High‐accuracy practical spline‐based 3D and 2D integral transformations in potential‐field geophysics

Potential, potential field and potential‐field gradient data are supplemental to each other for resolving sources of interest in both exploration and solid Earth studies. We propose flexible high‐accuracy practical techniques to perform 3D and 2D integral transformations from potential field components to potential and from potential‐field gradient components to potential field components in the space domain using cubic B‐splines. The spline techniques are applicable to either uniform or non‐uniform rectangular grids for the 3D case, and applicable to either regular or irregular grids for the 2D case. The spline‐based indefinite integrations can be computed at any point in the computational domain. In our synthetic 3D gravity and magnetic transformation examples, we show that the spline techniques are substantially more accurate than the Fourier transform techniques, and demonstrate that harmonicity is confirmed substantially better for the spline method than the Fourier transform method and that spline‐based integration and differentiation are invertible. The cost of the increase in accuracy is an increase in computing time. Our real data examples of 3D transformations show that the spline‐based results agree substantially better or better with the observed data than do the Fourier‐based results. The spline techniques would therefore be very useful for data quality control through comparisons of the computed and observed components. If certain desired components of the potential field or gradient data are not measured, they can be obtained using the spline‐based transformations as alternatives to the Fourier transform techniques.     

Multiridge Euler deconvolution

Potential field interpretation can be carried out using multiscale methods. This class of methods analyses a multiscale data set, which is built by upward continuation of the original data to a number of altitudes conveniently chosen. Euler deconvolution can be cast into this multiscale environment by analysing data along ridges of potential fields, e.g., at those points along lines across scales where the field or its horizontal or vertical derivative respectively is zero. Previous work has shown that Euler equations are notably simplified along any of these ridges. Since a given anomaly may generate one or more ridges we describe in this paper how Euler deconvolution may be used to jointly invert data along all of them, so performing a multiridge Euler deconvolution. The method enjoys the stable and high‐resolution properties of multiscale methods, due to the composite upward continuation/vertical differentiation filter used. Such a physically‐based field transformation can have a positive effect on reducing both high‐wavenumber noise and interference or regional field effects. Multiridge Euler deconvolution can also be applied to the modulus of an analytic signal, gravity/magnetic gradient tensor components or Hilbert transform components. The advantages of using multiridge Euler deconvolution compared to single ridge Euler deconvolution include improved solution clustering, increased number of solutions, improvement of accuracy of the results obtainable from some types of ridges and greater ease in the selection of ridges to invert. The multiscale approach is particularly well suited to deal with non‐ideal sources. In these cases, our strategy is to find the optimal combination of upward continuation altitude range and data differentiation order, such that the field could be sensed as approximately homogeneous and then characterized by a structural index close to an integer value. This allows us to estimate depths related to the top or the centre of the structure.     

火山岩油气藏重磁电震综合预测方法及应用

通过准噶尔盆地陆东地区数十口钻井资料的对比分析、归纳总结,提出了火山岩油气藏重磁电震综合预测方法.将正则化下延与延拓回返垂直二次导数串联形成了一个新的滤波器,该滤波器相当于首先通过正则化下延将位场曲面延拓至地下目的层段,降低火山岩埋深对磁J异常幅值的影响,然后利用延拓回返垂直二次导数对弱信号进行增强,不仅提高了位场异常...     

An Improved Analytic Signal Technique for the Depth and Structural Index from 2D Magnetic Anomaly Data

This paper presents a new inversion method for the interpretation of 2D magnetic anomaly data, which uses the combination of the analytic signal and its total gradient to estimate the depth and the nature (structural index) of an isolated magnetic source. However, our proposed method is sensitive to noise. In order to lower the effect of noise, we apply upward continuation technique to smooth the anomaly. Tests on synthetic noise-free and noise corrupted magnetic data show that the new method can successfully estimate the depth and the nature of the causative source. The practical application of the technique is applied to measured magnetic anomaly data from Jurh area, northeast China, and the inversion results are in agreement with the inversion results from Euler deconvolution of the analytic signal.     

Use of non-linear filtering for the regional–residual separation of potential field data

Regional–residual separation is essential in gravity and magnetic data interpretation and a variety of techniques have been proposed. Graphical determination of the regional allows geological information to be taken into account. Upward continuation can be used to obtain the regional field either empirically or using some hypothesis about the geology. In some cases, a matched filter can be designed and used to separate deep and shallow sources. Simple low pass filtering has also been used but without much success. Here we propose to use a non-linear filter approach to remove gravity and magnetic anomalies smaller than a given width. This technique attempts to mimic the graphical separation method. The results from synthetic models are presented as well as the results from a case study in eastern Canada and compared to regional gravity and magnetic anomalies obtained by other techniques. Contrary to the regional fields obtained by upward continuation, non-linear filtering does not have any physical meaning. However, its main advantage is that it gives a regional component of the gravity or magnetic field similar to the one obtained from a graphical separation.     

基于向上延拓的航空重力向下解析延拓解

位场向下与向上延拓之间存在固有的内在联系,向上延拓解算具有稳定可靠的优良特性,本文据此提出了借助向上延拓信息实现航空重力向下延拓稳定解算的两种方法,分别建立了点对点向下解析延拓模型和最小二乘向下解析延拓模型.其核心思想是,依据泰勒级数展开模型,将位场向下延拓解算过程转换为向上延拓计算和垂向偏导数解算两个步骤,通过第一步的处理有效抑制数据观测噪声对解算结果的干扰,通过第二步的处理成功实现向下延拓反问题的稳定解算,较好地解决了向下延拓解算固有的不适定性问题.分析研究了两种解析延拓模型的计算精度及适用条件,利用超高阶位模型EGM2008建立的模拟标准场数据对两种模型解算结果的合理性和有效性进行了数值验证,证明本文新方法实用易行,具有较高的应用价值.     

3D inversion of gravity data by separation of sources and the method of local corrections: Kolarovo gravity high case study

We present a novel methodology for 3D gravity/magnetic data inversion. It combines two algorithms for preliminary separation of sources and an original approach to 3D inverse problem solution. The first algorithm is designed to separate sources in depth and to remove the shallow ones. It is based on subsequent upward and downward data continuation. For separation in the lateral sense, we approximate the given observed data by the field of several 3D line segments. For potential field data inversion we apply a new method of local corrections. The method is efficient and does not require trial-and-error forward modeling. It allows retrieving unknown 3D geometry of anomalous objects in terms of restricted bodies of arbitrary shape and contact surfaces. For restricted objects, we apply new integral equations of gravity and magnetic inverse problems. All steps of our methodology are demonstrated on the Kolarovo gravity anomaly in the Danube Basin of Slovakia.     

Full gravity gradient tensors from vertical gravity by cosine transform

We present a method to calculate the full gravity gradient tensors from pre-existing vertical gravity data using the cosine transform technique and discuss the calculated tensor accuracy when the gravity anomalies are contaminated by noise. Gravity gradient tensors computation on 2D infinite horizontal cylinder and 3D ??Y?? type dyke models show that the results computed with the DCT technique are more accurate than the FFT technique regardless if the gravity anomalies are contaminated by noise or not. The DCT precision has increased 2 to 3 times from the standard deviation. In application, the gravity gradient tensors of the Hulin basin calculated by DCT and FFT show that the two results are consistent with each other. However, the DCT results are smoother than results computed with FFT. This shows that the proposed method is less affected by noise and can better reflect the fault distribution.     

位场曲化平的插值-迭代法

将起伏曲面B上的位场向下延拓至曲面最低点的平面A的插值-迭代法步骤是：1）将曲面B上的场值放置在水平面A上具有相同水平坐标的点上,作为A上的初值;2）用若干水平面切割B,从A的初值,用快速傅里叶变换法(FFT)向上延拓出这些平面的场值,用插值的方法从这些平面的场值计算曲面B的场值;3）根据B上的实测值与计算值的差值,对A上的值进行加权改正;4）重复步骤2）和3）,直到B上的差值小到可以忽略.这种插值-迭代法具有高的计算速度,比通常的FFT法延拓得更深,可以超过10倍点距.文中给出计算实例.     

位场数据曲化平的迭代法

位场数据曲化平是位场数据处理解释中的重要运算,但是它的计算量和计算的复杂性影响了它在许多处理和解释方法技术中的应用.本文提出一种位场数据曲化平的迭代方法,即通过把位场数据曲化平视为平面位场数据向上延拓的反问题,得到曲化平的线性积分方程,再把曲面上位场数据视为曲面平均高程面上的位场数据,利用向下延拓的波数域广义逆算法把平均高程面上的位场数据向下延拓到设定平面上,再根据曲面和其平均高程面的相对起伏对设定平面上的向下延拓数据进行起伏校正,最后再把所得平面上的位场数据向上延拓得到曲面上的位场数据,并进行迭代.把这种方法用于三维理论模型数据和实际磁场数据的曲化平处理均获得了理想的结果.     

Geophysical researches (gravity and magnetic) of the Eratosthenes Seamount in the eastern Mediterranean Sea

New free-air gravity and magnetic maps of the Eratosthenes Seamount and its vicinity were regenerated from potential field data. Stages of data processing are power spectrum, upward continuation, filtering on the free-air gravity anomaly data. RTP, pseudo-gravity transformation map, power spectrum, upward continuation, filtering, AS, and HGAS were applied on the magnetic data. A HGAS map shows the images and locations of the Eratosthenes magnetic body. Spectral analysis of the gravity and magnetic anomalies indicates that there is an elliptical elongated structure of the Eratosthenes Seamount in the width of approx. 86 km NW-SE orientation and in the length of 138 km NE-SW orientation, with a strike of N40°E and inclined to NW. It is considered that 22.49 ± 0.08 km obtained from power spectrum of the gravity data may be related to the crust thickness. Also, 15.67 ± 0.02 km obtained from power spectrum of the magnetic data is considered to be related to the magmatic basement of the Eratosthenes Seamount.     

优化滤波方法及其在中国大陆布格重力异常数据处理中的应用

在优选延拓法的理论基础上,研究提出基于格林等效层概念和维纳滤波器的优化滤波法,用于对重力异常数据进行去噪和分离.与传统向上延拓法和优选延拓法相比,优化滤波法分离异常与延拓高度无关,不需要已知延拓高度,具有一定的优势.理论重力模型数据的去噪和异常分离试验表明优化滤波法有效,异常分离效果优于传统向上延拓法和带通滤波法.利用优化滤波法对中国大陆重力异常数据去噪和异常分离,得到有效的布格重力异常和区域重力异常.以中国大陆深地震探测推断的莫霍面深度信息为约束,对区域重力异常数据进行密度界面约束反演,得到中国大陆莫霍面深度分布.本文方法为中国大陆深部探测和区域构造研究提供一定的技术支撑.     

三维重磁场“曲化平”的一个方法

本文从勘探地球物理的实际需要和特点出发,探讨了重磁位场解析延拓的数学提法,给出一套较为实用的数值解法。所论述的原理、方法适用于三维空间任意起伏地形条件下的重磁位场曲化平计算,可做“上延拓”计算,也可做“下延拓”计算。在一个比较复杂的理论模型上所做的方法检验表明“上延”和“下延”都取得了较好的效果。     

位场向下延拓的最小曲率方法

针对位场向下延拓的不适定性,我们将位场向下延拓视为向上延拓的反问题,提出以位场最小曲率作为约束条件来求解稳定的下延位场.我们将剖面位场向上延拓表达式用傅里叶矩阵的形式表示,以矩阵乘法形式给出延拓的表达式,同时向待反演的下延位场引入最小曲率约束,得到向下延拓的最小曲率解,并利用正交变换给出了更为简洁的频率域解.随后,利用Kronecker积将上述全部结果拓展至三维位场,给出了三维位场向下延拓的最小曲率解.此外,我们将位场数据的填充、扩充问题与向下延拓问题统筹考虑,提出一种新的向下延拓迭代格式,该算法面向实际资料处理需求、无须预扩充或填补数据.下延迭代时,对原始数据直接向下延拓,而空白部分利用上一次下延位场估计的上延值替代其空白值并对其向下延拓,直至获得最小曲率约束下稳定的向下延拓结果.同时,我们也讨论了利用改进L曲线和广义交叉验证(GCV)计算正则参数最优估计的问题.对理论模型和实际航空重力资料进行了向下延拓检验,处理结果表明位场向下延拓的最小曲率方法解能满足实际位场资料对向下延拓处理的需求,具有较高的下延精度.     

Interpretation of magnetic and gravity gradient tensor data using normalized source strength – A case study from McFaulds Lake,Northern Ontario,Canada

In this paper, we present a case study on the use of the normalized source strength (NSS) for interpretation of magnetic and gravity gradient tensors data. This application arises in exploration of nickel, copper and platinum group element (Ni‐Cu‐PGE) deposits in the McFaulds Lake area, Northern Ontario, Canada. In this study, we have used the normalized source strength function derived from recent high resolution aeromagnetic and gravity gradiometry data for locating geological bodies. In our algorithm, we use maxima of the normalized source strength for estimating the horizontal location of the causative body. Then we estimate depth to the source and structural index at that point using the ratio between the normalized source strength and its vertical derivative calculated at two levels; the measurement level and a height h above the measurement level. To discriminate more reliable solutions from spurious ones, we reject solutions with unreasonable estimated structural indices. This method uses an upward continuation filter which reduces the effect of high frequency noise. In the magnetic case, the advantage is that, in general, the normalized magnetic source strength is relatively insensitive to magnetization direction, thus it provides more reliable information than standard techniques when geologic bodies carry remanent magnetization. For dipping gravity sources, the calculated normalized source strength yields a reliable estimate of the source location by peaking right above the top surface. Application of the method on aeromagnetic and gravity gradient tensor data sets from McFaulds Lake area indicates that most of the gravity and magnetic sources are located just beneath a 20 m thick (on average) overburden and delineated magnetic and gravity sources which can be probably approximated by geological contacts and thin dikes, come up to the overburden.     

双复杂条件下带误差补偿的频率空间域有限差分法叠前深度偏移

油气勘探的重点正转向复杂地表条件和复杂地质条件的区域.双复杂条件下的叠前深度偏移方法是解决复杂地表条件和复杂地质构造成像的有效手段,基于"逐步累加"的"直接下延"法和"波场上延"法都是解决复杂地表成像的有效手段.波动方程的频率空间域有限差分深度偏移对介质速度横向变化有较强的适应性,适宜于复杂构造的偏移成像.然而,频率-空间域有限差分法求解时引入的误差影响了成像的质量,我们用了带误差补偿的频率-空间域有限差分偏移改善了起伏地表条件下的频空域有限差分偏移质量,对模型和实际资料进行了试算,得到了较好的成像效果.误差补偿可以在若干个外推步长上进行,通过对比分析,我们发现,相对于基于起伏地表的傅立叶有限差分法偏移来说,该方法在改善起伏地表条件下偏移成像质量的同时,也具有较高的运算效率.     

双复杂条件下的波动方程叠前深度偏移

复杂地表条件下的地震勘探越来越被人们所关注.双复杂条件下的叠前深度偏移方法是解决复杂地表条件和复杂地质构造成像的有效手段."波场上延"法能实现由非水平观测界面开始的偏移过程,解决复杂地表对地下构造成像的影响.复杂理论模型的试算以及实际资料处理表明,"波场上延"方法较好地克服了起伏地形对地下构造成像的影响,取得了令人满意的效果,实现了波动方程基准面校正和深度成像的有机结合.     

等效源法三维随机点位场数据处理和转换

为了实现曲面随机点位场数据的曲面延拓和转换,以磁异常位场数据为例,采用一组磁偶极子作为等效源,置于观测面下方的一个曲面上,把观测磁异常作为这组磁偶极子所产生磁异常的边界条件,通过求解线性方程组的方法反演磁偶极子磁矩的大小,再根据反演结果正演所要计算的磁异常数据,实现了曲面随机点磁异常位场数据的向上延拓、向下延拓、求导以及化极处理.在数据量较大时,为了提高反演计算的速度,把磁异常数据和磁偶极子分成若干小块,再利用各块磁异常数据分别反演该块数据下方磁偶极子的磁矩,并通过迭代计算来逐步取得更准确的反演结果.模型试验表明,磁异常位场数据向上延拓的均方根误差小于±2nT,向下延拓和化极也可以取得较高的精度,所提出的分块处理方法提高了延拓和转换的速度,实际资料处理给出了曲面随机点航磁异常数据向下延拓和化极的一个例子.     

21世纪重力与磁法勘探的展望

对21世纪重力与磁法勘探的仪器，数据处理技术，解释理论与方法，应用领域等方面的发展方向进行了分析与展望，发展航空标量，矢量，梯度重力测量和航空全梯度磁力测量，三分量磁力测量，提高综合信息采集能力；开展卫星重磁测量，综合卫星，航空，地面重磁测量资料研究地球结构与构造；发展高精度数据处理技术；重磁异常弱信号的提取，不同深度重磁异常的划分，低纬底变倾角化磁极以及位场面延拓；发展复杂条件下三维重磁场多参数综合反演可视化技术以及快速自动反演技术；探索磁性多参数的应用新领域，充分发挥磁法在环境污染调查中的作用并开拓应用新领域。     

Draping aeromagnetic data in areas of rugged topography

Expanding the magnetic field intensity measured at a constant altitude in a Taylor series allows the efficient continuation of such fields onto any given arbitrary surface. This is particularly useful for draping of constant altitude surveys in areas of rugged topography. The Taylor series approach allows the continuation to points below the level of the shallowest magnetic source present. Low-pass filtering is necessary to ensure the convergence of the series. The filtering parameters can be estimated from the power spectrum of the observed field and the maximum continuation distance. A synthetic data example shows that convergence of the series is slowest in areas of high vertical gradients, usually associated with body edges, and large (downward) continuation distances. The Taylor series method is used to drape data from a constant barometric altitude survey from central British Columbia (Canada) onto a surface with a constant terrain clearance. This survey is then joined to an adjacent survey flown in the draped mode. The resolution and amplitudes of the two surveys is seen to be comparable and results in a more coherent combined data set than that where no computational draping is done.