Efficient GOCE satellite gravity field recovery based on least-squares using QR decomposition

We develop and apply an efficient strategy for Earth gravity field recovery from satellite gravity gradiometry data. Our approach is based upon the Paige-Saunders iterative least-squares method using QR decomposition (LSQR). We modify the original algorithm for space-geodetic applications: firstly, we investigate how convergence can be accelerated by means of both subspace and block-diagonal preconditioning. The efficiency of the latter dominates if the design matrix exhibits block-dominant structure. Secondly, we address Tikhonov-Phillips regularization in general. Thirdly, we demonstrate an effective implementation of the algorithm in a high-performance computing environment. In this context, an important issue is to avoid the twofold computation of the design matrix in each iteration. The computational platform is a 64-processor shared-memory supercomputer. The runtime results prove the successful parallelization of the LSQR solver. The numerical examples are chosen in view of the forthcoming satellite mission GOCE (Gravity field and steady-state Ocean Circulation Explorer). The closed-loop scenario covers 1 month of simulated data with 5 s sampling. We focus exclusively on the analysis of radial components of satellite accelerations and gravity gradients. Our extensions to the basic algorithm enable the method to be competitive with well-established inversion strategies in satellite geodesy, such as conjugate gradient methods or the brute-force approach. In its current development stage, the LSQR method appears ready to deal with real-data applications.     

Distribution of vertical crustal movement rates in the Tohoku district, Japan, predicted by least-squares collocation

Least-squares prediction using an empirically deduced local covariance function was performed to investigate the temporal change in the rates of vertical crustal movements in the Tohoku district, Japan. Levelling data covering an area of approximately 450 × 275 km² observed between 1966 and 1995 were used and the results shown in the form of contour maps. Firstly we derived a covariance function of the rates of vertical crustal movement with a Gaussian form function. We used this function to estimate the spatial distribution of the rates of vertical crustal movements. By the present method, a steady tilt of the Tohoku district to the east, toward the Japan Trench and an areal uplift in the southwestern part were well reproduced. Moreover, a significant temporal change in vertical movement rates is clearly seen. Received: 15 July 1996      

应用最小二乘法建立钾长石释光测年标准生长曲线

标准生长曲线（Standardised Growth Curve,SGC）法的提出为高效快速地测定释光样品的等效剂量（D_e）值提供了可能。不同实验室的放射源剂量率、操作流程、仪器误差等的不同会导致SGC参数的不同。运用最小二乘法标准化实验剂量校正后的信号L_ri/T_ri*D_e得到一条本实验室的标准生长曲线,用来快速得到等效剂量值。通过R软件实现了最小二乘法标准化过程,建立了50、100、150、200、250℃激发温度下的钾长石的标准生长曲线,分析发现250℃下的SGC收敛性最好,由于此温度下的信号衰退率可忽略不计,因此,我们用250℃激发温度下的SGC方程来估计样品的等效剂量。比较250℃ SGC D_e和250℃ MET-pIRIR D_e的一致性,发现在0~500 Gy范围内用SGC法估计的D_e和MET-pIRIR法估计的D_e非常接近,表明了此范围内SGC法的可靠性,而在>600 Gy范围内出现较大差别。一方面,此范围内用于拟合SGC的数据点较少导致高剂量区域内拟合的方程参数不够准确;另一方面,600 Gy与SGC对应的饱和剂量水平值844.5 Gy已经接近,所以在>600 Gy区域测出的D_e很可能出现较大偏差。因此需在今后工作中逐步积累更多样品,完善SGC参数,使其也能胜任较老年龄样品。     

Least-squares self-coherency analysis of superconducting gravimeter records in search for the Slichter triplet

We develop a new approach for the spectral analysis of the superconducting gravimeter data to search for the spheroidal oscillation ₁S₁ of the Earth solid inner core. The new method, which we call least-squares (LS) self-coherency analysis, is based on the product of the least-squares spectra of segments of the time series under consideration. The statistical foundation of this method is presented in the new least-squares product spectrum theorem that establishes rigorously confidence levels for detecting significant peaks. We apply this approach along with a number of other innovative ideas to a 6-year long gravity series collected at the Canadian Superconducting Gravimeter Installation (CSGI) in Cantley, Canada, by splitting it into 72 statistically independent monthly records. Each monthly record is analysed spectrally and all monthly LS spectra are multiplied to construct the self-coherency spectrum of the 6-year gravity series. The self-coherency spectrum is then used to detect significant peaks in the band 3-7 h at various significant levels with the aim to identify a triplet of periods associated with the rotational/ellipsoidal splitting of ₁S₁ (Slichter triplet). From all the Slichter periods predicted by various researchers so far, Smylie's triplet appears to be the most supported one, albeit very weakly, both, before and after the atmospheric pressure effect is removed from the series. Using the viscous splitting law [Smylie, D.E., 1992. The inner core translational triplet and the density near Earth's center. Science 255, 1678-1682] as guide, we can also see one interesting and statistically significant triplet with periods A = {4.261 h, 4.516 h, 4.872 h}, which changes slightly to A′ = {4.269 h, 4.516 h, 4.889 h} after the atmospheric pressure correction is applied to the gravity series.     

Mapping and initial dilution estimation of an ocean outfall plume using an autonomous underwater vehicle

A monitoring mission to study the shape and estimate initial dilution of the S. Jacinto outfall plume using an autonomous underwater vehicle (AUV) was performed on July 30, 2002. In order to reduce the uncertainty about plume location and to concentrate the vehicle mission only in the hydrodynamic mixing zone, outputs of a near-field prediction model, based on effective real-time in situ measurements of current speed and direction and density stratification, were opportunistically used to specify in real time the mission transects. The surface characteristics of the outfall plume were found to be influenced strongly by the relatively weak stratification and low current velocities. Dilution was estimated using a temperature–salinity (TS-) diagram with initial mixing lines between wastewater and ambient waters. Effluent dilutions were at least 30:1 in this study. In order to efficiently map the plume dispersion we applied the least-squares collocation method technique. Our results demonstrate that AUVs can provide high-quality measurements of physical properties of effluent plumes in a quite effective manner and valuable considerations about the initial mixing processes under real oceanic conditions can be further investigated.     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

大连双井测汞震例分析

根据大连双井测汞震例，用最小二乘法原理，就汞测量预测地震进行了定理化研究，给出适宜本地区的经验预测公式，取得初步研究成果。     

A study on the combination of satellite, airborne, and terrestrial gravity data

Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earths gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjöberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjöberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.     

Best prediction in linear models with mixed integer/real unknowns: theory and application

In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).     

The Need for a Common Framework for Collection and Interpretation of Data in Platinum-Group Element Geochemistry

Platinum-group element (PGE) concentrations and the distribution of the metals in rocks serve as important tracers of mantle processes, as well as extraterrestrial input into crustal environments, but common standards regarding the gathering and presentation of PGE data have never been formalized. Effective modelling assumes that concentration data are within acceptable levels of precision, yet the practices used in some studies to determine precision do not adequately assess precision and, as a result, the uncertainties on PGE concentrations and PGE ratios are sometimes consistently underestimated. This article argues that replicate analyses of unknowns must be adopted more widely in order to overcome this problem. Related to the issue of uncertainties on PGE concentrations, is the issue of uncertainty associated with normalisation. Arguments have recently been put forward as to the significance of small positive or negative anomalies on chondrite nor-malized plots. At least four CI chondrite PGE datasets (of varying age and quality) are currently used for normalisation and significantly different patterns can be derived simply by using one dataset rather than another. This article is intended to open a debate within the PGE research community by asking whether more consistency needs to be applied in PGE analysis and in the subsequent interpretation of data. A rigorous assessment of the real uncertainties on PGE concentrations and the adoption of a standard CI chondrite PGE dataset, in order to eliminate bias from normalisation, are suggested to be central to this.