Damped Least-squares Research Articles

360° light field 3D display system based on a triplet lenses array and holographic functional screen

A 360° light field 3D display system is presented, which consists of a liquid crystal display, a novel triplet lenses array, and a holographic functional screen (HFS). The mapping relationship among pixels, 3D objects, and viewing positions are investigated. The aberration analysis of the single lens is carried out both in the simulation and the experiment, which shows that it cannot provide an excellent 3D image to the viewers. In order to suppress the aberrations, “the primary aberration theory” and “the damped least-squares method” are used for optical analysis and lens design. A 3D image with aberration correction can be viewed around the proposed display system.

Unidirectional coupler optimization of surface plasmon polaritons based on the damped least-squares method

Controlling the launching efficiency and the directionality of surface plasmon polaritons is a major goal for the development of plasmonic devices. This study presents the use of the damped least-squares method to optimize the geometry parameters of a surface plasmon polariton unidirectional coupler. A damped least-squares algorithm has been constructed and the performance of a surface plasmon polariton unidirectional coupler has been obtained with COMSOL LiveLink for MATLAB. We have optimized a quasi-periodic surface plasmon polariton unidirectional grating coupler, which consists of five groups of sub-wavelength groove-doublets with an extinction ratio of nearly 71 dB. Combination of the COMSOL and the damped least-squares method in a MATLAB environment is a practicable method for the design of a surface plasmon polariton unidirectional coupler. Moreover, the general design principles behind this study can readily be extended to other numeric algorithms and other plasmonic devices.

Near-surface shear-wave velocity estimation based on surface-wave inversion

Rayleigh waves always coexist with reflection waves in land seismic exploration. Many previous works have shown the value of Rayleigh waves in obtaining near-surface shear velocity, however, few of them show successful applications from exploration geophysics. With an aim of practical application of Rayleigh waves in exploration data, we discuss how to obtain an accurate and high-resolution dispersion curve, the inversion algorithm, and the comparison of velocity estimated from Rayleigh-wave inversion with refraction interpretation and tomographic inversion. We first analyze the resolution difference of f-v spectra calculated through f-k, τ-p, and phase-shift approaches and propose a two-step spectrum normalization scheme for accurate picking of the dispersion curve. We then adopt the weighted damped least-squares method as the inversion algorithm and find through synthetic data that the estimated shear velocity based on the algorithm is quite accurate and the inversion algorithm is fast and robust. Finally, we select seismic data from one land exploration project in China as an example for test, and find the inverted velocity from Rayleigh-wave dispersion has a good correlation with refraction interpretation and tomographic inversion.

Optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling

We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg–Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poisson's ratios. Moreover, only 3.7 grid-points per minimum shear wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates wavefields having smaller error under the same discretization setups. Profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.

Shear-wave velocity structure of the shallow sediments in the Bohai Sea from an ocean-bottom-seismometer survey

To investigate the seismic velocity structure of the shallow sediments in the Bohai Sea of China, we conducted a shear-wave velocity inversion of the surface-wave dispersion data from a survey of 12 ocean-bottom seismometers (OBSs) and 377 shots of a [Formula: see text] air gun. With OBS station spacing of approximately 5 km and air-gun shot spacing of approximately 190 m, high-quality Scholte-wave data were recorded by the OBSs within 0.4–5 km offset. We retrieved the Scholte-wave phase-velocity dispersion for the fundamental mode and first overtone in the frequency band of 0.9–3.0 Hz with the phase-shift method and inverted for the shear-wave velocity structure of the shallow sediments with a damped iterative least-squares algorithm. Pseudo-2D shear-wave velocity profiles with a depth of approximately 400 m revealed coherent features of relatively weak lateral velocity variation. We also estimated the uncertainty in shear-wave velocity structure based on the pseudo-2D profiles from six trial inversions with different initial models, which suggested a velocity uncertainty less than [Formula: see text] for most parts of the 2D profiles. The layered structure with little lateral variation may be attributable to the continuous sedimentary environment in the Cenozoic sedimentary basin of the Bohai Bay basin. The shear-wave velocity of 200–300 [Formula: see text] in the top 100 m of the Bohai seafloor may provide important information for offshore site response studies in earthquake engineering. Furthermore, the very low shear-wave velocity structure (150–600 [Formula: see text]) down to 400 m depth could produce a significant traveltime delay of approximately 1 s in the shear-wave arrivals, which needs to be considered to avoid serious bias in shear-wave traveltime tomographic models.

Obstacle Avoidance for Redundant Manipulators Utilizing a Backward Quadratic Search Algorithm

Obstacle avoidance can be achieved as a secondary task by appropriate inverse kinematics (IK) resolution of redundant manipulators. Most prior literature requires the time-consuming determination of the closest point to the obstacle for every calculation step. Aiming at the relief of computational burden, this paper develops what is termed a backward quadratic search algorithm (BQSA) as another option for solving IK problems in obstacle avoidance. The BQSA detects possible collisions based on the root property of a category of quadratic functions, which are derived from ellipse-enveloped obstacles and the positions of each link's end-points. The algorithm executes a backward search for possible obstacle collisions, from the end-effector to the base, and avoids obstacles by utilizing a hybrid IK scheme, incorporating the damped least-squares method, the weighted least-norm method and the gradient projection method. Some details of the hybrid IK scheme, such as values of the damped factor, weights and the clamping velocity, are discussed, along with a comparison of computational load between previous methods and BQSA. Simulations of a planar seven-link manipulator and a PUMA 560 robot verify the effectiveness of BQSA.

Improvement of Filed Curvature Aberration in a Projector Lens by Using Hybrid Genetic Algorithm With Damped Least Square Optimization

This paper improves the filed curvature aberration (FCA) in a projector lens by using a hybrid genetic algorithm (GA) with damped least square (DLS) optimization. A modern projector lens still exposes its rays on a large flat screen to form the image, with the result that the FCA is one of the key aberrations dominating the image quality. Most commercial optical software applies the DLS optimization focus on the minimum spot size to optimize the lens, simultaneously making the FCA hard to improve. Merging the GA optimization can involve the FCA in the optimization process. As a result, using the hybrid GA with DLS optimization not only effectively suppresses the FCA but also remarkably enhances the image resolution in a projector lens design.

Source-Type-Specific Inversion of Moment Tensors

The mapping of the fit of seismic moment tensor (MT) solutions in source-type space helps to characterize uncertainty and solution uniqueness. Current practice relies on the forward testing of a distribution of randomly generated MTs in source-type space, which is slow and does not necessarily recover the true maximum fit surface. We design an iterative damped least-squares inversion scheme to invert waveforms and/or P -wave first motions for best-fitting MT solutions for specific source types. An event associated with the sinkhole at the Napoleonville salt dome, Louisiana, an industrial quarry explosion, and an earthquake at The Geysers geothermal field, northern California, are presented as examples. We find that the inversion method is more accurate and successful than the random-search approach in recovering the region of best-fitting MT solutions or source types and is substantially faster. The approach also enables the determination of the best-fitting MT for specified source types such as pure double couples, tensile cracks, or explosions, as well as compound mechanisms in a single numerical framework. Online Material: Description and figures comparing inversion performance using spherical and Cartesian eigenvector parameterizations; figures showing the revised full moment tensor solution of event TE1 and additional network sensitivity solutions of event TE3.

Mapping deep-sea hydrothermal deposits with an in-loop transient electromagnetic method: Insights from 1D forward and inverse modeling

In transient electromagnetic (TEM) measurements, secondary fields that contain information on conductive targets such as hydrothermal mineral deposits in the seafloor can be measured in the absence of strong primary fields. A TEM system using a loop source is useful to the development of compact, autonomous instruments, which are well suited to submersible-based surveys. In this paper, we investigate the possibility of applying an in-loop TEM system to the detection of marine hydrothermal deposits through a one-dimensional modeling and inversion study. We examine step-off responses for a layered model and compare the characteristics of horizontal and vertical loop systems for detecting hydrothermal deposits. The feasibility study shows that TEM responses are very sensitive to a highly conductive layer. Time-domain target responses are larger and appear earlier in horizontal magnetic fields than in vertical ones, although the vertical field has 2–3 times larger magnitude than the horizontal one. An inverse problem is formulated with the Gauss–Newton method and solved with the damped and smoothness-constrained least-squares approach. The test example for a marine hydrothermal TEM survey demonstrated that the depth extent, conductivity and thickness of the highly conductive layer are well resolved.

Impedance inversion based on L1 norm regularization

Impedance is an important parameter in geophysics which is greatly relevant to petroleum reservoir. Traditional iterative inversion methods, such as damped least-squares inversion method, often lead to unreliable inversion results when there is strong noise in the seismic records. In this paper, we propose a new impedance inversion method called L1 norm regularization inversion method which has high anti-noise ability. The accuracy of impedances derived from the proposed method is much higher than those derived from damped least-squares inversion method using synthetic seismic record and field data. We set up a forward simulation model and add different intensities of uniform and Gaussian noise into the synthetic seismic records respectively. We find that when there is strong noise in the synthetic record, L1 norm regularization inversion method can obtain the inverted impedance with higher accuracy compared with damped least-squares inversion method. At last, it is found that impedance derived from L1 norm regularization method has a higher accuracy than impedance derived from damped-squares inversion method when there is strong noise both in synthetic record and field data. We conclude that the proposed inversion method shows higher anti-noise ability and can provide impedances with higher accuracy.

Lower Extremity Joint Angle Tracking with Wireless Ultrasonic Sensors during a Squat Exercise.

This paper presents an unrestrained measurement system based on a wearable wireless ultrasonic sensor network to track the lower extremity joint and trunk kinematics during a squat exercise with only one ultrasonic sensor attached to the trunk. The system consists of an ultrasound transmitter (mobile) and multiple receivers (anchors) whose positions are known. The proposed system measures the horizontal and vertical displacement, together with known joint constraints, to estimate joint flexion/extension angles using an inverse kinematic model based on the damped least-squares technique. The performance of the proposed ultrasonic measurement system was validated against a camera-based tracking system on eight healthy subjects performing a planar squat exercise. Joint angles estimated from the ultrasonic system showed a root mean square error (RMSE) of 2.85° ± 0.57° with the reference system. Statistical analysis indicated great agreements between these two systems with a Pearson's correlation coefficient (PCC) value larger than 0.99 for all joint angles' estimation. These results show that the proposed ultrasonic measurement system is useful for applications, such as rehabilitation and sports.

CYCLE LENGTH DEPENDENCE OF STELLAR MAGNETIC ACTIVITY AND SOLAR CYCLE 23

Solar cycle (SC) 23 was extraordinarily long with remarkably low magnetic activity. We have investigated whether this is a common behavior of solar-type stars. From the Ca ii H and K line intensities of 111 stars observed at Mount Wilson Observatory from 1966 to 1991, we have retrieved data of all 23 G-type stars and recalculated their cycle lengths using the damped least-squares method for the chromospheric activity index S as a function of time. A regression analysis was performed to find relations between the derived cycle length, Pavg, and the index for excess chromospheric emission, . As a noteworthy result, we found a segregation between young and old solar-type stars in the cycle length-activity correlation. We incorporated the relation for the solar-type stars into the previously known rule for stellar chromospheric activity and brightness to estimate the variation of solar brightness from SC 22 to SC 23 as (0.12 ± 0.06)%, much higher than the actual variation of total solar irradiance (TSI) ≤0.02%. We have then examined solar spectral irradiance (SSI) to find a good phase correlation with a sunspot number in the wavelength range of 170–260 nm, which is close to the spectral range effective in heating the Earth's atmosphere. Therefore, it appears that SSI rather than TSI is a good indicator of the chromospheric activity, and its cycle length dependent variation would be more relevant to the possible role of the Sun in the cyclic variation of the Earth's atmosphere.

Simultaneous Structure-Coupled Joint Inversion of Gravity and Magnetic Data Based on a Damped Least-Squares Technique

The structure-coupled joint inversion method of gravity and magnetic data is a powerful tool for developing improved physical property models with high resolution and compatible features; however, the conventional procedure is inefficient due to the truncated singular values decomposition (SVD) process at each iteration. To improve the algorithm, a technique using damped leastsquares is adopted to calculate the structural term of model updates, instead of the truncated SVD. This produces structural coupled density and magnetization images with high efficiency. A so-called coupling factor is introduced to regulate the tuning of the desired final structural similarity level. Synthetic examples show that the joint inversion results are internally consistent and achieve higher resolution than separated. The acceptable runtime performance of the damped least squares technique used in joint inversion indicates that it is more suitable for practical use than the truncated SVD method.

Meteorological time series forecasting with pruned multi-layer perceptron and two-stage Levenberg-Marquardt method

A multi-layer perceptron (MLP) defines a family of artificial neural networks often used in TS modelling and forecasting. Because of its 'black box' aspect, many researchers refuse to use it. Moreover, the optimisation (often based on the exhaustive approach where 'all' configurations are tested) and learning phases of this artificial intelligence tool (often based on the Levenberg-Marquardt algorithm - LMA) are weaknesses of this approach (exhaustively and local minima). These two tasks must be repeated depending on the knowledge of each new problem studied, making the process, long, laborious and not systematically robust. In this paper, a pruning process is proposed. This method allows, during the training phase, to carry out an inputs selecting method activating (or not) inter-nodes connections in order to verify if forecasting is improved. We propose to use iteratively the popular damped least-squares method to activate inputs and neurons. A first pass is applied to 10% of the learning sample to determine weights significantly different from 0 and delete other. Then, a classical batch process based on LMA is used with the new MLP. The validation is done using 25 measured meteorological TS and cross-comparing the prediction results of the classical LMA and the two-stage LMA.

Imaging Ultrasonic Dispersive Guided Wave Energy in Long Bones Using Linear Radon Transform

Multichannel analysis of dispersive ultrasonic energy requires a reliable mapping of the data from the time–distance (t–x) domain to the frequency–wavenumber (f–k) or frequency–phase velocity (f–c) domain. The mapping is usually performed with the classic 2-D Fourier transform (FT) with a subsequent substitution and interpolation via c = 2πf/k. The extracted dispersion trajectories of the guided modes lack the resolution in the transformed plane to discriminate wave modes. The resolving power associated with the FT is closely linked to the aperture of the recorded data. Here, we present a linear Radon transform (RT) to image the dispersive energies of the recorded ultrasound wave fields. The RT is posed as an inverse problem, which allows implementation of the regularization strategy to enhance the focusing power. We choose a Cauchy regularization for the high-resolution RT. Three forms of Radon transform: adjoint, damped least-squares, and high-resolution are described, and are compared with respect to robustness using simulated and cervine bone data. The RT also depends on the data aperture, but not as severely as does the FT. With the RT, the resolution of the dispersion panel could be improved up to around 300% over that of the FT. Among the Radon solutions, the high-resolution RT delineated the guided wave energy with much better imaging resolution (at least 110%) than the other two forms. The Radon operator can also accommodate unevenly spaced records. The results of the study suggest that the high-resolution RT is a valuable imaging tool to extract dispersive guided wave energies under limited aperture.

Crustal structure of Damavand volcano, Iran, from ambient noise and earthquake tomography

We investigated the shear wave velocity structure beneath the Damavand volcano, Iran, using analysis of Rayleigh wave empirical Green’s functions obtained from the cross-correlation of 1-year seismic ambient noise and 26 local earthquakes. After the observed phase, velocity dispersion curves were inverted to obtain phase velocity tomographic maps; the synthetic phase velocity dispersion curves were then estimated for each evenly spaced geographic grid point. Estimated synthetic phase velocities were inverted using the nonlinear damped least-squares inversion method to obtain a quasi-3D shear wave velocity for the study area. Analysis of obtained quasi-3D shear wave velocity model reveals the presence of three distinct low or high anomalies in the upper 5 km. A low-velocity layer (VS ~ 2.8 km/s) was observed in the upper ~2 km of the crust that includes the sedimentary sequence of carbonate, siliciclastic, and volcanic rocks. According to our results, there is an indication of a high-velocity layer (VS ≥ 3.0 km/s) at the depth range of 2 to 5 km, which may indicate the presence of dense, cooled magma. There is a clear indication of intrusion of the low-velocity anomaly/body into the higher velocity layer, in the depth range of approximately 3.0 to 4.5 km. The corresponding low-velocity body is interpreted as a hot magma chamber associated with the young eruption of Damavand. The chamber is located southwest of the crater, which stretches from the west to beneath the Damavand cone.

Inverse axial mounting stiffness design for lithographic projection lenses

In order to balance axial mounting stiffness of lithographic projection lenses and the image quality under dynamic working conditions, an easy inverse axial mounting stiffness design method is developed in this article. Imaging quality deterioration at the wafer under different axial vibration levels is analyzed. The desired image quality can be determined according to practical requirements, and axial vibrational tolerance of each lens is solved with the damped least-squares method. Based on adaptive interval adjustment, a binary search algorithm, and the finite element method, the axial mounting stiffness of each lens can be traveled in a large interval, and converges to a moderate numerical solution which makes the axial vibrational amplitude of the lens converge to its axial vibrational tolerance. Model simulation is carried out to validate the effectiveness of the method.

Multichannel filtering and reconstruction of ultrasonic guided wave fields using time intercept-slowness transform.

Multichannel ultrasonic axial-transmission data are multimodal by nature. As guided waves are commonly used in nondestructive material testing, wave field filtering becomes important because the analysis is usually limited to a few lower-order modes and requires their extraction. An application of the Radon transform to enhance signal-to-noise ratio and separate wave fields in ultrasonic records is presented. The method considers guided wave fields as superpositions of plane waves defined by ray parameters (p) and time intercepts (τ) and stacks the amplitudes along linear trajectories, mapping time-offset (t - x) data to a τ - p or Radon panel. The transform is implemented using a least-squares strategy with Cauchy-norm regularization that serves to enhance the focusing power. The method was verified using simulated data and applied to an uneven spatially sampled bovine-bone-plate data set. The results demonstrate the Radon panels show isolated amplitude clusters and the Cauchy-norm constraint provides a more focused Radon image than the damped least-squares regularization. Wave field separation can be achieved by selectively windowing the τ - p signals and inverse transformation, which is illustrated by the successful extraction of the A0 mode in bone plate. In addition, the method effectively attenuates noise, enhances the coherency of the guided wave modes, and reconstructs the missing records. The proposed transform presents a powerful signal-enhancement tool to process guided waves for further analysis and inversion.

3D frequency-domain finite-difference viscoelastic-wave modeling using weighted average 27-point operators with optimal coefficients

Experimental data suggest that the viscoelastic behavior of rocks is more easily and accurately described in the frequency domain than in the time domain, supporting the idea of simulating seismic wave propagation in the frequency domain. We evaluated weighted-averaged 27-point finite-difference operators for 3D viscoelastic wave modeling in the frequency domain. Within the proposed framework, we developed general equations for normalized phase velocities that can be used with arbitrary finite-difference operators. Three sets of weighting coefficients for second-order central finite-difference operators that minimize the numerical dispersion for up to five grid points per wavelength were found using a damped least-squares (LS) criterion as well as a global optimization scheme based on [Formula: see text]- and [Formula: see text]-norm criteria. The three sets produced very similar dispersion curves, and improvement provided by global optimization appeared marginal in this respect. We also evaluated a discrete form for the heterogeneous formulation of the 3D viscoelastic equations with a perfectly match layer (PML). Heuristic performance assessment of frequency-dependent PML absorption coefficients provided a simple rule giving good results for eight PMLs at all frequencies. The proposed formalism was implemented with a massively parallel direct solver. Modeling results were compared with an analytic solution and a time-domain finite-difference code, and they gave good agreement when using LS and [Formula: see text]-norm optimal coefficients. On the other hand, [Formula: see text]-norm coefficients produced noisy results, indicating that minimizing the difference between analytic and numerical phase velocities, although necessary, is not a sufficient condition to guarantee low-numerical noise. Finally, analysis of the computational resources required to factorize the impedance matrix revealed that the memory complexity of the factorization is [Formula: see text] for an [Formula: see text] grid, compared to [Formula: see text] for the viscoacoustic case.

Time series modeling with pruned multi-layer perceptron and 2-stage damped least-squares method

A Multi-Layer Perceptron (MLP) defines a family of artificial neural networks often used in TS modeling and forecasting. Because of its "black box" aspect, many researchers refuse to use it. Moreover, the optimization (often based on the exhaustive approach where "all" configurations are tested) and learning phases of this artificial intelligence tool (often based on the Levenberg-Marquardt algorithm; LMA) are weaknesses of this approach (exhaustively and local minima). These two tasks must be repeated depending on the knowledge of each new problem studied, making the process, long, laborious and not systematically robust. In this paper a pruning process is proposed. This method allows, during the training phase, to carry out an inputs selecting method activating (or not) inter-nodes connections in order to verify if forecasting is improved. We propose to use iteratively the popular damped least-squares method to activate inputs and neurons. A first pass is applied to 10% of the learning sample to determine weights significantly different from 0 and delete other. Then a classical batch process based on LMA is used with the new MLP. The validation is done using 25 measured meteorological TS and cross-comparing the prediction results of the classical LMA and the 2-stage LMA.