remote sensing

Technical Note Inversion of Geothermal Heat Flux under the Ice Sheet of , East

Lin Li 1, Xueyuan Tang 1,* , Jingxue Guo 1, Xiangbin Cui 1 , Enzhao Xiao 1, Khalid Latif 2, Bo Sun 1, Qiao Zhang 3 and Xiaosong Shi 1

1 Polar Research Institute of China, Shanghai 200136, China; [email protected] (L.L.); [email protected] (J.G.); [email protected] (X.C.); [email protected] (E.X.); [email protected] (B.S.); [email protected] (X.S.) 2 National Centre of Excellence in Geology, University of Peshawar, Peshawar 25130, Pakistan; [email protected] 3 Institute of Geomechanics, Chinese Academy of Geological Sciences, Beijing 100081, China; [email protected] * Correspondence: [email protected]

Abstract: 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, 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

Citation: Li, L.; Tang, X.; Guo, J.; Cui, that the Curie depth of the Princess Elizabeth Land is shallower than previously estimated, and the X.; Xiao, E.; Latif, K.; Sun, B.; Zhang, heat flux is consequently higher. Thus, the contribution of subglacial heat flux to the melting at the Q.; Shi, X. Inversion of Geothermal bottom of the ice sheet is likely greater than previously expected in this region. It further provides Heat Flux under the Ice Sheet of research clues for the formation of the developed subglacial water system in Princess Elizabeth Land. Princess Elizabeth Land, East Antarctica. Remote Sens. 2021, 13, Keywords: heat flux; Curie depth; aeromagnetic; subglacial geothermal; East Antarctica 2760. https://doi.org/10.3390/ rs13142760

Academic Editor: Hanwen Yu 1. Introduction The Antarctic continent is covered by an ice sheet, which is a potential main driver of Received: 4 June 2021 global sea-level changes. Heat flux of the bedrock significantly affects the melting of the ice Accepted: 9 July 2021 Published: 14 July 2021 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

Publisher’s Note: MDPI stays neutral freezing point, a higher heat flux causes the ice bottom to heat up, reducing its viscosity with regard to jurisdictional claims in and increasing its lubrication, and significantly accelerating the ice sheet flow [1]. Heat published maps and institutional affil- flux can also simulate past basic melting rates and help to explore the climate record of iations. 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 [4–6]. The local subglacial thermal conditions have a greater impact on local ice sheet movement, Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. which cannot be concealed by the low-resolution estimation results [7]. Recent studies This article is an open access article showed that the surface mass loss of the ice sheet in Princess Elizabeth Land (PEL) is not distributed under the terms and as obvious as it is at the bottom [8]. The main contributing factor of the ice sheet material conditions of the Creative Commons loss in this area may be obtained via ice sheet bottom analysis. Therefore, in order to more Attribution (CC BY) license (https:// accurately model ice velocities and their evolution, and to learn more about the geology in creativecommons.org/licenses/by/ PEL, it is necessary to obtain a more detailed understanding of the distribution of heat flux 4.0/). under the ice.

Remote Sens. 2021, 13, 2760. https://doi.org/10.3390/rs13142760 https://www.mdpi.com/journal/remotesensing Remote Sens. 2021, 13, x FOR PEER REVIEW 2 of 11

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. Remote Sens. 2021, 13, 2760 2 of 10 At present, there are three main methods for obtaining Antarctica heat flux infor- mation: (1) direct measurement of the geothermal gradient through ice boreholes or sedi- ments on the coast [6,9,10]; (2) inversion by multiple geophysical methods, such as seis- mology,At present, geomagnetism, there are three and mainglacier methods dynamics for obtaining [11–15]; Antarcticaand (3) using heat fluxice radar informa- and the tion: (1) direct measurement of the geothermal gradient through ice boreholes or sediments temperature gradient of the ice sheet or the ice bottom pressure melting point to calculate on the coast [6,9,10]; (2) inversion by multiple geophysical methods, such as seismology, geomagnetism,the distributions and of glacier heat flux dynamics [16,17]. [11 Due–15 ];to and the (3)rapid using flow, ice radarice sheet and evolution the tempera- is much turefaster gradient than that of theof bedrock. ice sheet However, or the ice bottomit is difficult pressure to calculate melting pointthe thermal to calculate conditions the of distributionsthe ice bottom of heatbased flux on [ 16the,17 temperature]. Due to the rapidgradient flow, of ice the sheet ice sheet evolution [17,18]. is much In addition, faster the thanice sheet that is of very bedrock. thick, However, and direct it measuremen is difficult tot calculate of geothermal the thermal heat flux conditions is very challenging of the ice[6]. bottom Seismological based on methods the temperature estimate gradient the subgla of thecial ice thermal sheet [structure17,18]. In addition,based on thethe icethermal sheetsensitivity is very of thick, seismic and directproperties. measurement The disadvanta of geothermalge of heatthis method flux is very is that challenging the uneven [6]. dis- Seismologicaltribution of seismic methods stations estimate leads the to subglacial the low resolution thermal structure of seismic based models, on the and thermal many seis- sensitivitymic parameters of seismic have properties. significant The uncertainties disadvantage [15]. of It this is, methodtherefore, is thata relatively the uneven common distributionand efficient of method seismic to stations invert leads the heat to the flux low through resolution aerial of seismicgeomagnetic models, observation. and many seismicUncertainty parameters surrounding have significant the uncertainties results obtained [15]. Itby is, the therefore, geomagnetic a relatively field common mainly origi- andnate efficient from the method coverage to invert of the the data heat set flux and through the reliability aerial geomagnetic of the inversion observation. method [1]. Avail- Uncertainty surrounding the results obtained by the geomagnetic field mainly orig- able internationally in the public domain, the Antarctic magnetic data, known as the AD- inate from the coverage of the data set and the reliability of the inversion method [1]. MAP2 data set, was released in 2018 and covers most of the Antarctica [19,20]. However, Available internationally in the public domain, the Antarctic magnetic data, known as the ADMAP2some areas data still set, have was releasedlarge data in 2018 gaps, and especially covers most in ofPEL, the Antarcticawhich was [19 inverted,20]. However, by satellite somemagnetic areas data still with have a large lower data resolution. gaps, especially In the Martos in PEL, Curie which depth was invertedmodel [1], by the satellite maximum magneticuncertainty data value with a(~8 lower km) resolution. in this region In the is Martos about Curie 23% depthof the model total [Antarctic1], the maximum mean value uncertainty(~34 km), which value is (~8 higher km) inthan this other region regions. is about 23% of the total Antarctic mean value (~34 km), which is higher than other regions. 2. Materials and Methods 2. Materials and Methods 2.1. Aeromagnetic Data 2.1. Aeromagnetic Data Through the operation of China’s first Antarctic fixed-wing aircraft observation plat- Through the operation of China’s first Antarctic fixed-wing aircraft observation plat- form,form, we we obtained obtained large-scale large-scale aeromagnetic aeromagnetic data data from from PEL. PEL. The The single single flight flight distance distance of of thethe geophysicalgeophysical and and glacier glacier survey, survey, carried carried out out by theby the Snow Snow Eagle Eagle 601 fixed-wing601 fixed-wing aircraft, aircraft, waswas approximately approximately 1700 1700 km km (6.5 (6.5 h of h endurance),of endurance), and and the totalthe total distance distance of the of survey the survey lines lines usedused in in this this study study were were about about 47,000 47,000 km (Figurekm (Figure1). During 1). During the aerial the survey, aerial flightsurvey, altitude flight alti- wastude maintained was maintained at 600 matabove 600 m the above ice surface. the ice surface.

FigureFigure 1. 1.Snow Snow Eagle Eagle 601 601 aeromagnetic aeromagnetic survey survey lines. lines.

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

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Remote Sens. 2021, 13, 2760 The CS-3 magnetometer was used to measure the magnetic field. It was installed3 of 10 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 magnetometercan be used for has detection high sensitivity in most and areas low noiseof the interference, world. The and magnetom its measurementeter system range consists canof a be Scintrex used for detectionCS-3 scalar in most magnetometer, areas of the world. an RMS The AARC510 magnetometer adaptive system aviation consists ofmagnetic a Scintrexreal-time CS-3 compensator, scalar magnetometer, and a Billingsley an RMS AARC510 TFM100-G2 adaptive carrier aviation magnetometer. magnetic real-time During a sur- compensator,vey, the CS-3 and magnetometer a Billingsley TFM100-G2measures the carrier total magnetometer.magnetic field During(with a a sampling survey, the interval CS-3of 1 magnetometers, flight speed measures of 300 km/h, the total and magnetic sampling field interval (with a samplingdistance intervalof ~83 m of along 1 s, flight the flight speedline), ofand 300 the km/h, AARC510, and sampling with a interval three-axis distance fluxgate of ~83 vector m along magnetometer, the flight line), provides and the com- AARC510,pensation with for the a three-axis influence fluxgate of the vectorflight directio magnetometer,n and altitude. provides compensation for the influenceThe ofgeomagnetic the flight direction diurnal and station altitude. (used for diurnal correction of the aeromagnetic sur- The geomagnetic diurnal station (used for diurnal correction of the aeromagnetic vey) uses an EREV-1 proton magnetometer, located at the airport near the Zhongshan survey) uses an EREV-1 proton magnetometer, located at the airport near the Zhongshan Station,Station, withwith an an absolute absolute accuracy accuracy of of less less than than 0.3 0.3 nT nT and and a samplinga sampling interval interval of 1of s. 1 s. In Inaddition, addition, a aperennial perennial geomagnetic geomagnetic observatory (with(with aa sampling sampling interval interval of of 1 s)1 ats) at the theZhongshan Zhongshan Station, Station, 10 10km km away away from from the the airp airport,ort, provides provides backup backup data data for for diurnal diurnal obser- observations.vations. WeWe used used the the GRIDSTCH GRIDSTCH module module of Oasisof Oasis Montaj Mo softwarentaj software for observational for observational data data fusionfusion with with ADMAP2. ADMAP2. This This method method provides provides smooth smooth blending blending without without over-smoothing over-smoothing high-frequencyhigh-frequency variations variations that that may may occur occur along along the suturethe suture path (Figurepath (Figure2). 2).

FigureFigure 2. 2. Comparison ofof aeromagneticaeromagnetic data data from from Snow Snow Eagle Eagle 601 601 observation observation platform platform with with AD- ADMAP2MAP2 magnetic magnetic data. data. The The (a)) ADMAP2ADMAP2 data data set set (note (note that that there there is no is data no indata the in blank the area),blank and area), and thethe ( b(b)) data data set set after after fusion fusion of of the the aeromagnetic aeromagnetic data data obtained obtained by Snow by Snow Eagle Eagle 601 fixed-wing 601 fixed-wing aircraft. aircraft. 2.2. Inversion Method 2.2. Inversion Method 2.2.1. Using Modified Centroid Method to Invert the Curie Depth 2.2.1.The Using geomagnetic Modified field Centroid is mainly Method produced to Invert in the the ferromagnetic Curie Depth strata that exist in the Earth’sThe geomagnetic crust. After reachingfield is mainly a certain produced temperature, in the the ferromagnetic ferromagnetic strata material that exist will in the favorEarth’s paramagnetism, crust. After reaching which will a significantlycertain temper lowerature, its magnetic the ferromagnetic field. At this material temperature will favor ◦ (580paramagnetism,C for ferromagnetic which will ore [significantly21]), the corresponding lower its magnetic lithospheric field. interface At this is temperature called the (580 Curie°C for isothermal ferromagnetic interface, ore which[21]), the is generally correspondi consideredng lithospheric to be the bottom interface of theis called magnetic the Curie strata.isothermal Magnetic interface, data canwhich invert is generally the large considered magnetic change to be the interface bottom and, of the thus, magnetic we can strata. determine the Curie depth [22]. Magnetic data can invert the large magnetic change interface and, thus, we can determine the Curie depth [22].

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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]: β − 1 h i ln(A(k)) = E − |k|Z − ln|k| + ln 1 − e−|k|(Zb−Zt) , (1) t 2 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, Zt, through the relationship with the slope of the straight line. Assuming F is the constant, then Equation (1) is approximated to: β − 1 ln(A(k)) ≈ F − |k|Z − ln|k|, (2) t 2 Second, we calculated the depth of the centroid of the deep field source. We deter- mined a fitting straight line from the low wavenumber region of the power spectrum, and calculated the centroid depth of the deep field source, Z0, through the relationship with the slope of the straight line. Assuming G is the constant: β − 1 ln(A(k)/k) ≈ G − |k|Z − ln|k|, (3) 0 2 Finally, the depth of the bottom interface of the deep field source was calculated by the following simple positional relationship, calculating the Curie depth Zb:

Zb = 2Z0 − Zt, (4) The uncertainty associated with this method is calculated by [25]: q 2 2 ∆Zb = 2 × ∆Z0 + ∆Zt , (5)

∆Zt, ∆Z0 and ∆Zb are the uncertainty of the top, the centroid, and the base of the magnetic source, respectively. The values of ∆Z0 and ∆Zt were computed using the standard deviation (σr) derived from the calculated power spectrum, and the linear fit in the wavenumber range was used to derive the slope (k2 − k1)[26]. This uncertainty (e) is: σ e = r , (6) k2 − k1 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 di- vision, 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]: s 1 n  2 R = ∑ A f (k) − Asyn(k) , (7) n i=1

where A f (k) is the observed power spectrum and Asyn(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. Remote Sens. 2021, 13, x FOR PEER REVIEW 5 of 11

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2.2.2. Calculating the Geothermal Heat Flux Based on Curie Depth 2.2.2.The Calculating relationship the Geothermal between the Heat Curie Flux depth Based and on Curiegeothermal Depth gradient can be approxi- matedThe by relationship the one-dimensional between the steady-state Curie depth heat and conduction geothermal equation, gradient which can be can approxi- calculate matedthe approximate by the one-dimensional geothermal heat steady-state flux of the heat Antarctic conduction subglacial equation, bedrock. which can calculate the approximateAssuming that geothermal there is heatno la fluxteral of heat the Antarcticproduction subglacial and conduction bedrock. in the bedrock un- der theAssuming ice sheet, that and there that is the no longitudinal lateral heatproduction thermal conductivity and conduction is constant, in the if bedrock is heat underflux, the is icedepth, sheet, andis thermal that the conductivity, longitudinal thermal and conductivityis temperature, is constant, then the if one-dimen-q is heat flux,sionalz is steady-state depth, λ is thermalheat conduction conductivity, equation and T willis temperature, be: then the one-dimensional steady-state heat conduction equation will be: () () = , (8) ∂T(z) q(z) = λ , (8) Assuming that is Curie depth, is ∂Curiez temperature, is surface tempera- ture,Assuming is the thatsurfaceZb is heat Curie production, depth, Tc. is Curie is the temperature, depth for heatT0. production, is surface temperature, and is the Hsurface0 is the geothermal surface heat heat production, flux, theh heatr is the conduction depth for heatequation production, will be and[27–29]:qs is the surface

geothermal heat flux, the heat conduction( ) equation will be [27–29]: (/) = + − (1 − ), (9) λ(T − T) H h2  c 0 0 r (−Zb/hr) qs = + H0hr − 1 − e , (9) Zb Zb 3.3. ResultsResults AccordingAccording toto thethe previousprevious approximateapproximate distributiondistribution ofof thethe Curie depth inversion re- resultssults for for magnetic magnetic data data in in the Antarctica [[1],1], we we divided divided the the observation observation area area into into 16 16 win- win- dowsdows and and calculated calculated the the Curie Curie depth depth separately separately in in each each window. window. The The size size of eachof each window window waswas setset to 400 ×× 400400 km, km, which which meant meant a 50% a 50% overlap overlap between between adjacent adjacent windows, windows, as shown as shownin Figure in Figure 3. The3 .calculation The calculation result result of each of each window window corresponded corresponded to the to thecenter center point. point. After Afterdata datafusion, fusion, each eachpiece piece of data of data was was processed processed according according to the to the larger larger spacing spacing grid grid of of AD- ADMAP2MAP2 (1.5 (1.5 km km spacing) spacing) to toensure ensure data data accuracy. accuracy.

FigureFigure 3. 3.Princess Princess Elizabeth Elizabeth Land Land block block divisions divisions for for calculation. calculation. Taking Taking the the marked marked position position (white (white circle)circle) as as the the center, center, the the size size of of each each window window is 400is 400× 400× 400 km, km, and and the the adjacent adjacent windows windows have have a 50% a 50% overlappingoverlapping coverage coverage rate. rate. There There are are 16 16 square square windows windows in totalin total covering covering PEL. PEL.

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InIn FigureFigure 4 4,,4, using usingusing window windowwindow 3 33 as asas an anan example, example,example, the thethe process processprocess of ofof artificially artificiallyartificially dividing dividingdividing the thethe wavenumberwavenumber ranges,ranges, thethe fittingfittingfitting statestate ofof theththee observed observed power power spectrum, spectrum, and and the the simulated simulated powerpower spectrumspectrum areare shown. shown.

33 33 ObservedObserved spectrum spectrum 22 22 SyntheticSynthetic spectrum spectrum 95%95% confidence confidence interval interval 11 11

00 00

-1-1 -1-1

-2-2 -2-2

-3-3 -3-3 00 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2 00 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2

((aa)) ((bb)) ((cc)) FigureFigure 4. 4. The The dividing dividing thethe the wavewave wave numbernumber number range range range and and and determination determination determination of of of the the the fractal fractal fractal parameters, para parameters,meters, using using using window window window 3 as3 3 as anas an example,an exam- exam- accordingple,ple, according according to the to to fitted the the fitted fitted curve. curve. curve. The ( aThe The) selected ( a(a) )selected selected high wavenumberhigh high wavenumber wavenumber range; range; (range;b) selected ( b(b) )selected selected low wavenumber low low wavenumber wavenumber range; andrange; range; (c) and degreeand ( c(c) ) ofdegreedegree fit between of of fit fit between between the observed the the observed observed power spectrumpower power sp sp andectrumectrum the and simulatedand the the simulated simulated power spectrum.power power spectrum. spectrum.

AccordingAccording to to thethe the radialradial radial powerpower power spectrumspectrum spectrum of of of PEL, PEL, PEL, the the the high high high wavenumber wavenumber wavenumber range range range is is gener-is gen- gen- erallyallyerally concentrated concentrated concentrated between between between 0.05 0.05 0.05 and and and 0.15, 0.15, 0.15, and and and the the the low low low wavenumberwavenumber wavenumber rangerange range isis is concentrated concentrated betweenbetween 0.01 0.010.01 and and 0.03. 0.03. 0.03. The The The difference difference difference between between between the thethe observed observed observed power power power spectrum spectrum spectrum and and theand the sim- sim- the ulatedsimulatedulated power power power spectrum spectrum spectrum R R was wasR was between between between 0.05 0.05 0.05 and and and 0.15. 0.15. 0.15. InIn the thethe process process of of of calculating calculating calculating the the the heat heat heat flux flux flux from from from the the the Curie Curie Curie depth, depth, depth, the the selection theselection selection of of geo- geo- of thermalgeothermalthermal parameters parameters parameters refers refers refers to to the the to previous theprevious previous inference inference inference results results results [1]. [1]. It [It1 is]. is generally It generally is generally believed believed believed that that ◦ 3 that the surface temperature, T , is ~0 C, the surface heat production, H , is ~2.5 mW/m33 , thethe surface surface temperature, temperature, , ,is is0 ~0 ~0 °C, °C, the the surface surface heat heat production, production, , ,is is0 ~2.5 ~2.5 mW/m mW/m, ,the the thermalthethermal thermal conductivity,conductivity, conductivity, , , isisλ ~2.8,~2.8 is ~2.8W/mK,W/mK, W/mK, andand thethe and depthdepth the depth forfor heatheat for production,production, heat production, , , isis ~8~8hr ,kmkm is ◦ ~8 km [30]. The Curie temperature, Tc, refers to a constant value of 580 C. According to the [30].[30]. TheThe CurieCurie temperature,temperature, , , refersrefers toto aa constantconstant valuevalue ofof 580580 °C.°C. AccordingAccording toto thethe CurieCurie depth depth results, results, 1616 areasareas werewere combinedcombined combined withwith with thethe the one-dimensionalone-dimensional one-dimensional steady-state steady-state heatheat conductionconduction equation equation toto calculatecalculate thethe geothermalgeothermal gradient gradient and and heat heat flux.flux. flux. Using Using window window 3 3 asas anan example,example, thethe calculationcalculation resultsresults ofof regionalregional geothermalgeothermal gradientgradient andand heatheat fluxfluxflux areare shownshown in in Figure Figure5 5. .5.

FigureFigure 5. 5. The The inversion inversion result result of of the the ground ground temperaturetemperaturtemperaturee gradient gradient using using window window 3 3 asas as anan an example.example. example.

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The combined calculation results of the Curie depth and the heat flux with the pre- vious calculationTheThe combined combined results calculation of the results all-Antarcticaresults of of the the Curie Cu magneticrie depth depth and field and the [1] heatthe are fluxheat compared with flux thewith previous in the Figures pre- 6 andviouscalculation 7. calculation results results of the of all-Antarctica the all-Antarctica magnetic magnetic field [1 ]field are compared[1] are compared in Figures in6 Figures and7. 6 and 7.

(a()a ) (b(b) ) FigureFigureFigure 6. Comparison6. 6.ComparisonComparison of ofCurie of Curie Curie depth depth maps. maps. The TheThe ( (a(aa))) previous previousprevious calculation calculation results results results of allof of Antarcticaall all Antarctica Antarctica [1]; and [1]; [1]; (band) and combining (b )( bcombining) combining our ourour calculationcalculation calculation results results results inin PELinPEL PEL (red (red (red frame frame frame area). area). area).

(a) (b) FigureFigure 7. 7.ComparisonComparison of of heat heat (fluxa flux) maps. maps. The The ( (aa)) previous previous calculation resultsresults ofof allall Antarctica Antarctica(b) [ 1[1],], and and (b ()b combining) combining our our Figurecalculationcalculation 7. Comparison results results in inof PEL PELheat (red (red flux frame frame maps. area). area). The Blue Blue (a) lines linesprevious indicate calculation thethe locationlocation results ofof rifts. rifts. of all Antarctica [1], and (b) combining our calculation results in PEL (red frame area). Blue lines indicate the location of rifts. TheThe Curie Curie depth depth range range of of East East Antarctica inin thethe MartosMartos model model was was from from 22 22 to to 63 63 km. km. From our inversion results, the Curie depth of PEL was observed to be shallower than From our inversion results, the Curie depth of PEL was observed to be shallower than previouslyThe Curie estimated depth range depths, of East most Antarctica of which varied in the in Martos the range model of 23–48 was from km, with22 to an 63 km. previously estimated depths, most of which varied in the range of 23–48 km, with an av- Fromaverage our inversion depth of aboutresults, 34 km,the Curie and a depth calculation of PEL error was of ±observed4 km. The to heatbe shallower flux map than erage depth of about 34 km, and a calculation error of ±4 km. The heat flux map showed previouslyshowed thatestimated Queen Marydepths, Land most (Label of which A in Figure varied7b) in had the the range same of magnitude 23–48 km, of with high an av- thatgeothermal Queen Mary heat Land flux as (Label the region A in closerFigure to 7b the) had western the same ice shelf, magnitude which isof consistenthigh geothermal with erageheat depthflux as ofthe about region 34 closer km, toand the a western calculation ice shelf, error which of ±4 iskm. consistent The heat with flux the map previous showed thatthe Queen previous Mary seismic Land model (Label inversion A in Figure results 7b [31) had,32]. the Overall, same the magnitude heat flux wasof high larger geothermal than seismicpreviously model estimated inversion in theresults eastern [31,32]. area ofOverall, PEL. Figure the heat7 indicates flux was values larger ranging than previously from 51 heat flux as the region closer to the western ice shelf, which is consistent with the previous estimatedto 84 mW/m in 2the, with eastern an average area of 66PEL. mW/m Figure2 and 7 anindicates error of values±7 mW/m ranging2, which from is close51 to to 84 seismicmW/mthe global model2, with continental inversionan average average results of 66 of mW/m [31,32]. 65 mW/m2 andOverall,2 [ 7an]. Byerror the comparison, ofheat ±7 fluxmW/m the was2 model, whichlarger of isMartosthan close previously etto a thel. estimatedglobal continental in the eastern average area of 65of mW/mPEL. Figure2 [7]. By 7 comparison,indicates values the model ranging of Martos from 51et al.to 84 mW/mhad a 2heat, with flow an valueaverage of 45–85 of 66 mW/m mW/m2 in2 and East an Antarctica error of [1],±7 mW/mwhich is2, slightlywhich islower close than to the globalour calculation continental result. average of 65 mW/m2 [7]. By comparison, the model of Martos et al. had a heat flow value of 45–85 mW/m2 in East Antarctica [1], which is slightly lower than our calculation result.

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had a heat flow value of 45–85 mW/m2 in East Antarctica [1], which is slightly lower than our calculation result.

4. Discussion Previous satellite altimetry studies showed that there is an undiscovered, large sub- glacial drainage network hidden under the ice sheet of PEL. A series of long, deep canyons from Gamburtsev Mountain extend to the coast of the (Label B in Figure7b). 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 (Figure7b) 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/m2 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.

5. Conclusions Based on the aeromagnetic data of PEL obtained by China’s fixed-wing aircraft ob- servation 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 evo- lution 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. Remote Sens. 2021, 13, 2760 9 of 10

Author Contributions: Conceptualization, L.L.; methodology, L.L.; software, L.L. and X.S.; valida- tion, X.T. and J.G.; formal analysis, E.X. and J.G.; investigation, J.G., X.C. and Q.Z.; resources, X.T., J.G. and X.C.; data curation, J.G.; writing—original draft preparation, L.L.; writing—review and editing, L.L., X.T., E.X. and K.L.; supervision, X.T. and B.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (Nos. 41941004, 41876230, and 41941006); the National key R&D Program of China (No. 2019YFC1509102); and the Shanghai Sailing Program (21YF1452100). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available upon request from the corresponding author. Acknowledgments: We acknowledge the Chinese National Antarctic Research Expedition and the University of Texas Institute for Geophysics in the undertaking of fieldwork and data acquisition. Conflicts of Interest: The authors declare no conflict of interest.

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