On the contributions of refined thermal expansion model to nonlinear variations in different GNSS height time series products

Abstract

The thermal expansion effect exhibited by the GNSS monument and their nearby bedrock due to surface temperature variations is an important factor affecting the nonlinear variation of GNSS height. However, current thermal expansion models either consider only the above-surface GNSS monuments or only non-seasonal temperature variations of the subsurface bedrock, and lack a comprehensive thermal expansion model. Furthermore, previous studies on the contribution of thermal expansion effects to nonlinear variations in GNSS height have been analyzed only against one set of GNSS time series products per single research, but in fact the GNSS time series provided by each agency varied considerably. In this study, we use a refined comprehensive thermal expansion model (TEVD_FSD) to evaluate its contribution to the nonlinear variations from several GNSS height products (SOPAC, JPL, and COMBINED) obtained using different data processing strategies. The results show that the most GNSS stations (about 95%) using the TEVD_FSD model exhibit an annual amplitude increase and phase lag, with an amplitude increase of up to 0.5 mm and phase lag of up to 13° compared with the finite element model, especially for inland and those with deeper GNSS monument stations. This phase lag improves its correlation with the GNSS height, which reduces the GNSS height value to improve the geophysical interpretation. The TEVD_FSD model estimates an annual amplitude of up to 7.5 mm, explaining at most 13.6% of the nonlinear variation in the COMBINED height. The COMBINED product exhibits a further WRMS reduction of up to 20% and 18.7% compared with the SOPAC and JPL products, respectively, which are likely due to the higher accuracy of the combined GNSS solution than of the independent data processing strategy. Our work indicates that differences in data processing strategies for GNSS height time series products significantly affect the interpretability of thermal expansion effects to nonlinear variations.

Appendix A: Refined comprehensive thermal expansion model

In this study, we consider that the temperature of the above-surface GNSS monuments is the same as the surface temperature. The vertical displacement of the above-surface monuments due to thermal expansion effects is (Yan et al. 2009, 2010):

$$\Delta {h}_{{\text{obse}}\_{\text{up}}}=\alpha \cdot L\cdot [T(t)-{T}{\prime}]$$

(A1)

where $\alpha$ is the coefficient of thermal expansion of GNSS monument; $L$ is the height of the GNSS monument above-surface; $T(t)$ is the surface temperature (°C) of the station at time t; ${T}^{\mathrm{^{\prime}}}$ is the average surface temperature (°C).

The surface temperature conducts heat to the subsurface bedrock by heat transfer, causing temperature changes in the subsurface monument and the bedrock (Yan et al. 2009, 2010):

$$\Delta T(y, t)={T}^{\mathrm{^{\prime}}}+\sum_{{\text{i}}=1}^{{\text{N}}}{A}_{{\text{i}}}{e}^{-{\text{y}}\sqrt{{\omega }_{{\text{i}}}/2k}}{\text{cos}}({\omega }_{{\text{i}}}t-y\sqrt{{\omega }_{{\text{i}}}/2k}-{\mathrm{\varphi }}_{{\text{i}}})$$

(A2)

where $k$ represents the thermal diffusivity; $y$ is the underground depth; $t$ is the time; $N$ is the number of decomposed harmonics of the surface temperature; ${A}_{{\text{i}}}$, ${w}_{{\text{i}}}$, and ${\mathrm{\varphi }}_{{\text{i}}}$ are the amplitude, angular frequency, and initial phase of the ith harmonic, respectively.

The thermal stress $\upvarepsilon$ is expressed as (Dong et al. 2002):

$$\upvarepsilon =-\frac{1+v}{1-v}\alpha \sum_{i=1}^{N}{A}_{i}{\text{exp}}\left(-y\sqrt{\frac{{w}_{{\text{i}}}}{2k}}\right){\text{cos}}({w}_{i}t-y\sqrt{\frac{{w}_{i}}{2k}}-{\varphi }_{i})$$

(A3)

where $v$ denotes the Poisson’s ratio. Other symbols are the same as in (A2).

We assume that the temperature subsurface varies with depth. The temperature is the same at the same depth. The subsurface monument is a fixed-shape rigid body and its temperature varies with depth, so we use the thermoelastic deformation modeling of a fixed-shape rigid body to estimate its thermal expansion effects (Li et al. 2023).

Combining the boundary conditions ($0<y<z$), the heat conduction equation, and the thermal stress equation, the thermal expansion-induced vertical displacement of the subsurface monument estimated by the thermoelastic deformation modeling of a fixed-shape rigid body is derived as:

$${\Delta h}_{{\text{obse}}\_{\text{down}}}=\frac{1+{v}_{1}}{1-{v}_{1}}{\alpha }_{1}\sum_{i=1}^{N}{A}_{i}\sqrt{\frac{{k}_{1}}{{w}_{i}}}\left[{\text{cos}}\left({w}_{i}t-{\varphi }_{i}-\frac{\pi }{4}\right)-{\text{exp}}\left(-z\sqrt{\frac{{w}_{i}}{2{k}_{1}}}\right){\text{cos}}\left({w}_{i}t-{\varphi }_{i}-z\sqrt{\frac{{w}_{i}}{2{k}_{1}}}-\frac{\pi }{4}\right)\right]$$

(A4)

where ${v}_{1}$, ${\alpha }_{1}$, and ${k}_{1}$ are the Poisson’s ratio, coefficient of thermal expansion, and thermal diffusivity of the subsurface monument, respectively; $z$ is the depth of the subsurface monument.

By integrating the thermal stress below the bedrock surface ($y\ge z$), the bedrock thermal expansion vertical displacement can be obtained, expressed as:

$${\Delta h}_{{\text{bedrock}}}=\frac{1+{v}_{2}}{1-{v}_{2}}{\alpha }_{2}\sum_{i=1}^{N}{A}_{i}\sqrt{\frac{{k}_{2}}{{w}_{i}}}{\text{exp}}\left(-z\sqrt{\frac{{w}_{i}}{2{k}_{2}}}\right){\text{cos}}\left({w}_{i}t-{\varphi }_{i}-z\sqrt{\frac{{w}_{i}}{2{k}_{2}}}-\frac{\pi }{4}\right)$$

(A5)

where ${v}_{2}$, ${\alpha }_{2}$, and ${k}_{2}$ are the Poisson’s ratio, coefficient of thermal expansion, and thermal diffusivity of the bedrock, respectively.

The comprehensive thermal expansion vertical displacement ${\Delta h}_{{{\text{TEVD}}}_{{\text{FSD}}}}$ is expressed as:

$${\Delta h}_{{{\text{TEVD}}}_{{\text{FSD}}}}=\Delta {h}_{{{\text{obse}}}_{{\text{up}}}}+{\Delta h}_{{{\text{obse}}}_{{\text{down}}}}+{\Delta h}_{{\text{bedrock}}}=\alpha \cdot h\cdot \left[T\left(t\right)-{T}^{\mathrm{^{\prime}}}\right]+\frac{1+{v}_{1}}{1-{v}_{1}}{\alpha }_{1}\sum_{i=1}^{N}{A}_{i}\sqrt{\frac{{k}_{1}}{{w}_{i}}}\left[{\text{cos}}\left({w}_{i}t-{\varphi }_{i}-\frac{\pi }{4}\right)-{\text{exp}}\left(-z\sqrt{\frac{{w}_{i}}{2{k}_{1}}}\right){\text{cos}}\left({w}_{i}t-{\varphi }_{i}-z\sqrt{\frac{{w}_{i}}{2{k}_{1}}}-\frac{\pi }{4}\right)\right]+\frac{1+{v}_{2}}{1-{v}_{2}}{\alpha }_{2}\sum_{i=1}^{N}{A}_{i}\sqrt{\frac{{k}_{2}}{{w}_{i}}}{\text{exp}}\left(-z\sqrt{\frac{{w}_{i}}{2{k}_{2}}}\right){\text{cos}}\left({w}_{i}t-{\varphi }_{i}-z\sqrt{\frac{{w}_{i}}{2{k}_{2}}}-\frac{\pi }{4}\right)$$

(A6)

where $\Delta {h}_{{\text{obse}}\_{\text{up}}}$, ${\Delta h}_{{\text{obse}}\_{\text{down}}}$, and ${\Delta {h}}_{{\text{bedrock}}}$ denote the thermal expansion vertical displacements of the above-surface and the subsurface monument as well as their nearby bedrock, respectively.