RELATED PROPERTIES OF MINIMUM-PHASE AND ZERO-PHASE TIME FUNCTIONS *

In this paper properties of the discrete zero-phase time function are derived and compared with related properties of the discrete minimum-phase time function. The two-sided minimum-length signal is introduced and it is derived that, for any given amplitude spectrum, the two-sided minimum-length signal and the signal with zero-phase spectrum are identical signals. A comparison is made between the one-sided minimum-length signal (minimum-phase signal) and the two-sided minimum-length signal (zero-phase signal). A computational scheme is discussed which determines the zero-phase correspondent of a given signal. A method is proposed to compute zero-phase least-square inverse filters. The efficiency of minimum-phase and zero-phase least-square inverse filters is shown on signals with different phase properties. A criterion is derived which determines whether a symmetric time function has the zero-phase property. The close relationship with the minimum-phase criterion is discussed. Finally the relationship between signal length and resolving power is illustrated on numerical examples.     

SIGNAL ADJUSTMENT OF VIBROSEIS AND IMPULSIVE SOURCE DATA*

In certain areas continuous Vibroseis profiling is not possible due to varying terrain conditions. Impulsive sources can be used to maintain continuous coverage. While this technique keeps the coverage at the desired level, for the processing of the actual data there is the problem of using different sources resulting in different source wavelets. In addition, the effect of the free surface is different for these two energy sources. The approach to these problems consists of a minimum-phase transformation of the two-sided Vibroseis data by removal of the anticipation component of the autocorrelation of the filtered sweep and a minimum-phase transformation of the impulsive source data by replacement of the recording filter operator with its minimum-phase correspondent. Therefore, after this transformation, both datasets show causal wavelets and a conventional deconvolution (spike or predictive) may be used. After stacking, a zero-phase transformation can be performed resulting in traces well suited for computing pseudo-acoustic impedance logs or for application of complex seismic trace analysis. The solution is also applicable to pure Vibroseis data, thereby eliminating the need for a special Vibroseis deconvolution. The processing steps described above are demonstrated on synthetic and actual data. The transformation operators used are two-sided recursive (TSR) shaping filters. After application of the above adjustment procedure, remaining signal distortions can be removed by modifying only the phase spectrum or both the amplitude and phase spectra. It can be shown that an arbitrary distortion defined in the frequency domain, i.e., a distortion of the amplitude and phase spectrum, is noticeable in the time section as a two-sided signal.     

DATA ACQUISITION AND PROCESSING OF CONVERTED pS-WAVES*

The earth's surface can be an effective means of generating converted pS-waves. Due to their nearly symmetrical ray path, conventional processing techniques can be used. As the wave is generated by reflection at the surface or at the base of surface layers one can expect a general filtering effect in the data for individual ray paths of a single shot gather. To balance the spectra of the traces a multiple-trace filter was used. This filter can be fully determined in the time domain using the prediction-error operators of the individual traces. The preferred mean spectrum to colour the traces was the geometric mean. As the process of spectral balancing requires a minimum-delay wavelet, the recording instrument was replaced by its corresponding minimum-phase equivalent. This process can also be carried out effectively in the time domain. Results of the application of minimum-delay transform and spectral balancing are discussed for single shot gathers and for the general improvement of the final stack.     

The simulation of far-field wavelets using frequency-domain air-gun array near-field wavelets

Air-gun arrays are used in marine-seismic exploration. Far-field wavelets in subsurface media represent the stacking of single air-gun ideal wavelets. We derived single air-gun ideal wavelets using near-field wavelets recorded from near-field geophones and then synthesized them into far-field wavelets. This is critical for processing wavelets in marine- seismic exploration. For this purpose, several algorithms are currently used to decompose and synthesize wavelets in the time domain. If the traveltime of single air-gun wavelets is not an integral multiple of the sampling interval, the complex and error-prone resampling of the seismic signals using the time-domain method is necessary. Based on the relation between the frequency-domain phase and the time-domain time delay, we propose a method that first transforms the real near-field wavelet to the frequency domain via Fourier transforms; then, it decomposes it and composes the wavelet spectrum in the frequency domain, and then back transforms it to the time domain. Thus, the resampling problem is avoided and single air-gun wavelets and far-field wavelets can be reliably derived. The effect of ghost reflections is also considered, while decomposing the wavelet and removing the ghost reflections. Modeling and real data processing were used to demonstrate the feasibility of the proposed method.     

TUTORIAL MINIMUM PHASE FOR CONTINUOUS TIME AND DISCRETE TIME FUNCTIONS*

The concept of minimum phase is clarified for geophysicists by collecting in one place the properties of minimum phase functions. The “earliest energy arrival” property in the time domain, the “minimum phase-slope property” in the frequency domain, and of the role of causal all-pass filters are demonstrated. The emphasis is placed on keeping the mathematics within the realm familiar to geophysicists and on making clear the somewhat imperfect match between physical continuous time functions and their associated discrete time representations.     

Klauder wavelet removal before vibroseis deconvolution

The spiking deconvolution of a field seismic trace requires that the seismic wavelet on the trace be minimum phase. On a dynamite trace, the component wavelets due to the effects of recording instruments, coupling, attenuation, ghosts, reverberations and other types of multiple reflection are minimum phase. The seismic wavelet is the convolution of the component wavelets. As a result, the seismic wavelet on a dynamite trace is minimum phase and thus can be removed by spiking deconvolution. However, on a correlated vibroseis trace, the seismic wavelet is the convolution of the zero-phase Klauder wavelet with the component minimum-phase wavelets. Thus the seismic wavelet occurring on a correlated vibroseis trace does not meet the minimum-phase requirement necessary for spiking deconvolution, and the final result of deconvolution is less than optimal. Over the years, this problem has been investigated and various methods of correction have been introduced. In essence, the existing methods of vibroseis deconvolution make use of a correction that converts (on the correlated trace) the Klauder wavelet into its minimum-phase counterpart. The seismic wavelet, which is the convolution of the minimum-phase counterpart with the component minimum-phase wavelets, is then removed by spiking deconvolution. This means that spiking deconvolution removes both the constructed minimum-phase Klauder counterpart and the component minimum-phase wavelets. Here, a new method is proposed: instead of being converted to minimum phase, the Klauder wavelet is removed directly. The spiking deconvolution can then proceed unimpeded as in the case of a dynamite record. These results also hold for gap predictive deconvolution because gap deconvolution is a special case of spiking deconvolution in which the deconvolved trace is smoothed by the front part of the minimum-phase wavelet that was removed.     

A hexagonal sampling grid for 3D recording and processing of 3D seismic data

3D seismic data are usually recorded and processed on rectangular grids, for which sampling requirements are generally derived from the usual 1D viewpoint. For a 3D data set, the band region (the region of the Fourier space in which the amplitude spectrum is not zero) can be approximated by a domain bounded by two cones. Considering the particular shape of this band region we can use the 3D sampling viewpoint, which leads to weaker sampling requirements than does the 1D viewpoint; i.e. fewer sample points are needed to represent data with the same degree of accuracy. The 3D sampling viewpoint considers regular nonrectangular sampling grids. The recording and processing of 3D seismic data on a hexagonal sampling grid is explored. The acquisition of 3D seismic data on a hexagonal sampling grid is an advantageous economic alternative because it requires 13.4% fewer sample points than a rectangular sampling grid. The hexagonal sampling offers savings in data storage and processing of 3D seismic data. A fast algorithm for 3D discrete spectrum evaluation and trace interpolation in the case of a 3D seismic data set sampled on a hexagonal grid is presented and illustrated by synthetic examples. It is shown that by using this algorithm the hexagonal sampling offers, approximately, the same advantage of saving 13.4% in data storage and computational time for 3D phase-shift migration.     

Evaluation of a simple,inexpensive, in situ sampler for measuring time‐weighted average concentrations of suspended sediment in rivers and streams

The accurate measurement of suspended sediment (<200 μm) in aquatic environments is essential to understand and effectively manage changes to sediment, nutrient, and contaminant concentrations on both temporal and spatial scales. Commonly used sampling techniques for suspended sediment either lack the ability to accurately measure sediment concentration (e.g., passive sediment samplers) or are too expensive to deploy in sufficient number to provide landscape‐scale information (e.g., automated discrete samplers). Here, we evaluate a time‐integrated suspended sediment sampling technique, the pumped active suspended sediment (PASS) sampler, which collects a sample that can be used for the accurate measurement of time‐weighted average (TWA) suspended sediment concentration and sediment particle size distribution. The sampler was evaluated against an established passive time‐integrated suspended sediment sampling technique (i.e., Phillips sampler) and the standard discrete sampling method (i.e., manual discrete sampling). The PASS sampler collected a sample representative of TWA suspended sediment concentration and particle size distribution of a control sediment under laboratory conditions. Field application of the PASS sampler showed that it collected a representative TWA suspended sediment concentration and particle size distribution during high flow events in an urban stream. The particle size distribution of sediment collected by the PASS and Phillips samplers were comparable and the TWA suspended sediment concentration of the samples collected using the PASS and discrete sampling techniques agreed well, differing by only 4% and 6% for two different high flow events. We should note that the current configuration of the PASS sampler does not provide a flow‐weighted measurement and, therefore, is not suitable for the determination of sediment loads. The PASS sampler is a simple, inexpensive, and robust in situ sampling technique for the accurate measurement of TWA suspended sediment concentration and particle size distribution.     

A SPATIAL ANALYSIS OF UPWARD CONTINUATION OF POTENTIAL FIELD DATA *

A spatial analysis of both continuous and discrete operators for upward continuation help us realize the problems and limitations which have been encountered before (Henderson 1960, Kontis 1971) but remained unsolved in practical application of upward continuation computation due to the finite length of data and operator in spatial domain. Various numerical examples show that an improvement of accuracy of continuation computations can be achieved through proper sampling and sufficient length of data.     

Time‐domain analysis of unbounded media using mixed‐variable formulations

Formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance when modelling unbounded domains. This can be done by means of fundamental solutions in space and time in connection with convolution integrals or by means of a frequency dependent boundary element representation, but for discrete frequencies Ω only. In this paper a method for interpolating discrete values of dynamic stiffness matrices by a continuous matrix valued rational function is proposed. The coupling between interface degrees of freedom is fully preserved. Another crucial point in soil–structure interaction analysis is how to implement an approximation in the spectral domain into a time‐domain analysis. Well‐known approaches for the scalar case are based on the partial‐fraction expansion of a scalar rational function. Here, a more general procedure, applicable to MDOF‐systems, for the transformation of spectral rational approximations into the time‐domain is introduced. Evaluation of the partial‐fraction expansion is avoided by using the so‐called mixed variables. Thus, unknowns in the time‐domain are displacements as well as forces. Copyright © 2001 John Wiley & Sons, Ltd.     

CHOICE OF SAMPLING STRATEGY AND ESTIMATION METHOD FOR CALCULATING NITROGEN AND PHOSPHORUS TRANSPORT IN SMALL LOWLAND STREAMS

As reliable estimates of stream nutrient transport are required for many purposes including trend analysis, mass balances and model development, the impact of sampling strategy and estimation method on the bias and precision of stream nitrogen (N) and phosphorus (P) transport calculations was evaluated. The study was undertaken in two catchments in eastern Denmark. Selection of the most accurate sampling strategy and estimation method, i.e. with the lowest root mean square error (RMSE) was based on random (Monte Carlo) runs for generating replicate data sets from an essentially complete record of the concentration of total N (TN), total P (TP), particulate P (PP) and dissolved P (DP) during a two-year period (June 1987 to June 1989). The evaluation comprised 13 different estimation methods and seven discrete sampling strategies involving three categories (regular, stratified and strata sampling). The regular sampling strategies were more accurate (lower RMSE) during high-flow periods than stratified sampling. The greatest improvement in RMSE for TN, TP, PP and DP transport was obtained when increasing the sampling frequency from 12 each year (monthly) to 18 (monthly in summer and fortnightly in winter) and 26 each year (fortnightly). The increase in accuracy (RMSE) was less when increasing the sampling frequency to 52 (weekly) or 104 (biweekly). Nearly all the methods evaluated underestimated the annual transport of TP and PP, whereas TN and DP were both under- and overestimated. The best method of estimating N and P transport when utilizing discrete sampling was both site- and time-dependent. The overall best and most reproducible (stream to stream, year to year) method for estimating annual transport of TN, TP, PP and DP was a linear interpolation method. When this method was used to derive estimates of annual TN and TP transport based on fortnightly sampling, the RMSE was 1.4–5.4 and 20.2–38.5%, respectively, in the Gelbæk stream and 1.1–4.9 and 10.5–15.0%, respectively, in the Gjern Å stream. Subdividing the hydrograph into two strata (low-flow and high-flow periods) and sampling these strata separately for calculating TP transport was superior to discrete sampling for the smaller of the two catchments. A combination of regular sampling (monthly) and pooled high-flow sampling (eight events out of a total of 43) reduced the RMSE of the annual TP load to 10.4%.     

Significance of ground motion time step in one dimensional site response analysis

The discrete nature of the numerical methods utilized in 1D site response analysis and calculation of the response spectra (e.g., frequency domain, Duhamel integral, and Newmark β methods) introduces time-step dependence in the resulting solution. Using an input ground motion with too large of a time-step leads to under-prediction of high-frequency characteristics of the system response due to limitations in the numerical solution of single and multiple degree of freedom systems. In order to reduce potential errors, using a sampling rate at least ten times greater than the maximum considered frequency is recommended. The preferred alternative is selection of input ground motions with a sufficiently small time step to avoid introducing numerical errors. However, where such motions are not available, then the time step of the ground motion can be reduced through interpolation. This paper demonstrates that the use of Fourier transform zero-padded interpolation is the preferred approach to obtain a ground motion with an adequate time step for the calculation of the elastic acceleration response spectra, and to analyze site response using either frequency or time domain methods.     

伊舒断裂带南、北段震源参数研究

利用时间城小地震地震图的方法，研究了震源参数与介质特性的差异。据初步研究结果，伊通－舒兰断裂南、北两段的震源应力降与Ｑ值存在差异。该方法是断层的震源深度范围内介质特性研究的补充，为断裂的分段提供一个重要方法     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

三维共偏移距叠前地震数据反假频射线束形成偏移成像

对稀疏/非规则采样或者低信噪比数据,射线束提取困难并伴随有假频产生,对叠加剖面和道集造成严重干扰.为了提升射线束偏移在稀疏和低信噪比地震数据采集中的成像效果,本文提出基于三角滤波的局部倾斜叠加波束形成偏移假频压制方法.射线束偏移首先将地震数据划分为超道集,经过部分NMO后转化为以射线束中心定义的共偏移距数据,倾斜叠加和反假频操作均在局部共中心点坐标上实现.时间域倾斜叠加是对地震数据的时移累加操作,三角低通滤波同样可以在时间域完成,在对地震数据进行因果和反因果积分后,亦为地震数据的时移累加.因此,三角低通滤波与倾斜叠加可在时间域结合同时完成,避免了频域滤波的正反傅里叶变换.本文在反假频公式中加入权重系数,用以对反假频的程度进行控制,达到分辨率和噪声压制的最佳折衷.以某海上三维实际数据为例,文中展示了反假频射线束形成对偏移叠加剖面和共成像点偏移距道集中的噪声进行了有效压制.     

Acoustic VTI modeling and pre-stack reverse-time migration based on the time–space domain staggered-grid finite-difference method

Reverse-time migration (RTM) is based on seismic numerical modeling algorithms, and the accuracy and efficiency of RTM strongly depend on the algorithm used for numerical solution of wave equations. Finite-difference (FD) methods have been widely used to solve the wave equation in seismic numerical modeling and RTM. In this paper, we derive a series of time–space domain staggered-grid FD coefficients for acoustic vertical transversely isotropic (VTI) equations, and adopt these difference coefficients to solve the equations, then analyze the numerical dispersion and stability, and compare the time–space domain staggered-grid FD method with the conventional method. The numerical analysis results demonstrate that the time–space domain staggered-grid FD method has greater accuracy and better stability than the conventional method under the same discretizations. Moreover, we implement the pre-stack acoustic VTI RTM by the conventional and time–space domain high-order staggered-grid FD methods, respectively. The migration results reveal that the time–space domain staggered-grid FD method can provide clearer and more accurate image with little influence on computational efficiency, and the new FD method can adopt a larger time step to reduce the computation time and preserve the imaging accuracy as well in RTM. Meanwhile, when considering the anisotropy in RTM for the VTI model, the imaging quality of the acoustic VTI RTM is better than that of the acoustic isotropic RTM.     

THE P400 WATERGUN SOURCE IN THE QUEST FOR IMPROVED RESOLUTION AT ALL DEPTHS*

The low-frequency response of the P400 watergun is an improvement over that of the S80 version. It has been further enhanced by deployment in a vertically staggered array. The notch in the amplitude spectrum at about 35 Hz due to the interference between the precursor and main implosion pulse has been virtually eliminated, and depth notches due to the free surface interface at the upper end of the spectrum have been greatly reduced. The spectral band is thus very broad and well-shaped and corresponds to an even shorter signature. After convolution with the streamer ghost, the amplitude of the implosion pulse from the composite far-field signature is more than 30 times the composite precursor amplitude. Thus the signal approaches the minimum-phase condition. Shot-generated noise scattered back by diffractors located at or near seabed is the single greatest impediment to increased penetration, especially in high resolution surveys. Such noise decays much less rapidly than signal, particularly so with broad band data. Furthermore, back-scattered interference coming from a cone of bearings between 15–45° with respect to the line is particularly damaging since in these directions the stack enhances the interference and confuses the primary velocity analysis. The watergun source patterns are therefore element-weighted to maximize directivity within these critical directions in the frequency band 50–155 Hz. When spatial aliasing is reduced by increased streamer spatial sampling and by discrete anti-alias filters during the record cycle,f-k filtering can be applied to the field records as an effective supplement to the source and receiver patterns without the aliased ‘wrapped around’ noise alignments destroying the upper end of the spectrum of genuine reflectors. Furthermore, increased spatial sampling is a pre-requisite in thef-k-migration process if the valuable higher frequencies of broad band data are to be migrated in steep-dip situations without aliasing. Thus, spatial resolution is a necessary complement to temporal resolution if maximum advantage of the broad-band P400 watergun source is to be realized.     

Knowledge gaps in our perceptual model of Great Britain's hydrology

There is a no lack of significant open questions in the field of hydrology. How will hydrological connectivity between freshwater bodies be altered by future human alterations to the hydrological cycle? Where does water go when it rains? Or what is the future space–time variability of flood and drought events? However, the answers to these questions will vary with location due to the specific and often poorly understood local boundary conditions and system properties that control the functional behaviour of a catchment or any other hydrologic control volume. We suggest that an open, shared and evolving perceptual model of a region's hydrology is critical to tailor our science questions, as it would be for any other study domain from the plot to the continental scale. In this opinion piece, we begin to discuss the elements of and point out some knowledge gaps in the perceptual model of the terrestrial water cycle of Great Britain. We discuss six major knowledge gaps and propose four key ways to reduce them. While the specific knowledge gaps in our perceptual model do not necessarily transfer to other places, we believe that the development of such perceptual models should be at the core of the debate for all hydrologic communities, and we encourage others to have a similar debate for their hydrologic domain.     

Investigation of the scale-dependent variability of radar-rainfall and rain gauge error covariance

There is a significant spatial sampling mismatch between radar and rain gauge data. The use of rain gauge data to estimate radar-rainfall error variance requires partitioning of the variance of the radar and rain gauge difference to account for the sampling mismatch. A key assumption in the literature pertaining to the error variance separation method used to partition the variance is that the covariance between radar-rainfall error and the error of rain gauges in representing radar sampling domain is negligible. Our study presents the results of an extensive test of this assumption. The test is based on empirical data and covers temporal scales ranging from 0.25 to 24 h and spatial scales ranging from 1 to 32 km. We used a two-year data set from two high quality and high density rain gauge networks in Oklahoma and excluded the winter months. The results obtained using a resampling procedure show that covariance can be considerable at large scales due to the significant variability. As the variability of the covariance rapidly increases with larger spatial and shorter temporal scales, applications of the error variance separation method at those scales require more caution. The variability of the covariance and one of its constituting variables, the variance ratio of radar and gauge errors, shows simple scaling behavior well characterized by a power-law.     

Residence and exposure times : when diffusion does not matter

Under constant hydrodynamic conditions and assuming horizontal homogeneity, negatively buoyant particles released at the surface of the water column have a mean residence time in the surface mixed layer of h/w, where h is the thickness of the latter and w (?>?0) is the sinking velocity Deleersnijder (Environ Fluid Mech 6(6):541–547, 2006a). The residence time does not depend on the diffusivity and equals the settling timescale. We show that this behavior is a result of the particular boundary conditions of the problem and that it is related to a similar property of the exposure time in a one-dimensional infinite domain. In 1-D advection–diffusion problem with a constant and uniform velocity, the exposure time—which is a generalization of the residence time measuring the total time spent by a particle in a control domain allowing the particle to leave and reenter the control domain—is also equal to the advection timescale at the upstream boundary of the control domain. To explain this result, the concept of point exposure is introduced; the point exposure is the time integral of the concentration at a given location. It measures the integrated influence of a point release at a given location and is related to the concept of number of visits of the theory of random walks. We show that the point exposure takes a constant value downstream the point of release, even when the diffusivity varies in space. The analysis of this result reveals also that the integrated downstream transport of a passive tracer is only effected by advection. While the diffusion flux differs from zero at all times, its integrated value is strictly zero.