EFFECTS OF DATA ERRORS AND ACCURACY IN SECOND DERIVATIVE COMPUTATION BY RATIONAL APPROXIMATION*

The effects of systematic (constant) and random errors in the observed data have been investigated analytically for rational approximation method of computing second derivative involving a summation of the products of the averages of the gravity field with the corresponding weight coefficients, both in numerator as well as in denominator. A theoretical gravity anomaly over three spheres has been analyzed to demonstrate the high accuracy in the approximation. Since the sums of the weight coefficients in numerator and denominator are zero and one respectively, the regional gravity anomaly, even though approximated by a constant value over the entire area under computation, can produce substantially large error in the calculated derivative value. This is happening because of the contribution of the regional field in the denominator. Thus, inspite of the high accuracy in rational approximation, the method has limited application to field cases where a combined gravity field consisting of regional and residual anomalies is usually used. Master curves are presented for the constant and random errors by which a rough estimate of the percentage of error in second derivative computation can be made provided one has some idea of the magnitudes of the regional field and random error.     

陕西省及邻近区域合网的测震精度

地震参数的测定精度是由多方面因素决定的。其中测震台纲的几何布局也起着较大的作用。本文试图用蒙特卡罗法计算陕西省及邻近区域台网的几何布局对测震精度产生的影响。自1965年以来,佐藤泰夫和D.Skoko采用直角座标系中的走时方程,先线性化处理,再采用蒙特卡罗法给出台站观测误差各种随机分布值,求解一系列线性方程组。他们     

Gravity changes and natural gas extraction in Groningen

Gravity observations made over the Groningen gasfield from 1978 onwards are available. The existing gravity data were reanalysed and a new survey was carried out. The observed gravity changes obtained during four surveys, spanning an 18-year period, were compared with the gravity effect due to natural gas extraction computed from the reservoir model and the production data. The random error in the gravity values is small enough to detect the effect of gas extraction after a few years (up to 2 μgal/year). The trends obtained from observed and calculated gravity changes agree well within their expected error margins after statistical data snooping. Due to the inadequate measurement set-up of the previous surveys, systematic errors present in the gravity data hampered the use of this data for refinement of the reservoir and the production models. However, with improved gravity monitoring, in particular a very systematic network set-up and well-calibrated instruments, this should now be feasible. By using the network analysis from the 1996 measurements for the planning of a network in space and time, most of the current problems can be avoided.     

LEAST-SQUARES FILTERS WITHOUT TRANSIENT ERRORS: AN EXAMINATION OF THE ERRORS IN LEAST-SQUARES FILTER DESIGN*

A new approach has been developed for the design of cross-equalization filters by the least-squares method. The filters estimated by this new exact method are subject to only two types of error: bias and random error. Cross-equalization filters estimated by a more conventional least-squares method are further subject to “transient error”. This type of error becomes important when designing filters from a data gate of a length comparable with the length of the filter, i.e., less than four times the length of the filter. The effect of altering various design parameters has been investigated for the new method. It has been found that the proportion of bias in the filter decreases as the effective filter length increases, whereas the random error in the filter decreases with increase in either the signal-to-noise ratio of the data or the ratio of the data duration to the filter length. The level of whitening applied to the auto-correlation matrix before inversion was not found to be a critical design parameter. Also, two techniques have been tested for reducing any anomalous d.c. component in the calculated filter.     

A linear approach to the influence of discharge measurement error on flood estimates

ABSTRACT

A linear approach is presented for analysing flood discharge series affected by measurement errors which are random in nature. A general model based upon the conditional probability concept is introduced to represent random errors and to analyse their effect on flood estimates. Flood predictions provided by quantiles are shown to be positively biased when performed from a sample of measured discharge. Though for design purposes such an effect is conservative, this bias cannot be neglected if the peak discharges are determined from stage measurements by means of the extrapolated tail of the rating curve for the gauging station concerned. Monte Carlo experiments, which have been carried out to analyse small sample effects, have finally shown that the use of the method of maximum likelihood is able to reduce the bias due to measurement errors in discharge data.     

Estimating external magnetic field differences at high geomagnetic latitudes from a single station

Providing an accurate estimate of the magnetic field on the Earth's surface at a location distant from an observatory has useful scientific and commercial applications, such as in repeat station data reduction, space weather nowcasting or aeromagnetic surveying. While the correlation of measurements between nearby magnetic observatories at low and mid‐latitudes is good, at high geomagnetic latitudes () the external field differences between observatories increase rapidly with distance, even during relatively low magnetic activity. Thus, it is of interest to describe how the differences (or errors) in external magnetic field extrapolation from a single observatory grow with distance from its location. These differences are modulated by local time, seasonal and solar cycle variations, as well as geomagnetic activity, giving a complex temporal and spatial relationship. A straightforward way to describe the differences are via confidence intervals for the extrapolated values with respect to distance. To compute the confidence intervals associated with extrapolation of the external field at varying distances from an observatory, we used 695 station‐years of overlapping minute‐mean data from 37 observatories and variometers at high latitudes from which we removed the main and crustal fields to isolate unmodelled signals. From this data set, the pairwise differences were analysed to quantify the variation during a range of time epochs and separation distances. We estimate the 68.3%, 95.4% and 99.7% confidence levels (equivalent to the 1σ, 2σ and 3σ Gaussian error bounds) from these differences for all components. We find that there is always a small non‐zero bias that we ascribe to instrumentation and local crustal field induction effects. The computed confidence intervals are typically twice as large in the north–south direction compared to the east‐west direction and smaller during the solstice months compared to the equinoxes.     

Optimum size and density of surface grid arrays for retrieving accurate shear-tensile fracturing of microearthquakes

Surface arrays became an important tool for monitoring the induced seismicity in hydraulic fracturing experiments and for assessing the impact of fluid injection on the fracturing process of microearthquakes. The layout of sensors plays a key role in this task because it controls the accuracy of event locations and retrieved seismic moment tensors. We simulate various configurations of grid sensor arrays characterized by a different number of sensors, array span, sensor spacing, depth of sources and various shear/tensile source mechanisms of events. The moment tensor inversion is carried out using synthetically calculated P-wave amplitudes with added random noise. A bias in the solutions is evaluated by errors in the double-couple percentage of inverted moment tensors because the double-couple errors inform us about the sensitivity of the network to detect the shear/tensile fracturing mode of induced microearthquakes. The results show that the accuracy of the double-couple percentage is mostly controlled by the offset-to-depth ratio R defined as the ratio of half of the network size to the event depth. The optimum value of R is in the range of 0.75–1.5 irrespective of the type of the focal mechanism. If 121 (11 × 11) sensors are distributed in a regular grid and recorded data are characterized by a 10% random noise, the double-couple error is less than 6%. This error increases, if R is not optimum or if the number of sensors is reduced. However, even sparse arrays with 49 (7 × 7) or 16 (4 × 4) sensors can yield a reasonable accuracy, provided the surface grid arrays are designed to have an optimum size.     

基于姿态变化的航空频率域电磁法仪器偏置实时校正方法研究

磁偶极子的航空频率域电磁法仪器在飞行测量的过程中由于仪器偏置的存在,且仪器偏置会随着外部气压、温度等环境因素以及收发线圈晃动的影响而呈现非线性变化,使得观测数据出现误差,因此需要对仪器偏置进行校正.而传统的在测线飞行前后将仪器抬至高空的"零场值"标定方法具有成本高、受测区环境影响大以及采用线性插值获取测线飞行过程中仪器偏置的精度低等缺点.本文根据仪器偏置与仪器姿态角变化无关的特性,通过测得仪器的姿态角信息,在满足重叠偶极子模型的条件下,实现对仪器偏置的高精度实时校正.模型仿真结果表明,在30m常规飞行高度下,该方法实时测得的仪器偏置精度接近于110m高空测得的精度;校正后仪器偏置的绝对误差与理论二次场的比值即相对误差小于5%,满足反演大地电导率的精度要求.该方法不仅减少了飞行的工作量,降低了飞行成本和飞行难度,而且可更加精确地获得测线飞行过程中仪器偏置的非线性变化值,提高航测数据的观测精度.     

The reliability of aeroplane attitude determination using the main geomagnetic field with application to tensor VLF data analysis

The attitude angles of an aeroplane used for geophysical measurements are necessary in order to calculate field components in a fixed reference frame. In this paper it is shown analytically how the measurement of static magnetic field components can be used to determine the attitude angles. Error analysis shows that when magnetic anomalies are small to moderate (less than 5% of the total field), the attitude angles can be determined to within a few degrees at high latitudes. For a given area the maximum error is linearly related to the magnitude of the anomaly. The technique is illustrated on a tensor VLF data set from an area in Sweden with local magnetic anomalies less than 1% of the regional total magnetic field. Attitude errors propagated into tensor VLF transfer functions are of the same order as their random errors.     

Cumulative experimental errors in pseudodynamic tests

In pseudodynamic tests, experimental feedback errors are accumulated in the step-by-step integration procedure. In this paper, the growth of cumulative experimental errors is examined. Approximate cumulative error bounds are derived for linear single- and multi-degree-of-freedom systems, based on realistic models of random and systematic feedback errors. These studies show that the rate of cumulative error growth with respect to the integration time step increases rapidly with the natural frequency of the specimen and the integration time interval used. Hence, the higher modes of a multi-degree-of-freedom system are more sensitive to experimental errors than the lower ones. Furthermore, it is shown that some systematic errors are extremely undesirable. Rational criteria for assessing the reliability of pseudodynamic test results are presented.     

Comparison of three techniques for locating a resonating magnetic field line

Three techniques for locating field lines in the magnetosphere that contain standing ULF pulsations are compared using dynamic spectra. The first technique compares ratios of the H- and D-components of the magnetic field at a single site; the second examines the ratios of the H-components at neighboring sites along a magnetic meridian; and the third displays the phase difference between H-components at neighboring sites. We find that the H:D ratio at a single station appears to detect magnetospheric standing waves but not their precise location. In contrast, the dual station H-ratio technique is sensitive to resonances local to the stations and has advantages over the widely used phase-gradient technique. In contrast to the latter technique calculating the H-power ratio does not require precise timing and provides two resonant locations, not one. We also find that the stations used need not be strictly confined to a single magnetic meridian. Resonance signatures can be detected with stations up to 1300 km in east–west separation. In our initial data near L=2 multiple-harmonic structure is generally not observed. The resonant wave period, when assumed to be the fundamental of the standing Alfven wave, gives densities in the range 3000–8000 amu/cm³. These mass densities agree with in situ observations at earlier epochs. The equatorial mass density varies most during the day (by over a factor of two for the case studied) at L=1.86 and much less (20%) at L=2.2. This is consistent with a constant upward flux of ions over this latitude range flowing into a flux tube whose volume increases rapidly with increasing L-value.     

Combined model state and parameter estimation with an ensemble Kalman filter in a North Sea station 1-D numerical model

The combined estimation of model state and parameters is investigated by using an ensemble Kalman filter in a 1-D numerical study of the temperature profile at a North Sea station located at 55°30??North and 0°55??East. This simplified model implementation allows to test many configurations for the sampling of the model error for the model state as well as for the parameters. Temperature profiles from thermistor data are assimilated. The influence of a vertically or non-vertically correlated model error is examined through the computation of correlation coefficients, root mean square errors, and model bias. As the CS station is located in the North Sea region stratified in summer, vertically non-correlated model error terms have a slight positive impact on the assimilative runs. The benefit of the combined estimation of the model state and parameters is examined by comparison of a simulation where the model parameters are not adjusted to simulations with the combined estimation of the model state either with one parameter or with a set of parameters related to the surface heat exchange. The optical attenuation coefficient, the sensible, and latent heat exchange coefficients are considered. The best results are obtained when a set of parameters are simultaneously adjusted.     

Source bias in epicentre determinations revisited

Most estimates of the epicentre of the Long Shot explosion (29 October 1965, Aleutian Islands) are biased to the north of the true epicentre, this bias being much larger than formal uncertainty estimates. The bias is interpreted by some as evidence that the P-wave speeds in the upper mantle in the source region are higher to the north than to the south; that is, the difference between estimated and true epicentre is a source bias. Others argue that much of the bias can be explained by station anomalies, i.e. differences in the wave speeds in the upper mantle at stations to the north relative to those to the south. Bias may also be introduced by measurement error if signal-to-noise ratio (SNR) is lower on average at stations to the south compared to those to the north and times are read late at the low SNR stations. Here I locate Long Shot and the other two Aleutian Island explosions (Milrow and Cannikin) using P times from small networks of stations all on continents. The results support the view that a large component of the bias in the epicentres of the three explosions is due to station anomalies and not source bias, but that for the Long Shot explosion (the lowest magnitude of the three explosions), measurement error due to systematic variations in the SNR between stations to the north and south also contributes to the bias.     

DIURNAL CORRECTIONS TO MAGNETIC SURVEYS—AN ASSESSMENT OF ERRORS*

The comparison of a group of continuous total field stations in Ireland with the record of the total field from Valentia Observatory enables the errors which would be involved in using the Observatory as a diurnal correction base to be examined. For the most part, for a single correction, these errors lie between ± 2 and ± 6 gammas. However, they have three notable characteristics: (i) they are not closely dependent on the magnetic ‘noisiness’ of the day, (ii) they are only marginally reduced by the application of station-base time differences and (iii), they are strongly related to differences between the amplitude of the daily magnetic variation at the station and at the base. A permanent geographical basis for the latter differences enables a system for the estimation of the errors to be proposed.     

Detailed analysis of the geomagnetic ground survey performed in middle-northern Croatia over the time interval 2003–2005

During the time interval 2003–2005 a ground survey of the total magnetic field intensity in the middle-northern part of Croatia was performed, in order to get a detailed insight of the field distribution over the region. A special effort was done to understand the uncertainty and errors in data reduction, namely different amplitudes of the diurnal variations and different secular variations at the ground stations with respect to the observatories. For data reduction to epoch 2004.5 the total field recordings from AQU, FUR, NCK and THY geomagnetic observatories have been used. We estimate the mean determination error to be one order of magnitude smaller than the maximal reduction error. Multiple linear regression and the method of empirical orthogonal functions were applied on the ground survey and observatory time series. This allowed us to explain the correlations among them and the variance of the total field. Furthermore, we identified two sources of serious artificial disturbances caused by DC leakage: the tramway system of the capital and one railway station in neighbouring Slovenia. Also, we estimate the region free of these disturbances. The magnitude of one of them is estimated to be more than five nT at the distance of about 10 km.     

Time variant analysis of geomagnetic signals describes the volcanic activity of Mt. Etna between early 2000 and late 2002

Volcanomagnetic anomalies have been mostly observed during strong eruptions. Our aim is to improve the geomagnetic data analysis to evidence the anomalies occurring in a larger time span, especially in the phases preceding the eruptive events. We developed a time variant statistical approach and applied it to the 2000–2002 Etna geomagnetic temporal series. It is based on an algorithm that statistically predicts the geomagnetic field at the station on the volcanic edifice by that recorded at the remote one. In such a way a number of significant changes in the time series (called statistical innovations), marking the local magnetic field change, were detected. The distribution of such statistical innovations accurately describes the Etna volcanic evolution: we note a progressive increase of the innovation occurrence as the eruptive cycles were approaching and only few and weak innovations at times between the various eruptive cycles. The significance of this analysis is further confirmed by the close agreement among the mean square prediction error, strain release and the volcanic activity behavior. On the contrary, the geomagnetic field at a single station or its difference at two stations do not have any clear correlation with other measured physical quantities. The complex pattern of the prediction error was also investigated by a multifractal analysis. We found that the Holder regularity increases with the intensification of the volcanic activity, implying that innovations tend to be less sporadic and correlated during the major volcanic phases.     

Magnetic cloud in the solar wind: A comparison with the classical model

The solar wind magnetic field distribution near the Earth has been studied and compared with the distribution anticipated according to the classical model. It has been indicated that a two-hump distribution of the IMF values discovered previously is not an artifact of averaging but reflects the actual structure of the magnetic field within the sector. In this case the magnetic field of polarity corresponding to the leading spot in the Northern Hemisphere is encountered more frequently. Not only the magnetic field magnitude but also the fields of either polarity increase with increasing activity. The distance between the peaks on the histogram of the magnetic field near the Earth increases from 6 to 10 nT. The quasi-22-year, 11-year, and quasibiennial (2.6 ± 0.3 years) cycles are observed in an alternate increase in the peaks, in the strength of the fields of either polarity, and in the ratio of the peaks to the occurrence frequency of zero values, respectively. The classical model is violated in approximately 25% of cases.     

一种地方与区域地震震源机制反演技术：广义极性振幅技术（一）——原理与数值实验

基于数十年来已有的研究进展,提出了一种地方和区域地震震源机制反演技术——广义极性振幅技术（GPAT）,并通过一系列数值实验检验了这种技术的可行性和抗干扰能力.首先,从地震波场概念出发,利用P波初动极性与广义震相振幅构建矢量,建立反演系统,并给出求解技术;然后,考虑影响反演结果的各种因素,包括台站布局、台站数目、随机噪声、震中位置误差、震源深度误差和速度模型误差,分别进行了单一因素影响测试;最后,同时考虑各种因素进行了综合测试.实验结果表明,GPAT是可行的,具有良好的抗干扰能力.需要强调的是,在众多影响因素中,速度模型误差对反演结果的影响最大.     

A computational study of nonresonant cross-field diffusion of energetic particles due to their interaction with interplanetary magnetic decreases

We present a new method of calculating cross-field diffusion of charged particles due to their interactions with interplanetary magnetic decreases (MDs) in high heliospheric latitudes. We use a geometric model that evaluates perpendicular diffusion to the ambient magnetic field as a function of particle's gyroradius, MD radius, ratio between fields outside and inside the MD, and a random impact parameter. We use Ulysses magnetic field data of 1994 to identify the MDs and get the empirical size and magnetic field decrease distribution functions. We let protons with energies ranging from 100 keV to 2 MeV interact with MDs. The MD characteristics are taken from the observational distribution functions using the Monte Carlo method. Calculations show that the increase in diffusion tends to saturate when particles' gyroradius becomes as large as MD radii, and that particles' gyroradius increases faster than diffusion distance as the energy of the particles is increased.     

Experimental errors of measurements of vertical geomagnetic gradients in the stratosphere

The errors of measurements of vertical geomagnetic gradients at altitudes of 20–40 km, using a balloon magnetic gradiometer with a 6-km-long measuring base oriented along gravity, have been studied in the work. The errors related to the deviation of the measuring base position relative to the vertical have been studied during the real balloon flight with the help of the navigation GPS receivers. The deviations of the measuring base within 5°, which can sometimes reach 15°, have been obtained. This results in a decrease in the magnetic gradient measurement accuracy due to the errors introduced in the specification of the normal magnetic field used to detect magnetic anomalies. To eliminate this error, a GPS receiver was built in each magnetometer in order to observe magnetometers during synchronous measurements and to correct the measurements for the normal magnetic field. It has been indicated that the effect of deviations of the measuring base position on the results is not more than 2% of the measured value at such organization of a gradiometer.