Study of Temperature Field Distribution in Topographic Bias Tunnel Based on Monitoring Data

In recent decades, numerous tunnels have been built in the cold region of China. However, the temperature field of topographically biased tunnels in the monsoon freeze zone has not been sufficiently studied. In this study, we monitored the temperature of the surrounding rock in two topographic bias sections of the Huitougou Tunnel and analyzed the results by fitting them to the monitoring results. The results showed that the temperature of the surrounding rock on both sides after tunnel excavation varied periodically in an approximate triangular function. As the distance from the cave wall increased, the annual average temperature of the surrounding rock did not change significantly, the amplitude decreased, and the delay time increased, while the annual maximum temperature decreased, and the annual minimum temperature increased. The heat generated by blasting, the heat of hydration of the primary and secondary lining, and the decorated concrete all caused a significant increase in the temperature of the surrounding rock within 4 m for a short period of time. Both construction and topographic factors led to asymmetry in the distribution of the surrounding rock temperature in different ways. The results of this paper are intended as a reference for other studies on temperature in deviated tunnels.

The topographic bias in gravimetric geoid determination revisited

The topographic potential bias at geoid level is the error of the analytically continued geopotential from or above the Earth’s surface to the geoid. We show that the topographic potential can be expressed as the sum of two Bouguer shell components, where the density distribution of one is spherical symmetric and the other is harmonic at any point along the normal to a sphere through the computation point. As a harmonic potential does not affect the bias, the resulting topographic bias is that of the first component, i.e. the spherical symmetric Bouguer shell. This implies that the so-called terrain potential is not likely to contribute significantly to the bias. We present three examples of the geoid bias for different topographic density distributions.

A numerical test of the topographic bias

In 1962 A. Bjerhammar introduced the method of analytical continuation in physical geodesy, implying that surface gravity anomalies are downward continued into the topographic masses down to an internal sphere (the Bjerhammar sphere). The method also includes analytical upward continuation of the potential to the surface of the Earth to obtain the quasigeoid. One can show that also the common remove-compute-restore technique for geoid determination includes an analytical continuation as long as the complete density distribution of the topography is not known. The analytical continuation implies that the downward continued gravity anomaly and/or potential are/is in error by the so-called topographic bias, which was postulated by a simple formula of L E Sjöberg in 2007. Here we will numerically test the postulated formula by comparing it with the bias obtained by analytical downward continuation of the external potential of a homogeneous ellipsoid to an inner sphere. The result shows that the postulated formula holds: At the equator of the ellipsoid, where the external potential is downward continued 21 km, the computed and postulated topographic biases agree to less than a millimetre (when the potential is scaled to the unit of metre).

On the topographic bias and density distribution in modelling the geoid and orthometric heights

It is well known that the success in precise determinations of the gravimetric geoid height (N) and the orthometric height (H) rely on the knowledge of the topographic mass distribution. We show that the residual topographic bias due to an imprecise information on the topographic density is practically the same for N and H, but with opposite signs. This result is demonstrated both for the Helmert orthometric height and for a more precise orthometric height derived by analytical continuation of the external geopotential to the geoid. This result leads to the conclusion that precise gravimetric geoid heights cannot be validated by GNSS-levelling geoid heights in mountainous regions for the errors caused by the incorrect modelling of the topographic mass distribution, because this uncertainty is hidden in the difference between the two geoid estimators.

The topographic bias in Stokes’ formula vs. the error of analytical continuation by an Earth Gravitational Model - are they the same?

Geoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth GravitationalModel (EGM) inevitably leads to a topographic bias, as the true disturbing potential at the geoid is not harmonic in contrast to its estimates. We show that this bias differs for the geoid heights represented by Stokes’ formula, an EGMand for the modified Stokes formula. The differences are due to the fact that the EGM suffers from truncation and divergence errors in addition to the topographic bias in Stokes’ original formula.

A Numerical Study of the Analytical Downward Continuation Error in Geoid Computation by EGM08

A Numerical Study of the Analytical Downward Continuation Error in Geoid Computation by EGM08Today the geoid can be conveniently determined by a set of high-degree spherical harmonics, such as EGM08 with a resolution of about 5'. However, such a series will be biased when applied to the continental geoid inside the topographic masses. This error we call the analytical downward continuation (DWC) error, which is closely related with the so-called topographic potential bias. However, while the former error is the result of both analytical continuation of the potential inside the topographic masses and truncation of a series, the latter is only the effect of analytical continuation.This study compares the two errors for EGM08, complete to degree 2160. The result shows that the topographic bias ranges from 0 at sea level to 5.15 m in the Himalayas region, while the DWC error ranges from -0.08 m in the Pacific to 5.30 m in the Himalayas. The zero-degree effects of the two are the same (5.3 cm), while the rms of the first degree errors are both 0.3 cm. For higher degrees the power of the topographic bias is slightly larger than that for the DWC error, and the corresponding global rms values reaches 25.6 and 25.3 cm, respectively, at nmax=2160. The largest difference (20.5 cm) was found in the Himalayas. In most cases the DWC error agrees fairly well with the topographic bias, but there is a significant difference in high mountains. The global rms difference of the two errors clearly indicates that the two series diverge, a problem most likely related with the DWC error.

Quality Estimates in Geoid Computation by EGM08

Quality Estimates in Geoid Computation by EGM08The high-degree Earth Gravitational Model EGM08 allows for geoid determination with a resolution of the order of 5'. Using this model for estimating the quasigeoid height, we estimate the global root mean square (rms) commission error to 5 and 11 cm, based on the assumptions that terrestrial gravity contributes to the model with an rms standard error of 5 mGal and correlation length 0:01° and 0:1°, respectively. The omission error is estimated to—0:7Δg [mm], where Δg is the regional mean gravity anomaly in units of mGal.In case of geoid determination by EGM08, the topographic bias must also be considered. This is because the Earth's gravitational potential, in contrast to its spherical harmonic representation by EGM08, is not a harmonic function at the geoid inside the topography. If a correction is applied for the bias, the main uncertainty that remains is that from the uncertainty in the topographic density, which will still contribute to the overall geoid error.