三维频率域航空电磁法的数值模拟及姿态影响和校正研究
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摘要
航空电磁法作为一种信息量大、工作效率高、成本低的地球物理勘探方法，经过几十年的发展，已经成为一种应用于普查找矿、地下水勘查、地质填图、环境工程等的常用勘探方法。直升机航空电磁勘测过程中，由于吊舱的姿态变化，在航空电磁数据中引入了姿态误差。
航空电磁数值模拟主要包括有限差分、有限元和积分方程方法。本文首先推导了二阶电场矢量的亥姆赫兹方程，接着采用交错网格有限差分方法对其进行离散化，并使用QMR算法求解线性方程组，得到各个网格节点及接收点处的二次电场值。然后，对接收点周围网格上的电场的旋度取数值近似，将得到的结果插值到感兴趣的点处，即可计算出磁场。
目前频率域直升机电磁系统常用的线圈系有三种，分别是水平共面线圈系（HCP）、垂直共面线圈系（VCP）和垂直同轴线圈系（VCX）。其中HCP和VCX是最常用的线圈系，主要应用于地质填图；而VCP线圈系在电性各向异性的应用中具有显著优势。本文使用一系列典型的三维地电模型（水平层状介质、直立板状体、倾斜板状体、组合正六面体模型），计算了不同线圈系（HCP、VCP和VCX）下的不同电阻率分布断面的航空电磁响应值，并分别分析了其基本特征。
对于水平层状介质模型，三种线圈系在不同的发射频率下二次磁场的实部和虚部异常曲线都是近似水平的；虚部异常值均高于实部异常值，并且实部和虚部异常值都随发射频率的升高而增大；VCP和VCX线圈系的异常曲线形态及数值几乎完全相同。这些特点都反映了地下介质是在x和y方向上（即水平方向上）是无限延伸的。对于直立板状体模型，所有频率、所有线圈系的二次磁场异常剖面曲线均以原点对称，并且随频率升高，虚部和实部的异常数值均逐渐变大；对于倾斜板状体模型，在三种线圈系和不同的发射频率下，其二次磁场异常剖面曲线实部及虚部的波峰（或波谷）的位置相比直立板状体模型的情况均明显向左移动，并且峰值都略有上升，而谷值都略有下降，通过这些特点，能判断出倾斜板状体的倾向；对于组合正六面体模型，各个线圈系在中低频时对两个正六面体的识别效果均不理想，只能勉强识别出左侧的低阻正六面体，HCP和VCP线圈系在高频时对右侧较高阻正六面体的识别均不太好，但能较好地识别出左侧的低阻正六面体，VCX线圈系在高频时能很好地同时识别出左侧低阻正六面体和右侧较高阻正六面体。
分析直升机电磁勘测数据时通常假定吊舱是沿直线且水平飞行的，但事实上，吊舱常常会发生一些姿态变化（如摆动、倾斜、偏航等）。其中摆动变化是由于吊舱与飞机的连接电缆左右晃动引起的；倾斜变化是由于飞机飞行速度发生变化而产生的；偏航变化是由于飞机飞行时吊舱受到侧向风的外力而产生的。当吊舱发生姿态变化时，线圈系的方向和位置均会发生改变。线圈系的位置变化主要影响发射和接收线圈的相对位置；而线圈系的方向变化主要影响磁偶极矩的大小。本文推导出了姿态变化后新的收发线圈位置坐标和磁偶极矩的大小，并使用三个典型模型（直立板状模型、倾斜板状模型和组合正六面体模型），计算了不同线圈系在发生不同姿态变化时的电磁响应，并与姿态变化前的电磁响应进行了对比。
对于不同的模型，HCP和VCX线圈系在发生摆动和倾斜变化时，只有高频情况下的异常曲线受影响较大，而中低频的异常曲线除个别偏移较大外，大部分都只发生微量的偏移；而对于VCP线圈系，摆动变化只对高频情况下的异常曲线影响较大，而对中低频异常曲线几乎没有影响；而各个频率下的VCP线圈系均对倾斜变化不敏感。
基于重叠偶极模型的姿态校正方法只考虑了线圈系方向上的变化，而忽略了线圈系位置上的变化。而实际上，当姿态变化的角度较大时，磁偶极子位置上的变化对电磁响应的影响是不能轻易忽略的，因此重叠偶极模型的近似必然会对校正效果产生一定的影响。在重叠偶极模型姿态校正方法的基础上，本文提出了一种新的姿态校正方法——方向-位置综合校正方法。该方法综合考虑了线圈系方向和位置变化对电磁响应的影响。该方法的研究步骤为：首先引入五种不同电性参数的均匀半空间模型，并计算了这些模型的线圈系在姿态变化前后的电磁响应比；接着，将总的姿态变化分割成两个独立的过程——方向变化和位置变化，并分别计算了线圈系在单独发生方向或位置变化时姿态变化前后的电磁响应比；然后，推导出了总姿态变化的电磁响应比与方向和位置单独变化时的电磁响应比的乘积近似成固定比例关系；最后，推导出了总校正因子的表达式。
对于发生摆动的HCP和VCP线圈系以及发生倾斜的HCP和VCX线圈系，其总姿态变化的电磁响应比与方向和位置单一变化电磁响应比的乘积的比值近似等于旋转角度的正割函数；而对于发生摆动的VCX线圈系以及发生倾斜的VCP线圈系，该比值为1。总姿态变化与方向变化和位置变化有关，总校正因子可以写成方向校正因子、位置校正因子与校正系数乘积的形式。方向校正因子可以通过分析不同线圈系之间的关系得到；而位置校正因子可以通过曲线拟合的方法得到。
均匀半空间模型的校正结果显示，无论何种模型参数、线圈系和姿态变化形式，方向-位置综合校正均能得到比重叠偶极校正更好的校正结果。直立板状体模型的校正结果显示，对于摆动变化的情况方向-位置综合校正能得到比重叠偶极校正更好的校正结果；而对于倾斜变化的校正，两者的校正误差则比较接近。倾斜板状体模型的校正结果与直立板状体模型的校正结果比较相似，说明重叠偶极和方向-位置综合校正方法对地下板状体的倾向不敏感。对组合正六面体模型发生摆动变化的校正结果显示，方向-位置综合校正方法在应用于此类问题时要明显优于重叠偶极校正方法；而对于倾斜变化的情况，重叠偶极和方向-位置综合校正方法均能取得较好的校正结果，并且两者都在各自某些特定的频率能发挥更好的作用。
Airborne electromagnetic method (AEM), as a geophysical exploration method ofinformative, high efficiency and low cost, after recent decades of development, has become acommon exploration method for the census prospecting, groundwater exploration,geological mapping, environmental&engineering, etc.. During the process ofhelicopter-borne electromagnetic survey, the bird's attitude change can introduce attitudeerror into airborne electromagnetic data.
In airborne electromagnetic simulation method, there are mainly including finitedifference method, finite element method and integral equation method. In this paper, wededuced the vector Helmholtz equation for the second order electric field whosediscretization form is achieved using the staggered grid finite difference method. The linearequations are solved using the QMR algorithm in order to get the valves of second electricfield at the grid points and receivers. The magnetic field is calculated by first taking anumerical approximation of the curl of the electric field on the grid surrounding the receiverand then interpolating the result to the point of interest.
Most frequency-domain helicopter-borne electromagnetic (HEM) systems currentlycomprise horizontal coplanar (HCP), vertical coaxial (VCX) and vertical coplanar (VCP) coilarrays. Among them the HCP and VCX coil arrays are often used for geologic mapping,and VCP coil array has remarkable advantage in the application of electrical anisotropymedium. In this paper, we use a series of typical3-D models (inc. horizontal layered medium,the vertical plate body, inclined plate body and combined column body models) to calculatethe airborne electromagnetic response of different coils (inc. HCP, VCP and VCX) underdifferent resistivity profile, and analyze their basic characteristics.
For horizontal layered medium model, anomalous curves of the real and imaginary partsof the second magnetic field are all nearly horizontal in different transmitting frequency, allfor these three different coils; the abnormal values of imaginary part are higher than those ofreal part respectively, and the abnormal values of the real and imaginary parts increase withthe increase of transmitter frequency; the abnormal curves and numerical value of VCP andVCX coil systems are almost the same. These characteristics reflect the underground mediumis infinite in x and y direction (i.e. horizontal direction). For vertical plate model, the anomaly profile curves of magnetic scattering field of these three coils is symmetrical about the originin all frequencies, and the abnormal values of the real and imaginary parts increase with theincrease of transmitter frequency. For inclined plate model, the crest (or trough) position ofthe magnetic scattering field anomaly profile curve is obviously to move to the left comparedto the vertical plate model, both for the real part and for the imaginary part, in addition, thevalue of crest increases slightly, but the value of trough decreases slightly. The tendency ofinclined plate can be declared with these tips. For a two cube combination model, these threecoil systems have poor capacities of recognizing the two cube-shaped bodies in low ormedium frequency, and barely identify the low resistivity cube on the left. The ability of HCPand VCP coil systems to recognize the high resistivity cube on the right is weak, but strongability for the low resistivity cube on the left, and VCX coil system is able to recognizesimultaneously both the low resistivity cube on the left and the high resistivity one on theright with high frequency.
The analysis of helicopter-borne electromagnetic survey data commonly assumes thatthe bird has flown straight and levelly. In actual fact, the bird exhibits some attitude changessuch as roll, pitch and yaw rotation. The roll is caused by the swing of the cable whichconnecting the bird and helicopter; the pitch is caused by the flight speed change; the yaw iscaused by the external lateral wind which forced on the bird. When the bird's attitude changes,the direction and position of the coil system will alter. The position change mainly affects therelative position of the transmiting and receiving coil, and the direction change mainly affectsthe size of the magnetic dipole moment. In this paper, we deduced the new positioncoordinates of the transmiting and receiving coils, and the expression of magnetic dipolemoment, when their attitude changed. Then, we used three typical models (vertical platemodel, inclined plate model and combined cube model) to calculate the electromagneticresponse of different coils with different attitude changes, and compared with theelectromagnetic response which attitude is unchanged.
For different models, while HCP and VCX coils changing in roll or pitch, abnormalcurve is affected in high frequency only, while most anomaly curves in low frequency areonly trace offset in addition to individual offset a relatively lot; But for the VCP coil system,only the anomaly curves in high frequency are affected, ones in low or medium frequencyhas no effect almost; and the VCP coil system in any frequency are not sensitive to pitchchanges.
The attitude correction method using a superposed dipole (SD) model only considers thechange of direction but ignore the position variation of the coil systems. In fact, when the angle variety is wider, the influence of the position change of electromagnetic dipole to theelectromagnetic response cannot be easily ignored, thus the approximation using thesuperposed dipole method is bound to have an influence on the correction effect. Based onthe superposed dipole correction method, this paper proposed a new attitude correctionmethod–direction-position (DP) correction method. This method considered the effect ofdirection and position changes on the electromagnetic response of the coil systems. The stepsof DP method are–(1) The uniformity half space models of five different electricalparameters are introduced firstly, and then the ratios, which are the electromagnetic responsebefore to after the attitude change, of the coil systems of these models are calculated;(2) Thetotal attitude changes is divided into two independent process-direction change and positionchange, and the electromagnetic response radios of the coil in a separate direction or positionchange are calculated;(3) Deduced an approximately constant relationship between theelectromagnetic response ratio of the total attitude effect and the product of theelectromagnetic response ratios when the direction effect and position changes separately;(4)Finally, derived the expression of total correction factor.
For the rolling HCP and VCP coil systems, the radio of their electromagnetic responseratio of total attitude change to the product of the electromagnetic response ratios when asingle direction and single position changes occurred is roughly equivalent to the secantfunction of the roll angle, also for the pitching HCP and VCX coil systems. However, for therolling VCX and pitching VCP coil systems, the ratio is equal to1. The total attitude changeis associated with both the direction and position changes, total correction factor can bewritten as the product of the direction, position correction factors and correction coefficient.Direction correction factor can be obtained by analyzing the relationship between differentcoil systems, and the position correction factor can be obtained using the curvet fittingmethod.
The correction results shows that, no matter which kind of model parameter, coil systemand attitude change chosen in homogeneous half space model, using the DP correctionmethod can get better correction result than using the SD method. The correction results ofthe vertical plate model shows that the DP correction method can get better correction resultthan the SD method for roll change, but the correction results using the two methods aresimilar to each other. The correction result of the inclined plate model is similar to that ofvertical plate model, which suggests that both the SD and DP correction methods were notsensitive to the tendency of plate underground. The correction results of roll change for thecombined cube model show that DP method is much better than SD method for the problems such as this; the SD and DP methods can obtain good correction results for pitch change, andboth of them can play a good role in their specific frequency.
航空电磁数值模拟主要包括有限差分、有限元和积分方程方法。本文首先推导了二阶电场矢量的亥姆赫兹方程，接着采用交错网格有限差分方法对其进行离散化，并使用QMR算法求解线性方程组，得到各个网格节点及接收点处的二次电场值。然后，对接收点周围网格上的电场的旋度取数值近似，将得到的结果插值到感兴趣的点处，即可计算出磁场。
目前频率域直升机电磁系统常用的线圈系有三种，分别是水平共面线圈系（HCP）、垂直共面线圈系（VCP）和垂直同轴线圈系（VCX）。其中HCP和VCX是最常用的线圈系，主要应用于地质填图；而VCP线圈系在电性各向异性的应用中具有显著优势。本文使用一系列典型的三维地电模型（水平层状介质、直立板状体、倾斜板状体、组合正六面体模型），计算了不同线圈系（HCP、VCP和VCX）下的不同电阻率分布断面的航空电磁响应值，并分别分析了其基本特征。
对于水平层状介质模型，三种线圈系在不同的发射频率下二次磁场的实部和虚部异常曲线都是近似水平的；虚部异常值均高于实部异常值，并且实部和虚部异常值都随发射频率的升高而增大；VCP和VCX线圈系的异常曲线形态及数值几乎完全相同。这些特点都反映了地下介质是在x和y方向上（即水平方向上）是无限延伸的。对于直立板状体模型，所有频率、所有线圈系的二次磁场异常剖面曲线均以原点对称，并且随频率升高，虚部和实部的异常数值均逐渐变大；对于倾斜板状体模型，在三种线圈系和不同的发射频率下，其二次磁场异常剖面曲线实部及虚部的波峰（或波谷）的位置相比直立板状体模型的情况均明显向左移动，并且峰值都略有上升，而谷值都略有下降，通过这些特点，能判断出倾斜板状体的倾向；对于组合正六面体模型，各个线圈系在中低频时对两个正六面体的识别效果均不理想，只能勉强识别出左侧的低阻正六面体，HCP和VCP线圈系在高频时对右侧较高阻正六面体的识别均不太好，但能较好地识别出左侧的低阻正六面体，VCX线圈系在高频时能很好地同时识别出左侧低阻正六面体和右侧较高阻正六面体。
分析直升机电磁勘测数据时通常假定吊舱是沿直线且水平飞行的，但事实上，吊舱常常会发生一些姿态变化（如摆动、倾斜、偏航等）。其中摆动变化是由于吊舱与飞机的连接电缆左右晃动引起的；倾斜变化是由于飞机飞行速度发生变化而产生的；偏航变化是由于飞机飞行时吊舱受到侧向风的外力而产生的。当吊舱发生姿态变化时，线圈系的方向和位置均会发生改变。线圈系的位置变化主要影响发射和接收线圈的相对位置；而线圈系的方向变化主要影响磁偶极矩的大小。本文推导出了姿态变化后新的收发线圈位置坐标和磁偶极矩的大小，并使用三个典型模型（直立板状模型、倾斜板状模型和组合正六面体模型），计算了不同线圈系在发生不同姿态变化时的电磁响应，并与姿态变化前的电磁响应进行了对比。
对于不同的模型，HCP和VCX线圈系在发生摆动和倾斜变化时，只有高频情况下的异常曲线受影响较大，而中低频的异常曲线除个别偏移较大外，大部分都只发生微量的偏移；而对于VCP线圈系，摆动变化只对高频情况下的异常曲线影响较大，而对中低频异常曲线几乎没有影响；而各个频率下的VCP线圈系均对倾斜变化不敏感。
基于重叠偶极模型的姿态校正方法只考虑了线圈系方向上的变化，而忽略了线圈系位置上的变化。而实际上，当姿态变化的角度较大时，磁偶极子位置上的变化对电磁响应的影响是不能轻易忽略的，因此重叠偶极模型的近似必然会对校正效果产生一定的影响。在重叠偶极模型姿态校正方法的基础上，本文提出了一种新的姿态校正方法——方向-位置综合校正方法。该方法综合考虑了线圈系方向和位置变化对电磁响应的影响。该方法的研究步骤为：首先引入五种不同电性参数的均匀半空间模型，并计算了这些模型的线圈系在姿态变化前后的电磁响应比；接着，将总的姿态变化分割成两个独立的过程——方向变化和位置变化，并分别计算了线圈系在单独发生方向或位置变化时姿态变化前后的电磁响应比；然后，推导出了总姿态变化的电磁响应比与方向和位置单独变化时的电磁响应比的乘积近似成固定比例关系；最后，推导出了总校正因子的表达式。
对于发生摆动的HCP和VCP线圈系以及发生倾斜的HCP和VCX线圈系，其总姿态变化的电磁响应比与方向和位置单一变化电磁响应比的乘积的比值近似等于旋转角度的正割函数；而对于发生摆动的VCX线圈系以及发生倾斜的VCP线圈系，该比值为1。总姿态变化与方向变化和位置变化有关，总校正因子可以写成方向校正因子、位置校正因子与校正系数乘积的形式。方向校正因子可以通过分析不同线圈系之间的关系得到；而位置校正因子可以通过曲线拟合的方法得到。
均匀半空间模型的校正结果显示，无论何种模型参数、线圈系和姿态变化形式，方向-位置综合校正均能得到比重叠偶极校正更好的校正结果。直立板状体模型的校正结果显示，对于摆动变化的情况方向-位置综合校正能得到比重叠偶极校正更好的校正结果；而对于倾斜变化的校正，两者的校正误差则比较接近。倾斜板状体模型的校正结果与直立板状体模型的校正结果比较相似，说明重叠偶极和方向-位置综合校正方法对地下板状体的倾向不敏感。对组合正六面体模型发生摆动变化的校正结果显示，方向-位置综合校正方法在应用于此类问题时要明显优于重叠偶极校正方法；而对于倾斜变化的情况，重叠偶极和方向-位置综合校正方法均能取得较好的校正结果，并且两者都在各自某些特定的频率能发挥更好的作用。
Airborne electromagnetic method (AEM), as a geophysical exploration method ofinformative, high efficiency and low cost, after recent decades of development, has become acommon exploration method for the census prospecting, groundwater exploration,geological mapping, environmental&engineering, etc.. During the process ofhelicopter-borne electromagnetic survey, the bird's attitude change can introduce attitudeerror into airborne electromagnetic data.
In airborne electromagnetic simulation method, there are mainly including finitedifference method, finite element method and integral equation method. In this paper, wededuced the vector Helmholtz equation for the second order electric field whosediscretization form is achieved using the staggered grid finite difference method. The linearequations are solved using the QMR algorithm in order to get the valves of second electricfield at the grid points and receivers. The magnetic field is calculated by first taking anumerical approximation of the curl of the electric field on the grid surrounding the receiverand then interpolating the result to the point of interest.
Most frequency-domain helicopter-borne electromagnetic (HEM) systems currentlycomprise horizontal coplanar (HCP), vertical coaxial (VCX) and vertical coplanar (VCP) coilarrays. Among them the HCP and VCX coil arrays are often used for geologic mapping,and VCP coil array has remarkable advantage in the application of electrical anisotropymedium. In this paper, we use a series of typical3-D models (inc. horizontal layered medium,the vertical plate body, inclined plate body and combined column body models) to calculatethe airborne electromagnetic response of different coils (inc. HCP, VCP and VCX) underdifferent resistivity profile, and analyze their basic characteristics.
For horizontal layered medium model, anomalous curves of the real and imaginary partsof the second magnetic field are all nearly horizontal in different transmitting frequency, allfor these three different coils; the abnormal values of imaginary part are higher than those ofreal part respectively, and the abnormal values of the real and imaginary parts increase withthe increase of transmitter frequency; the abnormal curves and numerical value of VCP andVCX coil systems are almost the same. These characteristics reflect the underground mediumis infinite in x and y direction (i.e. horizontal direction). For vertical plate model, the anomaly profile curves of magnetic scattering field of these three coils is symmetrical about the originin all frequencies, and the abnormal values of the real and imaginary parts increase with theincrease of transmitter frequency. For inclined plate model, the crest (or trough) position ofthe magnetic scattering field anomaly profile curve is obviously to move to the left comparedto the vertical plate model, both for the real part and for the imaginary part, in addition, thevalue of crest increases slightly, but the value of trough decreases slightly. The tendency ofinclined plate can be declared with these tips. For a two cube combination model, these threecoil systems have poor capacities of recognizing the two cube-shaped bodies in low ormedium frequency, and barely identify the low resistivity cube on the left. The ability of HCPand VCP coil systems to recognize the high resistivity cube on the right is weak, but strongability for the low resistivity cube on the left, and VCX coil system is able to recognizesimultaneously both the low resistivity cube on the left and the high resistivity one on theright with high frequency.
The analysis of helicopter-borne electromagnetic survey data commonly assumes thatthe bird has flown straight and levelly. In actual fact, the bird exhibits some attitude changessuch as roll, pitch and yaw rotation. The roll is caused by the swing of the cable whichconnecting the bird and helicopter; the pitch is caused by the flight speed change; the yaw iscaused by the external lateral wind which forced on the bird. When the bird's attitude changes,the direction and position of the coil system will alter. The position change mainly affects therelative position of the transmiting and receiving coil, and the direction change mainly affectsthe size of the magnetic dipole moment. In this paper, we deduced the new positioncoordinates of the transmiting and receiving coils, and the expression of magnetic dipolemoment, when their attitude changed. Then, we used three typical models (vertical platemodel, inclined plate model and combined cube model) to calculate the electromagneticresponse of different coils with different attitude changes, and compared with theelectromagnetic response which attitude is unchanged.
For different models, while HCP and VCX coils changing in roll or pitch, abnormalcurve is affected in high frequency only, while most anomaly curves in low frequency areonly trace offset in addition to individual offset a relatively lot; But for the VCP coil system,only the anomaly curves in high frequency are affected, ones in low or medium frequencyhas no effect almost; and the VCP coil system in any frequency are not sensitive to pitchchanges.
The attitude correction method using a superposed dipole (SD) model only considers thechange of direction but ignore the position variation of the coil systems. In fact, when the angle variety is wider, the influence of the position change of electromagnetic dipole to theelectromagnetic response cannot be easily ignored, thus the approximation using thesuperposed dipole method is bound to have an influence on the correction effect. Based onthe superposed dipole correction method, this paper proposed a new attitude correctionmethod–direction-position (DP) correction method. This method considered the effect ofdirection and position changes on the electromagnetic response of the coil systems. The stepsof DP method are–(1) The uniformity half space models of five different electricalparameters are introduced firstly, and then the ratios, which are the electromagnetic responsebefore to after the attitude change, of the coil systems of these models are calculated;(2) Thetotal attitude changes is divided into two independent process-direction change and positionchange, and the electromagnetic response radios of the coil in a separate direction or positionchange are calculated;(3) Deduced an approximately constant relationship between theelectromagnetic response ratio of the total attitude effect and the product of theelectromagnetic response ratios when the direction effect and position changes separately;(4)Finally, derived the expression of total correction factor.
For the rolling HCP and VCP coil systems, the radio of their electromagnetic responseratio of total attitude change to the product of the electromagnetic response ratios when asingle direction and single position changes occurred is roughly equivalent to the secantfunction of the roll angle, also for the pitching HCP and VCX coil systems. However, for therolling VCX and pitching VCP coil systems, the ratio is equal to1. The total attitude changeis associated with both the direction and position changes, total correction factor can bewritten as the product of the direction, position correction factors and correction coefficient.Direction correction factor can be obtained by analyzing the relationship between differentcoil systems, and the position correction factor can be obtained using the curvet fittingmethod.
The correction results shows that, no matter which kind of model parameter, coil systemand attitude change chosen in homogeneous half space model, using the DP correctionmethod can get better correction result than using the SD method. The correction results ofthe vertical plate model shows that the DP correction method can get better correction resultthan the SD method for roll change, but the correction results using the two methods aresimilar to each other. The correction result of the inclined plate model is similar to that ofvertical plate model, which suggests that both the SD and DP correction methods were notsensitive to the tendency of plate underground. The correction results of roll change for thecombined cube model show that DP method is much better than SD method for the problems such as this; the SD and DP methods can obtain good correction results for pitch change, andboth of them can play a good role in their specific frequency.
引文
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DAVIS A C, MACNAE J, HODGES G.2009. Predictions of bird swing from GPScoordinates[J]. Geophysics,74(6): F119-F126.
DAVIS A C, MACNAE J, ROBB T.2006. Pendulum motion in airborne HEM systems[J].Exploration Geophysics,37(4):355-362.
DAVIS A C.2007. Quantiative characterization of airborne electromagnetic systems[D].Australia: RMIT University.
DE GROOT-HEDLIN C.2006. Finite-difference modeling of magnetotelluric fields: Errorestimates for uniform and nonuniform grids[J]. Geophysics,71(3): G97-G106.
DESZCZ-PAN M, FITTERMAN D V, LABSON V F.1998. Reduction of inversion errors inhelicopter EM data using auxiliary information[J]. Exploration Geophysics,29(1/2):142-146.
FERNEYHOUGH A B.1985. The quantitative interpretation of airborne electromagneticdata[M]. Geophysics Laboratory, Department of Physics, University of Toronto.
FITTERMAN D V, YIN CHANGCHUN.2004. Effect of bird maneuver on frequency-domain helicopter EM response[J]. Geophysics,69(5):1203-1215.
FITTERMAN D V.1998. Sources of calibration errors in helicopter EM data[J]. ExplorationGeophysics,29(1/2):65-70.
FOUNTAIN D.1998. Airborne electromagnetic system-50years of development[J].Exploration geophysics,29(1/2):1-11.
FRASER D C.1978. Resistivity mapping with an airborne multicoil electromagneticsystem[J]. Geophysics,43(1):144-172.
FRASER D C.1981. Magnetite mapping with a multicoil airborne electromagnetic system[J].Geophysics,46(11):1579-1593.
FREUND R W, HOCHBRUCK M.1994a. On the use of two QMR algorithms for solvingsingular systems and applications in Markov chain modeling[J]. Numerical linearalgebra with applications,1(4):403-420.
FREUND R W, NACHTIGAL N M.1991. QMR: a quasi-minimal residual method fornon-Hermitian linear systems[J]. Numerische Mathematik,60(1):315-339.
FREUND R W, NACHTIGAL N M.1994b. An implementation of the QMR method basedon coupled two-term recurrences[J]. SIAM Journal on Scientific Computing,15(2):313-337.
FREUND R W, NACHTIGAL N M.1996. QMRPACK: A Package of QMR Algorithms[J].ACM Transactions on Mathematical Software (TOMS),22(1):46-77.
FREUND R W, SZETO T.1992. A quasi-minimal residual squared algorithm fornon-Hermitian linear systems[M]. Los Angeles: Department of Mathematics, Universityof California.
FREUND R W.1992. Conjugate gradient type methods for linear systems with complexsymmetric coefficient matrices[J]. SIAM Journal Scientific Statistical Computing,13(1),425-448.
HAAS C, LOBACH J, HENDRICKS S, et al.2009. Helicopter-borne maeasurements of seaice thickness, using a small and lightweight, digital EM system[J]. Journal of Appliedgeophysics,67(3):234-241.
HEROUX M A, VU P, YANG C.1991. A parallel preconditioned conjugate gradientpackage for solving sparse linear systems on a Cray Y-MP[J]. Applied NumericalMathematics,8(2):93-115.
HESTENES M R, STIEFEL E.1952. Methods of conjugate gradients for solving linearsystems[M]. NBS.
HOLLADAY J S, LO B, PRINSENBERG S K.1997. Bird orientation effects in quantitativeairborne electromagnetic interpretation of pack ice thickness sounding[C]//OCEANS'97.MTS/IEEE Conference Proceedings. IEEE,2:1114-1119.
HUANG HAOPING, FRASER D C.2000. Airborne resistivity and susceptibility mapping inmagnetically polarizable areas[J]. Geophysics,65(2):502-511.
HUANG HAOPING, FRASER D C.2001. Mapping of the resistivity, susceptibility, andpermittivity of the earth using a helicopter-borne electromagnetic system[J]. Geophysics,66(1):148-157.
HUANG HAOPING, FRASER D C.2002a. The use of quad-quad resistivity in helicopterelectromagnetic mapping[J]. Geophysics,67(2):459-467.
HUANG HAOPING, FRASER D C.2002b. Dielectric permittivity and resistivity mappingusing high freuency helicopter-borne electromagnetic data[J]. Geophysics,67(3):727-738.
KUZMIN P V, MORRISON E B.2011-5-24. Large airborne time-domain electromagnetictransmitter coil system and apparatus: U.S. Patent7,948,237[P].[2014-4-22].http://www.google.com/patents/US7948237#npl-citations/.
LI JING, ZENG ZHAOFA, LIU FENGHAN, et al.2013. Study of bird attitude effect andcorrections on frequency-domain helicopter EM system[C]//Proceedings ofSymposium on the Application of Geophysics to Engineering and EnvironmentalProblems2013, Society of Exploration Geophysicists and Environment and EngineeringGeophysical Society:1-10.
MACKIE R L, MADDEN T R, WANNAMAKER P E.1993. Three-dimensionalmagnetotelluric modeling using difference equations-Theory and comparisons tointegral equation solutions[J]. Geophysics,58(2):215-226.
MACKIE R L, SMITH J T, MADDEN T R.1994. Three-dimensional electromagneticmodeling using finite difference equations: The magnetotelluric example[J]. RodioScience,29(4):923-935.
NEWMAN G A, ALUMBAUGH D L.1995b. Frequency-domain modelling of airborneelectromagnetic responses using staggered finite differences[J]. GeophysicalProspecting,43(8):1021-1042.
NEWMAN G A, HOHMANN G W, ANDERSON W L.1986. Transient electromagneticresponse of the three-dimensional body in a layered earth[J]. Geophysics,51(8):1608-1627.
NEWMAN G A.1995a. Cross well electromagnetic inversion using integral and differentialequations[J]. Geophysics,60(3):899-911.
PALACKY G J, WEST G F.1991. Electromagnetic methods in applied geophysics:Application[J]. Airborne electromagnetic methods:811-879.
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