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On the topographic bias by harmonic continuation of the geopotential for a spherical sea-level approximation

  • Lars E. Sjöberg EMAIL logo
Published/Copyright: November 20, 2024
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Journal of Geodetic Science
From the journal Journal of Geodetic Science Volume 14 Issue 1

Abstract

Topography is a problem in geoid determination by the Stokes formula, a high degree Earth Gravitational Model (EGM), or for a combination thereof. Herein, we consider this problem in analytical/harmonic downward continuation of the external potential at point P to a geocentric spherical sea level approximation in geoid determination as well as to a sphere through the footpoint at the topography of the normal through P. Decomposing the topographic bias into a Bouguer shell component and a terrain component, we derive these components. It is shown that there is no terrain bias outside a spherical dome of base radius equal to the height H P of P above the sphere, and the height of the dome is about 0.4 × H P . In the case of dealing with an EGM, utilizing Molodensky truncation coefficients is one way to cope with the bias.

Keywords: analytical continuation; downward continuation; harmonic continuation; (quasi)geoid determination; topographic bias

1 Introduction

“Analytical continuation… means extending the domain, on which the function is defined, by the use of Taylor series” and “analytical continuation is frequently called harmonic continuation” (Hofmann-Wellenhof and Moritz 2015, Sect 8.6.4, with a reference to Kellog 1929, Chap. X). In physical geodesy, harmonic continuation means the external potential is mathematically assumed to be harmonic also inside topographic masses, which implies that its downward continuation (DWC) can be achieved by a Taylor series applied at the external computation point. Bjerhammar (1962, 1963) introduced the use of analytical continuation in physical geodesy. The use of Taylor series for the DWC of the external potential has been used for a long time in the KTH method of geoid determination. Then, the DWC into topographic masses implies a biased potential called topographic bias (TB, Sjöberg 2007). This method is called Least Squares Modification of Stokes formula by Additive corrections (LSMSA; e.g., Sjöberg and Bagherbandi 2017, Sect. 6.2.2), where the harmonic DWC is realized by using a Taylor series expansion ( T P ⁎ ) of the disturbing potential T P at the computation point P located at height H P to estimate the potential T g at the sea level along the normal to the reference ellipsoid through P:

(1) T P ⁎ = ∑ k = 0 ∞ ( − H P ) k k ! ∂ k T P ∂ H P k .

Then, the TB becomes

(2) TB ( T P ⁎ ) = T P ⁎ − T g = ( V P T ) ⁎ − V g T ,

where V P T and V g T are the topographic potentials at P and sea level, respectively.

It is convenient to decompose the TB into its components for the Bouguer shell or plate (TB B with thickness being equal to the height of the computation point) and the remaining topography, i.e., the terrain ( TB t ) :

(3) TB = TB B + TB t .

If the computation point P is located on or above the topography with a constant density ρ (multiplied by the gravitational constant G to the constant μ = G ρ ), the TB of the Bouguer shell and plate becomes (e.g., Sjöberg 2007):

(4a) TB B = 2 π μ H 2 + 2 3 H 3 R

and

(4b) TB B = 2 π μ H 2 ,

respectively, where H is the height of the Bouguer shell/plate and R is the sea level radius.

If the computation point P is located at the topographic surface, the decomposition as above is possible only if a region around P of lateral radius s ε > 0 has constant height, implying that the terrain is located outside this region. In practice, this assumption is most realistic in numerical geoid determination with topographic data available in finite compartments or blocks.

Sjöberg (2009) suggested that the terrain correction is not needed for topographic masses in the far zone beyond a lateral distance equal to the height of computation point ( H P ) , implying that the TB equals that of the Bouguer plate for a flat sea-level approximation.

Sjöberg (2023) and Sjöberg and Abrehdary (2023) demonstrated that the only region that may cause a terrain bias is in a dome with its base at the sea level and vertex at height H P /2 below point P. More precisely, there is no contribution to the TB from the region outside the part of a ball described by the circle

(5a) s 2 + ( H P + h ) 2 2 H P 2 = 1 ,

which rotates around the horizontal s-axis and is limited by 0 < s ≤ H P and height h limited by

(5b) 0 ≤ h ≤ ( 2 − 1 ) H P .

For more explanations, see Sjöberg and Abrehdary (2023, Sect. 4 with Figure 1) and Sjöberg (2023).

The main conclusion so far is that the terrain does not provide a potential bias, except possibly for masses located inside the above dome. We are not aware of other studies on the TB by analytical continuation. There is one related study by Wang (2023), who tested the bias for a cylinder and a cone, whose results partly disagree with ours. However, as his DWC technique is not analytical/harmonic, it deserves no further mention in this article.

In what follows, we will assume an arbitrary density distribution of the topography as well as a spherical approximation of the sea level in studying the TB in analytical continuation of the disturbing potential down to the geoid (Section 2) as well as to the topographic surface (e.g., for quasigeoid determination; Section 3). Moreover, in Section 4, we shortly discuss the possible TB in using an external type spherical harmonic series below the bounding sphere of the Earth, and, finally, Section 5 concludes the article.

2 TB at the sea level

For a spherical sea level and an arbitrary topographic density distribution, one may define the Bouguer shell potential as that of a radial symmetric density μ ( r ) , defined by the density along the vertical through the external computation point P, i.e.,

(6) V P B = ∫ R r Q μ ( r ) ∬ σ r 2 d r d σ r P 2 + r 2 − 2 r P r t = 4 π 1 r P ∫ R r Q μ ( r ) r 2 d r , if r P ≥ r ∫ R r Q μ ( r ) d r , if r P ≤ R ,

where Q denotes the surface point along the vertical through P, R is the sea-level radius and t = cos ψ P Q , with ψ P Q being the geocentric angle between P and Q. Hence, the TB of the Bouguer shell potential becomes

(7) TB B = ( V B ) ⁎ − V g B = 4 π ∫ R r s μ ( r ) r 2 R − r d r ,

where ( V B ) ⁎ and V g B denote the Bouguer shell potentials analytically downward continued from P to radius R and the actual potential at the sea level (geoid), respectively.

The remaining topographic potentials, the terrain potentials V P t and V g t at point P and geoid level, become:

(8a) V P t = ∬ σ 1 ∫ R r Q ν d r d σ l P

and

(8b) V g t = ∬ σ 1 ∫ R r Q ν d r d σ l 0 ,

respectively, where ν = μ − μ ( r ) is the terrain density (i.e., the difference between the actual density of the topography and that of the Bouguer shell) and

(8c) l P = r P 2 + r Q 2 − 2 r P r Q t and l 0 = R 2 + r Q 2 − 2 R r Q t .

Here, t = cos ψ , i.e., the cosine of the geocentric angle between points P and Q, ( r P , r Q , R ) are the geocentric radii of points (P, Q, P’), where P’ denotes the footpoint at the sea level of the normal through P. Finally, σ 1 = σ − σ 0 , where σ 0 is a finite cap around P ' (i.e., we assume ν = 0 inside σ 0 ).

If we denote the inverse distance between the computation point P and the integration point Q by A and the corresponding distance from the sea level to the integration point by B, one can relate A and B by the equation

(9a) A = 1 / l P = B 1 + q ,

where

(9b) B = 1 / l 0 and q = ( l P 2 − l 0 2 ) / l o 2 .

Moreover, if ∣ q ∣ < 1 , the square-root in equation (9a) can be expanded in a Taylor series, whose first term is B, and all other terms are polynomials in H P that vanish in the analytical DWC to the sea level by equation (1), implying that A ⁎ = B , i.e., there is no terrain bias ( TB t ) . On the other hand, if ∣ q ∣ ≥ 1 , there may be a bias.

The condition for ∣ q ∣ < 1 is met in two cases, which follow from equation (9b):

Case 1

If l P < l 0 , the condition always holds.

Case 2

If l P ≥ l 0 , the condition holds whenever l P ≤ 2 l 0 .

Next, we will explore which parts of the topography that are safely free from a T B t .

2.1 TB t exterior to the Bouguer shell

In Case 1, above the terrain is located exterior to the Bouguer shell through point P, and as ∣ q ∣ < 1 it follows that there is no TB t caused by this terrain. However, note that l P = l 0 is generally not included, but may cause a bias.

2.2 TB t inside the Bouguer shell

From Case 2 with l 0 < l P ≤ 2 l 0 , it follows from equation (8c) that there is no TB t if

(10) t < 2 R 2 + r Q 2 − r P 2 2 r Q ( R − H P ) .

Inserting r Q = R + H Q and r P = R + H P into equation (10) and introducing Δ H = H P − H Q , one obtains

(11) t < R 2 − R Δ H − ( H P 2 − H Q 2 ) / 2 R 2 − R Δ H − H P H Q = 1 + H P H Q − ( H P 2 − H Q 2 ) / 2 R 2 − R Δ H − H P H ,

or as t ≈ 1 − ψ 0 2 / 2 , where ψ 0 is the geocentric angle between P and Q, one obtains approximately to the order O ( 1 / R ) :

(12a) b 2 = R 2 ψ 0 2 > 2 H P 2 − ( H Q + H P ) 2

or

(12b) b 2 + ( H Q + H P ) 2 > 2 H P 2 ,

where b is the arclength along the sea level. That is, the TB t vanishes when condition (12b) is satisfied. ψ 0 is the limiting (minimum) geocentric angle for the unbiased DWC from point P to the sea level for terrain masses.

Changing the inequality (12b) to an exact equation:

(13) b 2 + ( H Q + H P ) 2 = 2 H P 2 .

One obtains the equation of a dome with its vertex at height ( 2 − 1 ) H P along the vertical through the computation point P and a base formed as a spherical cap of radius H P at the sea level. If the location of the terrain (at point Q) is outside the dome, there is no contribution to the TB, while inside the dome there may be a TB t , but only if the density deviates from that of the Bouguer plate. This result is in complete agreement with the result for a planar sea-level approximation, as shown by Sjöberg (2023) as well as by Sjöberg and Abrehdary (2023).

It also follows by setting H Q = 0 in equations (12a) and (13) that there will be no terrain bias outside a spherical cap with radius R ψ 0 and

(14) ψ 0 = H P / R .

3 TB of the DWC of the external potential to the topographic surface

We now consider the possible TB for the DWC of the potential from a point P located at the bounding sphere of the Earth down to a point P′ at the Earth’s surface (located at height H P ′ < H P ). In this case, there is no Bouguer shell bias ( TB B ) , as this is a free-air reduction. Also, analogous to equation (12a) and the discussion in the previous section, there will be no TB t for any terrain mass at point Q located outside the spherical dome described by the equation

(15a) b P ′ + ( H Q + Δ H P P ′ ) 2 = 2 Δ H P P ′ 2 ,

where

(15b) b P ′ ≈ ( R + H P ′ ) ψ 0 and Δ H P P ′ = H P − H P ′ .

This dome is located on the sphere of radius R + H P ′ , and its vertex is located at height ( 2 − 1 ) Δ H P P ′ above the sphere.

In agreement with the discussion for the DWC to the geoid, any terrain masses outside a cap with geocentric angle

(16) ψ 0 = ( H P − H P ′ ) / ( R + H P ′ )

will not generate a bias.

4 TB of a harmonic series of the exterior potential

  1. Frequently, the gravitational potential of the external type is expressed in a series of spherical harmonics that is analytically downward continued from, say, radius r P ≥ R e = radius of the bounding sphere of the topography, to radius R < R e by simply replacing radius r P by R for each degree. This is the result of the DWC from equation (1). Formally, the series may diverge when the number of terms in the series approaches infinity, and even if it converges already for a large number of terms, the result of the DWC may be poor due to increasing errors in higher degree harmonics. Here, we present two ways to reduce this error:

Method 1: Smooth each spherical harmonic A n m by c n A n m / ( c n + d c n ) , where c n and d c n are the degree variance and error degree variance of A n m , respectively. By truncating the series when the error degree variance reaches 100%, the mean square error becomes a minimum (Sjöberg 1986, Sjöberg and Bagherbandi 2017, Sect. 7.5.2.).

Method 2: Apply equation (1) at height P only for a limited number of terms. This is the “gradient method” practiced by Rapp (1997) to the first derivative and later by Hirt (2012) successfully to high degrees. This method could possibly be combined with the smoothing presented in Method 1.

Both proposed methods should suffer from the TBs discussed above. However, note that when the potential is downward continued to the topographic surface, there will be no TB B but possibly a TB t .

4.1 On the convergence of the series

If one excludes the potential generated by the topography in a sufficiently large cap below the computation point P (located at the bounding sphere), one can expect that the harmonic series will be convergent down to some internal sphere located at the sea level or topography below P (say, of approximate radii R and R + H Q, respectively).

Bucha and Kuhn (2020) proposed a method to obtain such a series by excluding the topographic contribution from a spherical cap around the computation point limited by the approximate spherical angle ψ = H int / R int , where H int and R int are the maximum topographic height above the sphere within the cap and its radius, respectively. Then, the modified harmonic series that excludes the contribution from the cap is simply obtained by multiplying each harmonic by its Molodensky truncation coefficient with argument ψ . However, the above angle ψ was adopted from Martinec (1998, p. 59, Equation 10), which dealt with the problem of DWC of topographic surface gravimetric data in a remove–restore technique by the Stokes–Helmert method. In the present application, the DWC is performed from the computation point P located at the bounding sphere, so that the cap size becomes ψ 0 ≈ H P / R and ψ 0 ≈ Δ H P P ′ / ( R + H P ′ ) ≈ Δ H P P ′ / R for DWC to the geoid and topography, respectively. Only this result is consistent with the results in Sections 2 and 3.

5 Conclusions and final remarks

We study the TB for analytical/harmonic DWC of the external (disturbing) potential to (a) a spherically shaped sea level and (b) to the topographic surface. We decompose the bias into that of the Bouguer shell (denoted TB B ) and the terrain bias (denoted ( TB t ) . In Case 1, the TB B is given by equations (4a) and (6) for a constant and radially symmetric density distribution, respectively, while in Case 2 there is no TB B .

From a practical point of view, we think it is reasonable to assume that there is a finite cap σ 0 > 0 around the computation point, where there is no terrain. Then, there is no terrain bias outside a spherical dome, as described at the end of Section 2. In Case 2, the corresponding dome is described at the end of Section 3.

In the case of using an Earth Gravitational Model with a finite set of spherical harmonics to determine the geoid or quasi-geoid, we believe that “the gradient method” is practical, which uses a truncated Taylor series applied at the bounding sphere. The series may be smoothed as discussed in Section 4, but this series may also suffer from the TB in the described spherical dome, which effect can be excluded by modifying the series with Molodensky’s truncation coefficients. However, in that case, the contribution to the (quasi)geoid from the topography inside the dome must be solved in a separate way (not discussed here).

  1. Funding information: Author states no funding involved.

  2. Author contribution: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.

  3. Conflict of interest: Author states no conflict of interest.

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Received: 2024-04-26
Revised: 2024-09-24
Accepted: 2024-09-27
Published Online: 2024-11-20

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jogs-2022-0180
Keywords for this article
analytical continuation; downward continuation; harmonic continuation; (quasi)geoid determination; topographic bias
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BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Displacement analysis of the October 30, 2020 (Mw = 6.9), Samos (Aegean Sea) earthquake
  3. Effect of satellite availability and time delay of corrections on position accuracy of differential NavIC
  4. Estimating the slip rate in the North Tabriz Fault using focal mechanism data and GPS velocity field
  5. On initial data in adjustments of the geometric levelling networks (on the mean of paired observations)
  6. Simulating VLBI observations to BeiDou and Galileo satellites in L-band for frame ties
  7. GNSS-IR soil moisture estimation using deep learning with Bayesian optimization for hyperparameter tuning
  8. Characterization of the precision of PPP solutions as a function of latitude and session length
  9. Possible impact of construction activities around a permanent GNSS station – A time series analysis
  10. Integrating lidar technology in artisanal and small-scale mining: A comparative study of iPad Pro LiDAR sensor and traditional surveying methods in Ecuador’s artisanal gold mine
  11. On the topographic bias by harmonic continuation of the geopotential for a spherical sea-level approximation
  12. Lever arm measurement precision and its impact on exterior orientation parameters in GNSS/IMU integration
  13. Book Review
  14. Willi Freeden, M. Zuhair Nashed: Recovery methodologies: Regularization and sampling
  15. Short Notes
  16. The exact implementation of a spherical harmonic model for gravimetric quantities
  17. Special Issue: Nordic Geodetic Commission – NKG 2022 - Part II
  18. A field test of compact active transponders for InSAR geodesy
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  21. Geodetic innovation in Chilean mining: The evolution from static to kinematic reference frame in seismic zones
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