Logarithmic Objective Function Research Articles

Deconvolution-Based Objective Functions for Full Waveform Inversion in the Laplace Domain

Objective functions are critical for successful seismic fullwaveform inversions (FWIs). Deconvolution-based objective functions yielded promising results in time- and frequency-domain inversions. In this study, we developed two deconvolution-based objective functions for Laplace-domain inversions. These objective functions are based on convolutional filters that transform one of the observed or modeled data into another. The objective functions are derived by forcing the convolutional filter to be a zero-lag delta function by updating the model parameters. We could obtain a relatively simple objective function in the Laplace domain since the convolution in the time domain is equivalent to the multiplication function in the Laplace domain and the Laplace transform of the delta function is unity. We compared the new deconvolution-based objective functions and the conventional logarithmic objective function using synthetic and Gulf of Mexico field data and obtained similar inversion results. We showed that the logarithmic objective function conventionally used in Laplace-domain FWIs can be understood in the framework of deconvolution-based objective functions, which explains the similar inversion results.

Handling Negative Values for the Logarithmic Objective Function in Acoustic Laplace-Domain Full-Waveform Inversion Using Real Variables

The Laplace-domain waveform inversion is a full-waveform inversion method that recovers large-scale subsurface models. The inversion updates subsurface model parameters to minimize the differences between the modeled and the observed wavefields in the Laplace domain. The inversion results can be used as an accurate initial model for subsequent high-resolution waveform inversions. Pure Laplace-domain wavefields can be obtained by transforming the time-domain signals using the Laplace transform of real variables. The real Laplace transform is mathematically identical to the Fourier transform using the imaginary angular frequency; however, the Laplace transform using only real variables is computationally more efficient than that using complex variables. The Laplace-transformed wavefields are real-valued signals, and thus, it is natural to use real values in the Laplace-domain waveform inversions. However, the real logarithm function in the logarithmic objective function cannot handle negative values. Inversions using complex logarithms can solve this problem, but they demand more memory and computations than those required for inversions using real variables only. We suggest a simple method to overcome the negative-value problem for the real logarithm in the objective function. By taking the absolute values of the negative signals in the logarithmic objective function, we can obtain inversion results from inversions using real variables only that are equivalent to those from inversions using complex variables. We demonstrate the proposed method using the Society of Exploration Geophysicists (SEG)/European Associations of Geoscientists & Engineers (EAGE) salt model and a field data set. The inversions using real variables only took less than 22% of the time of the inversion using complex variables in the numerical examples.

A study of areca nut leaf sheath fibers as a green sound-absorbing material

This study intends to explore the potential of areca nut leaf sheath (ALS) fibers as an efficient sound absorber. The ALS fibers were characterized to investigate their physical properties closely associated with their performance as an efficient sound absorber. ALS fibers were processed into non-woven form and were characterized for their acoustical characteristics. The sound absorption coefficient was measured on samples of different thicknesses as per ASTM E1050. The experimentally measured sound absorption coefficients were compared with the theoretically estimated ones, using the empirical models of Delany-Bazley, Miki, Garai-Pompoli, and Allard. The predicted and experimental results were compared in different frequency zones and for different thicknesses. The accuracy of empirical models was evaluated by analyzing the root mean squared error of sound absorption coefficients. The optimum thickness was identified by analyzing the noise reduction coefficient. A linear and a logarithmic objective function was framed with sound absorption coefficients sampled at 10 Hz, and in 1/12th octave band to further improve the accuracy of the empirical models by inverse acoustical characterization. Inverse acoustical characterization was performed using genetic algorithm on the results of optimum thickness. The inverse acoustical characteristics were used in the most accurate empirical model to estimate the characteristic impedance and complex wavenumber. This characteristic impedance and complex wavenumber were used to study the effect of air–gap behind the ALS sound absorbers. A maximum noise reduction coefficient of 0.78 on a 54 mm thick sample was reported in the presence of a 50 mm air–gap. An effective sound absorber from natural sources was the outcome of the present study.

A critique of the objective function utilized in calculating the Thrifty Food Plan.

The Thrifty Food Plan (TFP) is the basis of benefit allocations within the USDA’s Supplemental Nutrition Assistance Program (SNAP), which administers nearly $70 billion in benefits to over 42 million people annually. To produce the allocation of food within the TFP, the USDA uses a mathematical optimization model that solves for the daily apportionment across various food groups. The model is constrained by nutritional and consumption requirements to produce an “optimal” allocation. Despite the importance of the TFP, the computational solution developed by the USDA has received insufficient attention, with only a handful of articles written on the TFP optimization model. Here, we run three alternative objective functions that are simpler than the one used by USDA. Our first alternative objective function minimizes the sum of squared errors between the consumed market basket of goods and an allocated market basket of goods, the second alternative objective function minimizes the sum of the absolute value of the difference between the consumed market basket of goods and an allocated market basket of goods, and the third alternative objective function minimizes the weighted absolute deviation of allocations and actual consumption expressed as a proportion of observed consumption. A clear theoretical advantage of either of our methods is that they eliminate the need to arbitrarily set allocated consumption to nonzero values, as is the case for the logarithmic objective function used by USDA. In an operational sense, we find that our model formulations produce an allocation that fits actual consumption better than the objective function employed by the USDA.

On logarithmic fixed-charge transportation problem

The fixed-charge transportation problem (FCTP) is still a challenging problem in the field of mathematical programming. In this paper, we consider fixed-charge transportation problem with logarithmic objective function. In the absence of any suitable algorithm to obtain the solution of this type of nonlinear transportation problem, we discuss the advantage of polynomial approximation. There exists a major difference between the two problems that the variables in the polynomial transportation problem have no upper bound but in the logarithmic transportation problem they are bounded. Using the expansion of logarithm we show the resemblance between the structural behaviour of linear and fixed-charge transportation problems. We illustrate a numerical example in support of the developed method.

On logarithmic fixed-charge transportation problem

The fixed-charge transportation problem (FCTP) is still a challenging problem in the field of mathematical programming. In this paper, we consider fixed-charge transportation problem with logarithmic objective function. In the absence of any suitable algorithm to obtain the solution of this type of nonlinear transportation problem, we discuss the advantage of polynomial approximation. There exists a major difference between the two problems that the variables in the polynomial transportation problem have no upper bound but in the logarithmic transportation problem they are bounded. Using the expansion of logarithm we show the resemblance between the structural behaviour of linear and fixed-charge transportation problems. We illustrate a numerical example in support of the developed method.

Regularized Laplace–Fourier-Domain Full Waveform Inversion Using a Weighted l 2 Objective Function

Full waveform inversion (FWI) can be applied to obtain an accurate velocity model that contains important geophysical and geological information. FWI suffers from the local minimum problem when the starting model is not sufficiently close to the true model. Therefore, an accurate macroscale velocity model is essential for successful FWI, and Laplace–Fourier-domain FWI is appropriate for obtaining such a velocity model. However, conventional Laplace–Fourier-domain FWI remains an ill-posed and ill-conditioned problem, meaning that small errors in the data can result in large differences in the inverted model. This approach also suffers from certain limitations related to the logarithmic objective function. To overcome the limitations of conventional Laplace–Fourier-domain FWI, we introduce a weighted l 2 objective function, instead of the logarithmic objective function, as the data-domain objective function, and we also introduce two different model-domain regularizations: first-order Tikhonov regularization and prior model regularization. The weighting matrix for the data-domain objective function is constructed to suitably enhance the far-offset information. Tikhonov regularization smoothes the gradient, and prior model regularization allows reliable prior information to be taken into account. Two hyperparameters are obtained through trial and error and used to control the trade-off and achieve an appropriate balance between the data-domain and model-domain gradients. The application of the proposed regularizations facilitates finding a unique solution via FWI, and the weighted l 2 objective function ensures a more reasonable residual, thereby improving the stability of the gradient calculation. Numerical tests performed using the Marmousi synthetic dataset show that the use of the weighted l 2 objective function and the model-domain regularizations significantly improves the Laplace–Fourier-domain FWI. Because the Laplace–Fourier-domain FWI is improved, the frequency-domain FWI, in which the Laplace–Fourier-domain FWI result is used as the starting model, yields inversion result much closer to the true velocity.

Laplace-domain wave-equation modeling and full waveform inversion in 3D isotropic elastic media

Abstract The 3D elastic problem has not been widely studied because of the computational burden. Over the past few years, 3D elastic full waveform inversion (FWI) techniques in the time and frequency domains have been proposed by some researchers based on developments in computer science. However, these techniques still have the non-uniqueness and high nonlinearity problems. In this paper, we propose a 3D elastic FWI algorithm in the Laplace domain that can mitigate these problems. To efficiently solve the impedance matrix, we adopt a first-order absorbing boundary condition that results in a symmetric system. A conjugate gradient (CG) solver can be used because the Laplace-domain wave equation is naturally positive definite. We apply the Jacobi preconditioner to increase the convergence speed. We identify the permissible range of Laplace damping constants through dispersion analysis and accuracy tests. We perform the Laplace-domain FWI based on a logarithmic objective function, and the inversion examples are designed for a land setting, which means that the source is vertically excited and multi-component data are considered. The inversion results indicate that the inversion that uses only the vertical component performs slightly better than the multi-component inversion. This unexpected result is obtained partly because we use a vertically polarized source. We analyze the residuals and Frechet derivatives for each component to examine the characteristics of the Laplace-domain multi-component FWI. The results indicate that the residuals and Frechet derivatives for the horizontal component have a singularity problem. The numerical examples demonstrate that the singularity problem is related to the directivity of the displacement and to taking the logarithm of Laplace-domain wave fields. To avoid this singularity problem, we use a simple method that excludes the data near the singular region. Although we can use either simultaneous or sequential strategies to invert the Laplace-domain data, we apply a simultaneous inversion strategy in this paper. Nonetheless, the numerical examples demonstrate that our inversion algorithm yields reasonable long-wavelength velocity structures.

2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function

The wavefield in the Laplace domain has a very small amplitude except only near the source point. In order to deal with this characteristic, the logarithmic objective function has been used in many Laplace domain inversion studies. The Laplace-domain waveform inversion using the logarithmic objective function has fewer local minima than the time- or frequency domain inversion. Recently, the power objective function was suggested as an alternative to the logarithmic objective function in the Laplace domain. Since amplitudes of wavefields are very small generally, a power <1 amplifies the wavefields especially at large offset. Therefore, the power objective function can enhance the Laplace-domain inversion results. In previous studies about synthetic datasets, it is confirmed that the inversion using a power objective function shows a similar result when compared with the inversion using a logarithmic objective function. In this paper, we apply an inversion algorithm using a power objective function to field datasets. We perform the waveform inversion using the power objective function and compare the result obtained by the logarithmic objective function. The Gulf of Mexico dataset is used for the comparison. When we use a power objective function in the inversion algorithm, it is important to choose the appropriate exponent. By testing the various exponents, we can select the range of the exponent from 5 × 10−3 to 5 × 10−8 in the Gulf of Mexico dataset. The results obtained from the power objective function with appropriate exponent are very similar to the results of the logarithmic objective function. Even though we do not get better results than the conventional method, we can confirm the possibility of applying the power objective function for field data. In addition, the power objective function shows good results in spite of little difference in the amplitude of the wavefield. Based on these results, we can expect that the power objective function will produce good results from the data with a small amplitude difference. Also, it can partially be utilized at the sections where the amplitude difference is very small.

Robust Full Waveform Inversion Using Normalized Residual in the Frequency Domain

ABSTRACT A robust objective function for the full waveform inversion has been suggested in the frequency domain. The proposed objective function is defined as sum of complex absolute values of residual wavefields in the frequency domain. Generally, the full waveform inversion is extremely sensitive to the parameterization, frequency bandwidth, attenuation, noise and so on. Especially, noise is the most important factor in the full waveform inversion. In the frequency-domain waveform inversion, the attenuation of wavefields is one of the critical parameters for successful inversion. To verify the robustness of our algorithm, the proposed inversion scheme was tested in terms of the sensitivity to attenuation and noise. The comparison examples with both the conventional l 2-norm and the logarithmic objective function demonstrated that the proposed inversion algorithm is robust to noise and less sensitive to attenuation.

2D acoustic‐elastic coupled waveform inversion in the Laplace domain

ABSTRACTAlthough waveform inversion has been intensively studied in an effort to properly delineate the Earth's structures since the early 1980s, most of the time‐ and frequency‐domain waveform inversion algorithms still have critical limitations in their applications to field data. This may be attributed to the highly non‐linear objective function and the unreliable low‐frequency components. To overcome the weaknesses of conventional waveform inversion algorithms, the acoustic Laplace‐domain waveform inversion has been proposed. The Laplace‐domain waveform inversion has been known to provide a long‐wavelength velocity model even for field data, which may be because it employs the zero‐frequency component of the damped wavefield and a well‐behaved logarithmic objective function. However, its applications have been confined to 2D acoustic media.We extend the Laplace‐domain waveform inversion algorithm to a 2D acoustic‐elastic coupled medium, which is encountered in marine exploration environments. In 2D acoustic‐elastic coupled media, the Laplace‐domain pressures behave differently from those of 2D acoustic media, although the overall features are similar to each other. The main differences are that the pressure wavefields for acoustic‐elastic coupled media show negative values even for simple geological structures unlike in acoustic media, when the Laplace damping constant is small and the water depth is shallow. The negative values may result from more complicated wave propagation in elastic media and at fluid‐solid interfaces.Our Laplace‐domain waveform inversion algorithm is also based on the finite‐element method and logarithmic wavefields. To compute gradient direction, we apply the back‐propagation technique. Under the assumption that density is fixed, P‐ and S‐wave velocity models are inverted from the pressure data. We applied our inversion algorithm to the SEG/EAGE salt model and the numerical results showed that the Laplace‐domain waveform inversion successfully recovers the long‐wavelength structures of the P‐ and S‐wave velocity models from the noise‐free data. The models inverted by the Laplace‐domain waveform inversion were able to be successfully used as initial models in the subsequent frequency‐domain waveform inversion, which is performed to describe the short‐wavelength structures of the true models.

Sequentially ordered single-frequency 2-D acoustic waveform inversion in the Laplace-Fourier domain

SUMMARY In the conventional frequency-domain waveform inversion, either multifrequency simultaneous inversion or sequential single-frequency inversion has been implemented. However, most conventional frequency-domain waveform inversion methods fail to recover background velocity when low-frequency information is missing. Recently, new waveform inversion techniques in the Laplace and Laplace–Fourier domain have been proposed to recover background velocity structure from data with insufficient low-frequency information. In such techniques, however, all frequencies are inverted simultaneously, and this requires large computational resources and long computation times. In this paper, we propose a sequentially ordered single-frequency 2-D acoustic waveform inversion using the logarithmic objective function in the Laplace–Fourier domain. Our algorithm sequentially inverts single-frequency data in the Laplace–Fourier domain, thus reducing computational resources. Unlike most conventional waveform inversion methods requiring an initial velocity model close to the true model, we propose a one-step waveform inversion method in seeking to find a final velocity structure from the simple initial model through a hybrid combination of the Laplace domain inversion and the Fourier domain inversion. We adopt and evaluate the multiloop algorithm by modifying the double-loop algorithm commonly used in the conventional frequency-domain waveform inversion. Using the multiloop algorithm repeating loop over frequencies, the quality of the inversion results can be improved and the decision problem of the number of iterations for each frequency can be overcome effectively. Because the sequential order of the Laplace–Fourier frequencies in a 2-D plane should be assigned for inverting Laplace–Fourier frequency data consecutively, we present three different sequential orders of Laplace–Fourier frequencies while considering the multiscale and layer-stripping approach, and we compare the inversion results from the numerical experiments. We applied the sequentially ordered single-frequency 2-D acoustic waveform inversion in the full Laplace–Fourier domain to the synthetic seismic data produced from complex structure model and field data. A realistic model could be recovered in an efficient and robust manner, even using the two-layer homogeneous velocity model as an initial model. The inverted velocity model from the field data was validated by examining the migrated image from the pre-stack depth migration and the flattening of the common-image gathers or by comparing the synthetic shot gather with the real shot gather. The proposed one-step waveform inversion algorithm can be easily extended to the sequential inversion of 3-D acoustic or elastic data in the full Laplace–Fourier domain.

Comparison of waveform inversion, part 1: conventional wavefield vs logarithmic wavefield

ABSTRACTThis is the first in a series of three papers focused on using variants of a logarithmic objective function approach to full waveform inversion. In this article, we investigate waveform inversion using full logarithmic principles and compare the results with the conventional least squares approach. We demonstrate theoretically that logarithmic inversion is computational similar to the conventional method in the sense that it uses exactly the same back‐propagation technology as used in least‐squares inversion. In the sense that it produces better results for each of three numerical examples, we conclude that logarithmic inversion is also more robust. We argue that a major reason for the inherent robustness is the fact that the logarithmic approach produces a natural scaling of the amplitude of the residual wavefield by the amplitude of the modelled wavefield that tends to stabilize the computations and consequently improve the final result. We claim that any superiority of the logarithmic inversion is based on the fact that it tends to be tomographic in the early stage of the inversion and more dependent on amplitude differences in the latter stages.