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Estimating the lithospheric flexure of a plate with nonuniform flexural rigidity: a quantitative modelling approach
Earth, Planets and Space volume 73, Article number: 170 (2021)
Abstract
Lithospheric deformation is a fundamental process in plate tectonics. It is, therefore, critical to determine how the lithosphere responds to geological loads to better understand tectonic processes. The lithosphere can be modelled as the flexure of a thin, elastic plate over longterm (> 10^{5} yr) geological timescales. The partial differential equation for the flexure of an orthotropic plate is used indirectly to calculate theoretical admittance and coherence, which are then compared against the observed admittance and coherence to invert for the nonuniform flexural rigidity (or effective elastic thickness, T_{e}) of the plate. However, the process for accurately recovering variable lithospheric flexure remains unresolved, as the classical lithospheric model may overestimate the deflection of the plate. Here we adopt the classic lithospheric model with applied external and internal loads at the surface and Moho, respectively, and assume that the compensation material is denser than the mantle material beneath the Moho. The lithospheric flexure errors are derived mainly from the T_{e} and Moho recovery errors in this lithospheric model. Synthetic modelling is then performed to analyse the influence of the T_{e} and Moho errors. The analysis of synthetic modelling shows that: (1) the T_{e} errorinduced flexure errors exhibit a rippling pattern, and the rippling pattern is broader in high T_{e} regions; (2) the Moho errorinduced flexure errors mainly occur in the low T_{e} regions, and applying Airy isostasy theory in low T_{e} regions may still greatly overestimate the lithospheric deformation amplitude; and (3) the lithospheric flexure errors are dominated by the T_{e} and Moho errors in the high and low T_{e} regions, respectively.
Introduction
Our understanding of the mechanics of the Earth’s crust and lithosphere have been greatly advanced since the 1970s, via numerous modelling, including experimental examinations of the behaviour of physical bodies (e.g., Dorman and Lewis 1970, 1972; Lewis and Dorman 1970a, b) and derivations of the mathematics that describe lithospheric processes (e.g., McNutt 1980; Forsyth 1985; Burov and Diament 1992; Banks et al. 2001; Braitenberg et al. 2002; McKenzie 2003; Kirby and Swain 2009; Zhang et al. 2018a, b, c; Zhang et al. 2019a). These studies followed the principle that the lithosphere can be idealised if the main structural features are considered, thereby allowing simplified models to capture the key structures and geodynamics of the lithosphere. However, no information on the internal structure of the lithosphere has been used in these investigations, and its mechanical properties, such as the flexure of the tectonic plates, may be incorrectly estimated (Ribe 1982).
The lithospheric strength of tectonic plates reflects their resistance to vertical deformation in response to geological loads over longterm (> 10^{5} yr) geological timescales (Watts and Burov 2003). This assumption allows the lithosphere to be modelled as the flexure of a thin, elastic plate via the partial differential equation for the flexure of an orthotropic plate (Timoshenko and WoinowskyKrieger 1959). This equation is usually used indirectly to calculate theoretical admittance and coherence, which are then compared against observed ones to invert for the nonuniform flexural rigidity (or effective elastic thickness, T_{e}) of the plate (Kirby 2014). Numerous studies have used T_{e} to analyse lithospheric rheology and deformation (PérezGussinyé et al. 2009; Chen et al. 2015; Ji et al. 2017, 2020; Lu et al. 2020). However, this idealised term does not refer to an existing thickness or physical layer within the Earth, but instead corresponds to the thickness of an ideal elastic plate that undergoes the same deformation as the lithosphere under the same loads (Watts 2001). As T_{e} is a periphrastical parameter for understanding lithospheric rheology and deformation, directly modelling lithospheric flexure should provide key insights into the nature of tectonic evolution and dynamics.
Here we use the classic lithospheric model with applied external and internal loads at the surface and Moho, respectively, and assume that the compensation material is denser than the mantle material beneath the Moho (Fig. 1). The density contrast between the compensation material and mantle material beneath the Moho is set to 200 kg/m^{3}, based on the density contrast between the uppermost and basal lithospheric mantle (Kaban et al. 2016). Notably, in some special areas, loads may be taken from specific geological attributes instead of topographies. In the subduction zones, for instance, slab pull may be considered as the main loads and usually be handled as boundary loading (e.g., Zhang et al. 2014, 2021), applying the corresponding plate flexure models has made a great deal achievements (e.g., Zhang et al. 2018a, b, c; 2019a, b). Taking the topographies, however, as the loads has a widespread application in estimating plate flexure and should have a wider prospect. The model used in this study is still an idealised model that considers the vertical density variation in the lithosphere. Furthermore, we obtain T_{e} variations via the fan wavelet coherence method (Kirby and Swain 2004, 2008) and the Moho topography via a constrained gravity inversion method (Wu et al. 2017). Based on the finitedifference (FD) method shown by Kirby and Swain (2008), we reformat the equations to invert the lithospheric flexure of a twolayer lithospheric model. The errors associated with the lithospheric flexure estimation in such a lithospheric model are mainly derived from the T_{e} and Moho errors. Synthetic modelling is then performed to analyse the influence of the T_{e} and Moho errors on the lithospheric flexure estimation.
Methodology
The key to calculating lithospheric flexure via the FD method is to obtain accurate spatial distributions of the Moho and T_{e}. The Moho depth is usually recovered via gravity inversion (Wu et al. 2017). There are currently two main methods for mapping the spatial variations in T_{e} based on the coherence between the Bouguer gravity anomaly (BGA) and topography/bathymetry. One is the multitaper method (McKenzie and Fairhead 1997; PérezGussinyé et al. 2004), and the other is fan wavelet coherence method (Kirby and Swain 2004, 2008). The characteristic flexural wavelength is correlated with the strength of the lithosphere, and the flexural amplitude varies with both T_{e} and the magnitude of the load (Watts 2001). These variations make it difficult to choose the multitaper window size to recover T_{e}. Signals of features on a stronger lithosphere can be truncated when the selected window size is too small, resulting in an artificially low T_{e}. Conversely, T_{e} can be blended across multiple features, with a potential bias toward the largest feature in the window, when the selected window size is too large (PérezGussinyé et al. 2004, 2008; Kalnins and Watts 2009). The fan wavelet coherence method can alleviate these issues by allowing the T_{e} features to be located at different wavelengths, and is employed to estimate T_{e} in this study.
Gravity inversion for Moho topography
Joint inversions that involve gravity measurements and other geophysical constraints are generally employed to reduce the nonuniqueness of the bestfit models (Chappell and Kusznir 2008; Bai et al. 2014; Ji et al. 2018). Here we perform a threedimensional gravity inversion that is constrained by seismic observations (Wu et al. 2017) to recover an accurate Moho topography. The Moho topography that corresponds to the complete Bouguer gravity anomaly can be expressed as \(h\left( {\mathbf{r}} \right) = \, h_{e} \left( {\mathbf{r}} \right) + \Delta h\left( {\mathbf{r}} \right)\), where \(h_{e} \left( {\varvec{r}} \right)\) is the initial interface. The difference between the Moho and initial interface can then be obtained by substituting this difference into the gravity inversion equation (Oldenburg 1974):
where \(F\left[ {} \right]\) is the Fourier transform, \(\Delta g\) is the complete Bouguer gravity anomaly, \(\varsigma\) is Newton's gravitational constant, \(\rho\) is the density contrast across the interface, \({\varvec{k}}\) is the wave vector of the transformed function, \({\varvec{r}}\) is the entire x–y plane and \(z_{0}\) is the reference depth of the density interface, n is the iterations, whose value depends on the convergence rate of Eq. (1) and the error threshold.
We largely follow the procedure outlined in Wu et al. (2017), except for the implementation of Eq. (1) to determine the interface difference during each iteration of the inversion, to ensure that both the Moho depth and Bouguer gravity anomaly satisfy the inversion equation (Oldenburg 1974) after the Moho topography is recovered. The detailed process is as follows: (1) linearly fit the Moho depth values and the corresponding Bouguer gravity anomalies with the constraint points; (2) then, use the fitting parameters to form the initial Moho grid \(h_{e} \left( {\varvec{r}} \right)\) with the observed Bouguer gravity anomaly grid; (3) perform the iterative algorithm of Eq. (1); (4) the final Moho topography is obtained iteratively until the interface difference between adjacent iteration steps meets the error threshold, which is set to be 0.001 km in this study.
Thin plate flexure
For quick reference, we provide the key equations for estimating thin plate flexure, which can be easily calculated via the approach of Kirby and Swain (2008). We consider an elastic slab model that involves only the initial external (at the surface) and internal (at the Moho) loads, which are represented as \(\left[ {\left( {\rho_{c}  \rho_{f} } \right)gh_{i} } \right]\) and \(\left[ {\left( {\rho_{m}  \rho_{c} } \right)gw_{i} } \right]\), respectively. As the plate model is a linear system, the deflection can be divided into four components. The deflection of amplitude w_{T} at the Moho and h_{T} at the surface caused by the initial external load, w_{B} at the Moho and h_{B} at the surface caused by the initial internal load. Then, the final Moho topography after flexure is
and the final surface topography after flexure is
The final deflection amplitude v can then be expressed as
The relations of initial loads and deflection amplitude in the partial differential equation for flexure of an orthotropic plate are given by Timoshenko and WoinowskyKrieger (1959):
where the bending and twisting moments are
and where D is the flexural rigidity, \(\sigma\) is Poisson’s ratio. Furthermore, substituting the initial loads with Eq. (4), and simplifying Eq. (5) with the Laplace operator that gives (Timoshenko & WoinowskyKrieger 1959; Stark et al. 2003):
where all variables are functions of the (x, y) position, \(\Delta = \left( {\partial^{2} /\partial x^{2} } \right) + \left( {\partial^{2} /\partial y^{2} } \right)\) is the Laplace operator. The minus sign on the right side of Eq. (6) demonstrates the plate acts as resistance to the topographic deformation. When D reaches zero, Eq. (6) follows the Airy isostatic theory. The solution to Eq. (6) via the FD method is shown in Appendix A.
A larger Winkler foundation
A foundation that acts with a force proportional to the deflection at every point is known as a Winkler foundation (e.g., Huang and Thambiratnam 2001; Mofid and Noroozi 2009). An elastic foundation is usually expressed as \(\rho_{m} gv\) in linear elastic plate theory. However, the deflection predicted via linear theory is always too high; this nonlinearity decreases with increasing plate thickness, such that plates with T_{e} > 20 km are only slightly nonlinear (Ribe 1982). This overestimated deflection should be suppressed in ocean settings since T_{e} is generally low in these settings. Here we use a twolayer model (Fig. 1), whereby the internal load is confined to the Moho, such that the entire lithosphere is modelled with a lithospheric elastic foundation with higher density (\(\rho_{F}\)) than that of the uppermost layer of the mantle (\(\rho_{m}\)). Equation (6) is then changed to
The main advantages of this model are:

1.
The overestimated deflection of the lithosphere predicted via linear theory can be suppressed;

2.
The density of the mantle usually increases with depth, so the model fits real data better in the framework of lithosphericscale deflection;

3.
The main inversion function (Eq. (6)) in such a model can be converted to Eq. (7), which can be considered as a forward equation with a small Winkler foundation, thereby ensuring that stable calculation results are always obtainable.
Fan wavelet coherency method
The wavelet method for calculating the coherence between the topography and gravity data was first proposed by Stark et al. (2003), who employed a Gaussian window function as the transform kernel to estimate T_{e} in southern Africa. The continuous wavelet transform offers the ability to obtain the local phase and amplitude of the topography and gravity anomalies, whereas the Fourier transform requires wavenumber averaging of the spectral estimates. The Gaussian wavelet fails to reproduce the Fourier power spectrum exactly (Kirby 2005). Twodimensional Morlet wavelets, which are Gaussianmodulated complex exponential functions, can reproduce the Fourier spectrum (Kirby and Swain 2004). The spatial resolution of T_{e} can be improved by adjusting the central wavenumber of the Morlet wavelet (Kirby and Swain 2011). Only the square of the real component of the wavelet coherence needs to be employed to estimate T_{e} (Kirby and Swain 2008, 2009). The coherency, which is formed by obtaining auto and crossspectra at a series of azimuths over the 0°–180° range in a ‘fan’ and then averaging them (Kirby and Swain 2009), is computed as the real and positive coherence, which suffers from the problem of information loss. The squared real coherency can be expressed as a function of scale s and position x:
where B_{sxθ} and H_{sxθ} represent the complex wavelet coefficients of the Bouguer gravity anomaly and topography, respectively, the * indicates complex conjugation, and the angular brackets denote azimuthal averaging performed over the entire 180° azimuthal range. The continuous wavelet transform (CWT) is performed through the following equations:
Here, \(\widetilde{g}{(}s,{\varvec{x}},\theta {)}\) are the resulting ‘wavelet coefficients’, \(F^{  1}\) is the inverse 2D Fourier transform, \(\widehat{g}({\varvec{k}})\) is the 2D Fourier transform of a 2D spatially distributed signal \(g({\varvec{x}})\), \({\varvec{k}} = (u,v)\) is the 2D wavenumber, \(\widehat{\psi }({\varvec{k}})\) is the 2D Fourier transform of the 2D Morlet wavelet, scale s is also the dilation parameter, the “central wavenumber” \(\left {\left. {k_{0} } \right} \right. = \pi \sqrt {2/\ln 2} \approx 5.336\) is used in this study. Note that v only represents the frequency domain coordinate parameter in Eq. (9), but elsewhere in this article is the flexure.
Flexural components in Eqs. (2) and (3) can be replaced with equations including initial internal and external loads \(w_{i}\) and \(h_{i}\). As these two equations are linear, they form the socalled “loaddeconvolution equations” in the wavenumber domain with the assumption that T_{e} is uniform (Forsyth 1985; Kirby and Swain 2008). Following the formulation in Kirby and Swain (2011) and making the equations available for a twolayer model that
The parameters are
where G, H mean the observed Bouguer gravity anomaly and surface topography, \(\left {\left. {\mathbf{k}} \right} \right. = \left {\left. {k_{0} } \right} \right./s\) is the equivalent Fourier wavenumber, Z_{m} is the Moho depth, Z_{F} is the LAI depth, \(\varsigma\), E and \(\sigma\) are constant values (Table 1).
After Eq. (10) is solved, initial loads and flexural components can be obtained. Then, the predicted wavelet coherency is
where T is for surface load components, and B for subsurface load components. With the advantage of averaging being performed over the wavelet azimuth in the wavelet method, Eq. (12) can be further expressed as
where \(\left\langle {\left {\tilde{h}_{i} (s,{\mathbf{x}})} \right^{2} } \right\rangle_{180}\) and \(\left\langle {\left {\tilde{w}_{i} (s,{\mathbf{x}})} \right^{2} } \right\rangle_{180}\) mean the azimuthal averaged wavelet coefficients of initial surface and subsurface loads. By considering each local wavelet spectrum to be independent of adjacent spectra (the “decoupling” assumption), the wavelet coherency can be applied to approximately invert for local T_{e} (Stark et al. 2003; Kirby and Swain 2004). Then the final T_{e} distributions are obtained by finding the T_{e} that provides the minimum misfit between the observed and predicted coherencies at each grid point, using Brent’s method of 1D minimization (e.g., Press et al. 1992).
Synthetic modelling
Synthetic modelling is commonly used to test the accuracy of the wavelet method (Stark et al. 2003; Kirby and Swain 2004, 2008). Random fractal initial loads and T_{e} models are used as inputs to the flexural equation, which outputs the final surface topography (h) and Moho topography (w). We follow the same approach in our synthetic modelling. We employ Eq. (6) in the inversion since it is a thorough rearrangement of the flexural equation (Eq. (11)) in Kirby and Swain (2009) where the inputs are the final topography (h) and a random fractal initial internal load (w_{i}).
The equivalent topography, T_{e} and Moho topography, are required to estimate lithospheric flexure. The equivalent topography can be specifically calculated using a global elevation model, sediment thickness model and sediment thickness–density conversion formula (e.g., Sclater & Christie 1980; Ji et al. 2020), such that this random fractal initial load (h_{i}) and flexure (v) derived topography (h) can be used in the synthetic modelling. The influences of both the recovered T_{e} errors and inverted Moho errors on the FDderived lithospheric flexure are investigated to determine how these inputs shape the model results.
Model generation
We follow the synthetic modelling procedure of Macario et al. (1995), and use the midpoint displacement method (Peitgen and Saupe 1988) to generate twodimensional fractal Brownian surfaces that represent the initial external (surface, h_{i}) and internal (subsurface, w_{i}) loads. In addition, the T_{e} model is also obtained by applying the midpoint displacement method, as Kirby and Swain (2008) noted: "fractal T_{e} distributions are likely to be closer to the real Earth’s T_{e} structure." A fractal dimension of 2.5 is used for the surface (Fig. 2a) and subsurface loads (Fig. 2b), as well as the T_{e} model (Fig. 2c). The grids consist of 255 × 255 points at a 20km spacing and are scaled to possess a realistic amplitude. Furthermore, the T_{e} grid is lowpass filtered with a cutoff wavelength of 150 km. We choose a central wavenumber of 5.336 in the subsequent fan wavelet coherence inversion since we place greater importance on the T_{e} accuracy than its spatial resolution. We do not set a constant F (loading ratio), as in Kirby and Swain (2008). Although this approach may go against Forsyth’s fourth assumption about the proportionality of the power spectral densities of the initial loads (Forsyth 1985; Simons and Olhede 2013), the feasibility of the T_{e} inversion with multiple varying F interfaces via the fan wavelet coherence method has already been tested by Kirby and Swain (2009). The topography (Fig. 2d), Moho (Fig. 2e) and lithospheric flexure models (Fig. 2f) are obtained by applying the FD method to Eq. (7). Before applying the FD method, all the data sets were mirror extended to be 9 times of their original size to reduce the computation errors caused by boundary condition, and the extended boundaries were removed after all the calculation processes were completed. The Moho interface is then used to obtain the forward Bouguer gravity anomaly (Fig. 3), with the relevant parameters listed in Table 1. The Moho inherits highfrequency information from the initial subsurface. However, the modelled Bouguer gravity anomaly may suffer from signal degradation due to upward continuation of the signal since the observation surface is much higher than the Moho interface, resulting in attenuation of this highfrequency information.
Effect of T _{ e } errors
The fan wavelet coherence method is employed to invert for T_{e} (Fig. 4a) using the generated surface topography, Moho topography and forwardmodelled Bouguer anomaly grids, and to calculate its associated errors (Fig. 4b). Note that, all the grids of errors are calculated by subtracting the forwardmodelled grids from the inverted grids. This T_{e} inversion recovers well the main T_{e} features in the T_{e} model. The errors are randomly distributed across the entire map, but the trend of the local errors is consistent with the T_{e} gradient belts. This consistency is due mainly to the resolution of the transition wavelets, as well as two other adverse factors: (1) the decoupling assumption and (2) the assumption of uniform T_{e} in the loading equations (Kirby and Swain 2008).
We then used the generated topography, Moho topography and inverted T_{e} grids to estimate (Eq. (7)) the lithospheric flexure (Fig. 5a) and the associated errors (Fig. 5b). The longwavelength component of the two flexure maps is basically the same (Figs. 2f and 5a), and the amplitudes of the flexure errors (Fig. 5b) are up to one order of magnitude smaller than the T_{e} errors (Fig. 4b). The T_{e} errorinduced flexure errors exhibit a rippling pattern due to the deflection of a higher T_{e} plate possessing a lower amplitude and slower amplitude decay rate (see Fig. 6), although such a rippling pattern is truncated and/or deformed by the attitudes of T_{e} and total load. Furthermore, the range of the rippling pattern is affected by the T_{e} values and load amplitude, although it has a higher correlation with the T_{e} values in this synthetic modelling study, as the rippling pattern is obviously broader in the high T_{e} regions.
To further explore the relation between the flexure errors and the T_{e} errors, we simulated another data sets with a new T_{e} model, who possesses an opposite numerical distribution pattern against the first T_{e} model, and is notated as T_{e}^{r}. More exactly, T_{e}^{r} = maximum(T_{e}) + minimum(T_{e})T_{e} (Fig. 7a). With the same procedure described above, we obtain the maps of lithospheric flexure with T_{e}^{r} (Fig. 7b), inverted T_{e}^{r} grid (Fig. 8a), T_{e}^{r} errors due to the inversion (Fig. 8b), lithospheric flexure using the inverted T_{e}^{r} grid (Fig. 9a) and lithospheric flexure errors for the model inversion using the inverted T_{e}^{r} grid (Fig. 9b).
The high T_{e} region is concentrated in the central part of the T_{e}^{r} grid (Fig. 7a), this enhances the resistance of the central area to the applied loads and make the lithospheric flexure in the corresponding areas smoother (Fig. 8b). The inverted T_{e}^{r} grid still reflects well the long wavelength part of the T_{e}^{r} model, with the errors due to the inversion (Fig. 8b) much more concentrated at the central area. However, the consistency of the T_{e}^{r} errors and the T_{e}^{r} gradient belts is obvious as before. Figure 9 is dominated by the lithospheric flexure errors of rippling pattern, as significant errors are always surrounded by a series of errors in opposite sign. Though the difference between T_{e} and T_{e}^{r} grids is great, the response of lithospheric flexure is quite similar.
Effect of Moho errors
Bouguer gravity anomaly reflects the longwavelength Moho information. It follows that the shortwavelength component of the flexurederived Moho will be the main source of error (Fig. 10b) when we use the Bouguer anomaly to invert for the Moho topography (Fig. 10a). Although the maximum Moho error is 6 km, the Moho inversion yields reliable results as the rootmeansquare error is only 0.91 km. Furthermore, the even distribution of the Moho errors provides insights into how the Moho inversion errors impact the flexure errors.
We also used the generated topography, T_{e} model and recovered (inverted) Moho topography to estimate (Eq. (7)) the lithospheric flexure (Fig. 11a) and the associated errors (Fig. 11b). A comparison of the Moho and flexure results yields the following findings. (1) The Moho inversion errors are distributed evenly across the entire region, whereas the main flexure errors occur in the low T_{e} regions. (2) The Moho errors tend to induce flexure errors with similar magnitudes but opposite signs (see the white rectangles in Figs. 10b and 11b). (3) The plate still acts as a lowpass filter to the Moho disturbance, even in the low T_{e} regions, with an increase in the intensity of filtering, where there is an increase in the T_{e} values.
Although the flexure error in the low T_{e} regions, e.g., oceanic lithosphere, is sensitive to the Moho disturbance, this method is still suitable for investigations of the oceanic lithosphere. The reference depth of the oceanic Moho is shallow, which may allow us to retain more shortwavelength information, such that more accurate inversion results can be obtained. In addition, the FD methodderived flexure is suited for Airy isostasy theory in regions, where T_{e} reaches zero. However, we have to mention that, directly apply Airy isostasy theory in some low T_{e} regions may greatly overestimate the lithospheric deformation amplitude. For example, take the extreme point whose coordinate is (2.52, 2.54)*10^{3} km in the lower white rectangle in Fig. 10b, the maximum Moho error reaches 4.258 km and we may expect a corresponding lithospheric flexure error of \( (\rho_{m}  \rho_{c} )/(\rho_{F}  \rho_{m} )*4.258 =  9.5805\) km with the Airy isostasy theory (the coefficient \( (\rho_{m}  \rho_{c} )/(\rho_{F}  \rho_{m} )\) comes from Eq. (7) by setting D to be zero). However, this extreme point in Fig. 10b gets a value of − 2.520 km, the amplitude is only 26.3% to the expected value from the Airy isostasy theory. This huge overestimation clarifies the significance of applying the FD method even in low T_{e} regions when researching the vertical lithospheric/crustal deformation.
Integrated effects of the T _{ e } and Moho errors
Neither T_{e} nor the Moho can be recovered with absolute accuracy using real data sets. We therefore investigated the integrated effects of the T_{e} and Moho errors on the estimated lithospheric flexure errors to explicate the sources of the flexure errors. We used the generated topography, recovered T_{e} and inverted Moho topography to invert (Eq. (7)) for the lithospheric flexure (Fig. 12a) and calculate their associated errors (Fig. 12b). The longwavelength component of the lithospheric flexure is still well recovered, indicating that the FD method is stable when an integrated disturbance is applied. The lithospheric flexure errors are dominated by the T_{e} errors in the high T_{e} regions (Figs. 5b, 11b and 12b). Correspondingly, the flexure errors are dominated by the Moho errors in the low T_{e} regions, although the amplitudes of the flexure errors have been somehow changed in the low T_{e} regions (Figs. 11b and 12b).
Conclusions
Here, we employed the classic lithospheric model with an applied external load at the surface and internal load at the Moho, and assumed that the compensation material was denser than the mantle material beneath the Moho, in an attempt to estimate lithospheric flexure. The resultant lithospheric model is closer to the actual lithospheric structure, with effective suppression of the overestimated lithospheric deflection predicted via linear theory, and conversion of the main lithospheric flexure inversion function (Eq. (6)) into a forward equation with a small Winkler foundation (Eq. (7)). This lithospheric model, which based on T_{e} estimates from the fan wavelet coherence method and gravityderived Moho topography, yields a finitedifference solution of the partial differential equation that recalibrates the lithospheric flexure with high accuracy. Our main conclusions are as follows:

1.
The T_{e} errorinduced flexure errors exhibit a rippling pattern, and the rippling pattern is broader in high T_{e} regions.

2.
The Moho errorinduced flexure errors mainly occur in the low T_{e} regions, and applying Airy isostasy theory in low T_{e} regions may still greatly overestimate the lithospheric deformation amplitude.

3.
The flexure errors in the low and high T_{e} regions are dominated by the Moho and T_{e} errors, respectively.
Availability of data and materials
The data sets that were used, analysed and produced during the current study are available from the corresponding author upon reasonable request.
References
Bai Y, Williams SE, Müller RD, Liu Z, Hosseinpour M (2014) Mapping crustal thickness using marine gravity data: methods and uncertainties. Geophysics 79(2):G27–G36. https://doi.org/10.1190/geo20130270.1
Banks RJ, Francis SC, Hipkin RG (2001) Effects of loads in the upper crust on estimates of the elastic thickness of the lithosphere. Geophys J Int 145:291–299. https://doi.org/10.1046/j.0956540x.2001.01380.x
Braitenberg C, Ebbing J, Gotze HJ (2002) Inverse modelling of elastic thickness by convolution method—the eastern Alps as a case example. Earth Planet Sci Lett 202:387–404. https://doi.org/10.1016/S0012821X(02)007938
Burov EB, Diament M (1992) Flexure of the continental lithosphere with multilayered rheology. Geophys J Int 109:449–468. https://doi.org/10.1111/j.1365246X.1992.tb00107.x
Chappell AR, Kusznir N (2008) Threedimensional gravity inversion for Moho depth at rifted continental margins incorporating a lithosphere thermal gravity anomaly correction. Geophys J Int 174:1–13. https://doi.org/10.1111/j.1365246X.2008.03803.x
Chen B (2013) The effective elastic thickness over China and surroundings and its lithosphere dynamic implication, Doctoral thesis, China University of Geosciences, Wuhan. (in Chinese)
Chen B, Kaban MK, El Khrepy S, AlArifi N (2015) Effective elastic thickness of the Arabian plate: weak shield versus strong platform. Geophys Res Lett 42:3298–3304. https://doi.org/10.1002/2015GL063725
Dorman LM, Lewis BTR (1970) Experimental isostasy: 1. Theory of the determination of the Earth’s isostatic response to a concentrated load. J Geophys Res 75:3357–3365. https://doi.org/10.1029/JB075i017p03357
Dorman LM, Lewis BTR (1972) Experimental isostasy: 3. Inversion of the isostatic Green function and lateral density changes. J Geophys Res 77:3068–3077. https://doi.org/10.1029/JB077i017p03068
Forsyth DW (1985) Subsurface loading and estimates of the flexural rigidity of continental lithosphere. J Geophys Res 90:12623–12632. https://doi.org/10.1029/JB090iB14p12623
Ghali A, Neville AM, Brown TG (1989) Structural analysis: a unified classical and matrix approach. Chapman and Hall, New York, p 870
Huang M, Thambiratnam D (2001) Deflection response of plate on Winkler foundation to moving accelerated loads. Eng Struct 23:1134–1141. https://doi.org/10.1016/s01410296(01)000049
Ji F, Gao J, Li F, Shen Z, Zhang Q, Li Y (2017) Variations of the effective elastic thickness over the Ross Sea and Transantarctic Mountains and implications for their structure and tectonics. Tectonophysics 717:127–138. https://doi.org/10.1016/j.tecto.2017.07.011
Ji F, Li F, Gao JY, Zhang Q, Hao WF (2018) 3D density structure of the Ross Sea basins, West Antarctica from constrained gravity inversion and their tectonic implications. Geophys J Int 215:1241–1256. https://doi.org/10.1093/gji/ggy343
Ji F, Zhang Q, Zhou X, Bai Y, Li Y (2020) Effective elastic thickness of Zealandia and its implications for lithospheric deformation. Gondwana Res 86:46–59. https://doi.org/10.1016/j.gr.2020.05.008
Kaban MK, Stolk W, Tesauro M, El Khrepy S, AlArifi N, Beekman F, Cloetingh SAPL (2016) 3D density model of the upper mantle of Asia based on inversion of gravity and seismic tomography data. Geochem Geophys Geosyst 17:4457–4477. https://doi.org/10.1002/2016GC006458
Kalnins LM, Watts AB (2009) Spatial variations in effective elastic thickness in the Western Pacific Ocean and their implications for Mesozoic volcanism. Earth Planet Sci Lett 286:89–100. https://doi.org/10.1016/j.epsl.2009.06.018
Kirby JF, Swain CJ (2004) Global and local isostatic coherence from the wavelet transform. Geophys Res Lett 31:L24608. https://doi.org/10.1029/2004GL021569
Kirby JF (2005) Which wavelet best reproduces the Fourier power spectrum? Comput Geosci 31:846–864. https://doi.org/10.1016/j.cageo.2005.01.014
Kirby JF, Swain CJ (2006) Mapping the mechanical anisotropy of the lithosphere using a 2D wavelet coherence, and its application to Australia. Phys Earth Planet Inter 158:122–138. https://doi.org/10.1016/j.pepi.2006.03.022
Kirby JF, Swain CJ (2008) An accuracy assessment of the fan wavelet coherence method for elastic thickness estimation. Geochem Geophys Geosyst 9:Q03022. https://doi.org/10.1029/2007GC001773
Kirby JF, Swain CJ (2009) A reassessment of spectral Te estimation in continental interiors: The case of North America. J Geophys Res Solid Earth 114:B08401. https://doi.org/10.1029/2009JB006356
Kirby JF, Swain CJ (2011) Improving the spatial resolution of effective elastic thickness estimation with the fan wavelet transform. Comput Geosci 37:1345–1354. https://doi.org/10.1016/j.cageo.2010.10.008
Kirby JF (2014) Estimation of the effective elastic thickness of the lithosphere using inverse spectral methods: the state of the art. Tectonophysics 631:87–116. https://doi.org/10.1016/j.tecto.2014.04.021
Lewis BTR, Dorman LM (1970a) Correction to paper by Brian T. R. Lewis and LeRoy M. Dorman “Experimental isostasy: 2 An isostatic model for the U.S.A. derived from gravity and topographic data.” J Geophys Res 75:4482–4482. https://doi.org/10.1029/JB075i023p04482
Lewis BTR, Dorman LM (1970b) Experimental isostasy: 2. An isostatic model for the U.S.A. derived from gravity and topographic data. J Geophys Res 75:3367–3386. https://doi.org/10.1029/JB075i017p03367
Lu Z, Li CF, Zhu S, Audet P (2020) Effective elastic thickness over the Chinese mainland and surroundings estimated from a joint inversion of Bouguer admittance and coherence. Phys Earth Planet Inter 301:106456. https://doi.org/10.1016/j.pepi.2020.106456
Macario A, Malinverno A, Haxby WF (1995) On the robustness of elastic thickness estimates obtained using the coherence method. J Geophys Res Atmos 100:15163–15172. https://doi.org/10.1029/95JB00980
McKenzie D (2003) Estimating Te in the presence of internal loads. J Geophys Res Solid Earth 108(2438):B9. https://doi.org/10.1029/2002JB001766
McKenzie D, Fairhead D (1997) Estimates of the effective elastic thickness of the continental lithosphere from Bouguer and free air gravity anomalies. J Geophys Res Solid Earth 102(27523–27552):B12. https://doi.org/10.1029/97JB02481
McNutt M (1980) Implications of regional gravity for state of stress in the Earth’s crust and upper mantle. J Geophys Res Solid Earth 85:6377–6396. https://doi.org/10.1029/JB085iB11p06377
Mofid M, Noroozi M (2009) A plate on winkler foundation with variable coefficient. Sci Iran Trans A 16:249–255
Oldenburg DW (1974) The inversion and interpretation of gravity anomalies. J Geophysics 39:526–536. https://doi.org/10.1190/1.1440444
PérezGussinyé M, Lowry AR, Watts AB, Velicogna I (2004) On the recovery of effective elastic thickness using spectral methods: examples from synthetic data and from the Fennoscandian shield. J Geophys Res Solid Earth 109:B10409. https://doi.org/10.1029/2003JB002788
PérezGussinyé M, Lowry A, Phipps Morgan J, Tassara A (2008) Effective elastic thickness variations along the Andean margin and their relationship to subduction geometry. Geochem Geophys Geosyst 9:Q02003. https://doi.org/10.1029/2007GC001786
PérezGussinyé M, Metois M, Fernández M, Vergés J, Fullea J, Lowry AR (2009) Effective elastic thickness of Africa and its relationship to other proxies for lithospheric structure and surface tectonics. Earth Planet Sci Lett 287:152–167. https://doi.org/10.1016/j.epsl.2009.08.004
Peitgen HO, Saupe D (eds) (1988) The science of fractal images. SpringerVerlag, New York
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in Fortran 77. Cambridge University Press, Cambridge, U.K.
Ribe NM (1982) On the interpretation of frequency response functions for oceanic gravity and bathymetry. Geophys J Int 70:273–294. https://doi.org/10.1111/j.1365246X.1982.tb04968.x
Sclater JG, Christie P (1980) Continental stretching: an explanation of the postmidcretaceous subsidence of the Central North sea basin. J Geophys Res Solid Earth 85(6377–6396):B11. https://doi.org/10.1029/JB085iB11p06377
Simons FJ, Olhede SC (2013) Maximumlikelihood estimation of lithospheric flexural rigidity, initialloading fraction and load correlation, under isotropy. Geophys J Int 193:1300–1342. https://doi.org/10.1093/gji/ggt056
Stark CP, Stewart J, Ebinger CJ (2003) Wavelet transform mapping of effective elastic thickness and plate loading: validation using synthetic data and application to the study of southern African tectonics. J Geophys Res Solid Earth 108:B12. https://doi.org/10.1029/2001JB000609
Timoshenko SP, WoinowskyKrieger S (1959) Theory of plates and shells, 2nd edn. McGrawHill, New York
Watts AB, Burov EB (2003) Lithospheric strength and its relationship to the elastic and seismogenic layer thickness. Earth Planet Sci Lett 213:113–131. https://doi.org/10.1016/S0012821X(03)002899
Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, New York
Wu Z, Gao J, Ding W, Shen Z, Zhang T, Yang C (2017) The Moho depth of the South China Sea basin from threedimensional gravity inversion with constraint points and its characteristics. Chin J Geophys 60:368–383. https://doi.org/10.1002/cjg2.30053
Zhang F, Lin J, Zhan W (2014) Variations in oceanic plate bending along the Mariana trench. Earth Planet Sci Lett 401:206–214. https://doi.org/10.1016/j.epsl.2014.05.032
Zhang F, Lin J, Zhou Z (2019a) Flexural bending curvature and yield zone of subducting plates. Int Geol Rev. https://doi.org/10.1080/00206814.2019.1671237
Zhang F, Lin J, Zhou Z (2019b) Intratrench variations in flexural bending of the subducting Pacific Plate along the TongaKermadec Trench. Acta Oceanol Sin 38:81–90. https://doi.org/10.1007/s1313101914934
Zhang F, Lin J, Zhou Z, Yang H, Zhan W (2018a) Intra and intertrench variations in flexural bending of the Manila, Mariana and global trenches: implications on plate weakening in controlling trench dynamics. Geophys J Int 212:1429–1449. https://doi.org/10.1093/gji/ggx488
Zhang J, Sun Z, Xu M, Yang H, Zhang Y, Li F (2018b) Lithospheric 3D flexural modelling of subducted oceanic plate with variable effective elastic thickness along Manila Trench. Geophys J Int 215:2071–2092. https://doi.org/10.1093/gji/ggy393
Zhang J, Xu M, Sun Z (2018c) Lithospheric flexural modelling of the seaward and trenchward of the subducting oceanic plates. Int Geol Rev. https://doi.org/10.1080/00206814.2018.1550729
Zhang J, Zhang F, Lin J, Yang H (2021) Yield failure of the subducting plate at the Mariana Trench. Tectonophysics 814:228944. https://doi.org/10.1016/j.tecto.2021.228944
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This work is supported by the Second Institute of Oceanography, Ministry of Natural Resources (Grant No. JT1601), the National Natural Science Foundation of China (Grant Nos. 42076078, 42076047, 41676037, 91858213, 41776057, and 41761134051), the National Program on Global Change and Air Sea Interaction (Grant No. GASIGEOGE05) and the Institute of Crustal Dynamics, China Earthquake Administration (Grant No. ZDJ201930).
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MX processed and interpreted the data, produced the figures and wrote the main manuscript text. ZW assisted in the gravity inversion. FJ assisted in the application of the fan wavelet coherence method. AR, CL and JG contributed to the Lithospheric model discussion and manuscript revisions. All authors read and approved the final manuscript.
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Appendix A: Finitedifference method
Appendix A: Finitedifference method
The FD solution in Eq. (5) is mainly from Chen (2013), but we use the final loads instead of the initial loads on the right side of the equation. Equation (5) can be further simplified to
The differential operations in Eq. (A1) are approximated using central difference operators. A schematic diagram of the central difference grid (Ghali et al. 1989) is shown in Fig. 13, whereby a square grid is adopted, such that \(dx=dy\). The corresponding central difference operators for the firstorder, secondorder and Laplace differential operations are shown in Fig. 14.
The approximation to the differential operations is then undertaken with the central difference operators, followed by formula expansion and combination. The following coefficients of the indicated v_{i,j} terms for an isotropic plate can then be obtained:
where \(A = \frac{1  \sigma }{8}\left( {D_{i + 1,j + 1}  D_{i  1,j + 1}  D_{i + 1,j  1} + D_{i  1,j  1} } \right)\), \(dx = dy = \lambda\), and all of the coefficients must be divided by \(\lambda^{4}\). For Eq. (7), only the first term in Eq. (A2) should be changed to
The data sets of the study area can be meshed into M × N grids, and M × N deflection values need to be calculated. Linear algebraic equations that incorporate the coefficients in Eq. (A3) can be used to calculate the deflection values as follows:
where \(\overrightarrow {v}\) is the deflection vector (a column vector with M × N rows), R is the difference coefficient matrix (a matrix with M × N columns and M × N rows), and \(\overrightarrow {h}\) and \(\overrightarrow {w}\) are the final surface and subsurface topography vectors (column vectors with M × N rows), respectively. We employed periodic boundary conditions during the calculations, and 24 kinds of boundary conditions that should be considered (Eq. A2 and Fig. 13). Furthermore, compression of the sparse matrix R is recommended, since it contains very few nonzero elements.
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Xu, MJ., Wu, ZC., Ji, F. et al. Estimating the lithospheric flexure of a plate with nonuniform flexural rigidity: a quantitative modelling approach. Earth Planets Space 73, 170 (2021). https://doi.org/10.1186/s4062302101497y
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DOI: https://doi.org/10.1186/s4062302101497y
Keywords
 Loading of the earth
 Effective elastic thickness
 Lithospheric flexure
 Error analysis