1 Introduction

Although, as a seamount chain, it is exceeded in size only by the Hawaiian-Emperor Chain, little is known about the tectonic settings of the Louisville Ridge because of its remote location. The effective elastic thickness of the lithosphere (T e) is a fundamental parameter that is sensitive to the tectonic settings of a submarine feature. There have been few attempts at calculating the T e beneath features over the Louisville Ridge (Cazenave and Dominh 1984; Watts et al. 1988; Lyons et al. 2000)—probably because it has been surveyed by few ships, and high-accuracy depth and gravity data are sparse.

Constraining with sparsely distributed seas at geoid profiles, Cazenave and Dominh (1984) employed a three-dimensional (3D) forward modeling method to estimate T e over the Louisville Ridge. The geoid heights were calculated for different values of T e using bathymetric data. T e was recovered by minimizing the misfits between modeled geoid heights and seas at geoid profiles. The resolutions of both the geoid and the bathymetry that they used were relatively low. Watts et al. (1988) estimated T e over the ridge systematically using high-resolution ship bathymetry and gravity profiles perpendicular to the ridge. However, these two studies gave contradictory results. According to Cazenave and Dominh (1984), T e increased from southeast to northwest, while Watts et al. (1988) found the opposite trend. Lyons et al. (2000) tried to reconcile these results and introduced a 3D “bathymetry-predicting” method to estimate T e. In this method, the bathymetry around a seamount is predicted using high-resolution gravity anomalies derived from Geosat altimetric data for different T e. The predicted bathymetry is then compared with in situ ship soundings. T e is then recovered by minimizing the differences between the ship soundings and the predicted bathymetry. The results of Lyons et al. (2000) tend to agree with those of Cazenave and Dominh (1984) with respect to trend, showing increasing values from southeast to northwest.

The resolution and accuracy of altimetric gravity anomalies have improved dramatically in recent years (Sandwell and Smith 2009). Kalnins and Watts (2009) introduced the moving window admittance technique (MWAT) to determine the spatial variation of T e in the western Pacific, based on general bathymetric charts of the oceans (GEBCOs) and altimetric gravity anomaly data (SIO, version V16.1). In this method, T e is estimated by 3D spectral analysis for different window sizes (400 km × 400 km to 1400 km × 1400 km), and the final T e is the weighted mean of the results for different window sizes. GEBCO is the only grid that is not based on satellite altimetry data, but its accuracy is very low and it has the problem of “terrace” (Smith 1993).

In this study, using the MWAT method, a 1° × 1° T e model over the Louisville Ridge is re-calculated, with a new bathymetry model based on vertical gravity gradient anomaly data and ship soundings (BAT_VGG) (Hu et al. 2014). For comparison, GEBCO and the SIO V15.1 model are also used. The results show that BAT_VGG is superior to GEBCO and SIO V15.1 for estimating T e. The results are discussed in detail. The correlation between T e and the age of the lithosphere at the time of loading is re-evaluated based on our estimates of T e and the newest seamount ages from Clouard and Bonneville (2005) and Koppers et al. (2004, 2011) along the Louisville Ridge. The bathymetry and locations of seamount age samples are shown in Plate 1.

Plate 1
figure a

Bathymetry and free-air gravity anomalies over the Louisville Ridge system, southwest Pacific. Bathymetry data are taken from Hu et al. (2014) (BAT_VGG). The gravity anomalies are taken from SIO V20.1. Red dots and purple square denote sample locations from Clouard and Bonneville (2005) and Koppers et al. (2004, 2011, 2012), respectively, and the ages of these samples are shown in Table 5. Red star indicates the location of a hot-spot given by Wessel et al. (2006), and the black star shows the location from Koppers et al. (2011). Red diamond shows the location of the experiment described in Sect. 3, and the red boxes around it denote the extent of the data (the size of the windows) used for estimating T e by the MWAT method

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2 Theory

The theoretical basis for estimating T e is the flexural isostatic model (Watts 2001). Figure 1 illustrates a simple flexural crust model, in which h(x) is the seafloor topography and r(x) is the flexure of the Mohorovičić discontinuity (Moho) introduced by the seamount loading. The parameters of this simple model are summarized in Table 1.

Fig. 1
figure 1

A simple flexural isostatic model

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Table 1 Summary of parameters assumed for the simple flexural isostatic model
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In the frequency domain, according to the flexural isostatic model, the flexure of the Mohocan be obtained as

$$R(k) = - H(k)\frac{{(\rho_{\text{c}} - \rho_{\text{w}} )}}{{(\rho_{\text{m}} - \rho_{\text{c}} )}}\varPhi_{\text{e}} (k).$$
(1)

Here R(k) is the Fourier transform of r(x), H(k) is the Fourier transform of h(x), k = 2π/λ is the wavenumber, λ is the wavelength, ρ m, ρ c, and ρ w are the densities of mantle, crust, and seawater, respectively, and Φ e(k) is the flexural response function of the lithosphere, given by Walcott (1976)

$$\varPhi_{\text{e}} (k) = \left[ {\frac{{Dk^{4} }}{{(\rho_{\text{m}} - \rho_{\text{c}} )g}} + 1} \right]^{ - 1} ,$$
(2)

where g is the average acceleration due to gravity and \(D = ET_{\text{e}}^{3} /[12(1 - \upsilon^{2} )]\) is the flexural rigidity of the lithosphere (E is Young’s modulus and υ is Poisson’s ratio). Combining Eq. (4) in Parker (1973) with Eqs. (1) and (2), the gravity anomaly introduced by seafloor topography and the compensation mass is given by

$$\Updelta G(k) = 2\pi G\left( {\rho_{\text{c}} - \rho_{\text{w}} } \right){\text{e}}^{ - kd} \left[ {1 - \varPhi_{\text{e}} (k){\text{e}}^{ - kt} } \right]\times\left\{ {H(k) + \sum\limits_{n = 2}^{\infty } {\frac{{k^{n - 1} }}{n!}F\left[ {h^{n} (x)} \right]} } \right\},$$
(3)

where ΔG(k) is the gravity anomaly in the frequency domain, G is the universal gravitational constant, d is the mean water depth, t is the mean crustal thickness, and F indicates the Fourier transform. Discarding higher-order terms (n ≥ 2) in Eq. (3), the admittance relationship between seafloor topography and gravity anomaly data is

$$G(k) = 2\pi G\left( {\rho_{\text{c}} - \rho_{\text{w}} } \right){\text{e}}^{ - kd} \left[ {1 - \varPhi_{\text{e}} (k){\text{e}}^{ - kt} } \right]H(k).$$
(4)

Thus, we obtain the theoretical admittance, as given by Watts (2001):

$$Z(k) = 2\pi G\left( {\rho_{\text{c}} - \rho_{\text{w}} } \right){\text{e}}^{ - kd} \left[ {1 - \varPhi_{\text{e}} (k){\text{e}}^{ - kt} } \right].$$
(5)

The theoretical admittance curves are shown in Fig. 2 for different parameter values.

Fig. 2
figure 2

Theoretical admittance curves for T e = 3, 5, 10, and 25 km. Thick blue line denotes the uncompensated admittance between bathymetry and the gravity anomaly

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According to Fig. 2, at wavelengths shorter than 50 km, the theoretical admittance does not change significantly for different T e, since the topography is uncompensated at these wavelengths. The uncompensated theoretical admittance (Z uncom(k), shown by the thick blue line in Fig. 2) is given by

$$Z_{\text{uncom}} (k) = 2\uppi G\left( {\rho_{\text{c}} - \rho_{\text{w}} } \right){\text{e}}^{ - kd} .$$
(6)

3 Method

The MWAT method introduced by Kalnins and Watts (2009) was used in this study. T e is estimated by 3D spectral analysis for different window sizes (400 km × 400 km to 1400 km × 1400 km). The final T e is computed from a weighted mean of the results for different window sizes.

As an example, over the selected point as shown in Plate 1 (the red diamond at location 156°W, 46°S), the compensated and uncompensated theoretical admittances can be calculated using Eqs. (5) and (6). The observed admittance Z′(k) can be determined from observed gravity anomaly data, ΔG′(k), and the seafloor topography model B(k) as (McNutt 1979)

$$Z^{\prime} (k) = \frac{{\langle G^{\prime} (k) \cdot B^{*} (k)\rangle }}{{\langle B(k) \cdot B^{*} (k)\rangle }},$$
(7)

where * denotes the complex conjugate, and 〈·〉 indicates annular averaging of the spectral estimates. T e can be established by minimizing the root mean square (RMS) misfits between the observed and theoretical admittances. We calculated T e in two steps. First, in the 20–50 km wave band, the uncompensated theoretical admittance is calculated using Eq. (6) for different ρ c (2300–2900 kg/cm3) and d (mean model depth ± 500 m). The values of ρ c and d can be recovered area by area by fitting the theoretical and observed admittances. Second, at wavelengths longer than 50 km, using the recovered ρ c and d, the theoretical admittance can be calculated using Eq. (5) for different T e. We obtain the optimal T e when the RMS misfit is minimized.

Over the selected point (Plate 1, red diamond), for a window size of 10° × 10°, using the 3D spectral analysis method, the best estimated T e is as shown in Fig. 3. According to Fig. 3, physically plausible values for ρ c and d can be recovered, and the best fitted T e for the selected point is 11.5 km, while the minimal RMS misfit between the observed and theoretical admittances is 5.1 mGal/km.

Fig. 3
figure 3

The RMS misfit between theoretical and observed admittances for different T e (left), and comparison of theoretical and observed admittances (right). Black dots denote the observed admittance, and the red dashed, dotted, and dot-dashed lines indicate the theoretical admittances for T e = 9, 11.5, and 15 km, respectively. The observed admittance is calculated using the software package gravfft developed by Luis and Neves (2006)

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With the MWAT method, six windows from 400 km × 400 km to 1400 km × 1400 km are used to estimate T e. For different window sizes, different spectral samples are used by gravfft to calculate the observed admittance. At the selected point (156°W, 46°S), the result is as shown in Fig. 4 and Table 2.

Fig. 4
figure 4

Best fitted T e for different window sizes. Black, red, blue, green, purple, and yellow dashed lines indicate the results for window sizes from 400 km × 400 km to 1400 km × 1400 km, respectively

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Table 2 Estimated T e and spectral samples of the observed admittance for different window sizes
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4 Data and results

4.1 Data

In order to calculate T e with the MWAT method, seafloor topography and gravity anomaly grids are needed. In this study, we use gravity anomaly data from the Scripps Institution of Oceanography, University of California, San Diego (SIO version V20.1), which are derived from satellite altimetric observations (Sandwell and Smith 2009). Three kinds of bathymetry model, GEBCO, SIO V15.1, and BAT_VGG, are used, in order to test which is the best. GEBCO is the only grid that is not based on satellite altimetry data. It is a 1-min grid prepared from bathymetric contours of the world’s oceans and was originally available as a series of paper maps at 1:10 million scale and later as digital contours in the GEBCO Digital Atlas. These maps were contoured at 500-m depth intervals, by hand, from digital and analog ship soundings (Marks and Smith 2006). SIO V15.1 was released by the SIO and was derived from ship soundings and satellite altimetric gravity anomalies (Smith and Sandwell 1994). BAT_VGG was created using ship soundings and vertical gravity gradient anomalies (Hu et al. 2014). Both GEBCO and the SIO V15.1 model have been used to estimate oceanic lithospheric T e in some published papers (Kalnins and Watts 2009; Luis and Neves 2006). The accuracy of GEBCO is significantly lower than that of SIO V15.1. At the same time, however, some authors may doubt the results if SIO V15.1 is used to recover T e using the 3D spectral analysis technique, since the bathymetry is derived from gravity anomaly data in the 15–160 km wave band.

4.2 Results

In this study, a 1° × 1° T e model is calculated over the Louisville Ridge and the adjacent regions (180°E–230°E, 60°S–20°S). T e is estimated on 2091 grid nodes. A histogram of the distribution of the minimal RMS misfits between observed and theoretical admittances is shown in Fig. 5. The statistics of the recovered crustal density and minimal RMS misfits are given in Table 3.

Fig. 5
figure 5

Frequency distribution of minimal RMS misfits between observed and theoretical admittances when using a GEBCO, b SIO V15.1, and c BAT_VGG

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Table 3 Statistics of recovered crustal density and minimal RMS misfits between observed and theoretical admittances for different bathymetry models
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In Table 3, when using BAT_VGG to calculate T e, the mean of the recovered crustal density is 2.704 g/cm3, which is consistent with the mean crustal density from CRUST2.0 (about 2.772 g/cm3), the mean of the minimal RMS misfits is 5.834 mGal/km, 32.042 % of the RMS misfits are not larger than 5 mGal/km, and more than 99 % of the RMS misfits are not larger than 10 mGal/km. These results show that BAT_VGG is superior to the other two models when calculating T e using the MWAT method.

Frequency distribution histograms of T e are shown in Fig. 6 for the different bathymetry models used. According to Fig. 6c, most of the values of T e over the Louisville Ridge and its adjacent regions are less than 15 km.

Fig. 6
figure 6

Frequency distribution histograms of T e over the Louisville Ridge and the adjacent regions for different bathymetry models: a GEBCO, b SIO V15.1, and c BAT_VGG

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T e estimated using BAT_VGG is shown in Fig. 7, from which it can be seen that T e lies in the range 0–50 km, with a mean of 11.924 km and a standard deviation of 10.174 km. In the northwest of the study area, the estimated T e is clearly larger than elsewhere. The Louisville Ridge system has a medium value of T e (10–20 km). In general, T e over the basins both at the northeast and the southwest of the ridge is less than 10 km.

Fig. 7
figure 7

T e estimated using the bathymetry model BAT_VGG and the altimetric gravity anomalies from SIO V20.1 with the MWAT method. Black dots indicate locations where seamount ages are sampled

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In the study area, the T e of the lithosphere under 609 seamounts was estimated by Watts et al. (2006) using a bathymetry predicting method. The differences between their results and those of this study are shown in Fig. 8. Most of the absolute differences are less than 10 km, with a mean of −1.6 km and a standard deviation of 4.5 km.

Fig. 8
figure 8

Differences between the results for T e at some seamounts from Watts et al. (2006) and the results from this study

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Along the Louisville Ridge, T e of the lithosphere was estimated by Cazenave and Dominh (1984), Watts et al. (1988), and Lyons et al. (2000). For comparison, the MWAT method is used here to calculate T e of regions A-L of Lyons et al. (2000). The results are summarized in Table 4. According to Table 4 and Fig. 7, T e along the Louisville Ridge is usually less than 15 km, except for the Kermadec outer rise, where it is larger than 20 km. This may due to the dynamic effect of plate subduction and the use of the wrong window size in the MWAT method. For profiles 1–4 in Watts et al. (1988), near the trench, if a 4° × 4° window is used with the MWAT method, the best fitted T e will be 10.5, 8.5, 10, and 13 km, respectively, which are consistent with the values of T e given by Watts et al. In the southeast of the ridge, our results are consistent with those of Lyons et al. (2000). The values of T e on the ridge show no trend like that in the Hawaiian-Emperor seamount chain.

Table 4 Best fitting T e from Cazenave and Dominh (1984), Watts et al. (1988), Lyons et al. (2000), and this study
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5 Discussion and conclusions

Previous studies of oceanic lithospheric effective T e suggest that the strength of the lithosphere under seamounts and islands depends strongly on the age at the time of loading (Watts 1978, 2001; Calmant et al. 1990). The precise relationship recovered between T e and age at time of loading varies and there is no single isotherm that controls T e on a global scale (Kalnins and Watts 2009). Within the study area, we have collected 33 sampled seamounts whose ages are known (Clouard and Bonneville 2005; Koppers et al. 2004, 2011). The values of T e on these seamounts are interpolated from the T e model shown in Fig. 7. Seafloor ages under these seamounts are interpolated from Müller et al. (2008). The locations of the sampled seamounts and the estimated values of T e are collected in Table 5.

Table 5 Ages of seamounts and seafloor, and T e estimated over the Louisville Ridge
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According to the plate cooling model (Parsons and Sclater 1977; Stein and Stein 1992), the lithosphere will get colder and stronger further away from the mid-ocean ridge. Many studies have suggested that the lithospheric T e is to the first order determined by the age of lithosphere at the time of loading, and is given approximately by the depth to the 450 ± 150 °C isotherm (Watts 1978, 2001; Calmant et al. 1990).

Figure 9 shows the relationship between T e and the age of the lithosphere at the time of loading over the Louisville Ridge. The relationships over the Hawaiian-Emperor Chain and the Line Seamounts are also given for comparison.

Fig. 9
figure 9

T e versus the age of the oceanic lithosphere at the time of loading. Lines P&S and GDH1 represent the cooling plate models given by Parsons and Sclater (1977) and Stein and Stein (1992), respectively, blue triangles, red circles, and purple squares denote seamounts on the Hawaiian-Emperor Chain, the Louisville Ridge, and the Line Seamounts, respectively. The values of T e over the Hawaiian-Emperor Chain and the Line Seamounts were calculated by the authors using the same data and method as in this paper

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According to Fig. 9, in the study area, the dependence of T e on the age of the oceanic lithosphere at the time of loading is given mostly by the depth to the 150 °C–300 °C isotherm based on a cooling plate model. The values of T e over the Louisville Ridge are larger than those over the Line Ridge and lower than those over the Hawaiian-Emperor Chain. The T e of the lithosphere does not increase with the age of the lithosphere at the time of loading. These results indicate that T e is not controlled only by the age of the lithosphere at the time of loading. The eight samples in the northwest of the ridge show T e larger than 20 km. We attribute this to the dynamic effects of plate subduction. The most significant topography loads are the islands arc at the west of the trench. These loads are supported by plate subduction. But, when we calculate T e using MWAT method, it seems like these loads are supported by the strength of lithosphere. Therefore, we must be careful if the MWAT method is used to calculate the T e of the trench outer rise.