Tidal heating of a solid planetary body occurs by viscous dissipation, depending on its internal structure, and connected to its thermal and orbital states. Earlier calculations of the response of the Moon to tidal forces had considered lunar interior structure, but had not reproduced the geodetically observed dependence of the dissipation on the lunar tidal period. The attenuation of seismic waves in the deep lunar interior is consistent with the existence of a low-viscosity layer at the core-mantle boundary, which may explain the observed frequency dependence. Here I briefly introduce my/our own studies, especially a series of studies showing constraints on the deep lunar interior based on the tidal response parameters, and mention future studies to be done by myself/ourselves. In the first work, we numerically simulated the viscoelastic tidal response of the Moon that contains the low-viscosity layer at the core-mantle boundary, and compared the model results with geodetic observations available at that time (i.e., LLR, SELENE, and Chang'e 1). In our simulations, a layer with a viscosity of about 2×10¹⁶ Pa s leads to frequency-dependent tidal dissipation that explains tidal dissipation observations at both monthly and annual periods. Compared with the lunar asthenosphere, the inferred viscosity is extremely low, and suggests partial melting at the lunar core-mantle boundary. We also found that tidal dissipation is not evenly distributed over the lunar interior, but localized within the low-viscosity layer. This implies that this layer may act as a thermal blanket on the lunar core and has influenced the lunar thermal evolution. In the subsequent work, we revisited the constraints on the deep lunar interior with a possible low-viscosity zone at the core-mantle boundary obtained from our previous forward modeling. We compared the numerical model with several tidal parameters that have been improved or are newly determined by recent geodetic observations and analyses (i.e., LLR, GRAIL, and LRO). Our results are, in principle, consistent with these data, and suggest a low-viscosity layer with its outer radius of 540 ～560 km, which possibly extends into the region where deep moonquakes occur.

In a previous study, GPS (Global Positioning System) was presented as a new instrument for detecting Earth's free oscillation in three-direction displacement. We improve a method of the data stacking and perform two types of time-frequency analyses as additional tests, using the GEONET data after the 2011 Tohoku-oki earthquake analyzed by a kinematic PPP method. We find that a time-domain stacking notably reduces background noise in spectra, although the time-domain stacking seems improper at frequencies higher than 3.5 mHz. The STFT (short-time Fourier transform) time-frequency analysis and the CWT (continuous wavelet transform) time-frequency analysis reveal temporal changes of the spectral peaks for the Earth's free oscillation at frequencies higher than 1 mHz. In several fundamental modes of the Earth's free oscillation, the amplitudes of the spectral peaks tend to increase after their temporal decreases. The results of the analyses indicate that the stacked GPS data truly detected the low-frequency ground oscillation. But we additionally note that we might not be able to exclude effects of satellite clock error especially at very low frequencies (< 1 mHz).

Emerging engineering advancement of precise comparison of synchronously linked optical lattice clocks at a great distance, targeting to an accuracy equivalent to 1 cm in height difference, sheds new light on accurate geodetic leveling over long distances. Based on the theory of relativity, the rate difference between the clocks reflects the geopotential difference between the clock locations. Deriving the geopotential difference from the clock rate difference, however, requires removal of significant perturbation on the geopotential associated with time-variable tides, for example. Supposing the application of such “chronometric leveling" to coastal regions such as Japanese islands, we concern about the perturbation due not only to earth tides (ET), but to ocean tidal loading (OTL). Here, we evaluate the magnitudes of differential perturbation (DP) due to both ET and OTL separately on some realistic baselines taken from the Japanese leveling routes, with different distances ranging from a few tens to a thousand of kilometers. The results show that over a distance of 50 km the temporal changes in combined DP due to ET and OTL can reach the magnitude of 1 cm in terms of equivalent height difference (EHD); over a distance of 200 km OTL can produce DP with a maximum difference exceeding 1 cm EHD, and over a distance of 1000 km even root-mean-square amplitudes of OTL-induced DP can reach about 2 cm EHD. Therefore, future geodetic application of clock comparison to constraining a nationwide height datum over a long baseline must unavoidably take into consideration the removal of perturbation due not only to ET, but also to OTL.

　The standard deviation (SD) of leveling observation regulated by the public survey manual of Japan is calculated by the closing errors between fore and back. The closing errors are assumed to have no systematic error. The leveling observation data, however, show that the closing errors have the systematic error due to collimation change just after setting instrument or sinking staff. The Root Mean Square Error (RMSE) is more suitable for a measure of the leveling observation, because the RMSE estimates not only the variance by the random error but also the bias by the systematic error. The SD and the RMSE match under the condition that the systematic error itself becomes random.

　This paper insists that we should recognize the difference between the SD and the RMSE correctly, and that we should use “RMSE” instead of “SD” as a measure of the leveling observation.