Topographic and atmospheric effects on goce gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran

Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes’ (geopotential) coefficients. This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects, the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to the Earth’s surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is harmonic in the exterior to sea level. This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed.     

Three-Dimensional Magnetotelluric Inversion: An Introductory Guide for Developers and Users

In the last few decades, the demand for three-dimensional (3-D) inversions for magnetotelluric data has significantly driven the progress of 3-D codes. There are currently a lot of new 3-D inversion and forward modeling codes. Some, such as the WSINV3DMT code of the author, are available to the academic community. The goal of this paper is to summarize all the important issues involving 3-D inversions. It aims to show how inversion works and how to use it properly. In this paper, I start by describing several good reasons for doing 3-D inversion instead of 2-D inversion. The main algorithms for 3-D inversion are reviewed along with some comparisons of their advantages and disadvantages. These algorithms are the classical Occam’s inversion, the data space Occam’s inversion, the Gauss–Newton method, the Gauss–Newton with the conjugate gradient method, the non-linear conjugate gradient method, and the quasi-Newton method. Other variants are based on these main algorithms. Forward modeling, sensitivity calculations, model covariance and its parallel implementation are all necessary components of inversions and are reviewed here. Rules of thumb for performing 3-D inversion are proposed for the benefit of the 3-D inversion novice. Problems regarding 3-D inversions are discussed along with suggested topics for future research for the developers of the next decades.     

噪声对磁场向下延拓迭代法的计算误差影响分析

磁场向下延拓可以增强数据的分辨率.在现有的向下延拓方法中,迭代法具有较好的计算效果和较大的向下延拓高度.首先介绍了迭代法的基本原理,分析迭代法的特点;然后利用迭代法对不同噪声水平的模型数据和实测数据进行向下延拓计算,研究噪声对迭代法的计算误差影响.通过延拓结果和实际数据比较分析,客观评价了迭代法的计算精度、可靠性及适应性,对推广应用该方法具有实质性意义.     

航空重力测量数据向下延拓方法比较

首先介绍了空中重力异常向下延拓的直接代表法、正则化法、点质量法和基于球内Dirichlet问题的调和解法,分析了它们各自的特点.然后利用这些方法对中国大同航空重力测量数据进行向下延拓计算.通过对延拓结果与地面实测数据的比较分析,比较客观地评价了不同延拓方法的计算精度、可靠性及适应性.实际计算结果表明,直接代表法和正则化法的延拓结果具有较高的精度和可靠性,而球内Dirichlet问题调和解法的精度较差.     

位场向下延拓的改进迭代维纳滤波法

根据维纳滤波理论导出的位场向下延拓滤波器为最佳下延滤波器,但因其实现需要已知待求位场和噪声的功率谱而在实际应用中受到限制.针对该问题,本文首先提出一种基于位场径向平均功率谱的位场噪声水平估计方法,进而利用偏差准则求取正则化参数,实现位场正则化向下延拓;然后将位场正则化下延结果的功率谱作为待求位场功率谱的估计初值,采用带修正项的迭代维纳滤波方法来更新对待求位场功率谱的估计,最后提出本文的位场向下延拓改进迭代维纳滤波方法.基于理论重力模型数据及航磁实测数据进行了向下延拓对比试验,结果表明,改进迭代法具有较好的收敛性,且下延精度优于Tikhonov正则化法和递增型维纳滤波法.     

A geoid solution for airborne gravity data

Airborne gravity data is usually attached with satellite positioning of data points, which allow for the direct determination of the gravity disturbance at flight level. Assuming a suitable gridding of such data, Hotine’s modified integral formula can be combined with an Earth Gravity Model for the computation of the disturbing potential (T) at flight level. Based on T and the gravity disturbance data, we directly downward continue T to the geoid, and we present the final solution for the geoid height, including topographic corrections. It can be proved that the Taylor expansion of T converges if the flight level is at least twice the height of the topography, and the terrain potential will not contribute to the topographic correction. Hence, the simple topographic bias of the Bouguer shell yields the only topographic correction. Some numerical results demonstrate the technique used for downward continuation and topographic correction.     

Conditionality of inverse solutions to discretised integral equations in geoid modelling from local gravity data

The eigenvalue decomposition technique is used for analysis of conditionality of two alternative solutions for a determination of the geoid from local gravity data. The first solution is based on the standard two-step approach utilising the inverse of the Abel-Poisson integral equation (downward continuation) and consequently the Stokes/Hotine integration (gravity inversion). The second solution is based on a single integral that combines the downward continuation and the gravity inversion in one integral equation. Extreme eigenvalues and corresponding condition numbers of matrix operators are investigated to compare the stability of inverse problems of the above-mentioned computational models. To preserve a dominantly diagonal structure of the matrices for inverse solutions, the horizontal positions of the parameterised solution on the geoid and of data points are identical. The numerical experiments using real data reveal that the direct gravity inversion is numerically more stable than the downward continuation procedure in the two-step approach.     

Paleointensity and paleodirection of the geomagnetic field in the middle Miocene: Evidence from late cenozoic volcanites of primorye

We present the results of analyzing a representative collection of the middle Miocene 12.4–10.0 Ma basalts that compose the volcanic cover of the Shufan and Sovgavan plateaus, namely the Nikolo-L’vovsk (NL) and Sovetskaya Gavan (SG) volcanic fields. Preliminary data are obtained about the relicts of some volcanic edifices within the West and East Sikhote-Alin volcanic belts, namely the Shishlovskii, Malyshevo, and Truzhenik objects. It is established that the volcanic rocks from these localities are characterized by similar petrologic and magnetic properties. Thermal cleaning of the samples is carried out, and the coordinates of the paleomagnetic pole are determined as Λ = 190.2°E, Φ = 71.3°N for basalts of the Nokolo-L’vovsk area and Λ = 180.4°E, Φ = 71.9°N for rocks from the Sovgavan locality. These values are consistent with the data for coeval volcanics from other regions of Eurasia. Reliable determinations of the paleointensity H _pal for a representative collection of samples were obtained using the Thellier method. The corresponding values of the virtual dipole moment (VDM) are almost half its present-day value. The analysis of the Miocene VDM values available from the world database revealed a low average field 5.06 × 10²² Am² characterized by high variance σ = 2.13 × 10²² Am² at that time. The similarity of VDM values for the Miocene characterized by frequent inversions and for the Cretaceous Superchron supports the hypothesis of the lack of a correlation between the VDM values and the frequency of geomagnetic inversions.     

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.     

重力场向下延拓Milne法

重力场向下延拓能够突出局部和浅部的异常信息,分离叠加的异常特征.但是向下延拓通常具有过程不稳定、下延深度小、结果不准确等问题.针对向下延拓所存在的不足,本文利用重力场及其垂向一阶导数,基于辛普森(Simpson)求积公式,推导出重力场向下延拓米尔尼(Milne)公式.将本文向下延拓方法应用于模型数据,向下延拓模型结果及误差曲线表明,相对于向下延拓快速傅里叶变换(FFT)法和积分迭代法,向下延拓Milne法的深度更大,相对误差更小;相对模型值,向下延拓Milne法能够获得稳定且准确的结果.对加拿大乃查科(Nechako)盆地地区实测航空重力数据进行本文方法向下延拓验证,处理结果表明,相对于实测异常,本文方法向下延拓结果能够很好还原实测数据,并且在进一步向下延拓中反映原始异常的趋势,增强局部和细小异常信息.     

Wavefield continuation datuming using a near surface model

When topography and low velocity zone differences vary greatly, conventional vertical static time shifts will cause wavefield distortion and influence wave equation seismic imaging for seismic data acquired on a complex near surface. In this paper, we propose an approach to datum correction that combines a joint tomography inversion with wavefield continuation to solve the static problem for seismic data on rugged acquisition topography. First, the near surface model is obtained by refracted wave tomography inversion. Second, the wavefield of sources and receivers are continued downward and upward to accomplish datum correction starting from a flat surface and locating the datum above topography. Based on the reciprocal theorem, Huygens＇ and Fresnel principles, the location of sources and receivers, and regarding the recorded data on the surface as a secondary emission, the sources and receivers are upward-continued to the datum above topography respectively. Thus, the datum correction using joint tomography inversion and wavefield continuation with the condition of a complex near surface is accomplished.     

补偿向下延拓方法研究及应用

位场向下延拓是重、磁处理和解释的常用方法,但其不稳定性限制了其在资料处理及反演中的应用.本文基于补偿圆滑滤波思想以及空间域向下延拓迭代法,通过逐次补偿的办法实现位场的稳定向下延拓.同时,在频率域空间给出了该下延方法的频率域响应因子,并讨论了其低通滤波特性,理论模型和实际位场资料试验表明该方法向下延拓稳定性具有较高的延拓精度.将其应用于重力密度界面反演中,改进反演的稳定性,实际莫霍界面反演表明下延因子具备实用性.     

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.     

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.     

Space–time clustering of case–control data with residential histories: insights into empirical induction periods, age-specific susceptibility, and calendar year-specific effects

Our research group recently developed Q-statistics for evaluating space–time clustering in case–control studies with residential histories. This technique relies on time-dependent nearest-neighbor relationships to examine clustering at any moment in the life-course of the residential histories of cases relative to that of controls. In addition, in place of the widely used null hypothesis of spatial randomness, each individual’s probability of being a case is based instead on his/her risk factors and covariates. In this paper, we extend this approach to illustrate how alternative temporal orientations (e.g., years prior to diagnosis/recruitment, participant’s age, and calendar year) influence a spatial clustering pattern. These temporal orientations are valuable for shedding light on the duration of time between clustering and subsequent disease development (known as the empirical induction period), and for revealing age-specific susceptibility windows and calendar year-specific effects. An ongoing population-based bladder cancer case–control study is used to demonstrate this approach. Data collection is currently incomplete and therefore no inferences should be drawn; we analyze these data to demonstrate these novel methods. Maps of space–time clustering of bladder cancer cases are presented using different temporal orientations while accounting for covariates and known risk factors. This systematic approach for evaluating space–time clustering has the potential to generate novel hypotheses about environmental risk factors and provides insights into empirical induction periods, age-specific susceptibility, and calendar year-specific effects.     

Explicit formula for the geoid-quasigeoid separation

The explicit formula for the geoid-to-quasigeoid correction is derived in this paper. On comparing the geoidal height and height anomaly, this correction is found to be a function of the mean value of gravity disturbance along the plumbline within the topography. To evaluate the mean gravity disturbance, the gravity field of the Earth is decomposed into components generated by masses within the geoid, topography and atmosphere. Newton’s integration is then used for the computation of topography-and atmosphere-generated components of the mean gravity, while the combined solution for the downward continuation of gravity anomalies and Stokes’ boundary-value problem is utilized in computing the component of mean gravity disturbance generated by mass irregularities within the geoid. On application of this explicit formulism a theoretical accuracy of a few millimetres can be achieved in evaluation of the geoid-to-quasigeoid correction. However, the real accuracy could be lower due to deficiencies within the numerical methods and to errors within the input data (digital terrain and density models and gravity observations).     

A stable iterative downward continuation of potential field data

Downward continuation is a useful tool in the processing of potential field data, which can effectively enhance weak anomalies and identify overlap anomalies, but we all know that the computation of downward continuation is unstable, and easily distorts the true feature of potential field data. Because the computation of upward continuation and horizontal derivatives is stable, we proposed using the combination of upward continuation and horizontal derivative to accomplish the downward continuation of potential field data. The proposed method is demonstrated on synthetic potential field data, and the results show that the proposed method can finish the downward continuation of the data stably and precisely, and the precision of the proposed method is higher than the traditional method. We also apply it to real potential field data, and the results show that the proposed method accomplishes the downward continuation of the real data stably.     

重力异常视深度滤波及应用

为了解观测面以下位场场源体产生的异常,设计了一种波数域的滤波器,该滤波器可在无需岩石物性资料的前提下用于位场数据的反演．对位场资料用解析法和差分法上、下延拓,以消除浅表干扰源的影响,并最后下延至观测面以下,根据下延后场的极值确定场源的质心深度．理论模型的试算和实际资料的处理均获得了令人满意的效果．     

FULL WAVE EQUATION DOWNWARD CONTINUATION OF SEISMIC REFLECTION DATA*

A new method for suppressing multiple reflections in seismograms is developed. It is based on a downward continuation procedure which uses the full acoustic wave equation (hyperbolic form) as a downward continuation operator. We demonstrate that the downward continuation of the recorded wave field maps a reflectivity function without multiply reflected events. The method is applied successfully to individual traces of plane-wave decomposed (slant-stacked) synthetic and field data.     

2-D crustal Poisson’s ratio from seismic travel time inversion in Changbaishan Tianchi volcanic region

Based on the inversion method of 2D velocity structure and interface, the crustal velocity structures of P-wave and S-wave along the profile L ₁ are determined simultaneously with deep seismic sounding data in Changbaishan Tianchi volcanic region, and then its Poisson’s ratio is obtained. Calculated results show that this technique overcomes some defects of traditional forward calculation method, and it is also very effective to determine Poisson’s ratio distribution of deep seismic sounding profile, especially useful for study on volcanic magma and crustal fault zone. Study result indicates that there is an abnormally high Poisson’s ratio body that is about 30 km wide and 12 km high in the low velocity region under Tianchi crater. Its value of Poisson’s ratio is 8% higher than that of surrounding medium and it should be the magma chamber formed from melted rock with high temperature. There is a high Poisson’s ratio zone ranging from magma chamber to the top of crust, which may be the uprise passage of hot substance. The lower part with high Poisson’s ratio, which stretches downward to Moho, is possibly the extrusion way of hot substance from the uppermost mantle. The conclusions above are consistent with the study results of both tomographic determination of 3D crustal structure and magnetotelluric survey in this region. Foundation item: Key Project from China Earthquake Administration and the Project (95-11-02-01) from Ministry of Science and Technology (2001DIA10003). Contribution No. RCEG200401, Geophysical Exploration Center, China Earthquake Administration.