Wavenumber Components Research Articles

Robust full-waveform inversion based on automatic differentiation and differentiable dynamic time warping

AbstractFull waveform inversion is a methodology that determines high-resolution parameters. The widely used L2-norm misfit function has local minima if the low wavenumber components are not accurate. Suffering from a cycle skipping problem, the solution of waveform inversion will be trapped in the local minima. Dynamic time warping aims to find an optimal alignment between two signals, which is a more robust measure to avoid cycle-skipping challenges. However, the discontinuity makes the conventional dynamic time warping distance unsuitable for waveform inversion. We introduce a soft dynamic time warping distance as the misfit function, which is differentiable in that the inverted solution can converge to the accurate global minimum. We compare the convexity of the L2-norm and soft dynamic time warping distance and show that the soft dynamic time warping distance has a wider convexity range with different time shift and amplitudes. It can alleviate the half-wavelength limitation of the conventional L2-norm. We calculate the gradient using the automatic differentiation technique and the minibatch strategy and then analyse the alignment paths of different smooth parameters. A significant smooth parameter γ makes the Soft-DTW distance tending to the L2-norm, which generates new local minima. We recommend a small smooth parameter to ensure the convexity of the Soft-DTW distance. Numerical examples show that the soft dynamic time warping can effectively reconstruct the deep velocity parameters of the BG Compass and Marmousi models with noise robustness and lower dependence on the initial model.

Broadband muffler by merging negative density and negative compressibility

We present a broadband muffler using a metamaterial consisting of a membrane structure and a Helmholtz resonator. The membrane structure with ventilation exhibited negative effective density in the 290–435 Hz frequency range. The Helmholtz resonator exhibited negative effective compressibility in the 440–600 Hz range. We connected the two frequency ranges to continue the negativity by matching the end frequency of the negative effective density and the start frequency of the negative effective compressibility. The metamaterial muffler showed a wide range of imaginary components of wavenumbers from 290 to 600 Hz. The muffler exhibits 10 dB transmission loss or 10 % power transmission in the frequency range 275–500 Hz. The realization of a metamaterial muffler with a broad and low-frequency performance range is unprecedented when considering the resonant properties of the metamaterial.

Analytical study on vibration behaviors of pump-jet–shaft–submarine hull system in wavenumber–frequency​ domain

The pump-jet excitation is transmitted to submarine hull along rotor–shaft system and duct–stator system, which is one of the main causes of structural vibration of the hull. This paper proposes an analytical method for the coupled shaft–submarine hull system. The submarine hull is modeled as a jointed conical-orthogonally stiffened cylindrical–spherical shell coupled with several circular plates. The shaft is modeled with a three-dimensional beam and connected to the hull by bearings. A unified variational modeling procedure is developed to assemble each substructure. The hydrodynamic pressure acting on the rotor and the duct–stator system are respectively considered as single point force and multiple uniformly or non-uniformly spaced point forces acting on the shaft and the conical shell. The vibration displacements are analytically expanded with Fourier series in the circumferential direction. Thus, the wavenumber components that contribute to vibration responses are naturally separated and quantified. The convergence and accuracy of the proposed method are verified by the finite element method. The analytical study on vibration characteristics of the pump-jet–shaft–submarine hull system along distinct vibration transferring paths in circumferential wavenumber–frequency domain is conducted. The results indicate that the rotor excitation can excite more resonance peaks compared to the duct–stator excitation when neglecting the elasticity of the pump-jet propulsor. It mainly owns to the strong circumferential modal coupling and abundant modal contributions when the system is under the rotor excitation. Besides, the non-uniform distribution of external load contributes to the augmentation of resonance peaks compared to the responses under uniform load, which attributes to the participation of extra wavenumber components.

Three-dimensional wall heating elements effect on the instability in zero pressure gradient supersonic boundary layers

The streaky boundary layers have very important roles in laminar-turbulent transition. Streaks of appropriate size can influence stabilities in boundary layers. In this paper, the effect of steady streamwise elongated, spanwise periodic wall heating elements on the first mode instability in supersonic flat plate boundary layers was investigated. For the balance of the efficient and accuracy, the linearized Navier–Stokes equations are used to obtain the base flow and compared with compressible Navier–Stokes equations. A bi-global analysis tool is used for the instability analysis because the heating source has a much larger length-scale in the streamwise direction than that in the spanwise direction, and the streamwise velocity is much larger than the normal velocity and the spanwise one. Results indicated that the distortion caused by the three-dimensional surface heating elements could modify the first mode, resulting in a lower frequency but with an uncertain effect on the higher frequency modes. Additionally, the streaks make the lower spanwise wave number components of the even first mode disturbance in a three-dimensional supersonic boundary layer in the freestream. As a result, the spontaneous radiation of an acoustic wave to the far field was found for the even mode. These findings suggest that laminar-turbulence transition can be suppressed or enhanced by the three-dimensional wall heating.

Triggering of Whistler‐Mode Rising and Falling Tone Emissions in a Homogeneous Magnetic Field

AbstractWe perform a self‐consistent one‐dimensional electromagnetic particle simulation with a uniform magnetic field and open boundaries. The plasma environment consists of cold isotropic electrons, energetic electrons, and immobile ions. The energetic electrons are initialized with a subtracted‐Maxwellian distribution with temperature anisotropy. By oscillating external currents with a constant frequency 0.2 fce, where fce is the electron cyclotron frequency, a whistler‐mode wave is injected as a triggering wave from the center of the simulation system, and we investigated the process of interactions between the triggering wave and energetic electrons. We find that both rising‐tone and falling‐tone emissions are triggered through the formation of an electron hole and an electron hill in the velocity phase space consisting of a parallel velocity and the gyro‐phase angle of the perpendicular velocities. The rising‐tone emission varies from 0.2 fce to 0.4 fce, while the falling‐tone varies from 0.2 fce to 0.15 fce. The generation region of the rising‐tone triggered emission starts near the injection point of the triggering wave and moves upstream generating new subpackets. The generation region of the falling‐tone triggered emission also moves upstream generating new subpackets. The simultaneous formation of the electron hole and hill is identified by separating small and large wavenumber components corresponding to lower and higher frequencies, respectively, by applying the discrete Fourier transformation to the waveforms in space. Based on the simulation results of the whistler‐mode triggered emissions, we conclude that the mechanism of frequency variation of whistler‐mode chorus emissions works even in a uniform magnetic field.

Impact of the Low Wavenumber Structure in the Initial Vortex Wind Analyses on the Prediction of the Intensification of Hurricane Patricia (2015)

AbstractEnsemble clustering analysis was performed to explore the role of the initial hurricane vortex‐scale wind structure in the prediction of the intensification of Hurricane Patricia (2015). Convection‐allowing ensemble forecasts were classified into spin‐down (SPD) and spin‐up (SPU) groups. Specifically, 10 members with an intensification rate >0 m/s and 10 members with an intensification rate <0 m/s for the first 6 hr were defined as the SPD and SPU members. The result showed that the tangential winds outside the inner‐core region in the SPD members were weaker compared to the SPU members. Additionally, the SPD members had a weaker inflow near the surface and a weaker outflow between the heights of 8 and 12 km than the SPU members. The SPU members showed more significant azimuthal asymmetry than the SPD members in the surface, tangential and radial winds. Wavenumber analysis showed that the low wavenumber components dominated the differences between the SPD and SPU members. Numerical experiments were conducted to test the hypothesis generated by the clustering analysis. It was found that the storm's maximum wind speed (MWS) intensified during the first 6 hr of the model forecast if only the low wavenumber structure in the SPU members was included in the initial conditions, whereas it decayed during the first 6 hr if only the low wavenumber structure in the SPD members was included. This result confirms that the low wavenumber structure of the initial wind analyses was important in predicting the intensity changes of Hurricane Patricia (2015).

The viscoacoustic Green's function for the Helmholtz equation in a velocity gradient interface model

SUMMARY In viscoacoustically stratified media where the density is constant, 3-D wave propagation in the frequency–radial wavenumber domain is governed by the Helmholtz equation. In the case that the model is a velocity gradient interface where the squared velocity in depth is represented by a smooth Heaviside function of the Fermi–Dirac distribution type, the Helmholtz equation for a point source at arbitrary location is shown to have analytical solution for the Green's function. The velocity depth profile, which is a modification of the Epstein profile which has been thoroughly studied in different branches of physics, is described by four parameters: the velocities at minus and plus infinity, the reference depth of the gradient interface, and its smoothness. The Helmholtz equation is first solved in a model where the point source is absent. The solution to the source-free equation has four unknown constants that must be determined. The radiation conditions at minus and plus infinity and two conditions on the Green's function at the source depth allow the constants to be found. The Green's function solution can be represented in two mathematically equivalent algebraic forms involving ordinary hypergeometric functions. The first form allows a numerically stable implementation over all wavenumber components. The second form allows a physical, intuitive interpretation and is expressed mathematically as the sum of two terms. Each of the terms contains the product of constant-velocity reference phase shift functions and hypergeometric functions which take the role to adjust the amplitude and phase shift calculated by the reference phase shift functions to account for the depth varying velocity profile. Inverse Fourier transforms take the Green's function from the frequency–wavenumber domain to the frequency–space domain or time–space domain. The Green's function solution is valid for any sharpness of the interface. Selected numerical results are presented for the 1-D and 2-D Helmholtz equation to demonstrate the influence of the velocity gradient zone on the wavefield. The 1-D solution in an acoustic model is compared in time domain to the classical finite-difference wave propagation solution. For the purpose of interpretation of seismograms, we model for comparison the wavefield response in a model of two half-spaces in welded contact. For brevity, the latter model is referenced as the HS model.

Decoupled wave equation and forward modeling of qP wave in VTI media with the new acoustic approximation

Acoustic approximation has received wide attention in modeling and inversion for the anisotropic wave equation to avoid high computational cost and parameter trade-off in seismic inversion. However, it also is limited by the stability condition, such as the instability for the qP-wave equation in transversely isotropic media with [Formula: see text]. We have developed a new approach that decoupled wave equation and forward modeling of the qP wave in vertical transversely isotropic (VTI) media with the new acoustic approximation. To keep dispersion relations ([Formula: see text]) of the qSV wave in each direction equal to zero, we formulate the vertical S-wave velocity ([Formula: see text]) to be a function of the model parameters and wavenumber component ([Formula: see text]), rather than setting it to zero. Then, the corresponding dispersion relation of pure qP for the new acoustic approximation is derived. According to this decoupled dispersion relation, we obtain the decoupled wave equation of pure qP wave in VTI media by inverse Fourier transform. To solve the wave equation in the space domain efficiently, the operator [Formula: see text] in the wave equation is characterized in the space domain by its asymptotic approximation operator [Formula: see text]. From the qP wave equation in the time-space domain, we realize the forward modeling of the pure qP wave in VTI media with the finite-difference method. The dispersion relation analysis and numerical examples find that the decoupled qP-wave equation with the new acoustic approximation does not contain the degenerate qSV wave and is a pure qP-wave equation, which is in good agreement with the simulation results of the elastic wave equation and has high accuracy and is stable in VTI media with [Formula: see text] or [Formula: see text].

4D Seismic Monitoring with Diffraction-Angle-Filtering for Carbon Capture and Storage (CCS)

Carbon capture and storage (CCS) is a key technology that directly removes industry driven CO2 to achieve carbon neutrality. In the process of CCS, it is necessary to monitor whether injected CO2 is properly stored and not leaking. The behavior of CO2 can be investigated using a 4D seismic survey that compares seismic data before and after injection. We proposed a two-step monitoring with diffraction-angle filtering (DAF) to effectively locate the CO2 plume. Because DAF allows us to control wavenumber components, the gradient of full-waveform inversion (FWI), which is the first step, is composed of low-wavenumber components, and reverse time migration (RTM) for seismic imaging is carried out with high-wavelength components. To verify our method, we implemented FWI and RTM with and without DAF using the velocity model in the Volve oil field in the North Sea. Numerical examples show that the CO2 plume is properly detected from the difference between baseline and post-injected survey and the extension of the reflective boundary is improved compared to the results of the conventional method. With our proposed method, local minima problem is mitigated in FWI, and the boundaries between layers can be clearly distinguished in RTM.

A robust migration velocity analysis method based on adaptive differential semblance optimization

Migration velocity analysis (MVA) is an efficient tool to reconstruct low wavenumber components of the model. Compared with data domain methods, the MVA methods are implemented in image domain by processing imaging results directly through minimizing moveouts and improving the coherence of the common image gathers (CIGs), such as angle domain common image gathers (ADCIGs) or subsurface offset domain image gathers (ODCIGs). As one of the wave-equation based migration velocity analysis (WEMVA) methods, differential semblance optimization (DSO) can automatically detect the moveout existed in the CIGs and is convenient to implement. However, referring to the CIGs contaminated by spurious imaging artefacts caused by uneven illumination and irregular observation geometry, the traditional DSO method may produce poor velocity update with oscillations, which can prevent rapid convergence to a correct velocity model. To deal with this issue and extend the solution space, we introduce an adaptive DSO (ADSO) method by replacing the traditional image matching with energy distribution matching. By reducing dependence on the imaging results and introducing a variable weighting function in the calculation of the adjoint sources, the amplitude of the inverted model can be evenly distributed and free of artefacts. Three numerical examples show that the ADSO method is robust with little artefacts presenting in the gradients. Combined with the improved gradient, the ADSO can lead to a stable update.

Global Structure and Seasonal Variations of the Tidal Amplitude in Sporadic‐E Layer

AbstractThe Constellation Observing System for Meteorology, Ionosphere, and Climate radio occultation S4 indices measurements are used to investigate the global structure and seasonal variations of tidal components in the sporadic‐E (Es) layer. The diurnal tidal amplitude in the Es layer intensity and occurrence rate shows a wave‐4 longitudinal structure, which is due to the diurnal tidal components of zonal wave numbers s = −3 (eastward, DE3). The wave‐4 longitudinal pattern is also observed in the semidiurnal tide during spring and winter, which mostly results from the combination of semidiurnal components of zonal wave numbers s = −2 (eastward, SE2), s = 2 (westward, SW2). In addition, the semidiurnal tidal amplitude in Es layers presents obvious seasonal variability, and it dominates in the northern hemisphere during spring and summer and in the southern hemisphere during autumn and winter. Furthermore, the total semidiurnal tidal amplitude in Es layers in the northern hemisphere is greater than that in the southern hemisphere. Another important finding is that the diurnal tidal pattern at the lower altitude (below 102 km) Es layer is different from that of the higher altitude (above 102 km) Es layer and is also different from that of wind, which supports that the wind shear theory is inapplicable for the Es layer formation at low altitude proposed by many researchers.

Investigation on the MLT tidal variability during September 2019 minor sudden stratospheric warming

Tidal variability in the mesosphere and lower thermosphere (MLT) during September 2019 Southern hemisphere minor sudden stratospheric warming (SSW) is investigated utilizing ground-based meteor radar wind observations from the equatorial, extratropical, middle, and high latitude stations and global reanalysis dataset. The polar warming is found to move from the mesosphere to the stratosphere until the peak warming day (PWD) of the SSW. The diurnal and semidiurnal tides at individual observational sites do not exhibit any consistent response during the observational interval, but a notable and consistent variability in some specific zonal wavenumber components, i. e., DW1 (migrating diurnal tide), DE3 (nonmigrating eastward wavenumber 3 diurnal tide), and SW2 (migrating semidiurnal tide) is found in the global reanalysis dataset. Incidentally, the warming event occurs during Spring equinox when a dominant seasonal change in the tidal activities generally takes place and hence seasonal variability is also looked into while identifying the SSW impact during the observational interval. It is found that the seasonal broad changes in the DW1, DE3, and SW2 amplitudes can be explained by the variability in the tidal sources, i.e., water vapor, convective activity, ozone, etc during the observational period. However, the extracted short-term variability in the global tidal modes on removing seasonal trend reveals noticeable response in connection with the warming event. The deseasoned amplitude of the DW1 significantly enhances around the PWD at most of the present latitudes. The deseasoned DE3 amplitude responds significantly in the middle atmosphere at low latitudes during the warming phase. The deseasoned SW2 exhibit clear enhancement around the PWD at all the latitudes. However, the deseasoned tidal features do not seem to correlate well with that of the source species unlike the seasonal ones that imply involvement of complex processes during the warming event, seeking further future investigations in this regard.

Two-step full waveform inversion of diving and reflected waves with the diffraction-angle-filtering-based scale-separation technique

SUMMARYFull waveform inversion (FWI) is a highly non-linear optimization problem that aims to reconstruct high-resolution subsurface structures. The success of FWI in reflection seismology relies on appropriate updates of low-wavenumber background velocity structures, which are generally driven by the diving waves in conventional FWI. On the other hand, the reflected waves mainly contribute to updating high-wavenumber components rather than low-wavenumber components. To extract low-wavenumber information from the reflected waves in addition to the diving waves, we propose a two-step FWI strategy that separates a given model into the reflectivity and background velocity models and then alternately update them using the scale-separation technique based on diffraction-angle filtering (DAF; which was proposed to effectively control wavenumber components of the FWI gradient). Our strategy first inverts the high-wavenumber reflectivity model by suppressing energy at large diffraction angles, which are necessary to compute the reflection wave paths (i.e. the rabbit-ears-shaped kernels) for low-wavenumber updates in the subsequent stage. Then, we extract low-wavenumber components due to the diving (banana-shaped kernels) and reflected waves (rabbit-ears-shaped kernels) from the gradient by suppressing energy at small diffraction angles. Our strategy is similar to reflection waveform inversion (RWI) in that it separates a given model into high- and low-wavenumber components and uses the rabbit-ears-shaped kernels for low-wavenumber updates. The main difference between our strategy and RWI is that our strategy adopts the DAF-based scale-separation technique in the space domain, which makes our algorithm of using both the banana-shaped and rabbit-ears-shaped kernels computationally attractive. By applying our two-step inversion strategy to the synthetic data for the Marmousi-II model and the real ocean-bottom cable data from the North sea, we demonstrate that our method properly reconstructs low-wavenumber structures even if initial models deviate from the true models.

Decadal changes of wintertime poleward heat and moisture transport associated with the amplified Arctic warming

The Arctic warming, especially during winter, has been almost twice as large as the global average since the late 1990s, which is known as the Arctic amplification. Yet linkage between the amplified Arctic warming and the midlatitude change is still under debate. This study examines the decadal changes of wintertime poleward heat and moisture transports between two 18-year epochs (1999–2016 and 1981–1998) with five atmospheric reanalyses. It is found that the wintertime Arctic warming induces an amplification of the high latitude stationary wave component of zonal wavenumber one but a weakening of the wavenumber two. These stationary wave changes enhance poleward heat and moisture transports, which are conducive to further Arctic warming and moistening, acting as a positive feedback onto the Arctic warming. Meanwhile, the Arctic warming reduces atmospheric baroclinicity and thus weakens synoptic eddy activities in the high latitudes. The decreased transient eddy activities reduce poleward heat and moisture transports, which decrease the Arctic temperature and moisture, acting as a negative feedback onto the Arctic warming. Since the increased poleward heat transport by stationary waves is nearly cancelled by the decreased transport by transient eddies, the total poleward heat transport contributes little to the Arctic warming. However, the total poleward moisture transport increases over most areas of the high latitudes that is dominated by the increased transport by stationary waves, which provides a significant net positive feedback onto the Arctic warming and moistening. Such a poleward moisture transport feedback may be particularly crucial to the amplified Arctic warming during winter when the ice-albedo feedback vanishes.

Electron Bernstein wave aided beat wave of Hermite-cosh-Gaussian laser beam absorption in a collisional nanocluster plasma

In this paper, we investigate the electron Bernstein wave aided beat wave of Hermite-cosh-Gaussian laser beam absorption in a collisional nanocluster plasma. The interactions of electric field profile of Hermite-cosh-Gaussian laser beams with electron clouds of nanoclusters cause the beat wave. The nonlinear ponderomotive force is generated through the beat wave of two Hermite-cosh-Gaussian laser beams. The time average power absorption per electron per unit area per unit time by electron Bernstein wave aided electric field of laser beat wave is derived. The graphical discussions were promised to achieve extreme absorption via spatial shape of Hermite-cosh-Gaussian laser beam and resonance condition of beat wave to surface plasmon frequency. The electron Bernstein waves parameters such as thermal velocity, cyclotron frequency and perpendicular component of wavenumber is also leading with beat wave for control and enhancement of absorption process in plasma embedded with nanocluster. The absorption is controlled by tuning the laser beam width, mode index, collisional frequency, clustered radius, and density.

Prediction of Flow Velocity from the Flexural Vibration of a Fluid-Conveying Pipe Using the Transfer Function Method

This study presents a method to predict the flow velocity in a fluid-conveying pipe using vibration signals from the pipe surface. The flexural vibration of a fluid pipe is investigated through wave propagation. The wavenumbers and mode shapes of the pipe are determined based on its mechanical properties and flow velocities. The transient components of wavenumbers at low frequencies vary and converge on all values at higher frequencies as the flow velocity is increased. While the stationary fluid pipe exhibits symmetrical mode shapes, pipes with increasing flow velocities exhibit an asymmetric mode shape distribution skewed on one side of the axis. The resonant frequencies shift to the low frequency side as the flow velocity increases. The analytical results of the vibration analysis are used in the transfer function method to predict the flow velocities. To validate the accuracy of the prediction method, numerical vibration signals simulated by the finite element model are used. The actual input flow velocity is compared with the numerical results regarding the same to gauge the accuracy of the prediction method. This method can be used to monitor the flow rate without using flow meters, and thus protect pipelines from sudden malfunction.

Multi-frequency localized wave energy for delamination identification using laser ultrasonic guided wave

Delamination is the most common and dangerous failure mode for multilayered structures. Delamination defects of different shapes and sizes have different sensitivity to guided wave of different frequencies and modes. So that it is necessary to study the application of multi-frequency methods for achieving detection. In this study, the algorithm of multi-frequency localized wave energy is present using laser ultrasonic guided waves for delamination identification. Localized wave energy is acoustic energy in space under specific wavenumber. New wavenumber components occur in damaged composite plates and its localized wave energy can be used for delamination identification. The localized wave energy is not only related to mode conversion caused by the decrease of structural thickness above the delamination, but also the scattering waves in delamination region. The scattering waves make acoustic energy redistributed and it is enhanced at specific spatial position. The discovery has been verified in simulation and experiment. Multi-frequency experimental results show lower noises and more discernible profile of delamination region in two specimens, including medial and non-medial delamination. In the case of medial delamination, the actual dispersion curve is closer to the dispersion curve of upper laminate at high frequency; in the case of non-medial delamination, the actual dispersion curve is similar to the ideal situation ignoring the effect of epoxy resin. Based on the actual dispersion curves, two critical parameters of proper frequencies and filter threshold are selected for delamination identification using laser ultrasonic guided wave.

New Directional Wave Satellite Observations: Towards Improved Wave Forecasts and Climate Description in Southern Ocean

AbstractIn spite of continuous improvements of ocean wave models in the last decades, large errors still remain in particular under strongly forced conditions, often encountered in the Southern Ocean, where strong westerly winds generate some of the fiercest waves on Earth in almost unlimited fetch conditions. The newly launched China‐France Oceanography SATellite (CFOSAT) provides directional spectra of ocean waves for both wind seas and swells. Compared to Synthetic Aperture Radar (SAR), it can resolve shorter wavelengths in all directions, which dominate in non‐fully developed wind waves. Here, the assimilation of these CFOSAT wavenumber components is proved to bring more accurate predictions of wave growth compared to the assimilation of significant wave height alone. A notable reduction of model bias is found in the Southern Ocean, especially in the Pacific Ocean sector. Results further exhibit a downward shift of the wave age, consistent with theoretical wave growth curves.

Waves in screeching jets

Abstract

Diffraction-angle filtering of gradient for acoustic full-waveform inversion

Seismic full-waveform inversion (FWI) estimates subsurface velocity structures by reducing data misfit between observed and modeled data. Simultaneous matching of transmitted and reflected waves in seismic FWI causes different updates of different wavenumber components of a given model depending on the diffraction angle between incident and diffracted rays. Motivated by the inverse-scattering imaging condition and elastic FWI, we have applied a diffraction-angle filtering technique in acoustic FWI, which enables us to separate transmission and reflection energy in the partial derivative wavefields. Diffraction-angle filtering is applied to the virtual source, which is the model parameter perturbation acting as a source for the partial derivative wavefields. Diffraction-angle filtering consists of two diffraction-angle filters (DAFs), DAF-I and DAF-II. The former is derived from the particle acceleration of the incident wavefields and suppresses energies at either small or large diffraction angles by changing the sign of the weighting factor. DAF-I is exactly identical to the conventional inverse scattering approach. The DAF-II is derived from the artificial shear strain of the incident P-wave and additionally suppresses energies at intermediate diffraction angles. With this mechanism, we can design various types of diffraction-angle filtering to control the updates of wavenumber components of the misfit gradient with respect to the P-wave velocity. Using the synthetic Marmousi-II data and real ocean-bottom seismic data from the North Sea, we determined that diffraction-angle filtering enables us to control low-, intermediate-, and high-wavenumber components of the gradient direction.