### Displacement Green’s function sensitivities

We introduce the sensitivity kernel *K* (Ito and Simons 2011; Martens et al. 2016a) to quantify the sensitivities of displacement Green’s functions to linear perturbations of the density, bulk modulus, and shear modulus:

$${\rm K}_{p}^{l} = \frac{{G(\theta ,m + \Delta m_{p}^{l} ) - G(\theta ,m)}}{{\Delta m_{p}^{l} }} = \frac{\Delta G}{{\Delta m_{p}^{l} }},$$

(11)

where *p* denotes the perturbed parameter, i.e., the density, shear modulus or bulk modulus; *l* denotes the perturbed layer; *∆m*^{l}_{p} denotes the linear variation of the perturbed parameter *p* in layer *l* of the original Earth model *m*; and *∆G* denotes the variations between the perturbed Green’s functions *G(m* + *∆m*^{l}_{p}*)* and original Green’s functions *G(m)*. According to Tarantola and Albert (2005) and Martens et al. (2016a), we denote the three physical quantities in common log space, so a + 1% linear perturbation *∆m*^{l}_{p}, i.e., log10(1.01), is equal to 0.0043.

When we compute the sensitivity kernels of displacement Green’s functions to one certain parameter (one of the two elastic moduli or the density), we keep the other two parameters fixed. We perturb the density, bulk modulus, and shear modulus separately in an original Earth model to derive three perturbed Earth models and then compute the displacement Green’s functions with the perturbed Earth models and original Earth model. The CRUST1.0 (Laske et al. 2013) model is combined with PREM (Dziewonski and Anderson 1981) as the original Earth model in our study. Since the sediment layers in the CRUST1.0 model are thin and their contributions to the OTL displacements are negligible, we average the sediment layers with the upper crust to become the outermost layer in each model (Guo et al. 2004). Taiwan Island can be divided into six regions (Fig. 1) based on the CRUST1.0 model, which correspond to the six sets of structures shown in Fig. 2.

As shown in Figs. 3, 4, the sensitivity patterns can be distinguished between the two elastic moduli. The sensitivity patterns of the shear modulus exhibit transitions between positive values and negative values, and for the bulk modulus, they are mostly positive. We also find that the displacement Green’s functions are more sensitive to perturbations of the elastic moduli in the near-surface layers, even though these layers are much thinner than the deeper layers in the upper mantle. As the depth of the perturbed layers increases, the sensitivities are still significant in the low-velocity zone (LVZ) and gradually decrease from the transition zone to the lower mantle at depths ranging from 670 to 771 km. However, the sensitivities are restored in the lower mantle at depths from 771 to 2741 km since the thickness of the lower mantle layer is massive, which means a high number of perturbations. Moreover, the deeper the layers in which we perturb the elastic moduli, the greater the angular distances the peak sensitivities show. In particular, the perturbation depths of the shear modulus are approximately equal to the angular distances of the sensitivity transitions converted into kilometers, as shown in Fig. 3.

The sensitivities of the horizontal displacement Green’s functions to density perturbations are always negative in both the near and the far fields, and the sensitivities of the vertical displacement Green’s functions exhibit positive sensitivities in the far field. The density sensitivity can therefore be discerned from the sensitivities of the elastic moduli in both the near and the far fields, and the density sensitivity can be discerned in the vertical and horizontal components. In addition, the density sensitivity kernels are weak in the lithosphere above 80 km depth and in the lower mantle at depths from 2741 to 2891 km. However, the sensitivity kernels become large in the lower mantle at depths from 771 to 2741 km. Obviously, the density kernels are sensitive to the layer thickness, not the perturbation depth.

In this section, we compute the displacement Green’s functions relative to the center of mass of the solid Earth (CE) reference frame, as well as the center of mass of the entire Earth system (CM), which includes the solid Earth and surface loads, reference frame (Blewitt 2003). As the sensitivities of the displacement Green’s functions in the CE frame to linear perturbations of the two elastic moduli are consistent with the corresponding sensitivities in the CM frame, their sensitivities are independent of the reference frame. However, the displacement Green’s functions are more sensitive to density perturbations at certain far fields in the CM frame, as shown in Additional file 1: Figure S1, than to those in the CE frame, especially to perturbations in much thicker layers.

For completeness, we investigate the sensitivities of the displacement Green’s functions to perturbations of the two elastic moduli and density in the other models (TW2, TW3, TW4, and TW5) in the CE frame. The sensitivity characteristics of the displacement Green’s functions for one certain parameter (one of the two elastic moduli or the density) are consistent among these models, although there are some differences in their sensitivity patterns. Not surprisingly, these differences between TW1 and the other models are mainly found in the thicknesses of the perturbed layers, and the displacement Green’s functions are generally more sensitive to thicker layers with the same name in different models. As we mentioned above, perturbations of the elastic moduli at deeper depths result in the peak sensitivity corresponding to a larger angular distance. In addition, the influences caused by the magnitudes of the three physical quantities on the sensitivity are negligible.

Since the thickness of the perturbed layer can affect the sensitivity kernel, we further perturb the layers with a fixed thickness of 20 km in TW1 at depths from 0 to 700 km. Clearly, the peak sensitivities are much smaller when perturbations of the elastic moduli occur in deeper layers, and the corresponding angular distances are larger, which indicate that perturbations of the elastic moduli at certain depths give rise to variations in the displacement Green’s functions at specific angular ranges. We note in Fig. 5 that the high sensitivity of the displacement Green’s functions can arise from perturbations of the elastic moduli only within tens of kilometers near the surface. The sensitivity kernels corresponding to perturbations of the bulk modulus are great above 40 km depth, and for the shear modulus, they are approximately 40 km deep for the horizontal components and extend to approximately 100 km deep for the vertical component. Although the sensitivity kernels for the shear modulus exhibit transitions at certain angular distances, beneath 60 km depth, the displacement Green’s functions are generally more sensitive to perturbations of the shear modulus, especially for the vertical component. The density kernels are weak in thin perturbed layers and basically remain consistent in the near field.

### OTL displacement sensitivities

Analyzing the OTL displacement sensitivity to the upper mantle and crustal structure can provide necessary knowledge of the independent constraints on the three physical quantities (density, shear modulus, and bulk modulus), even the structural depth. OTL displacements at the surface depend not only on the angular distances from observation stations to mass loads, as in the case of the displacement Green’s functions assuming a point mass load, but also on specific positions since ocean tides exhibit complex distributions at both spatial and temporal scales. We further investigate the sensitivity characteristics of the OTL displacements on Taiwan Island (with a resolution of 0.25° × 0.25°) to linear perturbations of the three physical quantities, especially the corresponding relations between the depth of the structural perturbations and the angular distance from the station to the coastline.

We begin with the sensitivity characteristics of OTL displacements to linear perturbations of the bulk modulus, shear modulus, and density in different layers of the original Earth model. Since the sensitivity characteristics of OTL displacements to perturbations of one certain quantity in these models (TW1, TW2, TW3, TW4, and TW5) are consistent and the strongest responses of the OTL displacements shown in Figure 6 appear in the TW1 region, we only show the results of TW1. For the other models, the sensitivity patterns show slight graphic shifts of the peak sensitivities and small amplitude differences (the most sensitive model is TW4, as shown in Additional file 1: Figure S3). Because the vector discrepancies of the modeled M_{2} OTL displacements to +1% linear perturbations in the lithosphere are smaller than 0.03 mm, which can be negligible with the present accuracy, we only show the sensitivity patterns for +10% linear perturbations of the three physical quantities. First, we compute the variations in the modeled M_{2} OTL displacements to +10% linear perturbations of the density, bulk modulus, and shear modulus separately in each crust and mantle layer of TW1. As OTL displacements are more sensitive to the upper mantle and crustal structure (Farrell 1972; Ito and Simons 2011; Yuan et al. 2013), we only show the results for the perturbed layers above the transition zone (400–600 km depth) in Figure 7. Then, we calculate the sensitivities of the modeled M_{2} OTL displacements based on Eq. (12) to perturbations of the three physical quantities. The sensitivity patterns shown in Additional file 1: Figure S2 corresponding to +10% linear perturbations of the density, bulk modulus, and shear modulus are consistent with the patterns of the vector discrepancies shown in Figure 7:

$${\rm K}_{U,p}^{l} (r,H) = \frac{{U(r,H,G(m + \Delta m_{p}^{l} )) - U(r,H,G(m))}}{{\Delta m_{p}^{l} }} = \frac{\Delta U}{{\Delta m_{p}^{l} }},$$

(12)

where *K*^{l}_{U,p} is the sensitivity of OTL displacements to parameter *p* perturbed linearly in layer *l*; *U* denotes the OTL displacement modeled as Eq. (10); *r* denotes the location of the station relative to the mass load; *H* denotes the ocean tide model; *G* denotes the displacement Green’s functions; ∆*U* denotes the vector difference of OTL displacements considering both amplitudes and phases; and *∆m*^{l}_{p} denotes the variation of the perturbed parameter *p* in perturbed layer *l*.

As shown in Fig. 7, the modeled M_{2} OTL displacements are also sensitive to linear perturbations of the elastic moduli but not to those of the density. In particular, the horizontal OTL displacements are more sensitive to perturbations of the elastic moduli than the vertical displacements, except to those of the shear modulus in the upper mantle. We note that the horizontal displacement Green’s functions vary at wider angular ranges than the vertical displacement Green’s functions shown in Fig. 3, which may explain the higher sensitivity of the horizontal OTL displacements. The regions in which the horizontal components are sensitive to perturbations of the elastic moduli in the upper mantle can extend to most of Taiwan Island and even exhibit a uniform pattern. Since the LVZ is 80–220 km deep, considering the specific relation between the angular range of the peak sensitivity and the perturbation depth, which is near or exceeds the maximum distance across Taiwan Island, when we perturb deeper layers beneath the LVZ, their sensitivities still exhibit a uniform pattern. It is worth noting that since the sensitivities of the displacement Green’s functions for the shear modulus exhibit transitions at certain angular distances, as shown in Figs. 3, 4, as we perturb the shear modulus from the upper crust to the region above the LVZ (LID), the regions of peak sensitivities gradually move inland from western Taiwan, which reveals that the shear modulus at certain depths mainly controls the OTL responses on the surface within the specific distance ranges from stations to the coastline.

The most sensitive regions of the modeled M_{2} OTL displacements are mainly located in northern and western Taiwan Island. In fact, Fig. 6 shows that these regions are exactly where the responses of OTL displacements for the M_{2} tidal harmonic are strong. The variations of the OTL displacements in the horizontal components in response to + 10% perturbations of the bulk modulus can reach the sub-millimeter level except in the lower crust and for the north component in the region above the transition zone, consistent with the magnitudes of the estimation accuracy of GPS tidal displacements (Yuan and Chao 2012; Yuan et al. 2013), and for the vertical component, the variations can reach the sub-millimeter level in the upper mantle. The variations of the OTL displacements in the east component in response to perturbations of the shear modulus are also at the sub-millimeter level except in the lower crust and the region above the transition zone, and for the vertical component, the variations increase dramatically in the LVZ and the region above the transition zone. Because the thicknesses of the LVZ and the region above the transition zone are thicker than the crustal layers, the responses of OTL displacements to perturbations of the elastic moduli are stronger. Here, we reiterate that the variations of the OTL displacements controlled by perturbations of one certain physical quantity in a specific layer cannot be added numerically to the variations of the OTL displacements caused by perturbations of the other physical quantities in any layer.

Since the layer thickness can affect the sensitivity, we normalize the sensitivity of OTL displacements by the corresponding layer thickness to attenuate this effect. Moreover, by implementing this operation, we can more clearly show the sensitivity kernels to linear perturbations of a single physical quantity in different layers. Figure 8 shows that the unit sensitivities of the modeled M_{2} OTL displacements at 24.5°N on Taiwan Island to perturbations of the elastic moduli from the upper crust to the LVZ are separated for the horizontal components, and the sensitivities gradually decrease with increasing perturbation depth, especially for the bulk modulus. The unit sensitivities to perturbations of the bulk modulus in the crustal layers dramatically change in the vertical component. It is worth noting that the peaks and troughs of the unit sensitivities to the shear modulus perturbations, which correspond to specific angular distances, significantly translate in the vertical component. The density sensitivities are weak. We also analyze the sensitivities of the modeled M_{2} OTL displacements to linear perturbations of the three physical quantities in layers with a fixed thickness of 20 km, and the effects of the perturbed depth on the sensitivities are very clear above 80 km for perturbations of the elastic moduli, especially for the shear modulus within the specific ranges of angular distances.

### Sensitivity comparisons between Earth models and ocean tide models

We further discuss the general sensitivity of OTL displacements to different Earth models. The Earth models used in our study include TW1, AK135 (Kennett et al. 1995), AK135-F (Kennett et al. 1995; Montagner and Kennett 1996), STW105 (Kustowski et al. 2008), Huang 1D (Huang et al. 2014), and 1066A (Gilbert et al. 1975). Figure 9 depicts the structural deviations of the different Earth models relative to TW1 in log space above 800 km depth. We note that the structural deviations are mainly above 220 km depth. Due to the magnitudes of the three physical quantities and different layer thicknesses, 1066A and Huang 1D are significant, especially for Huang 1D in the upper mantle. The deviations in the two elastic moduli are significantly larger than those in the density. In the crustal layers, the deviations of the elastic moduli can exceed 0.2 in log space, which corresponds to a 58% perturbation, and deviations form Huang 1D in the upper mantle reach 0.1 in log space, which corresponds to a 26% perturbation. Most deviations are less than 0.03 in log space, which corresponds to a 7% perturbation.

Then, we compute the displacement Green’s functions in the CE frame using these Earth models, and their respective differences are still relative to TW1. The sediment layers in TW1 and Huang 1D, as well as the ocean layers in STW105, AK135-F, and PREM, are correspondingly averaged with their outermost crustal layers. Figure 10 shows that the differences in the displacement Green’s functions between 1066A and TW1 in the very near field (angular distances smaller than 0.1°) can be attributed to structural deviations above approximately 13 km depth, and the differences among AK135-F, PREM, STW105, 1066A, and TW1 at angular distances from 0.1° to 0.4° probably result from structural deviations above 40 km depth. Deviations in the structure at depths from 80 to 350 km cause differences in the displacement Green’s functions between Huang 1D and TW1 at angular distances from 0.4° to 3°.

We model the M_{2} OTL displacements based on the six Earth models and calculate the vector and RMS differences of the OTL displacements among these different Earth models. For the comparison, we also calculate the vector and RMS differences of the OTL displacements among five ocean tide models, namely, DTU10 (Cheng and Andersen 2011), EOT20 (Hart-Davis et al. 2021), HAMTIDE11A (Zahel 1995; Taguchi et al. 2014), FES2014b (Lyard et al. 2006; Carrère et al. 2016), TPXO9-atlas (Egbert and Erofeeva 2002; Egbert et al. 2010), and FESJB, which combines FES2014b with the NAO99Jb regional model (Matsumoto et al. 2000), and we attempt to compare the sensitivity level of OTL displacements to the Earth models with the sensitivity level of OTL displacements to these ocean tide models. The RMS differences of the modeled M_{2} OTL displacements between different ocean tide models and FESJB are shown in Fig. 11, as are the RMS differences between different Earth models and TW1. The RMS differences of the modeled M_{2} OTL displacements with the results between different ocean tide models and FES2014b are shown in Fig. 12, as are the results between different Earth models and PREM.

Figures 11 and 12 show that the RMS differences of the modeled M_{2} OTL displacements for the horizontal components, especially the east component, among the different Earth models are generally larger than those among the ocean tide models, which indicates that the OTL displacements for the M_{2} tidal harmonic in the horizontal components are generally more sensitive to the Earth’s structure than to the errors in those ocean tide models. However, the sensitivity levels of OTL displacements to different Earth models in the vertical component are relatively low, except for the Huang 1D and 1066A models.