Hotine’s modified kernel for normal height determination

Abstract:

For gravity field modelling, gravity disturbance inputs play a role once they are effortlessly determined through coordinates derived from the Global Navigation Satellite System (GNSS) and gravity acceleration measurements. With this anomalous quantity, the fixed GBVP can be directly solved using Hotine’s function. This paper intends to present the Hotine-modified kernel using Vaníček and Kleuberg’s (1987) approach and the software developed. This methodology uses non-gridded residual gravity disturbances as input. Normal height values were determined for the evaluation process, first using Hotine’s function and second recovering the gravity potential from a geoid model determined by Least Squares Collocation (LSC). The same data was used for both approaches, and the solutions were compared in eight Brazilian stations, including one IHRS. Hotine’s solutions demonstrated consistent convergence in the order of -9 cm (MGIN) and 14 cm (MGUB), RBMC stations, using the LSC method. When evaluating benchmarks, the Hotine method shows a difference of 12 cm at PPTE, 8 cm at MGUB, and 14 cm at MGIN.

Keywords:
Kernel; Hotine; Numerical Integration, Height Systems

1. Introduction

With the implementation of the International Height Reference System/Frame (IHRS/F) (IAG, 2015), the importance of regional gravity field modeling was evident in providing high-resolution solutions. In this sense, the Geodetic Boundary Value Problem (GBVP), a requirement to define anomalous quantities, can be formulated in distinct ways according to the available input and the interested variables. Usually, the most used GBVP is solving an integral formulation in the scalar-free or fixed approach, introducing the Stokes or Hotine functions. Conversely, Least Squares Collocation (LSC) has been offered advantages by combining the distinct variables.

Between the GBVP methodologies, some parameters in these routines are subject to change and adapt to the study region, such as the kernel, gridding interpolation inputs, and adjustment of the covariance function. An important aspect of these methodologies is the kernel modification not only to guarantee a property spectral transferred for the residual anomalous quantities but also due to its advantages in high pass filtering properties (Featherstone 2003). Also, kernel modification is done in alignment with the degree and order of GGM used in the remotion of the Long Wavelength Component (LWC) on the gravity anomaly or the gravity disturbance. Some characteristics may be evaluated for a reliable modification, such as function convergence, the integration cap size, and the expansion coefficient of the series of spherical harmonic functions used.

Among the strategies for kernel modification, Vaníček and Kleusberg’s (1987) methodology has presented satisfactory results in the Stokes kernel. On the other hand, some adaptations of the referred modification in the Hotine kernel are presented by Sjöberg and Eshagh (2009) and Novák (2003) without the inclusion of the Global Geopotential Model (GGM). Featherstone (2013), in its hybrid Hotine’s kernel modification, used Vaníček and Kleusberg’s (1987) approach using GGM truncated at degree 50. This segment, recently, was also the subject of study by Sakil et al. (2021) for quasi-geoid modeling.

In fact, towards the standardization procedures to realize the IHRS, the challenge remains to distinguish the dependency of the result on the methodology and the input data (Sánchez et al. 2021). In terms of strategy, a well-adapted kernel can provide reliable solutions. On the other hand, gravity disturbances, with their geodetic coordinates (latitude, longitude, and height), allow the fixed GBVP to be solved directly. About this input format, it is frequently given at discrete locations or average blocks. The latter uses conventional techniques such as polynomial, trigonometric interpolation, and spline interpolation, which offer certain advantages. In contrast, in locations where gravity data is well-distributed, using raw point data directly in computations is beneficial, as it avoids potential signal degradation or data alteration inherent in the gridding procedure (Sakil et al. 2021).

Since the IHRS/F study group seeks to distinguish the contributions of methodology and database in the solution, comparing normal heights derived from a well-adapted kernel based on the Hotine function with those obtained from gravity potential recovery using an LSC geoid model, it may contribute to the ongoing IHRS/F research, especially when both strategies were computed with the same input data.

This paper presents a numerical integration approach using the modified kernel of Hotine, adapted with the Vaníček and Kleusberg strategy. As input, non-gridded gravity disturbances residual data were used for the Hotine methodology to determine the disturbing potential. The method has been validated by a comparative analysis of normal height in eight stations in Brazil. Subsequently, the input data was selected at a 200 km radius around the station. A 5x5-minutes grid with the same database was used to compute the LSC geoid model for southeast Brazil. After that, the results of the Hotine strategy were evaluated with normal height values computed through gravity potential recovering from a recent geoid model, determined by Least Squares Collocation (LSC) (Bottoni and Barzaghi 1993).

The paper is structured as follows: Section 2 presents the introduction, reviewing the theoretical concepts of Hotine’s kernel modification. Section 3 describes the characteristics of the study area and the methodology for Hotine’s Numerical Integration. Section 4 outlines the methods for gravity potential recovery. Section 5 evaluates the convergence of the kernel modification. The results are presented and discussed in Section 6, followed by final considerations and recommendations for future studies in Section 7.

2. Theoretical Aspects of the Modified Hotine Kernel

The Hotine formula, presented by the British geodesist and surveyor, is given by (1) to determine the disturbing potential. After applying Bruns’ formula to the expression (Exp) for geoid undulation (Hotine 1969, Exp. 29.53), also known as Hotine-Koch (Hofmann-Wellenhof and Moritz 2006, Exp. 2-367), we obtain:

$T=\frac{R}{4\pi }\underset{\sigma }{\iint }H\left(\psi \right)\delta gsin\partial \sigma$ (1)

where, R is the radius based on the normal ellipsoid, is the gravity disturbance, and the Hotine’s kernel H (ψ) is given by (Hotine 1969, Exp. 29.17):

$H\left(\psi \right)=\csc \left(\frac{\psi }{2}\right)-\mathrm{}\mathrm{l}\mathrm{n}\mathrm{}\left(1+\mathrm{c}\mathrm{s}\mathrm{c}\left(\frac{\psi }{2}\right)\right)$ (2)

It can be expressed by the Legendre Polynomial series (Hotine, 1969, Exp. 29.17):

$H\left(\psi \right)=\sum _{n=0}^{\infty }\frac{2n+1}{n+1}{P}_n\left(cos\psi \right)$ (3)

With P _n (cosψ) being the Legendre polynomial of degree n.

When the GGM is considered to represent the gravitational field with the LWC for gravity disturbances (δg _n ), the kernel is modified based on the degree (n) and order (m) used. Thus, δg _n is given by the expansion of spherical harmonic function, starting from degree and order n=m=2 up to the maximum degree and order used (l). Besides, the remove-compute-restore (RCR) principle is applied to terrestrial gravity disturbances (δg ^terr ) removing the LWC, resulting in its residuals (δg ^res ) related to a geodetic coordinate latitude, longitude and height (α, ψ, h) or with its position in terms of the azimuth, spherical distance, and height (φ, λ, h). The medium and short-wavelength component of digital terrain elevation information (δg ^topo ) can be used as well, yielding in the following:

$\delta g^{res}\left(\alpha ,\psi ,h\right)=\delta g^{terr}\left(\alpha ,\psi ,h\right)-\sum _{n=2}^l\delta g_n\left(\alpha ,\psi ,h\right)-\delta g^{topo}\left(\alpha ,\psi ,h\right)=\sum _{n=n+1}^{\infty }\delta g_n\left(\alpha ,\psi ,h\right)$ (4)

The integration interval will be restricted to a spherical cap radius ψ ₀ according to the GGM degree and order selected. Thus, the Hotine expression with the modified kernel ($\delta H_l^{mod}$) in order to get the residual disturbing potential will be:

${T(\varphi ,\lambda ,h)}^{res}=\frac{R}{4\pi }\underset{\psi =0}{\overset{{\psi }_0}{\int }}\underset{\alpha =0}{\overset{360}{\int }}{\delta g}^{res}(\alpha ,\psi ,h){\delta H}_n^{mod}sin\left(\psi \right)\partial \psi \partial \alpha$ (5)

here, the geodetic latitude, longitude, and height (φ, λ, h) refers to the station of interest, where the disturbing potential will be determined, while the positions of the residual gravity disturbances are expressed in terms of azimuth, spherical distance, and height (α, ψ, h). The $\delta H_n^{mod}\left(\right)$ is the modified kernel, obtained by (6), subtracting $\delta H_n^{*}\left(\right)$ from δH _n (ψ):

$\delta H_n^{mod}\left(\psi \right)=\delta H_n\left(\psi \right)-\delta H_n^{*}\left(\psi \right)$ (6)

with δH _n (ψ) being the original kernel H(ψ), Exp (3), reduced from the kernel related to the degree and order of GGM, H _n (ψ):

$\delta H_n\left(\psi \right)=H\left(\psi \right)-H_n\left(\psi \right)$ (7)

with H _n (ψ) representation in terms of Legendre polynomials:

$H_n\left(\psi \right)=\sum _{n=2}^l\frac{2n+1}{n+1}{P}_n\left(\psi \right)$ (8)

In exp (6), $\delta H_n^{*}\left(\right)$ is linked to the coefficients t _n related to the rate of the kernel concerning the spectral series. This procedure is known as the minimizing L2 norm, and it is used by Vaníček and Kleusberg (1987) in a spheroidal kernel for the Stokes function (Featherstone 2003), adapted now for Hotine kernel. The equation can be represented by:

${\delta H}_n^{*}\left(\psi \right)=\sum _{n=0}^l\frac{2n+1}{2}{t_{n}P}_{n}cos\left(\psi \right)$ (9)

The determination of t _n , with the Legendre polynomials of degree n and order m (0 ≤ n ≤ l and 0 ≤ m ≤ l), is given by a system of linear equations using the least square method (Vaníček and Kleusberg 1987, pp. 91), with the relation:

$\sum _{m=0}^l\frac{\left(2m+1\right)}{2}{e}_{nm}\left({\psi }_0\right)t_m=Q_n^l\left({\psi }_0\right)$ (10)

here,

$e_{nm}\left({\psi }_0\right)=\underset{\psi ={\psi }_0}{\overset{\pi }{\int }}{P}_n\left(cos\psi \right)P_m\left(cos\psi \right)sin\psi \partial \psi$ (11)

and,

$Q_n^l\left({\psi }_0\right)=Q_n\left({\psi }_0\right)-\sum _{j=2}^l\frac{\left(2j+1\right)}{\left(j-1\right)}{e}_{nj}\left({\psi }_0\right)$ (12)

where j is auxiliar index (0 ≤ j ≤l), and the Molodensky truncation coefficients:

$Q_n\left({\psi }_0\right)=\underset{\psi ={\psi }_0}{\overset{\pi }{\int }}H(\psi )P_n\left(cos\psi \right)sin\psi \partial \psi$ (13)

The kernel modification performed to receive the GGM, as shown for Hotine, often focuses on significantly reducing errors in gravity field modeling and accelerating the convergence rate of error series. It also offers potentially high pass filtering properties (Vaníček and Featherson 1998). Among many forms of modification, the most known and used, generally applied in the Stokes kernel, are Wong and Gore (1969), Meissl (1971), and Vaníček and Kleusberg (1987).

3 Study Area and Methodology for Hotine’s Numerical Integration

The methodology was applied at eight stations located in different cities. In São Paulo state, the selected stations were Presidente Prudente (PPTE), São José do Rio Preto (SJRP), São Carlos (EESC), and Botucatu (SPBO), and in Minas Gerais state, Monte Claros (MGMC), Inconfidentes (MGIN), Governador Valadares (MGGV), and Uberlândia (MGUB). These locations were selected based on their spatial distribution and proximity to the Local Gravity Reference System (LGR) and the Brazilian Continuous GNSS Monitoring Network (Rede Brasileira de Monitoramento Contínuo do Sistema GNSS - RBMC) stations (Figure 1).

Recently, the stations of MGMC and MGUB have had densification gravity campaigns to improve data coverage. These efforts were the object of important projects such as the National Council for Scientific and Technological Development (Process number: CNPQ 420555/2016-1) and the Center for Geodetic Studies (CENEGEO). The used gravity database is compiled and treated by Escola Politécnica da Universidade de São Paulo (EPUSP), gathering data from different institutions, including the Brazilian Institute of Geography and Statistics (Instituto Brasileiro de Geografia e Estatística - IBGE), CENEGEO, the National Agency of Petroleum, Natural Gas, and Biofuels ( Agência Nacional do Petróleo, Gás Natural e Biocombustíveis - ANP), Petrobras, and the Mineral Resources Research Company (Companhia de Pesquisa de Recursos Minerais - CPRM).

The computation, using Hotine’s integral and the geoid recovery process, was performed using SIRGAS2000 coordinates, epoch 2000.4, for RBMC. Good-quality benchmark markers (BMs), with a standard deviation of less than 7 cm for normal height, gravity acceleration values, and GNSS coordinates, located near the stations, have been selected to carry out the computations and assess discrepancies with the High Precision Altimetry Network (Rede Altimétrica de Alta Precisão - RAAP). The BM has a normal height determined by IBGE, which uses spirit leveling referred to Imbituba datum, based on mean sea level, as a reference. These normal heights rely on the adjustment of the network in terms of geopotential numbers, adopting the gravity acceleration measurement on the BM (IBGE 2019). Unfortunately, some stations, such as MGGV and MGIN, do not have a proper BM nearby. In this way, the closest BM was selected.

Figure 1:
Study Area.

A summary of the station’s height information, including their normal height (Hn), geodetic height (h), and the number of gravimetric points used for each RBMC and RAAP station, in a radius of 200 km, is shown in Table 1.

Table 1:
Study Stations details (m).

STATION	RBMC		RAAP
	h	nº points	BM	h	Hn	nº points	RBMC dist
PPTE	431.049	5989	3139Z	431.049	436.367	5989	0.0
EESC	824.587	6402	2887J	929.261	935.696	6479	7,400.0
SJRP	535.909	5802	2988C	520.755	528.010	5805	200.0
SPBO	803.122	6709	2875N	872.222	877.809	6810	10,300.0
MGGV	180.118	2091	1169D	692.162	702.436	2363	118,600.0
MGIN	883.720	7725	2803L	821.207	824.043	7566	46,700.0
MGMC	618.164	2864	668Z	649.593	661.806	2858	2,200.0
MGUB	869.241	4822	2165V	893.214	902.924	5183	50,000.0

The gravity points were selected from the database in a radius of approximately 200 km (1.8°) from the IHRF station. The low frequencies of gravity disturbances were obtained using the XGM2019 (Zingerle et al. 2020) expanded to degree and order 100 due to the spatial resolution compatibility, calculated following Torge and Müller 2012 (Spatial Resolution = 180/ degree and order), or using a database around a radius of approximately 200 km (~ 20000 km/degree and order) (Doğanalp 2022, Exp. 18). The atmospheric correction (Tenzer et al. 2010) was applied to the database, as well as the zero-tide concept (Mäkinen 2021). Subsequently, the treatment of the permanent tide was applied using the approach of Sánchez et al. (2021).

Regarding the Hotine software, the preliminary routine for the Stokes kernel was developed at University of New Brunswick and later modified at University of São Paulo for use with Hotine’s kernel. The software computes the Molodensky truncation coefficient, Exp (13), using M. Paul’s algorithm. This strategy computes a finite fixed number of terms without approximation, using recurrence relations and the properties of Legendre polynomials (Paul 1973, pp. 421). Thereafter, the computed coefficients are inserted into the HOTINE-PTG-UP-TK.F routine to determine the residual disturbing potential, using Hotine integral, Exp (5), and the IHRS quantities.

4. Determination of Normal Height by Recovering the Gravity Potential using a Geoid Model

The LSC method (Bottoni and Barzaghi 1993), used for the geoid model, consists of a mathematical prediction technique that applies statistical information on the covariance function, considering the observation errors (Torge 2001, pp. 232). The covariance function model is employed to fit the empirical covariance function, which describes the behavior of the anomalous field, reflecting the variation magnitude and roughness from a statistical perspective. In this process, the input data was the residual gravity disturbances, and the XGM2019 provided the LWC developed up to degree and order (n and m) 100. The Hotine method was chosen for standardization.

The model data grid was determined by averaging the residual gravity disturbances at 5 minutes resolution without filling the gravimetric voids. Also, the data was corrected from the atmospheric effects (Tenzer et al., 2010) and transformed into the zero-tide concept (Mäkinen, 2021). The terrain effects were not considered, aiming to standardize the solutions.

It is worth mentioning that the geoid was computed once this surface is related to the gravity potential concept within the IHRF/IHRS framework. On the other hand, the Brazilian Geodetic System (BGS) is based on normal height (IBGE, 2019). Thus, the geoid height was converted following the geoid recovery process of Sánchez et al. (2021), and the normal height in the mean-tide concept was determined. The computation step for the normal height solution based on the geoid model to recover the gravity potential (Hn_GM2GP) is summarized in Figure 2.

Figure 2:
Normal height determination based on the geoid model to recover the gravity potential.

The provisory gravity potential W _prov was computed using the gravity potential of the Earth, the geoid undulation, and the geodetic height of the station. Since the calculus starts on the geoid surface and ends on the physical surface, the mean gravity of the station was considered to average the density of mass, along with the topographic correction (TC).

The correction ∆W ^ITRF (φ), related to the International Terrestrial Reference Frame (ITRF) coordinates, was computed for the station, and the gravity potential in the zero-tide concept was obtained. By following the IAG 2015 resolution, the quantities should be related to the mean tidal system/mean crust to include the time average of the tide-generating potential generated by masses outside the Earth. Thus, the correction zero-tide to mean-tide concept W _ZT2MT was determined according to Mäkinen (2021). Both latter corrections were also added to the Hotine method.

In the final step, the normal height $H_n^{LSC}$ is obtained dividing by a mean normal gravity acceleration $\stackrel{-}{{\gamma }_Q}$ using GRS80 ellipsoid parameters (MORITZ, 2000), where f is the flattening of the ellipsoid, m is a form factor, a is the semi-major axis, h _p is the geodetic height, and ζ_GGM is the height anomaly.

5. Evaluation of Convergence for Hotine’s Kernel Modification Function

The combination of terrestrial gravity data and GGM, using the RCR technique, is essential for consistently determining Earth’s gravity field quantities. These spectra relation requires modifying the kernel to derive a representative function around the integration point. For this purpose, the spatial resolution, corresponding to the area of data availability for the point integration, had to be compatible with the GGM degree and order. As a result, the kernel is modified according to the n=m chosen to reduce anomalies or gravity disturbances using the RCR procedure. Subsequently, the function’s convergence toward zero is observed near the chosen cap radius (ψ ₀ ), following the behavior of the selected Legendre polynomials. In summary, the faster the convergence, the lower the leakage of high-frequency data.

For the modified Hotine’s kernel analysis, the function is observed in four situations: in the case of n=m=30, compatible with a distance of 6 degrees (180°/30=6°); n=m =60, with a distance of 3°; n=m =100, with a distance of 1.8°; and, n=m =200, with a distance of 0.9°. Figure 3 illustrates the classical Wong-Gore kernel (in blue), the non-modified Hotine kernel (in orange), and the modified Hotine kernel (in green).

Figure 3:
Representation of Wong-Gore, Hotine’s non-modified and Hotine’s modified kernel (a- a distance of 6° and n=m =30; b- a distance of 3° and n=m =60; c- a distance of 1.8° and n=m =100; d- a distance of 0.9° and n=m =200.

Notably, the convergence to zero of Hotine’s modified function is compatible with ψ ₀ , opposite to the normal and Wong-Gore (W-G) functions after the truncation. Therefore, the Hotine’s curves indicate the consistency of the modified kernel with the four integration intervals. On the contrary, W-G maintains instability and allows the spectral leakage effect. The results obtained by testing the cap radius of 6° and 3° also show proper behaviors. Besides, the convergence to zero near 1.8° (n=m=100) and 0.9°, n=m=200, goes to infinity smother compared to the normal and W-G methodologies.

Truncating the cap radius to 1.8° shows agreement in terms of function convergence and could adequately represents the Earth’s gravity field behavior approximately 200 km of the station.

6. Numerical Results for Normal Heights

The normal heights computed at different stations using the modified Hotine integral with the appropriate kernel through numerical integration (Hn_Hotine) and those obtained by recovering the gravity potential from a geoid model (Hn_GM2GP) are presented in Table 2.

Table 2:
Normal height solutions at RBMC and BM (m) stations.

RBMC				RAAP
STATION	Hn_Hotine	Hn_GM2GP	h	BM	Hn IBGE	Hn_Hotine	Hn_GM2GP	h
PPTE	436.612	435.874	431.049	3139Z	436.725	436.613	435.873	431.049
EESC	830.812	830.535	824.587	2887J	936.054	935.452	935.154	929.261
SJRP	542.970	542.610	535.909	2988C	528.368	527.814	527.454	520.755
SPBO	808.754	808.247	803.122	2875N	878.167	877.713	877.208	872.222
MGGV	189.927	190.358	180.118	1169D	702.794	702.447	702.224	692.162
MGIN	887.449	886.517	883.720	2803L	824.401	824.261	824.014	821.207
MGMC	630.558	630.650	618.164	668Z	662.164	661.971	662.100	649.593
MGUB	879.569	879.433	869.241	2165V	903.282	903.202	902.483	893.214

It is worth mentioning that although the methodologies used the same database, divergences exist between them. The Hn_GM2GP solution, derived from gridded data averaged from residual gravity disturbances at a 5-minutes grid, modifies the original data. In contrast, the Hn_Hotine proposal was applied to point data computation, which is expected to more faithfully represent the gravity field. Additionally, the LSC used in the Hn_GM2GP solution is a stochastic methodology, while the Hotine integral is a deterministic approach.

Analyzing the difference between the two techniques at the RBMC stations in São Paulo, where gravity coverage is adequate, the larger differences between the Hn_Hotine and Hn_GM2GP methods are in PPTE and SPBO, 74 cm and 50 cm. In these stations, there are more gravimetric voids compared with EESC and SJRP, where the difference is 28 cm and 36 cm. In this sense, the gridding process may also affect the convergence of the solutions.

In Minas Gerais, the differences between the solutions are MGGV -43 cm, MGIN 93 cm, MGMC -10 cm, and MGUB 14 cm. Although the proper densification around the stations for reliable results is a requirement, it is observed that the difference between methodologies may be related to the abrupt change of normal height values and the presence of important gravity disturbances in the area. MGIN, as an example, despite having fewer gravimetric voids compared to MGGV, is located in the Sea Mountain Range. It reinforces the importance of a number higher than 5,000 points and homogeneous spatial distribution around the station, especially in rough topography.

The 2018 RAAP realization presented significant changes in the height values due to the transition from normal-orthometric to normal heights and the adjustment of the network. IBGE was the first to adopt a methodology aligned with the IAG resolution for the IHRS/IHRF. Prior to that, the leveling heights were corrected using the normal gravity field. Furthermore, the inconsistency in the spirit leveling line near São Paulo was removed, which resulted in a discontinuity in the normal heights (IBGE, 2018, pp. 35). The discrepancy between these realizations is around 30 cm in Minas Gerais; while in São Paulo, it can vary from 30 cm to -60 cm (IBGE, 2018, pp. 36).

The solutions Hn_Hotine and Hn_GM2GP are related to gravity potential W ₀ = 62636853.40 m²/s², which was conventionally adopted worldwide. Meanwhile, the leveling network is based on sea records, presenting a gap with respect to . In this sense, a study by Guimarães et al. (2025) computed the gravity potential for the Imbituba datum as 62,636,849.87 m²/s²with standard deviation of ± 0.224 m²/s². It indicated an offset of 0.358 m which should be added to the normal heights referred to Imbituba, when compared with the W ₀ value. Thus, the IBGE values are presented in Table 2, considering the mentioned offset.

The objective of the mentioned comparison is to assess the discrepancy between the globally computed gravity potential and physical realization of Brazilian Vertical datum. Such evaluation is crucial for the unification of the height systems between countries. Concerning adding the offset to the BM normal heights, the normal height determined by IBGE differs from the Hotine by about 11 cm, 60 cm, 55 cm, and 45 cm for PPTE, EESC, SJRP, and SPBO, respectively. In Minas Gerais, these discrepancies are 35 cm, 14 cm, 19 cm, and 8 cm in MGGV, MGIN, MGMC, and MGUB, respectively. On the other hand, the mean difference between Hn_GM2GP and BM is 68 cm. Figure 4 shows the difference at the BM, considering the offset at the datum height.

Figure 4:
Normal height difference: 4a) Hn from the BM minus Hn_GM2GP; and 4b) Hn from the BM minus Hn_Hotine.

Based on the analysis of the RBMC normal heights, the Hn_Hotine solutions are also consistent with those of Hn_GM2GP in terms of disturbing potential (Table 3). It is crucial to recognize that the gravity potential recovery methodology begins in the geoid and ends on the physical surface at a normal height. In this sense, to compare the disturbing potential (T _P ) from the Hn_GM2GP, it is necessary to relate it to the conventional gravity potential W ₀ , considering the zero-order degree term and discrepancy between the normal gravity potential.

Table 3:
Disturbing Potential (m²/s²).

Stations	Hn_Hotine	Hn_GM2GP
PPTE	-50.417	-47.462
EESC	-56.034	-57.525
SJRP	-64.969	-65.687
SPBO	-50.223	-48.674
MGGV	-92.412	-98.730
MGIN	-31.397	-27.388
MGMC	-117.075	-116.088
MGUB	-96.179	-90.815

However, in the gravity potential recovering from a geoid, the complete mass density may not be fully addressed compared to the inverse process of Bouguer reduction. The expression to determine W _P using the geoid height, the computation of the mean gravity acceleration value () is required. In this process, the terrain effects are considered by the effect of topographic irregularities concerning a Bouguer plate. For this, a free-air gradient should be regarded for an upward continuation of the quantities from the geoid to the physical surface, where the normal height is related. Additionally, the lack of knowledge of the mass density would be reflected in the upward continuation of the quantities from the geoid to the physical surface, where the normal height is related.

7. Concluding Remarks

The kernel modification determines the convergency of the function in the numerical integration process, ensuring an adequate spectral transference of the small wavelength components. It enhances the precision of any solution and refines the Earth’s gravity field representation. The presented Hotine integral modification provides advantages. Firstly, it uses the gravity disturbances as input data, which can be determined independently of gravity reductions. Secondly, the raw discrete data avoids losing important gravity field signs during the grid procedures.

This paper shows the Hotine-modified kernel using the Vaníček and Kleusberg (1987) strategy. The numerical evaluation was performed by computing the normal height in eight stations in São Paulo and Minas Gerais states. The comparison was made by recovering the gravity potential from a geoid model, determined by the LSC, using the Sanchez et al. (2021) approach.

The Hotine solutions have demonstrated good agreement with the LSC for well-densified stations, exhibiting discrepancies of merely 28 cm, 9 cm and 14 cm, as observed at the EESC, MGMC and MGUB RBMC stations. In the same analysis for the BM, Hotine method shows a mean difference of 31 cm for the eight stations, while LSC presents 68 cm in the same comparison with the BM.

Regarding the discrepancies, an offset was evident when the LSC method was compared to Hotine and IBGE. The formulation for the gravity recovery process from a geoid may have omitted an essential gradient to translate the geoid for the physical surface, where the normal height is related. As a next step, the computation could be performed using a precise Digital Terrain Model to adequately determine the terrain effect, taking into account the complete reverse Bouguer reduction with the free-air gradient.

The results presented in this paper show consistency with traditional methods, such as LSC. Additionally, it would be valuable to test the Hotine methodology in other areas, with well-known precision from the gravity and normal height database. A region with gravity densification better than 5-minutes could enhance the effectiveness of point data in the numeric integration, supported by a reliable modified kernel.

ACKNOWLEDGMENT

We thank René Forsberg and C.C. Tscherning for providing the GRAVSOFT package and Riccardo Barzaghi for the Fast Collocation routines. Also, the Association of Geodesy for the financial support for participating in the 14th International School on “The Determination and Use of the Geoid” and to all the professors involved. We also thank the Coordination for the Improvement of Higher Education Personnel - Brazil (CAPES) - Financial Code 001, for the financial support, and the National Council for Scientific and Technological Development (CNPq grant number 303239/2022-0 300113/2022-6) and the Minas Gerais Research Foundation (FAPEMIG grant number: APQ-00891-24) for the support of the research in Minas Gerais state. We would also like to thank IBGE for providing the data, as well as all the institutions involved.

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• DATA AVAILABILITY
  The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Publication Dates

• Publication in this collection
  05 Dec 2025
• Date of issue
  2025

History

• Received
  31 Mar 2025
• Accepted
  24 Oct 2025