitted surfaces and the validation points was about $2\times {10}^{-8}$ [i.e., the misclosure error of Eq. (9)], which shows the goodness of fit. In Table 4, the fitted ellipsoid specifications are listed.

As known from geometry, the equation of a triaxial ellipsoid readsand the normal line to the ellipsoid at an arbitrary point $P(x_0,y_0,z_0)$ can be found from

It is well known in the literature that the normal line to the geoid is called a vertical. Then, we call the curvature-skewness angles of these lines the vertical skewness. This angle between the verticals at Points $P_1$ and $P_2$ then reads

In Table 5, the curvature-skewness angles between some points in Kebnekaise are listed (for the rest of the areas, please see the Supplemental Materials, Tables S11–S13).

It is clear from Table 5 that it is not possible to ignore the effect of the curvature-skewness angle (even if the vertical angles are observed reciprocally). This means that the plumb lines at adjacent points are not parallel, even in small-scale geodetic networks. Therefore, one should always be aware of not relying upon the vertical axis of a theodolite to establish a valid up-component of the coordinate system everywhere in the network. This issue is more complex because a local geoid takes sharp changes. Therefore, vertical angle observation is best avoided, and methods like that proposed by Shirazian et al. (2021) should be used as the observation and planning strategy.