Inversion of Geothermal Heat Flux under the Ice Sheet of Princess Elizabeth Land, East Antarctica

: Antarctic geothermal heat ﬂux is a basic input variable for ice sheet dynamics simulation. It greatly affects the temperature and mechanical properties at the bottom of the ice sheet, inﬂuencing sliding, melting, and internal deformation. Due to the fact that the Antarctica is covered by a thick ice sheet, direct measurements of heat ﬂux are very limited. This study was carried out to estimate the regional heat ﬂux in the Antarctic continent through geophysical inversion. Princess Elizabeth Land, East Antarctica is one of the areas in which we have a weak understanding of geothermal heat ﬂux. Through the latest airborne geomagnetic data, we inverted the Curie depth, obtaining the heat ﬂux of bedrock based on the one-dimensional steady-state heat conduction equation. The results indicated that the Curie depth of the Princess Elizabeth Land is shallower than previously estimated, and the heat ﬂux is consequently higher. Thus, the contribution of subglacial heat ﬂux to the melting at the bottom of the ice sheet is likely greater than previously expected in this region. It further provides research clues for the formation of the developed subglacial water system in Princess Elizabeth Land. aeromagnetic data from PEL. The single ﬂight distance of the geophysical and glacier survey, carried out by the Snow Eagle 601 ﬁxed-wing aircraft, was approximately 1700 km (6.5 h of endurance), and the total distance of the survey lines used in this study were about 47,000 km (Figure 1). During the aerial survey, ﬂight altitude was maintained at 600 m above the ice surface.


Introduction
The Antarctic continent is covered by an ice sheet, which is a potential main driver of global sea-level changes. Heat flux of the bedrock significantly affects the melting of the ice sheet bottom, which may influence the ice sheet mass balance. Geothermal heat flux (GHF) is an important boundary condition of ice sheet dynamical models. For the areas below freezing point, a higher heat flux causes the ice bottom to heat up, reducing its viscosity and increasing its lubrication, and significantly accelerating the ice sheet flow [1]. Heat flux can also simulate past basic melting rates and help to explore the climate record of old ice cores [2]. Studies showed that East Antarctica has a smaller heat flux than the West Antarctica. This may be one of the reasons why the West Antarctic ice sheet is changing more rapidly [3]. However, some drilling results showed that the observed heat flux in some areas far exceeded the generally estimated heat flux by the GHF model [4][5][6]. The local subglacial thermal conditions have a greater impact on local ice sheet movement, which cannot be concealed by the low-resolution estimation results [7]. Recent studies showed that the surface mass loss of the ice sheet in Princess Elizabeth Land (PEL) is not as obvious as it is at the bottom [8]. The main contributing factor of the ice sheet material loss in this area may be obtained via ice sheet bottom analysis. Therefore, in order to more accurately model ice velocities and their evolution, and to learn more about the geology in PEL, it is necessary to obtain a more detailed understanding of the distribution of heat flux under the ice.
At present, there are three main methods for obtaining Antarctica heat flux information: (1) direct measurement of the geothermal gradient through ice boreholes or sediments on the coast [6,9,10]; (2) inversion by multiple geophysical methods, such as seismology, geomagnetism, and glacier dynamics [11][12][13][14][15]; and (3) using ice radar and the temperature gradient of the ice sheet or the ice bottom pressure melting point to calculate the distributions of heat flux [16,17]. Due to the rapid flow, ice sheet evolution is much faster than that of bedrock. However, it is difficult to calculate the thermal conditions of the ice bottom based on the temperature gradient of the ice sheet [17,18]. In addition, the ice sheet is very thick, and direct measurement of geothermal heat flux is very challenging [6]. Seismological methods estimate the subglacial thermal structure based on the thermal sensitivity of seismic properties. The disadvantage of this method is that the uneven distribution of seismic stations leads to the low resolution of seismic models, and many seismic parameters have significant uncertainties [15]. It is, therefore, a relatively common and efficient method to invert the heat flux through aerial geomagnetic observation.
Uncertainty surrounding the results obtained by the geomagnetic field mainly originate from the coverage of the data set and the reliability of the inversion method [1]. Available internationally in the public domain, the Antarctic magnetic data, known as the ADMAP2 data set, was released in 2018 and covers most of the Antarctica [19,20]. However, some areas still have large data gaps, especially in PEL, which was inverted by satellite magnetic data with a lower resolution. In the Martos Curie depth model [1], the maximum uncertainty value (~8 km) in this region is about 23% of the total Antarctic mean value (~34 km), which is higher than other regions.

Aeromagnetic Data
Through the operation of China's first Antarctic fixed-wing aircraft observation platform, we obtained large-scale aeromagnetic data from PEL. The single flight distance of the geophysical and glacier survey, carried out by the Snow Eagle 601 fixed-wing aircraft, was approximately 1700 km (6.5 h of endurance), and the total distance of the survey lines used in this study were about 47,000 km ( Figure 1). During the aerial survey, flight altitude was maintained at 600 m above the ice surface. to more accurately model ice velocities and their evolution, and to learn more about the geology in PEL, it is necessary to obtain a more detailed understanding of the distribution of heat flux under the ice.
At present, there are three main methods for obtaining Antarctica heat flux information: (1) direct measurement of the geothermal gradient through ice boreholes or sediments on the coast [6,9,10]; (2) inversion by multiple geophysical methods, such as seismology, geomagnetism, and glacier dynamics [11][12][13][14][15]; and (3) using ice radar and the temperature gradient of the ice sheet or the ice bottom pressure melting point to calculate the distributions of heat flux [16,17]. Due to the rapid flow, ice sheet evolution is much faster than that of bedrock. However, it is difficult to calculate the thermal conditions of the ice bottom based on the temperature gradient of the ice sheet [17,18]. In addition, the ice sheet is very thick, and direct measurement of geothermal heat flux is very challenging [6]. Seismological methods estimate the subglacial thermal structure based on the thermal sensitivity of seismic properties. The disadvantage of this method is that the uneven distribution of seismic stations leads to the low resolution of seismic models, and many seismic parameters have significant uncertainties [15]. It is, therefore, a relatively common and efficient method to invert the heat flux through aerial geomagnetic observation.
Uncertainty surrounding the results obtained by the geomagnetic field mainly originate from the coverage of the data set and the reliability of the inversion method [1]. Available internationally in the public domain, the Antarctic magnetic data, known as the AD-MAP2 data set, was released in 2018 and covers most of the Antarctica [19,20]. However, some areas still have large data gaps, especially in PEL, which was inverted by satellite magnetic data with a lower resolution. In the Martos Curie depth model [1], the maximum uncertainty value (~8 km) in this region is about 23% of the total Antarctic mean value (~34 km), which is higher than other regions.

Aeromagnetic Data
Through the operation of China's first Antarctic fixed-wing aircraft observation platform, we obtained large-scale aeromagnetic data from PEL. The single flight distance of the geophysical and glacier survey, carried out by the Snow Eagle 601 fixed-wing aircraft, was approximately 1700 km (6.5 h of endurance), and the total distance of the survey lines used in this study were about 47,000 km ( Figure 1). During the aerial survey, flight altitude was maintained at 600 m above the ice surface.  The CS-3 magnetometer was used to measure the magnetic field. It was installed on the tail of the aircraft to avoid magnetic interference from the metal fuselage. The CS-3 magnetometer has high sensitivity and low noise interference, and its measurement range can be used for detection in most areas of the world. The magnetometer system consists of a Scintrex CS-3 scalar magnetometer, an RMS AARC510 adaptive aviation magnetic real-time compensator, and a Billingsley TFM100-G2 carrier magnetometer. During a survey, the CS-3 magnetometer measures the total magnetic field (with a sampling interval of 1 s, flight speed of 300 km/h, and sampling interval distance of~83 m along the flight line), and the AARC510, with a three-axis fluxgate vector magnetometer, provides compensation for the influence of the flight direction and altitude.
The geomagnetic diurnal station (used for diurnal correction of the aeromagnetic survey) uses an EREV-1 proton magnetometer, located at the airport near the Zhongshan Station, with an absolute accuracy of less than 0.3 nT and a sampling interval of 1 s. In addition, a perennial geomagnetic observatory (with a sampling interval of 1 s) at the Zhongshan Station, 10 km away from the airport, provides backup data for diurnal observations.
We used the GRIDSTCH module of Oasis Montaj software for observational data fusion with ADMAP2. This method provides smooth blending without over-smoothing high-frequency variations that may occur along the suture path ( Figure 2). The CS-3 magnetometer was used to measure the magnetic field. It was installed on the tail of the aircraft to avoid magnetic interference from the metal fuselage. The CS-3 magnetometer has high sensitivity and low noise interference, and its measurement range can be used for detection in most areas of the world. The magnetometer system consists of a Scintrex CS-3 scalar magnetometer, an RMS AARC510 adaptive aviation magnetic real-time compensator, and a Billingsley TFM100-G2 carrier magnetometer. During a survey, the CS-3 magnetometer measures the total magnetic field (with a sampling interval of 1 s, flight speed of 300 km/h, and sampling interval distance of ~83 m along the flight line), and the AARC510, with a three-axis fluxgate vector magnetometer, provides compensation for the influence of the flight direction and altitude.
The geomagnetic diurnal station (used for diurnal correction of the aeromagnetic survey) uses an EREV-1 proton magnetometer, located at the airport near the Zhongshan Station, with an absolute accuracy of less than 0.3 nT and a sampling interval of 1 s. In addition, a perennial geomagnetic observatory (with a sampling interval of 1 s) at the Zhongshan Station, 10 km away from the airport, provides backup data for diurnal observations.
We used the GRIDSTCH module of Oasis Montaj software for observational data fusion with ADMAP2. This method provides smooth blending without over-smoothing high-frequency variations that may occur along the suture path ( Figure 2).

Using Modified Centroid Method to Invert the Curie Depth
The geomagnetic field is mainly produced in the ferromagnetic strata that exist in the Earth's crust. After reaching a certain temperature, the ferromagnetic material will favor paramagnetism, which will significantly lower its magnetic field. At this temperature (580 °C for ferromagnetic ore [21]), the corresponding lithospheric interface is called the Curie isothermal interface, which is generally considered to be the bottom of the magnetic strata. Magnetic data can invert the large magnetic change interface and, thus, we can determine the Curie depth [22].

Using Modified Centroid Method to Invert the Curie Depth
The geomagnetic field is mainly produced in the ferromagnetic strata that exist in the Earth's crust. After reaching a certain temperature, the ferromagnetic material will favor paramagnetism, which will significantly lower its magnetic field. At this temperature (580 • C for ferromagnetic ore [21]), the corresponding lithospheric interface is called the Curie isothermal interface, which is generally considered to be the bottom of the magnetic strata. Magnetic data can invert the large magnetic change interface and, thus, we can determine the Curie depth [22]. We used the centroid method to calculate the Curie depth. This method is based on the existence of a linear relationship between the spectral power density and specific wave number ranges. Compared with the defractal spectral method adopted by the Martos model, this method has very similar results under the same parameters [23]. In this method, it is assumed that the low wavenumber area of the power spectrum is caused by the deep field source, and the high wavenumber area is caused by the shallow field source. The Curie depth is considered to be the bottom interface of the deep field source, and, thus, it can be obtained through power spectrum analysis [24]: where A(k) is the radially averaged amplitude spectrum, and E is the constant related to the magnetization direction and geomagnetic field direction. First, we calculated the top surface depth of the deep field source. Then, we determined a fitting straight line from the high wavenumber region of the power spectrum, and calculated the depth of the deep field source top interface, Z t , through the relationship with the slope of the straight line. Assuming F is the constant, then Equation (1) is approximated to: Second, we calculated the depth of the centroid of the deep field source. We determined a fitting straight line from the low wavenumber region of the power spectrum, and calculated the centroid depth of the deep field source, Z 0 , through the relationship with the slope of the straight line. Assuming G is the constant: Finally, the depth of the bottom interface of the deep field source was calculated by the following simple positional relationship, calculating the Curie depth Z b : The uncertainty associated with this method is calculated by [25]: ∆Z t , ∆Z 0 and ∆Z b are the uncertainty of the top, the centroid, and the base of the magnetic source, respectively. The values of ∆Z 0 and ∆Z t were computed using the standard deviation (σ r ) derived from the calculated power spectrum, and the linear fit in the wavenumber range was used to derive the slope (k 2 − k 1 ) [26]. This uncertainty ( ) is: For the division of the deep and the shallow field sources in the wavenumber range, manual intervention must be processed using digital methods, instead of automatic division, to reduce the error. Since the fractal parameters β are unknown, in the case of wavenumber range participating in the matching has been determined, the observed power spectrum and the random magnetization model power spectrum can be optimally matched by continuously adjusting β (with an adjustment interval of 0.1). Only then will the β reflect the geological structure more truly.
The goodness of fit is evaluated using the following equation [23]: where A f (k) is the observed power spectrum and A syn (k) is the model power spectrum. The smaller the R value, the better the fit. Manual adjustment is required in the process of adjusting the wavenumber range and β, which can reduce the uncertainty related to β. The relationship between the Curie depth and geothermal gradient can be approximated by the one-dimensional steady-state heat conduction equation, which can calculate the approximate geothermal heat flux of the Antarctic subglacial bedrock.
Assuming that there is no lateral heat production and conduction in the bedrock under the ice sheet, and that the longitudinal thermal conductivity is constant, if q is heat flux, z is depth, λ is thermal conductivity, and T is temperature, then the one-dimensional steady-state heat conduction equation will be: Assuming that Z b is Curie depth, T c . is Curie temperature, T 0 . is surface temperature, H 0 is the surface heat production, h r is the depth for heat production, and q s is the surface geothermal heat flux, the heat conduction equation will be [27][28][29]:

Results
According to the previous approximate distribution of the Curie depth inversion results for magnetic data in the Antarctica [1], we divided the observation area into 16 windows and calculated the Curie depth separately in each window. The size of each window was set to 400 × 400 km, which meant a 50% overlap between adjacent windows, as shown in Figure 3. The calculation result of each window corresponded to the center point. After data fusion, each piece of data was processed according to the larger spacing grid of ADMAP2 (1.5 km spacing) to ensure data accuracy. The relationship between the Curie depth and geothermal gradient can be approximated by the one-dimensional steady-state heat conduction equation, which can calculate the approximate geothermal heat flux of the Antarctic subglacial bedrock.
Assuming that there is no lateral heat production and conduction in the bedrock under the ice sheet, and that the longitudinal thermal conductivity is constant, if is heat flux, is depth, is thermal conductivity, and is temperature, then the one-dimensional steady-state heat conduction equation will be: Assuming that is Curie depth, is Curie temperature, is surface temperature, is the surface heat production, is the depth for heat production, and is the surface geothermal heat flux, the heat conduction equation will be [27][28][29]:

Results
According to the previous approximate distribution of the Curie depth inversion results for magnetic data in the Antarctica [1], we divided the observation area into 16 windows and calculated the Curie depth separately in each window. The size of each window was set to 400 × 400 km, which meant a 50% overlap between adjacent windows, as shown in Figure 3. The calculation result of each window corresponded to the center point. After data fusion, each piece of data was processed according to the larger spacing grid of AD-MAP2 (1.5 km spacing) to ensure data accuracy.  In Figure 4, using window 3 as an example, the process of artificially dividing the wavenumber ranges, the fitting state of the observed power spectrum, and the simulated power spectrum are shown.
Remote Sens. 2021, 13, x FOR PEER REVIEW 6 of 11 In Figure 4, using window 3 as an example, the process of artificially dividing the wavenumber ranges, the fitting state of the observed power spectrum, and the simulated power spectrum are shown.  According to the radial power spectrum of PEL, the high wavenumber range is generally concentrated between 0.05 and 0.15, and the low wavenumber range is concentrated between 0.01 and 0.03. The difference between the observed power spectrum and the simulated power spectrum R was between 0.05 and 0.15.
In the process of calculating the heat flux from the Curie depth, the selection of geothermal parameters refers to the previous inference results [1]. It is generally believed that the surface temperature, , is ~0 °C, the surface heat production, , is ~2.5 mW/m 3 , the thermal conductivity, , is ~2.8 W/mK, and the depth for heat production, , is ~8 km [30]. The Curie temperature, , refers to a constant value of 580 °C. According to the Curie depth results, 16 areas were combined with the one-dimensional steady-state heat conduction equation to calculate the geothermal gradient and heat flux. Using window 3 as an example, the calculation results of regional geothermal gradient and heat flux are shown in Figure 5.   According to the radial power spectrum of PEL, the high wavenumber range is generally concentrated between 0.05 and 0.15, and the low wavenumber range is concentrated between 0.01 and 0.03. The difference between the observed power spectrum and the simulated power spectrum R was between 0.05 and 0.15.
In the process of calculating the heat flux from the Curie depth, the selection of geothermal parameters refers to the previous inference results [1]. It is generally believed that the surface temperature, T 0 , is~0 • C, the surface heat production, H 0 , is~2.5 mW/m 3 , the thermal conductivity, λ, is~2.8 W/mK, and the depth for heat production, h r , is 8 km [30]. The Curie temperature, T c , refers to a constant value of 580 • C. According to the Curie depth results, 16 areas were combined with the one-dimensional steady-state heat conduction equation to calculate the geothermal gradient and heat flux. Using window 3 as an example, the calculation results of regional geothermal gradient and heat flux are shown in Figure 5. In Figure 4, using window 3 as an example, the process of artificially dividing the wavenumber ranges, the fitting state of the observed power spectrum, and the simulated power spectrum are shown. According to the radial power spectrum of PEL, the high wavenumber range is generally concentrated between 0.05 and 0.15, and the low wavenumber range is concentrated between 0.01 and 0.03. The difference between the observed power spectrum and the simulated power spectrum R was between 0.05 and 0.15.
In the process of calculating the heat flux from the Curie depth, the selection of geothermal parameters refers to the previous inference results [1]. It is generally believed that the surface temperature, , is ~0 °C, the surface heat production, , is ~2.5 mW/m 3 , the thermal conductivity, , is ~2.8 W/mK, and the depth for heat production, , is ~8 km [30]. The Curie temperature, , refers to a constant value of 580 °C. According to the Curie depth results, 16 areas were combined with the one-dimensional steady-state heat conduction equation to calculate the geothermal gradient and heat flux. Using window 3 as an example, the calculation results of regional geothermal gradient and heat flux are shown in Figure 5.  The Curie depth range of East Antarctica in the Martos model was from 22 to 63 km. From our inversion results, the Curie depth of PEL was observed to be shallower than previously estimated depths, most of which varied in the range of 23-48 km, with an average depth of about 34 km, and a calculation error of ±4 km. The heat flux map showed that Queen Mary Land (Label A in Figure 7b) had the same magnitude of high geothermal heat flux as the region closer to the western ice shelf, which is consistent with the previous seismic model inversion results [31,32]. Overall, the heat flux was larger than previously estimated in the eastern area of PEL. Figure 7 indicates values ranging from 51 to 84 mW/m 2 , with an average of 66 mW/m 2 and an error of ±7 mW/m 2 , which is close to the global continental average of 65 mW/m 2 [7]. By comparison, the model of Martos et al. had a heat flow value of 45-85 mW/m 2 in East Antarctica [1], which is slightly lower than our calculation result. The Curie depth range of East Antarctica in the Martos model was from 22 to 63 km. From our inversion results, the Curie depth of PEL was observed to be shallower than previously estimated depths, most of which varied in the range of 23-48 km, with an average depth of about 34 km, and a calculation error of ±4 km. The heat flux map showed that Queen Mary Land (Label A in Figure 7b) had the same magnitude of high geothermal heat flux as the region closer to the western ice shelf, which is consistent with the previous seismic model inversion results [31,32]. Overall, the heat flux was larger than previously estimated in the eastern area of PEL. Figure 7 indicates values ranging from 51 to 84 mW/m 2 , with an average of 66 mW/m 2 and an error of ±7 mW/m 2 , which is close to the global continental average of 65 mW/m 2 [7]. By comparison, the model of Martos et al. had a heat flow value of 45-85 mW/m 2 in East Antarctica [1], which is slightly lower than our calculation result. The Curie depth range of East Antarctica in the Martos model was from 22 to 63 km. From our inversion results, the Curie depth of PEL was observed to be shallower than previously estimated depths, most of which varied in the range of 23-48 km, with an average depth of about 34 km, and a calculation error of ±4 km. The heat flux map showed that Queen Mary Land (Label A in Figure 7b) had the same magnitude of high geothermal heat flux as the region closer to the western ice shelf, which is consistent with the previous seismic model inversion results [31,32]. Overall, the heat flux was larger than previously estimated in the eastern area of PEL. Figure 7 indicates values ranging from 51 to 84 mW/m 2 , with an average of 66 mW/m 2 and an error of ±7 mW/m 2 , which is close to the global continental average of 65 mW/m 2 [7]. By comparison, the model of Martos et al. had a heat flow value of 45-85 mW/m 2 in East Antarctica [1], which is slightly lower than our calculation result.

Discussion
Previous satellite altimetry studies showed that there is an undiscovered, large subglacial drainage network hidden under the ice sheet of PEL. A series of long, deep canyons from Gamburtsev Mountain extend to the coast of the West Ice Shelf (Label B in Figure 7b). These canyons contain large amounts of subglacial water and lakes [33]. The aerial ice radar detection results of Snow Eagle 601 also confirmed this conclusion [34,35]. The North-South rift zones were formed during the Cenozoic crustal uplift and through volcanism in the area [7,36,37]. The rift system is believed to be an area where the geothermal heat flux is significantly increased [38]. In our calculation results, there is a high-low-high change trend in this area. In the process of rift formation, the influence of complex tectonism on the surrounding strata, such as the formation of metamorphic rocks caused by ductile shear zone and nappe structure, can be used to explain the origin of high and low heat flux in this area [39,40]. The high heat flux region described by this study (Figure 7b) may represent the product of orogeny and rift extension, as well as possible "thermal events" [41] during the evolution of the Antarctic continent [42,43].
Furthermore, there is a clear correlation between the subglacial water system and the heat flux [44,45]. Temperature profile analysis in ice boreholes showed that the bedrocks are warm at sites where subglacial water exists [18]. Geothermal heat flux of 120 ± 20 mW/m 2 can cause basal melting of up to 6 ± 1 mm/a [46]. Jamieson believed that the canyons reduced the likelihood of rapid ice flow due to the fact that rather than concentrating the water at one point and enhancing the sliding of the ice bottom, they capturing the water body so it cannot lubricate the ice-bed interface over a wide area [33]. The detailed distribution of water bodies under ice need to be analyzed by calculation of ice radar reflectivity. The heat flux distribution in this study may encourage further explanation of the distribution and cause of the subglacial water system in this area.
In general, the geomagnetic method is an efficient method for estimating heat flux. The inversion of the Curie depth, however, induces error due to the assumption of structural division, and it becomes difficult to divide the low wavenumber range, which represents the deep magnetic field source. The calculation of heat flux may also contain some errors due to the uneven heat production. The survey line spacing in the southeastern part of the study area close to Vostok is relatively large. It is greater than 100 km at the widest point, which may cause an error of more than 10 km in the estimation of the Curie depth. In order to improve the calculation results, it is necessary to carry out further survey lines of airborne geophysical observation and deeper borehole measurements. An integrated study of the regional geological structure of PEL in combination with geophysical inversion would provide more accurate information about the subglacial thermal environment in this region.

Conclusions
Based on the aeromagnetic data of PEL obtained by China's fixed-wing aircraft observation platform, and by using the modified centroid method and the one-dimensional steady-state heat conduction equation, we obtained geothermal heat flux data on PEL's ice sheet bottom. The Curie depth result was shallower than previously estimated and the heat flux was higher, showing that the contribution of subglacial heat flux to the melting of the ice sheet in this area is greater than previously calculated, which may be an important reason for the development of the subglacial water system in PEL.
Our calculation results showed that the high geothermal heat flux is better correlated with the melting of the bottom ice and the distribution of water in this area. GHF inference provided relatively accurate boundary conditions for simulating the dynamics and evolution of the ice sheet, which may help in estimating the bottom melting rate of the East Antarctic ice sheet and its impact on sea-level rise.

Data Availability Statement:
The data presented in this study are available upon request from the corresponding author.