Detection of multiple fracture sets using observations of shear‐wave splitting in microseismic data

The hydrocarbon industry is moving increasingly towards tight sandstone and shale gas resources – reservoirs that require fractures to be produced economically. Therefore, techniques that can identify sets of aligned fractures are becoming more important. Fracture identification is also important in the areas of coal bed methane production, carbon capture and storage (CCS), geothermal energy, nuclear waste storage and mining. In all these settings, stress and pore pressure changes induced by engineering activity can generate or reactivate faults and fractures. P‐ and S‐waves are emitted by such microseismic events, which can be recorded on downhole geophones. The presence of aligned fracture sets generates seismic anisotropy, which can be identified by measuring the splitting of the S‐waves emitted by microseismic events. The raypaths of the S‐waves will have an arbitrary orientation, controlled by the event and geophone locations, meaning that the anisotropy system may only be partly illuminated by the available arrivals. Therefore to reliably interpret such splitting measurements it is necessary to construct models that compare splitting observations with modelled values, allowing the best fitting rock physics parameters to be determined. Commonly, splitting measurements are inverted for one fracture set and rock fabrics with a vertical axis of symmetry. In this paper we address the challenge of identifying multiple aligned fracture sets using splitting measured on microseismic events. We analyse data from the Weyburn CCS‐EOR reservoir, which is known to have multiple fracture sets, and from a hydraulic fracture stimulation, where it is believed that only one set is present. We make splitting measurements on microseismic data recorded on downhole geophone arrays. Our inversion technique successfully discriminates between the single and multiple fracture cases and in all cases accurately identifies the strikes of fracture sets previously imaged using independent methods (borehole image logs, core samples, microseismic event locations). We also generate a synthetic example to highlight the pitfalls that can be encountered if it is assumed that only one fracture set is present when splitting data are interpreted, when in fact more than one fracture set is contributing to the anisotropy.     

基于Low rank近似的黏弹性介质衰减补偿微地震逆时定位与裂缝成像

真实地下介质具有黏弹性,地震波在传播过程中会发生耗散与频散.忽视黏弹性介质的吸收衰减效应,逆时延拓过程中地震波将会出现振幅减弱、相位失真等现象,无法准确定位震源真实位置,因此需要对黏弹性介质中传播的波场进行衰减补偿,并通过采用合适的成像算子对微地震震源进行定位与裂缝成像.本文基于耗散与频散解耦的分数阶黏弹性波动方程模拟...     

基于异向波速模型的微震定位改进

针对波速分层的区域岩体,在异向波速模型的基础上,对垂向上的应力波按岩体波速值大小作分段区别,推导震源应力波走时关系式,建立分层速度定位目标函数,基于此提出一种由参数准备、层速度反演、微震定位三个模块组成的分层速度定位模型SV,并采用遗传算法进行优化求解.然后,对分层速度定位模型在已构建微震监测系统的白鹤滩水电站左岸岩质边坡进行验证.微震事件重定位结果表明,分层速度定位模型定位微震事件的最大、最小和平均偏离层内错动带程度指标较单一速度模型分别降低了57.17%、36.51%和57.35%,证明了定位模型在波速分层的区域岩体微震定位应用中比单一速度定位模型更加合理可靠.     

Seismic anisotropy in a hydrocarbon field estimated from microseismic data

The study of seismic anisotropy in exploration seismology is gaining interest as it provides valuable information about reservoir properties and stress directions. In this study we estimate anisotropy in a petroleum field in Oman using observations of shear‐wave splitting from microseismic data. The data set was recorded by arrays of borehole geophones deployed in five wells. We analyse nearly 3400 microearthquakes, yielding around 8500 shear‐wave splitting measurements. Stringent quality control reduces the number of reliable measurements to 325. Shear‐wave splitting modelling in a range of rock models is then used to guide the interpretation. The difference between the fast and slow shear‐wave velocities along the raypath in the field ranges between 0–10% and it is controlled both by lithology and proximity to the NE‐SW trending graben fault system that cuts the field formations. The anisotropy is interpreted in terms of aligned fractures or cracks superimposed on an intrinsic vertical transversely isotropic (VTI) rock fabric. The highest magnitudes of anisotropy are within the highly fractured uppermost unit of the Natih carbonate reservoir. Anisotropy decreases with depth, with the lowest magnitudes found in the deep part of the Natih carbonate formation. Moderate amounts of anisotropy are found in the shale cap rock. Anisotropy also varies laterally with the highest anisotropy occurring either side of the south‐eastern graben fault. The predominant fracture strikes, inferred from the fast shear‐wave polarizations, are consistent with the trends of the main faults (NE‐SW and NW‐SE). The majority of observations indicate subvertical fracture dip (>70° ). Cumulatively, these observations show how studies of shear‐wave splitting using microseismic data can be used to characterize fractures, important information for the exploitation of many reservoirs.     

Source-independent waveform inversion for attenuation estimation in anisotropic media

In previous publications, we presented a waveform-inversion algorithm for attenuation analysis in heterogeneous anisotropic media. However, waveform inversion requires an accurate estimate of the source wavelet, which is often difficult to obtain from field data. To address this problem, here we adopt a source-independent waveform-inversion algorithm that obviates the need for joint estimation of the source signal and attenuation coefficients. The key operations in that algorithm are the convolutions (1) of the observed wavefield with a reference trace from the modelled data and (2) of the modelled wavefield with a reference trace from the observed data. The influence of the source signature on attenuation estimation is mitigated by defining the objective function as the ℓ₂-norm of the difference between the two convolved data sets. The inversion gradients for the medium parameters are similar to those for conventional waveform-inversion techniques, with the exception of the adjoint sources computed by convolution and cross-correlation operations. To make the source-independent inversion methodology more stable in the presence of velocity errors, we combine it with the local-similarity technique. The proposed algorithm is validated using transmission tests for a homogeneous transversely isotropic model with a vertical symmetry axis that contains a Gaussian anomaly in the shear-wave vertical attenuation coefficient. Then the method is applied to the inversion of reflection data for a modified transversely isotropic model from Hess. It should be noted that due to the increased nonlinearity of the inverse problem, the source-independent algorithm requires a more accurate initial model to obtain inversion results comparable to those produced by conventional waveform inversion with the actual wavelet.     

层状TI介质中微地震定位和各向异性速度结构同时反演

微地震监测被广泛应用于非常规油气资源的水力压裂作业、油藏描绘和水驱前缘监测工程中.微地震定位采用的初始速度模型一般是基于地震测井记录和射孔数据建立,该速度模型的不准确性易引起定位误差.为降低这种定位误差,本文发展了一种微地震定位和各向异性速度结构同时反演的方法.研究对象为1-D的层状TI介质,其中对称轴方向任意.利用改进的分区多步最短路径算法计算qP、qSV和qSH波的到达时间和射线路径,结合共轭梯度法求解带约束的阻尼最小二乘问题.数值模拟结果表明,该算法能同时进行各向异性速度结构模型（每层的Thomsen参数和界面深度）和微震震源参数（空间坐标和发震时刻）的反演,并且对随机噪声不敏感,有利于实际工程应用.

     

Approximate traveltime inversion in downhole microseismic monitoring

In downhole microseismic monitoring, accurate event location relies on the accuracy of the velocity model. The model can be estimated along with event locations. Anisotropic models are important to get accurate event locations. Taking anisotropy into account makes it possible to use additional data – two S-wave arrivals generated due to shear-wave splitting. However, anisotropic ray tracing requires iterative procedures for computing group velocities, which may become unstable around caustics. As a result, anisotropic kinematic inversion may become time consuming. In this paper, we explore the idea of using simplified ray tracing to locate events and estimate medium parameters. In the simplified ray-tracing algorithm, the group velocity is assumed to be equal to phase velocity in both magnitude and direction. This assumption makes the ray-tracing algorithm five times faster compared to ray tracing based on exact equations. We present a set of tests showing that given perforation-shot data, one can use inversion based on simplified ray-tracing even for moderate-to-strong anisotropic models. When there are no perforation shots, event-location errors may become too large for moderately anisotropic media.     

Stability of source mechanisms inverted from P‐wave amplitude microseismic monitoring data acquired at the surface

We study the stability of source mechanisms inverted from data acquired at surface and near‐surface monitoring arrays. The study is focused on P‐wave data acquired on vertical components, as this is the most common type of acquisition. We apply ray modelling on three models: a fully homogeneous isotropic model, a laterally homogeneous isotropic model and a laterally homogeneous anisotropic model to simulate three commonly used models in inversion. We use geometries of real arrays, one consisting in surface receivers and one consisting in ‘buried’ geophones at the near‐surface. Stability was tested for two of the frequently observed source mechanisms: strike‐slip and dip‐slip and was evaluated by comparing the parameters of correct and inverted mechanisms. We assume these double‐couple source mechanisms and use quantitatively the inversion allowing non‐double‐couple components to measure stability of the inversion. To test the robustness we inverted synthetic amplitudes computed for a laterally homogeneous isotropic model and contaminated with noise using a fully homogeneous model in the inversion. Analogously amplitudes computed in a laterally homogeneous anisotropic model were inverted in all three models. We show that a star‐like surface acquisition array provides very stable inversion up to a very high level of noise in data. Furthermore, we reveal that strike‐slip inversion is more stable than dip‐slip inversion for the receiver geometries considered here. We show that noise and an incorrect velocity model may result in narrow bands of source mechanisms in Hudson's plots.     

煤矿冲击地压的微地震监测研究

为了研究煤矿冲击地压与岩层在三维空间破裂之间的关系,进而探索依据岩层破裂规律预测和预报冲击地压的可能性,文中采用自行研制的防爆型微地震定位监测（MS）系统,基于定位原理,监测了山东华丰煤矿冲击地压煤层（四层煤）及其解放层（六层煤）开采过程中的岩层破裂过程和二次应力场分布变化的过程,得到了如下结论：冲击地压的发生与岩层破裂密切相关,四层煤下顺槽处于六层煤顶板破裂区的外边缘时,正处于高应力区内,在此处掘进容易引发冲击地压,必须将六层煤下顺槽位置向实体煤侧移动20 m以上,或将四层煤下顺槽位置内移20 m以上,才能消除四层煤的冲击地压;六层煤和四层煤开采时,工作面前方断层活化的距离分别为250 m和350 m左右,根据这一距离,及时对断层带进行卸压处理,可以消除由断层带引发的冲击地压;监测显示了工作面周围岩层的三维破裂形态和范围,为矿井确定防水煤柱的高度提供了可靠的依据;监测证明了厚层砾岩的破裂、断层活化、采场附近关键层的破裂是引起冲击地压的主要原因,证明了所研制的硬件和定位软件具有较高的精度和实用性,可以在煤矿和边坡、隧道等领域应用.     

Comparison of migration‐based location and detection methods for microseismic events

Microseismic monitoring in the oil and gas industry commonly uses migration‐based methods to locate very weak microseismic events. The objective of this study is to compare the most popular migration‐based methods on a synthetic dataset that simulates a strike‐slip source mechanism event with a low signal‐to‐noise ratio recorded by surface receivers (vertical components). The results show the significance of accounting for the known source mechanism in the event detection and location procedures. For detection and location without such a correction, the ability to detect weak events is reduced. We show both numerically and theoretically that neglecting the source mechanism by using only absolute values of the amplitudes reduces noise suppression during stacking and, consequently, limits the possibility to retrieve weak microseismic events. On the other hand, even a simple correction to the data polarization used with otherwise ineffective methods can significantly improve detections and locations. A simple stacking of the data with a polarization correction provided clear event detection and location, but even better results were obtained for those data combined with methods that are based on semblance and cross‐correlation.     

背景为各向异性的含裂缝岩石频散和衰减计算方法研究

裂缝广泛分布于各类储层岩石中, 并且会显著提高储层的渗流能力.因此, 裂缝的评价和表征对于提高油气产能具有重要意义.由于裂缝与背景介质之间的波致流会显著影响地震波的频散和衰减特性, 所以地震勘探是评价裂缝性储层的有效手段.裂缝地震定量表征的前提是要基于含裂缝岩石中波致流对频散和衰减的影响建立含裂缝岩石物理特性与地震性质的关系.然而, 目前相关的理论研究大部分基于各向同性背景这一假设, 难以有效应用于常见的各向异性储层.本文针对背景为各向异性的含裂缝岩石提出了频散和衰减的计算方法.该方法首先将含裂缝岩石中的各向异性背景介质等效为层状背景介质; 然后, 通过分析不同频率下层状含裂缝岩石中的流体压力分布, 理论计算了两个特定的中间频率并求解得到两个中间频率下的弹性参数; 进一步, 以计算得到的两个特定中间频率以及高低频极限下的弹性参数为基础, 应用数值方法求解得到弛豫函数中的未知参数, 最终实现了背景为各向异性含裂缝岩石中频散和衰减的理论模拟.通过将理论预测结果与实验测量和数值模拟结果进行对比, 验证了该方法在背景为各向异性含不同分布裂缝岩石中的有效性.本文提出的方法考虑了常见的各向异性背景对含裂缝岩石频散和衰减的影响, 因而在裂缝性储层的地震勘探中具有广泛的应用前景.

     

Convolutional neural networks for automated microseismic detection in downhole distributed acoustic sensing data and comparison to a surface geophone array

Distributed acoustic sensing is a growing technology that enables affordable downhole recording of strain wavefields from microseismic events with spatial sampling down to ∼1 m. Exploiting this high spatial information density motivates different detection approaches than typically used for downhole geophones. A new machine learning method using convolutional neural networks is described that operates on the full strain wavefield. The method is tested using data recorded in a horizontal observation well during hydraulic fracturing in the Eagle Ford Shale, Texas, and the results are compared to a surface geophone array that simultaneously recorded microseismic activity. The neural network was trained using synthetic microseismic events injected into real ambient noise, and it was applied to detect events in the remaining data. There were 535 detections found and no false positives. In general, the signal-to-noise ratio of events recorded by distributed acoustic sensing was lower than the surface array and 368 of 933 surface array events were found. Despite this, 167 new events were found in distributed acoustic sensing data that had no detected counterpart in the surface array. These differences can be attributed to the different detection threshold that depends on both magnitude and distance to the optical fibre. As distributed acoustic sensing data quality continues to improve, neural networks offer many advantages for automated, real-time microseismic event detection, including low computational cost, minimal data pre-processing, low false trigger rates and continuous performance improvement as more training data are acquired.     

微破裂向量扫描在丰探15井压裂微震实时监测中的应用

浙江油田苏北采油厂于2017-07-28～29对丰探15井近4 km深的泰二段第三段进行了水力压裂,分成试压和主压.我们同时应用微破裂向量扫描(Vector Scanning,VS)实施了地面微震实时监测,目的是评估监测质量、实时监测的可行性、与压裂效果.我们严格执行了安静处布台和有效去噪等应用VS的必要条件,最大限度地提高了信噪比,平均的最小信噪比为2.35%;输出结果可靠.为今后更有效地设计和控制压裂,通过电信网和互联网,我们实施了野外采集数据的实时传输和遥控数据处理解释,观察了每分钟裂缝带的发展;实时监测系统可行.主压的最终X型破裂明显是在试压裂缝的基础上,由试压的300 m长,扩展到400～500 m,并加密了缝网;井域最大主压应力方位被限制在NE(50～80)±5°.本次压裂微震活动有明显的间歇性,较大微震群发生前积累能量所需时间,由初期的10 min内,逐渐延长到最后的百分钟.微震活动由压裂点扩展到远处的顺序清晰,这可能同此井附近的介质较为完整有关.每分钟的压裂微震活动状态一般在5～15 min内可报告给压裂施工团队.通过这次典型的水力压裂微震实时监测,我们验证了:VS的原...     

时窗能量特征法拾取微地震波初始到时的可行性研究

微地震（MS）波初始到时的自动拾取是MS监测数据处理的关键技术之一,也是实现MS震源自动定位的技术难点.本文在MS震源定位结果反演与推断的研究基础上,对不同类型MS波的到时点特征进行了分析与描述,并对不同时窗长度下能量特征值的变化规律进行了研究,提出了控制时窗移动范围和确定时窗长度自适应参数的具体方法,利用建立的MS波初始到时点特征的模式识别库,对拾取的到时进行模式归类、定量评价和匹配,提高了自动拾取结果的可靠性.研究结果表明,对典型的信噪比高的MS波,到时自动拾取的结果与手工拾取的结果基本一致;对无量纲大振幅的MS波,到时自动拾取结果的可靠性要高于手工拾取,对信噪比低和到时点不清晰的MS波自动拾取的可靠性较低.     

Constrained tomography of realistic velocity models in microseismic monitoring using calibration shots

The knowledge of the velocity model in microseismic jobs is critical to achieving statistically reliable microseismic event locations. The design of microseismic networks and the limited sources for calibration do not allow for a full tomographic inversion. We propose optimizing a priori velocity models using a few active shots and a non‐linear inversion, suitable to poorly constrained systems. The considered models can be described by several layers with different P‐ and S‐wave velocities. The velocities may be constant or have 3D gradients; the layer interfaces may be simple dipping planes or more complex 3D surfaces. In this process the P‐ and S‐ wave arrival times and polarizations measured on the seismograms constitute the observed data set. They are used to estimate two misfit functions: i) one based on the measurement residuals and ii) one based on the inaccuracy of the source relocation. These two functions are minimized thanks to a simulated annealing scheme, which decreases the risk of converging to a local solution within the velocity model. The case study used to illustrate this methodology highlights the ability of this technique to constrain a velocity model with dipping layers. This was performed by jointly using sixteen perforation shots recorded during a multi‐stage fracturing operation from a single string of 3C‐receivers. This decreased the location inaccuracies and the residuals by a factor of six. In addition, the retrieved layer dip was consistent with the pseudo‐horizontal trajectories of the wells and the background information provided by the customer. Finally, the theoretical position of each calibration shot was contained in the uncertainty domain of the relocation of each shot. In contrast, single‐stage inversions provided different velocity models that were neither consistent between each other nor with the well trajectories. This example showed that it is essential to perform a multi‐stage inversion to derive a better updated velocity model.     

Seismic attenuation and velocity dispersion in fractured rocks: The role played by fracture contact areas

The presence of fractures in fluid‐saturated porous rocks is usually associated with strong seismic P‐wave attenuation and velocity dispersion. This energy dissipation can be caused by oscillatory wave‐induced fluid pressure diffusion between the fractures and the host rock, an intrinsic attenuation mechanism generally referred to as wave‐induced fluid flow. Geological observations suggest that fracture surfaces are highly irregular at the millimetre and sub‐millimetre scale, which finds its expression in geometrical and mechanical complexities of the contact area between the fracture faces. It is well known that contact areas strongly affect the overall mechanical fracture properties. However, existing models for seismic attenuation and velocity dispersion in fractured rocks neglect this complexity. In this work, we explore the effects of fracture contact areas on seismic P‐wave attenuation and velocity dispersion using oscillatory relaxation simulations based on quasi‐static poroelastic equations. We verify that the geometrical and mechanical details of fracture contact areas have a strong impact on seismic signatures. In addition, our numerical approach allows us to quantify the vertical solid displacement jump across fractures, the key quantity in the linear slip theory. We find that the displacement jump is strongly affected by the geometrical details of the fracture contact area and, due to the oscillatory fluid pressure diffusion process, is complex‐valued and frequency‐dependent. By using laboratory measurements of stress‐induced changes in the fracture contact area, we relate seismic attenuation and dispersion to the effective stress. The corresponding results do indeed indicate that seismic attenuation and phase velocity may constitute useful attributes to constrain the effective stress. Alternatively, knowledge of the effective stress may help to identify the regions in which wave induced fluid flow is expected to be the dominant attenuation mechanism.     

基于机器学习和台阵相关性的水力压裂微地震事件自动识别及到时拾取

利用密集台阵对水力压裂微地震进行监测将有助于优化储层压裂、揭示断层活化.为满足密集台阵海量采集数据的处理需求, 本文建立了一种综合运用多种机器学习方法和台阵相关性的、无需人工干预的自动处理流程, 从而能够快速得到高质量的密集台阵震相到时目录.该综合策略包括: (1)利用迁移学习在连续波形中快速检测地震事件; (2)利用U型神经网络PhaseNet自动拾取P波、S波震相; (3)利用三重线性剔除法, 结合密集台阵到时相关性剔除异常到时数据和地震事件; (4)利用K-means和SVM两类机器学习算法, 进一步区分发震时刻接近的多个地震事件, 减小事件漏拾率.通过将该流程应用于四川盆地长宁—昭通页岩气开发区微地震监测数据, 并将自动处理结果与人工拾取结果进行比对发现, 二者在震级测定、定位以及走时成像结果等方面具有很好的一致性, 表明本文处理流程结果精度可达到手动处理精度.本文结果为密集台阵地震监测数据的高效、高精度处理提供了新思路.

     

Effect of stress interactions on anisotropic P-SV-wave dispersion and attenuation for closely spaced cracks in saturated porous media

Generally, local stress induced by individual crack hardly disturbs their neighbours for small crack densities, which, however, could not be neglected as the crack density increases. The disturbance becomes rather complex in saturated porous rocks due to the wave-induced diffusion of fluid pressures. The problem is addressed in this study by the comparison of two solutions: the analytical solution without stress interactions and the numerical method with stress interactions. The resultant difference of effective properties can be used to estimate the effect of stress interactions quantitatively. Numerical experiments demonstrate that the spatial distribution pattern of cracks strongly affects stress interactions. For regularly distributed cracks, the resulting stress interaction (shielding or amplification) shows strong anisotropy, depending on the arrangement and density of cracks. It has an important role in the estimation of effective anisotropic parameters as well as the incident-angle-dependency of P- and SV-wave velocities. Contrarily, randomly distributed cracks with a relative small crack density generally lead to a strong cancellation of stress interactions across cracks, where both the numerical and analytical solutions show a good agreement for the estimation of effective parameters. However, for a higher crack density, the incomplete cancellation of stress interactions is expected, exhibiting an incidence-angle dependency, slightly affecting effective parameters, and differentiating the numerical and analytical solutions.     

Effective anisotropic velocity model from surface monitoring of microseismic events

We develop a methodology to obtain a consistent velocity model from calibration shots or microseismicity observed on a buried array. Using a layered 1D isotropic model derived from checkshots as an initial velocity model, we invert P‐wave arrival times to obtain effective anisotropic parameters with a vertical axis of symmetry (VTI). The nonlinear inversion uses iteration between linearized inversion for anisotropic parameters and origin times or depths, which is specific to microseismic monitoring. We apply this technique to multiple microseismic events from several treatments within a buried array. The joint inversion of selected events shows a largely reduced RMS error indicating that we can obtain robust estimates of anisotropic parameters, however we do not show improved source locations. For joint inversion of multiple microseismic events we obtained Thomsen anisotropic parameters ε of 0.15 and δ of 0.05, which are consistent with values observed in active seismic surveys. These values allow us to locate microseismic events from multiple hydraulic fracture treatments separated across thousands of metres with a single velocity model. As a result, we invert the effective anisotropy for the buried array region and are able to provide a more consistent microseismicity mapping for past and future hydraulic fracture stimulations.     

The estimation of spectra from time‐frequency transforms for use in attenuation studies

We introduce the signal dependent time–frequency distribution, which is a time–frequency distribution that allows the user to optimize the tradeoff between joint time–frequency resolution and suppression of transform artefacts. The signal‐dependent time–frequency distribution, as well as the short‐time Fourier transform, Stockwell transform, and the Fourier transform are analysed for their ability to estimate the spectrum of a known wavelet used in a tuning wedge model. Next, the signal‐dependent time–frequency distribution, and fixed‐ and variable‐window transforms are used to estimate spectra from a zero‐offset synthetic seismogram. Attenuation is estimated from the associated spectral ratio curves, and the accuracy of the results is compared. The synthetic consisted of six pairs of strong reflections, based on real well‐log data, with a modeled intrinsic attenuation value of 1000/Q = 20. The signal‐dependent time–frequency distribution was the only time–frequency transform found to produce spectra that estimated consistent attenuation values, with an average of 1000/Q = 26±2; results from the fixed‐ and variable‐window transforms were 24±17 and 39±10, respectively. Finally, all three time–frequency transforms were used in a pre‐stack attenuation estimation method (the pre‐stack Q inversion algorithm) applied to a gather from a North Sea seismic dataset, to estimate attenuation between nine different strong reflections. In this case, the signal‐dependent time‐frequency distribution produced spectra more consistent with the constant‐Q model of attenuation assumed in the pre‐stack attenuation estimation algorithm: the average L1 residuals of the spectral ratio surfaces from the theoretical constant‐Q expectation for the signal‐dependent time‐frequency distribution, short‐time Fourier transform, and Stockwell transform were 0.12, 0.21, and 0.33, respectively. Based on the results shown, the signal‐dependent time‐frequency distribution is a time–frequency distribution that can provide more accurate and precise estimations of the amplitude spectrum of a reflection, due to a higher attainable time–frequency resolution.