Complicated Wave Field Research Articles

Angular Integral Autocorrelation for Speed Estimation in Shear-Wave Elastography

The utilization of a reverberant shear-wave field in shear-wave elastography has emerged as a promising technique for achieving robust shear-wave speed (SWS) estimation. However, many types of estimators cannot accurately measure SWS within such a complicated 3D wave field. This study introduces an advanced autocorrelation estimator based on angular integration known as the angular integral autocorrelation (AIA) approach to address this issue. The AIA approach incorporates all the autocorrelation data from various angles during measurements, resulting in enhanced robustness to both noise and imperfect distributions in SWS estimation. The effectiveness of the AIA estimator for SWS estimation is first validated using a k-Wave simulation of a stiff branching tube in a uniform background. Furthermore, the AIA estimator is applied to ultrasound elastography experiments, magnetic resonance imaging (MRI) experiments, and optical coherence tomography (OCT) studies across a range of different excitation frequencies on tissues and phantoms, including in vivo scans. The results verify the capacity of the AIA approach to enhance the accuracy of SWS estimation and the signal-to-noise ratio (SNR), even within an imperfect reverberant shear-wave field. Compared to simple autocorrelation approaches, the AIA approach can also successfully visualize and define lesions while significantly improving the estimated SWS and SNR in homogeneous background materials and providing improved elastic contrast between structures within the scans. These findings demonstrate the robustness and effectiveness of the AIA approach across a wide range of applications, including ultrasound elastography, magnetic resonance elastography (MRE), and optical coherence elastography (OCE), for accurately identifying the elastic properties of biological tissues in diverse excitation scenarios.

Remotely imaging seismic ground shaking via large-N infrasound beamforming

Seismic ground motion creates low-frequency atmospheric sound (infrasound) that is detectable at remote sensor arrays. However, earthquake infrasound signal analysis is complicated by interference between multiple waves arriving at sensors simultaneously, reducing the accuracy and detail of ground motion detection. Here we show that individual waves in complicated wavefields can be resolved by recording infrasound on large-N arrays and processing with CLEAN beamforming. Examining both a local (ML3.5, purely tropospheric infrasound propagation) and regional earthquake (ML6.5, upper-atmospheric returns), we detect infrasound from tens of km away and up to several hundred km away respectively. Source regions span arcs of approximately 90°, indicating that although detection bias does occur (most likely from atmospheric winds) the recorded infrasound sources are widely dispersed and not simply epicentral. Infrasound-based remote detection of ground motion over wide areas can complement point measurements by seismometers and spur innovations in earthquake research and real-time hazard monitoring.

An application of internal multiple prediction and primary reflection retrieval to salt structures

Irregular salt bodies complicate the seismic wavefield propagation and cause multipathing, leading to intricate internal multiples that cannot be easily suppressed during data processing. To investigate the performance of internal multiple removal in salt bodies, we compare two types of Marchenko-based internal multiple removal strategies. One focuses on internal multiple prediction (IMP) based on Marchenko redatuming and seismic interferometry. With the retrieved upgoing and downgoing Green’s functions, internal multiples can be reconstructed using convolutional interferometry. The second method emphasizes primary reflection retrieval (PRR) by capturing primary reflections without adaptive subtraction. It is derived from a modified method based on the Gel’fand-Levitan-Marchenko equation. Testing with the Pluto salt model, we find that for the IMP the critical point is obtaining the purely scattering components of the upgoing and downgoing Green’s functions from the complicated wavefield. Our complex salt model indicates that the predicted internal multiples’ adaptive subtraction also introduces an extra challenge for attenuation quality. We develop a layer-by-layer adaptive subtraction scheme for the IMP to obtain primary reflections. The constant-time truncation operator in the time domain makes the PRR independent of additional model information, which can exclude internal multiples from the primary reflections. However, selecting an appropriate smoothing window for truncation requires further investigation. We evaluate the weaknesses of these methods and enhance each method individually. Our numerical results indicate that these improved schemes can correctly handle internal multiples in salt structure scenarios. In the addition, we study the robustness of the IMP when an erroneous background velocity model is used to estimate the direct arrival and the effects of sampling on the virtual receiver/source lines in IMP.

MR Elastography With Optimization-Based Phase Unwrapping and Traveling Wave Expansion-Based Neural Network (TWENN).

Magnetic Resonance Elastography (MRE) can characterize biomechanical properties of soft tissue for disease diagnosis and treatment planning. However, complicated wavefields acquired from MRE coupled with noise pose challenges for accurate displacement extraction and modulus estimation. Using optimization-based displacement extraction and Traveling Wave Expansion-based Neural Network (TWENN) modulus estimation, we propose a new pipeline for processing MRE images. An objective function with Dual Data Consistency (Dual-DC) has been used to ensure accurate phase unwrapping and displacement extraction. For the estimation of complex wavenumbers, a complex-valued neural network with displacement covariance as an input has been developed. A model of traveling wave expansion is used to generate training datasets for the network with varying levels of noise. The complex shear modulus map is obtained through fusion of multifrequency and multidirectional data. Validation using brain and liver simulation images demonstrates the practical value of the proposed pipeline, which can estimate the biomechanical properties with minimal root-mean-square errors when compared to state-of-the-art methods. Applications of the proposed method for processing MRE images of phantom, brain, and liver reveal clear anatomical features, robustness to noise, and good generalizability of the pipeline.

Frequency-domain elastic wave modeling for vertical transversely isotropic media with an average-derivative optimal method

Based on an average-derivative method (ADM), it is feasible to develop an optimal elastic wave finite-difference nine-point scheme for vertical transversely isotropic (VTI) media in the frequency domain. The optimization coefficients of ADM can be determined by minimizing phase-velocity dispersion error. Nevertheless, for the complex VTI media, the point-by-point optimization, which only considers a single anisotropic parameter pair, will require high computational cost. To address this issue, we take into account the other two low-cost schemes: the overall simultaneous optimization in terms of various anisotropy parameter pairs and the interpolation based on the completely computed optimization coefficients. Applying the 58 measured anisotropy parameter pairs of sedimentary rocks, we test the performances of point-by-point optimization, the overall optimization, interpolation, and classical nine-point method (CM). The results indicate that the number of grid points per qS wavelength of the interpolation scheme is closest to that of the point-by-point optimization scheme and much lower than that of the overall optimization scheme and CM. In addition, the surface of solid earth is a strong impedance-discontinuity surface (free surface), resulting in the more complicated wavefields. To realize the full-waveform simulation for elastic VTI media, it is necessary to implement the free-surface boundary condition in detail. Based on the equivalence between the stress-image and adaptive parameter-modified schemes for implementing the free-surface boundary condition in the time domain, we give a new frequency-domain free-surface expression for the VTI media. We still call this expression the stress-image scheme. Numerical experiments demonstrate that (1) the optimal ADM can produce more accurate results than CM at almost the same computational cost, (2) the interpolation scheme has almost identical accuracy with the point-by-point optimization scheme, and (3) the stress-image scheme is feasible for the implementation of free-surface boundary condition in a VTI medium.

Determination of Concrete Formwork Removal Time Based on Ultrasound Reflection

Concrete is vastly used in construction due to its high compressive strength, and its ease of shaping. Formworks are used to render the desired shape to the material during its change from a liquid to a solid state, until the concrete develops the required strength. Removing the formwork early can cause excessive deformations or even failure; however, postponing removal also has a delaying effect in construction with economic consequences. In practice, the removal of the formworks is based on the development of the compressive strength vs. the self-weight and temporary loads related to the construction. This study aims to monitor the hardening of concrete columns by means of the reflection of ultrasonic waves. As concrete hardens, its mechanical impedance increases influencing the reflection that can be received by one-sided non-invasive measurements. By suitably selecting the stiffness of the ultrasonic buffer material, the moment that the concrete obtains a specific stiffness can be safely determined for the first time. Results demonstrate sensitivity of the technique on real scale formworks, as it can pinpoint the moment in which the formwork can be removed, while numerical simulations help in the understanding of the complicated wave field and validate the methodology.

Investigation of Time-Lapse Changes with DAS Borehole Data at the Brady Geothermal Field Using Deconvolution Interferometry

Distributed acoustic sensing (DAS) has great potential for monitoring natural-resource reservoirs and borehole conditions. However, the large volume of data and complicated wavefield add challenges to processing and interpretation. In this study, we demonstrate that seismic interferometry based on deconvolution is a convenient tool for analyzing this complicated wavefield. We also show the limitation of this technique, in that it still requires good coupling to extract the signal of interest. We extract coherent waves from the observation of a borehole DAS system at the Brady geothermal field in Nevada. The extracted waves are cable or casing ringing that reverberate within a depth interval. These ringing phenomena are frequently observed in the vertical borehole DAS data. The deconvolution method allows us to examine the wavefield at different boundary conditions and separate the direct waves and the multiples. With these benefits, we can interpret the wavefields using a simple 1D string model and monitor its temporal changes. The velocity of this wave varies with depth, observation time, temperature, and pressure. We find the velocity is sensitive to disturbances in the borehole related to increasing operation intensity. The velocity decreases with rising temperature. The reverberation can be decomposed into distinct vibration modes in the spectrum. We find that the wave is dispersive and the fundamental mode propagates with a large velocity. This interferometry method can be useful for monitoring borehole conditions or reservoir property changes using densely-sampled DAS data.

Efficient scattering-free wavefront transformation with power flow conformal bianisotropic acoustic metasurfaces

Bianisotropic metasurfaces have enabled highly efficient wavefront transformation. However, a passive and lossless bianisotropic metasurface must conserve local power at every point over the metasurface, hindering its application in handling complicated wavefields other than plane waves. In this paper, the power flow-conformal design methodology and bianisotropic unit cells are combined to comply with such a restriction. Our proposed approach provides a general recipe for arbitrary wavefront transformation with maximum power efficiency. As a demonstration, a transmission-type acoustic metasurface was designed to focus 3000 Hz plane wave airborne sound in the near field with theoretically unitary power efficiency. The metasurface was validated by both numerical simulation and experiment.

Theoretical analysis and numerical simulation of acoustic waves in gas hydrate-bearing sediments* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11974018 and 11734017) and the Strategic Pilot and Technology Special Fund of the Chinese Academy of Sciences, China (Grant No. XDA14020303).

Based on Carcione–Leclaire model, the time-splitting high-order staggered-grid finite-difference algorithm is proposed and constructed for understanding wave propagation mechanisms in gas hydrate-bearing sediments. Three compressional waves and two shear waves, as well as their energy distributions are investigated in detail. In particular, the influences of the friction coefficient between solid grains and gas hydrate and the viscosity of pore fluid on wave propagation are analyzed. The results show that our proposed numerical simulation algorithm proposed in this paper can effectively solve the problem of stiffness in the velocity–stress equations and suppress the grid dispersion, resulting in higher accuracy compared with the result of the Fourier pseudospectral method used by Carcione. The excitation mechanisms of the five wave modes are clearly revealed by the results of simulations. Besides, it is pointed that, the wave diffusion of the second kind of compressional and shear waves is influenced by the friction coefficient between solid grains and gas hydrate, while the diffusion of the third compressional wave is controlled by the fluid viscosity. Finally, two fluid–solid (gas-hydrate formation) models are constructed to study the mode conversion of various waves. The results show that the reflection, transmission, and transformation of various waves occur on the interface, forming a very complicated wave field, and the energy distribution of various converted waves in different phases is different. It is demonstrated from our studies that, the unconventional waves, such as the second and third kinds of compressional waves may be converted into conventional waves on an interface. These propagation mechanisms provide a concrete wave attenuation explanation in inhomogeneous media.

Seismic modelling and its application on designing the seismic survey parameters, a case study of an Iranian oil field

ABSTRACTSpecifying survey parameters is more challenging when dealing with complex structures such as faults due to the complicated wavefield behaviour involved in these structures. Seismic modelling enables geophysicists to have a better understanding of the subsurface image before seismic acquisition, processing and interpretation. In this regard, seismic survey modelling is employed to construct a model close to the real structure, and to obtain very realistic synthetic seismic data. Therefore, a survey designer can investigate the effects of various survey parameters such as shot and receiver positions, and receiver line bearing on the resulting image using illumination and resolution analyses. The primary objective of this study is to apply seismic modelling to designing seismic survey parameters such as receiver line bearing, wavelets and their frequencies to analyse the resolution and illumination of the faulted target. A ray-based seismic model was built using 2D seismic data, geological reports and well logs. A pre-stack depth migration simulator was used to evaluate the effect of source wavelets with different peak frequencies (20, 60 and 80 Hz) on the resulting seismic image. This simulation indicated that a source wavelet with a peak frequency of 20 Hz could not image the target fault plane, but higher peak frequencies (60 and 80 Hz) show higher resolution and better recognition of the fault plane. Therefore, for any wavelet frequency, it is necessary to assess the ability of the fault illumination. The results proved that a survey designer could enhance the image of fault planes in a seismic section by applying ray-based analyses, concerning illumination and resolution of the fault image.

Broadband transmission-type coding metamaterial for wavefront manipulation for airborne sound

The recent advent of coding metamaterials, as a new class of acoustic metamaterials, substantially reduces the complexity in the design and fabrication of acoustic functional devices capable of manipulating sound waves in exotic manners by arranging coding elements with discrete phase states in specific sequences. It is therefore intriguing, both physically and practically, to pursue a mechanism for realizing broadband acoustic coding metamaterials that control transmitted waves with a fine resolution of the phase profile. Here, we propose the design of a transmission-type acoustic coding device and demonstrate its metamaterial-based implementation. The mechanism is that, instead of relying on resonant coding elements that are necessarily narrow-band, we build weak-resonant coding elements with a helical-like metamaterial with a continuously varying pitch that effectively expands the working bandwidth while maintaining the sub-wavelength resolution of the phase profile that is vital for the production of complicated wave fields. The effectiveness of our proposed scheme is numerically verified via the demonstration of three distinctive examples of acoustic focusing, anomalous refraction, and vortex beam generation in the prescribed frequency band on the basis of 1- and 2-bit coding sequences. Simulation results agree well with theoretical predictions, showing that the designed coding devices with discrete phase profiles are efficient in engineering the wavefront of outcoming waves to form the desired spatial pattern. We anticipate the realization of coding metamaterials with broadband functionality and design flexibility to open up possibilities for novel acoustic functional devices for the special manipulation of transmitted waves and underpin diverse applications ranging from medical ultrasound imaging to acoustic detections.

Using Mirror Migration of OBS Data to Image the Deepwater Area of South China Sea

Ocean bottom seismograph (OBS) is widely used in investigating deep crustal structure, which is characterized by a large amount of data information and abundant frequency components because of its multi-component acquisition. OBS is seldom used in deepwater oil and gas exploration and basin research due to the high cost. The complicated seismic wave field is caused by the complex seabed topography, basin and oil and gas structure in deepwater area, which increases the difficulty of image processing. In addition to reflection imaging, we utilize the multiple of OBS data to make accurate imaging and have achieved desirable results in a deep sea area in South China Sea in this paper. Firstly, the original P and Z components of OBS data are processed by wave field separation to obtain the upgoing wave filed and downgoing wave filed. Secondly, its image velocity filed is constructed. Finally, downgoing wave data is used to image (called mirror migration). Compared with conventional migration, the mirror migration can clearly image the seabed and provide better illumination for shallow layer below the seafloor in the case of sparse nodes, which is proved by the migration results of theoretical and real data in this paper.

Improving fault image by determination of optimum seismic survey parameters using ray-based modeling

In complex structures such as faults, salt domes and reefs, specifying the survey parameters is more challenging and critical owing to the complicated wave field behavior involved in such structures. In the petroleum industry, detecting faults has become crucial for reservoir potential where faults can act as traps for hydrocarbon. In this regard, seismic survey modeling is employed to construct a model close to the real structure, and obtain very realistic synthetic seismic data. Seismic modeling software, the velocity model and parameters pre-determined by conventional methods enable a seismic survey designer to run a shot-by-shot virtual survey operation. A reliable velocity model of structures can be constructed by integrating the 2D seismic data, geological reports and the well information. The effects of various survey designs can be investigated by the analysis of illumination maps and flower plots. Also, seismic processing of the synthetic data output can describe the target image using different survey parameters. Therefore, seismic modeling is one of the most economical ways to establish and test the optimum acquisition parameters to obtain the best image when dealing with complex geological structures. The primary objective of this study is to design a proper 3D seismic survey orientation to achieve fault zone structures through ray-tracing seismic modeling. The results prove that a seismic survey designer can enhance the image of fault planes in a seismic section by utilizing the proposed modeling and processing approach.

Relationship of scattering phase shifts to special radiation force conditions for spheres in axisymmetric wave-fields.

When investigating the radiation forces on spheres in complicated wave-fields, the interpretation of analytical results can be simplified by retaining the s-function notation and associated phase shifts imported into acoustics from quantum scattering theory. For situations in which dissipation is negligible, as taken to be the case in the present investigation, there is an additional simplification in that partial-wave phase shifts become real numbers that vanish when the partial-wave index becomes large and when the wave-number-sphere-radius product vanishes. By restricting attention to monopole and dipole phase shifts, transitions in the axial radiation force for axisymmetric wave-fields are found to be related to wave-field parameters for traveling and standing Bessel wave-fields by considering the ratio of the phase shifts. For traveling waves, the special force conditions concern negative forces while for standing waves, the special force conditions concern vanishing radiation forces. An intermediate step involves considering the functional dependence on phase shifts. An appendix gives an approximation for zero-force plane standing wave conditions. Connections with early investigations of acoustic levitation are mentioned and some complications associated with viscosity are briefly noted.

Long-range propagation and associated variability of internal tides in the South China Sea

The variability of internal tides during their generation and long-range propagation in the South China Sea (SCS) is investigated by driving a high-resolution numerical model. The present study clarifies the notably different processes of generation, propagation and dissipation between diurnal and semidiurnal internal tides. Internal tides in the SCS originate from multiple source sites, among which the Luzon Strait is dominant, and contributes approximately 90% and 74% of the baroclinic energy for M2 and K1, respectively. To the west of the Luzon Strait, local generation of K1 internal tides inside the SCS is more energetic than the M2 tides. Diurnal and semidiurnal internal tides from the Luzon Strait radiate into the SCS in a north-south asymmetry but with different patterns because of the complex two-ridge system. The tidal beams can travel across the deep basin and finally arrive at the Vietnam coast and Nansha Island more than 1000-1500 km away. During propagation, M2 internal tides maintain a southwestward direction, whereas K1 exhibit complicated wave fields because of the superposition of waves from local sources and island scattering effects. After significant dissipation within the Luzon Strait, the remaining energy travels into the SCS and reduces by more than 90% over a distance of ∼1000 km. Inside the SCS, the K1 internal tides with long crests and flat beam angles are more influenced by seafloor topographical features and thus undergo apparent dissipation along the entire path, whereas the prominent dissipation of M2 internal tides only occurs after their arrival at Zhongsha Island. This article is protected by copyright. All rights reserved.

Wave propagation in a solar quiet region and the influence of the magnetic canopy

Aims. We seek indications or evidence of transmission/conversion of magnetoacoustic waves at the magnetic canopy, as a result of its impact on the properties of the wave field of the photosphere and chromosphere. Methods. We use cross-wavelet analysis to measure phase differences between intensity and Doppler signal oscillations in the Halpha, CaII H, and G-band.We use the height of the magnetic canopy to create appropriate masks to separate internetwork (IN) and magnetic canopy regions. We study wave propagation and differences between these two regions. Results. The magnetic canopy affects wave propagation by lowering the phase differences of progressive waves and allowing the propagation of waves with frequencies lower than the acoustic cut-off. We also find indications in the Doppler signals of Halpha of a response to the acoustic waves at the IN, observed in the CaII H line. This response is affected by the presence of the magnetic canopy. Conclusions. Phase difference analysis indicates the existence of a complicated wave field in the quiet Sun, which is composed of a mixture of progressive and standing waves. There are clear imprints of mode conversion and transmission due to the interaction between the p-modes and small-scale magnetic fields of the network and internetwork.

Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions and numerical simulations

Special soliton analytical solutions are seeked for the so‐called double dispersion equation (or Porubov's equation), which describes the propagation of longitudinal waves in an elastic rod. Using the F‐expansion method, a new interesting class of traveling solitary waves is obtained. As a byproduct, the results obtained by other authors for some special cases are regained. Some numerical simulations are also performed. Splitting of various initial pulses during propagation into a sequence of solitary waves is considered. The dependence of the amplitude and the velocity of the solitary waves on Poisson's ratio is discussed in detail. Collisions between some obtained solitary waves are also presented. The simulation results are compared with the obtained exact analytical solutions—the latter reflect the perfect balance between the nonlinearity and dispersion. The comparison indicates that the obtained exact solutions are useful for describing more complicated wave fields studied by numerical simulations. It also indicates that the numerical results for auxetic materials are in better agreement with theoretical solutions than in the case of ordinary materials. Furthermore, one can conclude that the Secant pulse generates solitary waves closer to the analytical predictions than a Gaussian pulse. This is similar to the case of variational method, where the Secant trial function is also more proper than the Gaussian one.

Theoretical analysis of a row of piles as passive barriers and an equivalent in-filled trench model

A theoretical study on the ground vibration isolation efficiency by a row of piles as passive barrier in a three-dimensional context was presented. The analysis was accomplished with the aid of integral equations governing Rayleigh wave scattering, used to predict the complicated Rayleigh wave field generated by a number of irregular scatters embedded in an elastic half-space. Then, the passive isolation effectiveness of a row of piles for screening Rayleigh wave was studied in detail. The effects of relevant parameters on the screening effectiveness were investigated and analyzed from the perspective of equivalence with in-filled trench. The results show that using a row of rigid piles as wave barrier is more effective than that of flexible ones, and an optimum reduction of vibration can be achieved either by increasing the size of piles or by decreasing the net spacing between the piles. Finally, based on the derived integral equation for Rayleigh wave scattering, the principle of equivalent modeling of the barrier of piles by an in-filled trench is put forward, which simplifies the analysis of vibration isolation by a row of piles.

Time-lapse wave-equation migration velocity analysis

Time-lapse (4D) analysis of seismic data acquired at different stages of hydrocarbon production or gas/fluid injection has been very successful at imaging detailed reservoir changes. Conventional time-domain analysis of 4D data sets usually assumes a linear perturbation about a reference baseline earth model. However, this assumption is violated when production/injection significantly alters the subsurface generating large 4D velocity changes, time shifts, and complicated 4D wavefield coda, necessitating a more robust 4D analysis involving prestack wave-equation depth migration and velocity analysis. We address these situations by extending conventional 3D wave-equation migration velocity analysis (WEMVA) based on one-way wave-equations and single-scattering theory to 4D velocity estimation using a “parallel” inversion approach involving parallel solution of two separate inversion problems. Recognizing that the 4D WEMVA strategy requires precomputed baseline/monitor image-difference volumes, we develop an approximate 4D WEMVA technique that replaces these differences with a single weight function derived from the smooth background time-lapse image difference. We demonstrate the usefulness of the parallel and an approximate 4D WEMVA approach using a synthetic time-lapse [Formula: see text] geosequestration experiment that requires inverting for a thin-layer velocity change derived from [Formula: see text] injection in an analogue North Sea reservoir. The parallel 4D WEMVA solutions generate an excellent high-resolution velocity estimates, whereas the approximate methods recover lower-resolution estimates with magnitudes that must be rescaled through a post-inversion gradient line-search.

Time-lapse wave-equation migration velocity analysis

Time-lapse (4D) analysis of seismic data acquired at different stages of hydrocarbon production or gas/fluid injection has been very successful at imaging detailed reservoir changes. Conventional time-domain analysis of 4D data sets usually assumes a linear perturbation about a reference baseline earth model. However, this assumption is violated when production/injection significantly alters the subsurface generating large 4D velocity changes, time shifts, and complicated 4D wavefield coda, necessitating a more robust 4D analysis involving prestack wave-equation depth migration and velocity analysis. We address these situations by extending conventional 3D waveequation migration velocity analysis (WEMVA) based on one-way wave-equations and single-scattering theory to 4D velocity estimation using a “parallel”inversion approach involving parallel solution of two separate inversion problems. Recognizing that the 4D WEMVA strategy requires precomputed baseline/monitor image-difference volumes, we develop an approximate 4D WEMVA technique that replaces these differences with a single weight function derived from the smooth background time-lapse image difference. We demonstrate the usefulness of the parallel and an approximate 4D WEMVA approach using a synthetic time-lapse CO2 geosequestration experiment that requires inverting for a thin-layer velocity change derived from CO2 injection in an analogue North Sea reservoir. The parallel 4D WEMVA solutions generate an excellent highresolution velocity estimates, whereas the approximate methods recover lower-resolution estimates with magnitudes that must be rescaled through a post-inversion gradient line-search.