Robust H∞ control for aseismic structures with uncertainties in model parameters

This paper presents a robust H∞ output feedback control approach for structural systems with uncertainties in model parameters by using available acceleration measurements and proposes conditions for the existence of such a robust output feedback controller. The uncertainties of structural stiffness, damping and mass parameters are assumed to be norm-bounded. The proposed control approach is formulated within the framework of linear matrix inequalities, for which existing convex optimization techniques, such as the LMI toolbox in MATLAB, can be used effectively and conveniently. To illustrate the effectiveness of the proposed robust H∞ strategy, a six-story building was subjected both to the 1940 El Centro earthquake record and to a suddenly applied Kanai-Tajimi filtered white noise random excitation. The results show that the proposed robust H∞ controller provides satisfactory results with or without variation of the structural stiffness, damping and mass parameters.     

Lp-NORM DECONVOLUTION1

The purpose of deconvolution is to retrieve the reflectivity from seismic data. To do this requires an estimate of the seismic wavelet, which in some techniques is estimated simultaneously with the reflectivity, and in others is assumed known. The most popular deconvolution technique is inverse filtering. It has the property that the deconvolved reflectivity is band-limited. Band-limitation implies that reflectors are not sharply resolved, which can lead to serious interpretation problems in detailed delineation. To overcome the adverse effects of band-limitation, various alternatives for inverse filtering have been proposed. One class of alternatives is L_p-norm deconvolution, L₁norm deconvolution being the best-known of this class. We show that for an exact convolutional forward model and statistically independent reflectivity and additive noise, the maximum likelihood estimate of the reflectivity can be obtained by L_p-norm deconvolution for a range of multivariate probability density functions of the reflectivity and the noise. The L_∞-norm corresponds to a uniform distribution, the L₂-norm to a Gaussian distribution, the L₁-norm to an exponential distribution and the L₀-norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the L_p-norm deconvolution problem is solved best by minimizing the L₀-norm of the reflectivity and the L₂-norm of the noise. However, the L₀-norm is difficult to implement in an algorithm. From a practical point of view, the frequency-domain mixed-norm method that minimizes the L₁norm of the reflectivity and the L₂-norm of the noise is the best alternative. L_p-norm deconvolution can be stated in both time and frequency-domain. We show that both approaches are only equivalent for the case when the noise is minimized with the L₂-norm. Finally, some L_p-norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible L_p-norm deconvolution methods is narrowed down to three methods with p= 1 and/or 2. Given the assumptions of sparsely distributed reflectivity and Gaussian noise, we conclude that the mixed L₁norm (reflectivity) L₂-norm (noise) performs best. However, the problems inherent to single-trace deconvolution techniques, for example the problem of generating spurious events, remain. For practical application, a greater problem is that only the main, well-separated events are properly resolved.     

Early detection of brine and CO2 leakage through abandoned wells using pressure and surface-deformation monitoring data: Concept and demonstration

In this paper, we develop a methodology for early detection of potential CO₂ leakage from geological storage formations using pressure and surface-deformation anomalies. The basic idea is based on the fact that leakage-induced pressure signals travel much faster than the migrating CO₂; thus such anomalies may be detected early enough for risk management measures taking effect in avoiding substantial CO₂ leaks. The early detection methodology involves automatic inversion of anomalous brine leakage signals with efficient forward pressure and surface-deformation modeling tools to estimate the location and permeability of leaky features in the caprock. We conduct a global sensitivity analysis to better understand under which conditions pressure anomalies can be clearly identified as leakage signals, and evaluate signal detectability for a broad parameter range considering different detection limits and levels of data noise. The inverse methodology is then applied to two synthetic examples of idealized two-aquifer-and-one aquitard storage systems, with an injection well and a leaky well, for different monitoring scenarios. In Example 1, only pressure data at the monitoring and injection wells are used for leakage detection. Our results show that the accuracy of leakage detection greatly depends on the level of pressure data noise. In Example 2, joint inversion of pressure and surface-deformation measurements significantly improves the speed of convergence toward the true solution of the leakage parameters and enables early leakage detection. In both examples, successful detection is achieved when two monitoring wells are appropriately placed within up to 4 km from the leaky well.     

基于气枪震源信号和背景噪声的2016年呼图壁6.2级地震前后波速变化特征初步研究

本文选取呼图壁气枪发射台周边流动台记录的2016年6月至2017年12月气枪震源信号和背景噪声信号,分别采用互相关时延检测法和移动窗互谱法研究2016年12月呼图壁6.2级地震前后波速变化特征,得到以下初步结果:呼图壁6.2级地震前后,距6.2级地震60 km的流动台记录的气枪震源信号未发现明显的波速变化,距6.2级地...     

基于小波变换的大尺度岩体结构微震监测信号去噪方法研究

为将小波去噪方法应用于大尺度岩体结构微震监测信号的去噪研究,首先在MATLAB环境下进行仿真,验证了使用Symlet6小波进行小波去噪的可行性;利用4种自适应阈值规则对含噪信号进行去噪对比,结果表明4种阈值去噪后的信号在均方差较小的情况下都极大地提高了信号的信噪比,有效地去除了噪声,对不同的含噪信号,无偏似然原则阈值去...     

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt→0, the space step size Δx=Q/(2S₀BC]_k) where Q is discharge, S₀ is bed slope, B is channel width and C_k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs. © 1997 John Wiley & Sons, Ltd.     

Estimating magnetic dike parameters using a non‐linear constrained inversion technique: an example from the Särna area,west central Sweden

In this paper, we describe a non‐linear constrained inversion technique for 2D interpretation of high resolution magnetic field data along flight lines using a simple dike model. We first estimate the strike direction of a quasi 2D structure based on the eigenvector corresponding to the minimum eigenvalue of the pseudogravity gradient tensor derived from gridded, low‐pass filtered magnetic field anomalies, assuming that the magnetization direction is known. Then the measured magnetic field can be transformed into the strike coordinate system and all magnetic dike parameters – horizontal position, depth to the top, dip angle, width and susceptibility contrast – can be estimated by non‐linear least squares inversion of the high resolution magnetic field data along the flight lines. We use the Levenberg‐Marquardt algorithm together with the trust‐region‐reflective method enabling users to define inequality constraints on model parameters such that the estimated parameters are always in a trust region. Assuming that the maximum of the calculated g_zz (vertical gradient of the pseudogravity field) is approximately located above the causative body, data points enclosed by a window, along the profile, centred at the maximum of g_zz are used in the inversion scheme for estimating the dike parameters. The size of the window is increased until it exceeds a predefined limit. Then the solution corresponding to the minimum data fit error is chosen as the most reliable one. Using synthetic data we study the effect of random noise and interfering sources on the estimated models and we apply our method to a new aeromagnetic data set from the Särna area, west central Sweden including constraints from laboratory measurements on rock samples from the area.     

On integral approach to regional gravity field modelling from satellite gradiometric data

Solution of the gradiometric boundary value problems leads to three integral formulas. If we are satisfied with obtaining a smooth solution for the Earth’s gravity field, we can use the formulas in regional gravity field modelling. In such a case, satellite gradiometric data are integrated on a sphere at satellite level and continued downward to the disturbing potential (geoid) at sea level simultaneously. This paper investigates the gravity field modelling from a full tensor of gravity at satellite level. It studies the truncation bias of the integrals as well as the filtering of noise of data. Numerical studies show that by integrating T _zz with 1 mE noise and in a cap size of 7°, the geoid can be recovered with an error of 12 cm after the filtering process. Similarly, the errors of the recovered geoids from T _xz,yz and T _{xx-yy, 2xy} are 13 and 21 cm, respectively.     

Signal-to-noise-ratio improvement of a linear seismic array

Summary The aim of this paper is improvement in signal-to-noise-ratio (SNR) through the use of a linear seismic array. The noise is considered to be composed of a coherent part with nature similar to that of incoming signals and an incoherent part having a constant power all across the array. All possible values of seismometer positions which minimise the coherent noise power are derived. Advantages of the delayed-summation processing technique, in so far as it enables us to obtain superior values of the signal-to-noise-ratio improvement, and furnishes unique values of signal wavelengths and signal amplitudes as compared to the simple-summation technique which gives relatively inferior values ofSNR, and enables us to obtain the signal wave-lengths as solutions of a trigonometric equation are brought out analytically. Being suitable for field work due to its simple nature, the simplesummation technique is studied in detail. Upper limit on the value of array aperture for actual improvement of all the expected signals are found. Requirements to be met for achieving the optimum value of theSNR improvement are deduced.     

Comparative study of monthly inflow prediction methods for the Three Gorges Reservoir

Due to the complexity of influencing factors and the limitation of existing scientific knowledge, current monthly inflow prediction accuracy is unable to meet the requirements of various water users yet. A flow time series is usually considered as a combination of quasi-periodic signals contaminated by noise, so prediction accuracy can be improved by data preprocess. Singular spectrum analysis (SSA), as an efficient preprocessing method, is used to decompose the original inflow series into filtered series and noises. Current application of SSA only selects filtered series as model input without considering noises. This paper attempts to prove that noise may contain hydrological information and it cannot be ignored, a new method that considerers both filtered and noises series is proposed. Support vector machine (SVM), genetic programming (GP), and seasonal autoregressive (SAR) are chosen as the prediction models. Four criteria are selected to evaluate the prediction model performance: Nash–Sutcliffe efficiency, Water Balance efficiency, relative error of annual average maximum (RE_max) monthly flow and relative error of annual average minimum (RE_min) monthly flow. The monthly inflow data of Three Gorges Reservoir is analyzed as a case study. Main results are as following: (1) coupling with the SSA, the performance of the SVM and GP models experience a significant increase in predicting the inflow series. However, there is no significant positive change in the performance of SAR (1) models. (2) After considering noises, both modified SSA-SVM and modified SSA-GP models perform better than SSA-SVM and SSA-GP models. Results of this study indicated that the data preprocess method SSA can significantly improve prediction precision of SVM and GP models, and also proved that noises series still contains some information and has an important influence on model performance.     

香港GPS基准站坐标序列特征分析

利用香港GPS连续运行参考站网络2001年1月至2007年8月的观测资料,全面深入地分析了12个基准站坐标序列特征.本文采用主成分空间滤波算法去除公共误差,来提高坐标序列的信噪比,并采用最大似然估计准则定量估计滤波后坐标序列的噪声特性,计算了地球表面质量负荷（包括大气、非潮汐海洋、积雪和土壤水）对香港GPS基准站坐标序列的影响.研究结果表明：香港GPS基准站坐标序列具有高度的空间相关性,其公共误差具有较强的季节性变化特征;地表质量负荷变化引起的香港地壳形变可以解释公共误差序列中约为3mm的垂向周年变化,经过质量负荷改正后的公共误差序列与高阶电离层误差高度相关;滤波后坐标序列的噪声特性可以用可变白噪声加闪烁噪声模型来描述,顾及闪烁噪声所计算的速度误差要比只考虑可变白噪声计算的速度误差大2~6倍;基准站间存在达1.5 mm/yr的相对水平运动,揭示香港地区存在活动断层;部分基准站坐标具有明显的振幅为1～2 mm本地季节性变化,所有测站的残差序列也表现出强烈的季节性变化.     

Stochastic analysis of neutrally stratified cirrus layer

Two cirrus cloud systems observed during the winter of 2001 at the Southern Great Plains site of the Atmospheric Radiation Measurements program in Oklahoma, USA are studied because of the distinct neutrally stratified layers formed within the clouds. Observations are obtained with 35 GHz millimeter-wave radar and backscattering cross-section η(t) signals within radar-reflectivity restricted sublayers of the clouds are analyzed. The neutrally stratified layers of cirrus clouds are known to be associated with the existence of generating cells. The statistics of radiative properties within the neutrally stratified layers is obtained to be non-Gaussian and time-dependent. The purpose of this research is to derive a model of the cloud-generating cells layer in cirrus based on the statistics of observations. The Fokker–Planck equation approach provides suitable framework to treat non-Gaussian, time-dependent probability density functions (pdfs) such as those found for the η(t) signals. It is shown that the deviations from Gaussianity of radiative properties of the neutrally stratified generating cells layer in cirrus can be modeled by linear stochastically perturbed dynamics with multiplicative noise statistics. Because the multiplicative noise is often identified with state-dependent variations of stochastic feedbacks from unresolved system components it is expected that derived stochastic model will be useful for parameterization of cirrus in global circulation models (GCMs).     

Seasonal snow cover decreases young water fractions in high Alpine catchments

Estimation of young water fractions (F_yw), defined as the fraction of water in a stream younger than approximately 2–3 months, provides key information for water resource management in catchments where runoff is dominated by snowmelt. Knowing the average dependence of summer flow on winter precipitation is an essential context for comparing regional drought severity and provides the hydrological template for downstream water users and ecosystems. However, F_yw estimation based on seasonal signals of stable isotopes of oxygen and hydrogen has not yet explicitly addressed how to parsimoniously include the seasonal shift of water input from snow. Using experimental data from three high-elevation, Alpine catchments (one dominated by glacier and two by snow), we propose a framework to explicitly include the delays induced by snow storage into estimates of F_yw. Scrutinizing the key methodological choices when estimating F_yw from isotope data, we find that the methods used to construct precipitation input signals from sparse isotope samples can significantly impact F_yw. Given this sensitivity, our revised procedure estimates a distribution of F_yw values that incorporates a wide range of possible methodological choices and their uncertainties; it furthermore compares the commonly used amplitude ratio approach to a direct convolution approach, which circumvents the assumption that the isotopic signals have a sine curve shape, an assumption that is generally violated in snow-dominated environments. Our new estimates confirm that high-elevation Alpine catchments have low F_yw values, spanning from 8 to 11%. Such low values have previously been interpreted as the impact of seasonal snow storage alone, but our comparison of different F_yw estimation methods suggests that these low F_yw values result from a combination of both snow cover effects and longer storage in the subsurface. In contrast, in the highest elevation, glacier dominated catchment, F_yw is 3–4 times greater compared to the other two catchments, due to the lower storage and faster drainage processes. A future challenge, capturing spatio-temporal snowmelt isotope signals during winter baseflow and the snowmelt period, remains to improve constraints on the F_yw estimation technique.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

基于SW统计量的自适应时频峰值滤波压制地震勘探随机噪声研究

由于金属矿区地震记录中随机噪声性质复杂且信噪比低,常规降噪方法难以达到预期的滤波效果.时频峰值滤波(TFPF)方法是实现低信噪比地震勘探记录中随机噪声压制的有效方法,但其在复杂地震勘探随机噪声下时窗参数优化问题仍难以解决.本文充分利用地震勘探噪声的统计特性,结合Shapiro-Wilk(SW)统计量辨识地震勘探记录中的微弱有效信号,提出基于SW统计量的自适应时频峰值滤波降噪方法(S-TFPF).在S-TFPF方案中,对于有效信号集中区,S-TFPF方法根据信号频率特征,选择有利于信号保持的较短时窗长度;对于噪声集中区,按噪声方差自适应增加时窗长度,增强随机噪声压制能力.S-TFPF应用于合成记录和共炮点记录的滤波结果表明,与传统时频峰值滤波方法相比,S-TFPF方法可以有效抑制低信噪比地震勘探记录中的随机噪声,更好地恢复出同相轴.     

Approximate Stochastic Self-Similarity of Envelopes of High-Frequency Teleseismic P-Waves from Large Earthquakes

A wavetrain of high-frequency (HF) P waves from a large earthquake, when recorded at a distant station, looks like a segment of modulated noise, with its duration close to the duration of rupture. These wavetrains, with their bursts and fadings, look much more intermittent than a segment of common stationary random noise. We try to describe quantitatively this bursty behavior. To this end, variogram and spectral analyses are applied to time histories of P-wave envelopes (squared-amplitude or instant-power signals) in six HF bands of 1-Hz width. Nine M _w = 7.6–9.2 earthquakes were examined, using, in total, 232 records and 992 single-band traces. Variograms of integrated instant power are approximately linear on a log–log scale, indicating that the correlation structure of the instant-power signal is approximately self-similar. Also, estimates of the power spectrum of the instant-power signal look approximately linear on a log–log scale. Log–log slopes of the variograms and spectra deliver estimates of the Hurst exponent H that are mostly in the range 0.6–0.9, markedly above the value H = 0.5 of uncorrelated (white-noise) signals. The preferred estimate over the entire data set is H = 0.83, still, this estimate may include some bias, and must be treated as preliminary. The inter-event scatter of H estimates is about 0.04, reflecting individual event-to-event variations of H. Many of the average log–log spectral plots show slight concavity that perturbs the approximately linear slope; this is a secondary effect that seems to be mostly related to the limited bandwidth of the data. Evidence is given in support of the idea that the observed approximately self-similar correlation structure of the P-wave envelope originates in a similar structure of the body wave instant-power signal radiated by the source, so that the propagation-related distortions can be regarded as limited. The facts presented suggest that the space–time organization of the earthquake rupture process is multiscaled and bears significant fractal features; it deviates from the brittle-crack model with its two well-separated characteristic scales. Phenomenologically, the high-frequency body-wave radiation from an earthquake source can be thought of as a product of stationary noise and the square root of a positive random envelope function with a power-law spectrum. From the viewpoint of applications, the self-similarity of body wave envelopes provides a useful constraint for earthquake source models used to simulate strong ground motions.     

Parametric calibration of the Hargreaves–Samani equation for use at new locations

The Hargreaves–Samani (HS) evapotranspiration equation is very useful for the on‐site irrigation management in data‐short situations such as small and midsize farms and landscaped areas. Although much work has been performed to improve the precision of the evapotranspiration (ET_o) estimates for use at new locations, the results have not been consistent and many have not been confirmed by other works. The purpose of this study was to review and to evaluate the seven most promising parameters used for the calibration of the HS evapotranspiration equation, using two different regions: California and Bolivia. The results of this study show that annual correlations between HS and Penman–Monteith can be misleading because the correlation is poor in the humid months and improves progressively along the dry season until the first rains. The average monthly wind speed can be used for both spatial and seasonal calibration of the HS equation, especially during the irrigation season. Elevation and precipitation can be used to calibrate the HS equation when no reference ET_o values are available at nearby stations. The monthly value of K_T calculated from solar radiation follows a parabolic function along the year and should not be used for improving the estimates of the HS equation because the clearness index produces better results than actual solar radiation measurements. The results also indicate that the use of distance to coast, temperature range and temperature parameter does not improve the precision of the HS equation. Copyright © 2012 John Wiley & Sons, Ltd.     

REDUCTION OF HARMONIC DISTORTION IN VIBRATORY SOURCE RECORDS*

Due to non-linear effects, the swept frequency signals (sweeps) transmitted into the subsurface by vibrators are contaminated by harmonics. Upon correlation of the recorded seismograms, these harmonics lead to noise trains which are particularly disturbing in the case of down-sweeps. The method described in this paper—which can be regarded as a generalization of Sorkin's approach to the suppression of even order harmonics—allows elimination, from the final vibratory source seismogram, of harmonics of the sweep up to any desired order. It requires that not one single signal but rather a series of M signals is employed where each signal has an initial phase differing from that of the previous one of the series by the phase angle 2πM. Prior to stacking, the seismograms generated with the different signals have to be brought into the form they would have if they had been generated with the same signal. The method seems also to be capable of reducing the correlation noise if sign-bit recording techniques are used.     

Scientific achievements from the first phase (1997–2003) of the Global Geodynamics Project using a worldwide network of superconducting gravimeters

This paper aims to review the main scientific achievements which were obtained in the first phase (1997–2003) of the Global Geodynamics Project (GGP) consisting of a worldwide network of superconducting gravimeters (SG) of about 20 instruments. We show that the low noise levels reached by these instruments in various frequency bands allow us either to investigate new signals of very small amplitude or to better determine other signals previously seen. We first report new results in the long-period seismic band with special emphasis on the detection of the ₂S₁ normal mode and the splitting of the fundamental spheroidal mode ₂S₀ after the magnitude 8.4 Peru earthquake in 2001. We also discuss briefly the ‘hum’, which consists of a sequence of fundamental normal modes existing between 2 and 7 mHz even in the lack of any seismic excitation, and was first discovered on the Syowa (in Antarctica) instrument in 1998. We will comment on the search for the Slichter mode ₁S₁ of degree 1 which is associated with a translational motion of the inner core inside the liquid core. Atmospheric effects are reviewed from the local to the global scale and the improvement due to pressure loading computations on residual gravity signals is shown. An interesting study exhibiting the gravity consequence due to sudden rainfall and vertical mass motion in the atmosphere (without ground pressure change) is presented. The precision of the SGs leads to some convincing results in the tidal domain, concerning the fluid core resonance effect (free core nutation (FCN)) on diurnal tides or various loading effects (linear, non-linear) from the oceans. In particular, SGs gravity measurements are shown to be useful validating tools for ocean tides, especially if they are small and/or confined to coastal regions. The low instrumental drift of the SGs also permits to investigate non-tidal effects in time-varying gravity, especially of annual periodicity. Hydrology has also a signature which can be seen in SG measurements as shown by several recent studies. At even lower frequency, there is the Chandler motion of 435-day period which leads to observable gravity changes at the Earth's surface. We finally report on the progress done in the last years in the problem of calibrating/validating space satellite data with SG surface gravity measurements.     

THE ESTIMATION OF SIGNAL SPECTRA AND RELATED QUANTITIES BY MEANS OF THE MULTIPLE COHERENCE FUNCTION *

A seismic trace recorded with suitable gain control can be treated as a stationary time series. Each trace, χ_j(t), from a set of traces, can be broken down into two stationary components: a signal sequence, α_j(t) *s(t—τ_j), which correlates from trace to trace, and an incoherent noise sequence, n_j(t), which does not correlate from trace to trace. The model for a seismic trace used in this paper is thus χ_j(t) =α_j(t) * s(t—τ_j) +n_j(t) where the signal wavelet α_j(t), the lag (moveout) of the signal τ_j, and the noise sequence n_j(t) can vary in any manner from trace to trace. Given this model, a method for estimating the power spectra of the signal and incoherent noise components on each trace is presented. The method requires the calculation of the multiple coherence function γ_j(f) of each trace. γ_j(f) is the fraction of the power on traced at frequency f that can be predicted in a least-square error sense from all other traces. It is related to the signal-to-noise power ratio ρ_j(f) by where K_j(f) can be computed and is in general close to 1.0. The theory leading to this relation is given in an Appendix. Particular attention is paid to the statistical distributions of all estimated quantities. The statistical behaviour of cross-spectral and coherence estimates is complicated by the presence of bias as well as random deviations. Straightforward methods for removing this bias and setting up confidence limits, based on the principle of maximum likelihood and the Goodman distribution for the sample multiple coherence, are described. Actual field records differ from the assumed model mainly in having more than one correctable component, components other than the required sequence of reflections being lumped together as correlated noise. When more than one correlatable component is present, the estimate for the signal power spectrum obtained by the multiple coherence method is approximately the sum of the power spectra of the correlatable components. A further practical drawback to estimating spectra from seismic data is the limited number of degrees of freedom available. Usually at least one second of stationary data on each trace is needed to estimate the signal spectrum with an accuracy of about 10%. Examples using synthetic data are presented to illustrate the method.