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SUMMARY We quantitatively estimate the spurious annual vertical deformations due to the mismodelling of tropospheric delays over Japan, through a numerical simulation. For this purpose, we simulate the GPS data for present-day satellite distributions, using realistic cumulative tropospheric delays that are calculated using ray tracing and numerical weather models. To obtain the spurious deformations, the simulated GPS data are then analysed by using the same GPS processing software and the same analysis strategies as those used in the analysis of the observed GPS data. In the estimated spurious annual vertical deformations, it is found that the amplitudes of the deformations increase toward the north, reaching up to 3 mm at around 45 ◦ N, whereas the phase lags are uniform throughout Japan and vary by around 220 ◦ , indicating spurious subsidence in the middle of February. In the western part of Japan, the observed annual vertical deformations are less affected by spurious annual vertical deformations and more likely caused by the atmospheric and oceanic loading. In the northern part of Japan, we observe large (up to 8 mm) annual vertical deformations with phase lags corresponding to subsidence in winter; subsidence in winter is interpreted to be the effect of snow loading. However, some part of the observed subsidence is found to be spurious due to the mismodelling of the tropospheric delays. These spurious signals may be mitigated by using mapping functions that are constructed by using direct ray tracing and numerical weather models in the GPS analysis.

SUMMARY We developed a semi-analytical method to evaluate the effect of the second-order ionospheric correction on GPS positioning. This method is based on the semi-analytical positioning error simulation method developed by Geiger and Santerre in which, assuming the continuous distribution of the satellites, a normal equation is formed to estimate the positioning error taking all the contributions of the ranging error by the visible satellites into account. Our method successfully reproduced the averaged time-series of three IGS sites which is comparable to the rigorous simulation. We then evaluated the effect of the ionospheric error on the determination of the reference frame. We evaluated the additional Helmert parameters that are required for the ionospheric effect. We found that the ionospheric effect can lead to annual scale changes of 0.1 ppb, with an offset of 1.8 mm and a semi-annual oscillation of 1 mm in the z-direction. However, these values are too small to explain the current deviations between the GPS-derived reference frame and the ITRF reference frame. Next, we estimated the apparent scale changes due to the ionospheric error in the GEONET coordinate time-series in Japan. We could qualitatively reproduce the observed semi-annual scale changes peaking at the equinoxes and having asymmetrical amplitudes between the vernal and autumnal equinoxes.

Summary In magnetotelluric (MT) studies, the observed response function (the MT impedance) usually suffers from galvanic distortions due to near-surface inhomogeneities. Removal of these effects is essential to obtain an accurate model of the subsurface electrical conductivity structure. Galvanic distortion is usually expressed by a simple real tensor multiplying the undistorted regional impedance. The problem still remains of how to solve the ensuing linear equations in order to determine the distortion tensor and then to obtain the undistorted impedance. The methods presented and widely applied in previous works assume two-dimensionality for the undistorted impedance. This paper proposes a method that employs a relationship between the spatial derivatives of the horizontal electric field and the vertical geomagnetic component, which can be directly derived from Faraday’s law. The identity derived from the relationship is written using the vertical magnetic transfer function, the impedance, and the spatial derivatives of the impedance and horizontal magnetic transfer functions. The present method determines the real distortion tensor so that the identity is satisfied. Therefore, the method has two major advantages: (1) galvanic distortion that is to be removed from the impedance tensor is clearly defined, and (2) the method is applicable even when the regional structure is 3-D.