Accurate Forward Model Research Articles

Fast retrieval of XCO2 over east Asia based on Orbiting Carbon Observatory-2 (OCO-2) spectral measurements

Abstract. The increase in greenhouse gas concentrations, particularly CO2, has significant implications for global climate patterns and various aspects of human life. Spaceborne remote sensing satellites play a crucial role in high-resolution monitoring of atmospheric CO2. However, the next generation of greenhouse gas monitoring satellites is expected to face challenges, particularly in terms of computational efficiency in atmospheric CO2 retrieval and analysis. To address these challenges, this study focuses on improving the speed of retrieving the column-averaged dry-air mole fraction of carbon dioxide (XCO2) using spectral data from the Orbiting Carbon Observatory-2 (OCO-2) satellite while still maintaining retrieval accuracy. A novel approach based on neural network (NN) models is proposed to tackle the nonlinear inversion problems associated with XCO2 retrievals. The study employs a data-driven supervised learning method and explores two distinct training strategies. Firstly, training is conducted using experimental data obtained from the inversion of the operational optimization model, which is released as the OCO-2 satellite products. Secondly, training is performed using a simulated dataset generated by an accurate forward calculation model. The inversion performance and prediction performance of the machine learning model for XCO2 are compared, analyzed, and discussed for the observed region over east Asia. The results demonstrate that the model trained on simulated data accurately predicts XCO2 in the target area. Furthermore, when compared to OCO-2 satellite product data, the developed XCO2 retrieval model not only achieves rapid predictions (<1 ms) with good accuracy (1.8 ppm or approximately 0.45 %) but also effectively captures sudden increases in XCO2 plumes near industrial emission sources. The accuracy of the machine learning model retrieval results is validated against reliable data from Total Carbon Column Observing Network (TCCON) sites, demonstrating its ability to effectively capture CO2 seasonal variations and annual growth trends.

Investigation of acoustoelastic surface acoustic waves in prestressed media

Dispersion of broadband surface acoustic waves (SAWs) is often used for the nondestructive evaluation of near-surface elastic properties of materials. The accurate forward model of SAW propagation is essential for inversion. In this paper, we present a combining semi-analytical finite element (SAFE) and perfectly matched layer method for accurately predicting the phase velocity dispersion, attenuation and polarization modes of SAWs in any prestressed anisotropic layered half-space. Equations of motion are formulated based on the Biot's theory of small deformations influenced by initial stress, so it does not matter whether the residual stress results from hyperelastic transformations. The simulated results are firstly validated by the solutions from the exact three-dimensional elasticity theory-based method. The SAFE method is then used to discuss the effect of material anisotropy and residual stress on the SAWs. Finally, the method is applied to two practical examples. The differences and similarities of two common types of acoustoelastic theories are also discussed.

The SSR Brightness Temperature Increment Model Based on a Deep Neural Network

The SSS (sea surface salinity) is an important factor affecting global climate changes, sea dynamic environments, global water cycles, marine ecological environments, and ocean carbon cycles. Satellite remote sensing is a practical way to observe SSS from space, and the key to retrieving SSS satellite products is to establish an accurate sea surface brightness temperature forward model. However, the calculation results of different forward models, which are composed of different relative permittivity models and SSR (sea surface roughness) brightness temperature increment models, are different, and the impact of this calculation difference has exceeded the accuracy requirement of the SSS inversion, and the existing SSR brightness temperature increment models, which primarily include empirical models and theoretical models, cannot match all the relative permittivity models. In order to address this problem, this paper proposes a universal DNN (deep neural network) model architecture and corresponding training scheme, and provides different SSR brightness temperature increment models for different relative permittivity models utilizing DNN based on offshore experiment data, and compares them with the existing models. The results show that the DNN models perform significantly better than the existing models, and that their calculation accuracy is close to the detection accuracy of a radiometer. Therefore, this study effectively solves the problem of SSR brightness temperature correction under different relative permittivity models, and provides a theoretical support for high-precision SSS inversion research.

A switching Gaussian process latent force model for the identification of mechanical systems with a discontinuous nonlinearity

Abstract An approach for the identification of discontinuous and nonsmooth nonlinear forces, as those generated by frictional contacts, in mechanical systems that can be approximated by a single-degree-of-freedom model is presented. To handle the sharp variations and multiple motion regimes introduced by these nonlinearities in the dynamic response, the partially known physics-based model and noisy measurements of the system’s response to a known input force are combined within a switching Gaussian process latent force model (GPLFM). In this grey-box framework, multiple Gaussian processes are used to model the unknown nonlinear force across different motion regimes and a resetting model enables the generation of discontinuities. The states of the system, nonlinear force, and regime transitions are inferred by using filtering and smoothing techniques for switching linear dynamical systems. The proposed switching GPLFM is applied to a simulated dry friction oscillator and an experimental setup consisting of a single-storey frame with a brass-to-steel contact. Excellent results are obtained in terms of the identified nonlinear and discontinuous friction force for varying: (i) normal load amplitudes in the contact; (ii) measurement noise levels, and (iii) number of samples in the datasets. Moreover, the identified states, friction force, and sequence of motion regimes are used for evaluating: (1) uncertain system parameters; (2) the friction force–velocity relationship, and (3) the static friction force. The correct identification of the discontinuous nonlinear force and the quantification of any remaining uncertainty in its prediction enable the implementation of an accurate forward model able to predict the system’s response to different input forces.

Second-order symplectic PRK scheme for ground-penetrating radar simulations in dispersive materials

An accurate and effective forward model is helpful in interpreting measured ground-penetrating radar (GPR) data. In this paper, an explicit second-order symplectic partitioned Runge–Kutta scheme (SPRK) with Higdon's absorbing boundary condition is developed to simulate GPR wave propagation in dispersive media. For this purpose, the first-order Debye dispersion model is successfully applied to the second-order SPRK method. A single-layer dispersive medium model and a three-layer pavement structure model are simulated, in order to verify the accuracy and efficiency of the second-order SPRK method, as well as to clarify the effect of dispersion on the GPR echo signal. The numerical simulation results demonstrate that the proposed algorithm not only can maintain the simulation accuracy, but can also save about 19% of the calculation time, when compared with the FDTD algorithm, in a dispersive medium. We also obtain the GPR profile of a pavement structure with a circular cavity disease and an underground pipe structure in a dispersive medium, and the simulation results show that the dispersion characteristics of the materials lead to high absorption of the GPR electromagnetic waves. Finally, field experiment data are used to further verify the applicability of the algorithm. The numerical simulations and experiments indicate that the dispersion of the medium must be considered in GPR numerical simulation, and the authenticity of the simulated image must be ensured.

Calibration of a Finite Element Forward Model in Eddy Current Inspection

We report on the use of a novel constrained optimisation algorithm for calibrating the finite element model in an eddy current inspection application. An accurate finite element forward model is often important in such eddy current applications for training neural networks or as part of an iterative solver. However, the subject of calibration of the model has not received much attention in the literature to date. We consider a multi-frequency, eddy current depth profiling application, which is important for non-destructive testing in the nuclear industry. In the optimisation algorithm, we use a Levenberg-Marquardt algorithm and a bisection search to ensure constraints are satisfied, coupled with a truncated Gradient and Hessian method. We calibrate two types of finite element models, one using a filament representation for the coils and the other using a full 3D approach. The results show the feasibility of using the constrained non-linear optimisation algorithm for tuning the finite element model parameters. The mean signal-noise ratio after tuning on a truncated spectrum was 29.47 dB for the 3D model and 28.96 dB for the filament; in contrast the mean SNR using the measured coil parameters was 1.48 dB and 4.98 dB for the two uncalibrated models, respectively. The results show that a filament model is competitive with a 3D model of the coils (within the nuclear graphite application); therefore, a computationally faster filament model can be used with minimal effect on accuracy.

Fast design and optimization method for an ultra-wideband perfect absorber based on artificial neural network acceleration

In recent years, ultra-wideband perfect absorbers (UWPAs) based on metamaterial nanostructures have been widely studied due to their excellent performance. However, designing and optimizing the absorber quickly and accurately remains challenging in the broad-band range. In this work, by adopting the genetic algorithm combined with artificial neural network (ANN) acceleration, an effective method with the global search ability and extremely fast optimization speed is developed, which only requires an accurate forward ANN model and offers tremendous controls to the designer. In the context of the optimization of the nano-multilayers absorber, the most of mean absolute errors (MAEs) of ANN between the actual spectral absorptance and the predicted spectral absorptance are less than 1E-2, demonstrating the outstanding consistency. What's more, the optimization speed is four orders of magnitude faster than conventional electromagnetic simulation software, 1.2 × 105 search calculations take only 573 s. Meanwhile, we also show that the hybridization between multilayer cavity plasmonic resonant modes is the main cause of the ultra-wideband absorptance. This method significantly increases the viability of more complex ultra-wideband perfect absorber designs and provides opportunities to manipulate light-matter interactions in the future.

Addressing CT metal artifacts using photon-counting detectors and one-step spectral CT image reconstruction.

The constrained one-step spectral CT image reconstruction (cOSSCIR) algorithm with a nonconvex alternating direction method of multipliers optimizer is proposed for addressing computed tomography (CT) metal artifacts caused by beam hardening, noise, and photon starvation. The quantitative performance of cOSSCIR is investigated through a series of photon-counting CT simulations. cOSSCIR directly estimates basis material maps from photon-counting data using a physics-based forward model that accounts for beam hardening. The cOSSCIR optimization framework places constraints on the basis maps, which we hypothesize will stabilize the decomposition and reduce streaks caused by noise and photon starvation. Another advantage of cOSSCIR is that the spectral data need not be registered, so that a ray can be used even if some energy window measurements are unavailable. Photon-counting CT acquisitions of a virtual pelvic phantom with low-contrast soft tissue texture and bilateral hip prostheses were simulated. Bone and water basis maps were estimated using the cOSSCIR algorithm and combined to form a virtual monoenergetic image for the evaluation of metal artifacts. The cOSSCIR images were compared to a "two-step" decomposition approach that first estimated basis sinograms using a maximum likelihood algorithm and then reconstructed basis maps using an iterative total variation constrained least-squares optimization (MLE+TV ). Images were also compared to a nonspectral TV reconstruction of the total number of counts detected for each ray with and without normalized metal artifact reduction (NMAR) applied. The simulated metal density was increased to investigate the effects of increasing photon starvation. The quantitative error and standard deviation in regions of the phantom were compared across the investigated algorithms. The ability of cOSSCIR to reproduce the soft-tissue texture, while reducing metal artifacts, was quantitativelyevaluated. Noiseless simulations demonstrated the convergence of the cOSSCIR and MLE+TV algorithms to the correct basis maps in the presence of beam-hardening effects. When noise was simulated, cOSSCIR demonstrated a quantitative error of -1 HU, compared to 2 HU error for the MLE+TV algorithm and -154 HU error for the nonspectral TV +NMAR algorithm. For the cOSSCIR algorithm, the standard deviation in the central iodine region of interest was 20 HU, compared to 299 HU for the MLE+TV algorithm, 41 HU for the MLE+TV +Mask algorithm that excluded rays through metal, and 55 HU for the nonspectral TV +NMAR algorithm. Increasing levels of photon starvation did not impact the bias or standard deviation of the cOSSCIR images. cOSSCIR was able to reproduce the soft-tissue texture when an appropriate regularization constraint value wasselected. By directly inverting photon-counting CT data into basis maps using an accurate physics-based forward model and a constrained optimization algorithm, cOSSCIR avoids metal artifacts due to beam hardening, noise, and photon starvation. The cOSSCIR algorithm demonstrated improved stability and accuracy compared to a two-step method of decomposition followed byreconstruction.

MCMC Method of Inverse Problems Using a Neural Network—Application in GPR Crosshole Full Waveform Inversion: A Numerical Simulation Study

Ground-penetrating radar (GPR) crosshole tomography is widely applied to subsurface media images. However, the inadequacies of ray methods may limit the resolution of crosshole radar images, since the ray method is a type of high-frequency approximation. To solve this problem, the full waveform method is introduced for GPR inversion. However, full waveform inversion is computationally expensive. In this paper, we introduce a trained neural network that can be evaluated very quickly to replace a computationally intensive forward model. Additionally, the forward error of the trained neural network can be statistically analyzed. We demonstrate a methodology for a full waveform inversion of crosshole ground-penetrating radar data using the Markov chain Monte Carlo (MCMC) method. An accurate forward model based on Maxwell’s equations is replaced by a quickly trained neural network. This method achieves a high computation efficiency, which is four orders of magnitude faster than the accurate forward model. The inversion result of the synthetic waveform data shows a good performance of the trained neural network, which greatly improves the calculation efficiency.

Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation

In radio occultation (RO) data processing and data assimilation, the forward model (FM) is used to calculate bending angle (BA) from refractivity (N). The accuracy and precision of forward modeled BA are affected by refractivity profiles and FM methods, including Abel integral algorithms (direct, exp, exp_T, linear) and methods of interpolating refractivity during integral (log-cubic spline and log-linear). Experiment 1 compares these forward model methods by comparing the difference and relative difference (RD) of the experimental value (forward modeled ECMWF analysis) and the true value (BA of FY3D RO data). Results suggested that the exp with log-cubic spline (log-cubic) interpolation is the most accurate FM because it has better integral accuracy (less than 2%) to inputs, especially when the input is lower than an order of magnitude of 1 × 10−2 (that is, above 60 km). By contrast, the direct induced a 10% error, and the improvement of exp T to exp is limited. Experiment 2 simulated the exact errors of an FM (exp) based on inputs on different vertical resolutions. The inputs are refractivity profiles on model levels of three widely used analyses, including ECMWF 4Dvar analysis, final operational global analysis data (FNL), and ERA5. Results demonstrated that based on exp and log-cubic interpolation, BA on model level of ECMWF 4Dvar has the highest accuracy, whose RD is 0.5% between 0–35 km, 4% between 35–58 km, and 1.8% between 58–80 km. By contrast, the other two analyses have low accuracy. This paper paves the way to better understanding the FM, and simulation errors on model levels of three analyses can be a helpful FM error reference.

Wideband Microwave Sensor for Downhole Water-Cut Monitoring

In the petroleum industry, downhole water-cut loggings are essential for reservoir management. Previously, most of them are based on permittivity measurements by low-frequency capacitance methods. However, they fail to work in aging oil fields with high water cuts and highly mineralized subterranean water. In this article, a nonresonant self-calibrated wideband microwave sensor is proposed to solve this problem. Taking advantage of the Cerenkov-like enhanced radiation of coaxial leaky-wave antennas, sufficient signals can be received even when the fluid under test (FUT) is highly conductive. An accurate wideband forward model has been established with no empirical or undetermined quantities. Complex permittivities and water fractions of the FUT can be retrieved without calibrations, making the sensor especially suitable for the downhole working conditions with unknown temperature drifts. NaCl solutions with different concentrations (0–130 g/L), oil–water emulsions with different water fractions (70%–100%), and ethanol–water solutions have been tested in the frequency band ranging from 1 to 8 GHz. Dielectric constants and conductivities of the NaCl solutions range from 40 to 80 and from 0 to 15 S/m, respectively, covering the main measurement ranges of high-water-cut monitoring.

Model reduction in acoustic inversion by artificial neural network.

In ultrasound tomography, the speed of sound inside an object is estimated based on acoustic measurements carried out by sensors surrounding the object. An accurate forward model is a prominent factor for high-quality image reconstruction, but it can make computations far too time-consuming in many applications. Using approximate forward models, it is possible to speed up the computations, but the quality of the reconstruction may have to be compromised. In this paper, a neural network-based approach is proposed that can compensate for modelling errors caused by the approximate forward models. The approach is tested with various different imaging scenarios in a simulated two-dimensional domain. The results show that with fairly small training datasets, the proposed approach can be utilized to approximate the modelling errors, and to significantly improve the image reconstruction quality in ultrasound tomography, compared to commonly used inversion algorithms.

Deep learning-based method to accurately estimate breast tissue optical properties in the presence of the chest wall.

.Significance: In general, image reconstruction methods used in diffuse optical tomography (DOT) are based on diffusion approximation, and they consider the breast tissue as a homogenous, semi-infinite medium. However, the semi-infinite medium assumption used in DOT reconstruction is not valid when the chest wall is underneath the breast tissue.Aim: We aim to reduce the chest wall’s effect on the estimated average optical properties of breast tissue and obtain accurate forward model for DOT reconstruction.Approach: We propose a deep learning-based neural network approach where a convolution neural network (CNN) is trained to simultaneously obtain accurate optical property values for both the breast tissue and the chest wall.Results: The CNN model shows great promise in reducing errors in estimating the optical properties of the breast tissue in the presence of a shallow chest wall. For patient data, the CNN model predicted the breast tissue optical absorption coefficient, which was independent of chest wall depth.Conclusions: Our proposed method can be readily used in DOT and diffuse spectroscopy measurements to improve the accuracy of estimated tissue optical properties.

Planning in RTS Games with Incomplete Action Definitions via Answer Set Programming

Standard game tree search algorithms, such as minimax or Monte Carlo Tree Search, assume the existence of an accurate forward model that simulates the effects of actions in the game. Creating such model, however, is a challenge in itself.One cause of the complexity of the task is the gap in level of abstraction between the informal specification of the model and its implementation language. To overcome this issue, we propose a technique for the implementation of forward models that relies on the Answer Set Programming paradigm and on well-established knowledge representation techniques from defeasible reasoning and reasoning about actions and change. We evaluate our approach in the context of Real-Time Strategy games using a collection of StarCraft scenarios.

S11 Parameter Calculation of Frequency Selective Surface Based on Deep Learning

This paper proposes the FSS-CNN network model as a forward predictor, replacing the function of the Maxwell equation solver of commercial software. The predictor is different from the numerical optimization method, and the data-driven method based on machine learning (ML) can be expressed and generalize complex functions or data to discover unknown relationships between a large number of variables. In the frequency range of 2∼18GHz, the S11 parameter prediction of the corresponding metal pixel pattern can be easily realized by an accurate forward neural network model. The MSE reaches the level of less than 0.1 and the time consumption is less than 0.07s, which meets the requirements of fast, efficient and automatic calculation.

Comparative Study of Markov Chain With Recurrent Neural Network for Short Term Velocity Prediction Implemented on an Embedded System

Short-term prediction models for an ego-vehicle's speed contributes to the improvement of vehicle safety, driveability, and fuel economy. To achieve these desired outcomes, an accurate forward speed prediction model and its successful implementation in a real system is a prerequisite. This paper compares six velocity prediction models based on two types of data-driven models, a Markov chain and a Recurrent Neural Network (RNN), by implementing them in an embedded system to evaluate their prediction accuracy and execution time. The inputs to each model are the driving information acquired on a specific route, such as internal vehicle information, relative speed and distance to the vehicle in the front of the ego-vehicle, and ego-vehicle's location estimated by the GPS signal along with the B-spline roadway model. The proposed prediction models predict the velocity profile of the ego-vehicle up to the prediction horizon of 150 m. The parameters of the proposed models have been optimized using Hyper-parameter Optimization via Radial basis function and Dynamic coordinate search. By applying real driving data, the Markov chain-based models show slightly lower prediction accuracy but shorter execution time than those of the RNN-based models.

Predicting chemical shifts with graph neural networks.

Inferring molecular structure from Nuclear Magnetic Resonance (NMR) measurements requires an accurate forward model that can predict chemical shifts from 3D structure. Current forward models are limited to specific molecules like proteins and state-of-the-art models are not differentiable. Thus they cannot be used with gradient methods like biased molecular dynamics. Here we use graph neural networks (GNNs) for NMR chemical shift prediction. Our GNN can model chemical shifts accurately and capture important phenomena like hydrogen bonding induced downfield shift between multiple proteins, secondary structure effects, and predict shifts of organic molecules. Previous empirical NMR models of protein NMR have relied on careful feature engineering with domain expertise. These GNNs are trained from data alone with no feature engineering yet are as accurate and can work on arbitrary molecular structures. The models are also efficient, able to compute one million chemical shifts in about 5 seconds. This work enables a new category of NMR models that have multiple interacting types of macromolecules.

Radiative transfer in the [formula omitted] A-band - a fast and accurate forward model based on the [formula omitted]-distribution approach

An efficient method based on accurate transmissivity calculations to account for gaseous absorption in Monte Carlo / quasi-Monte Carlo radiative transfer codes is presented. The modeling approach is based on an improved non-uniform transmission formulation, called the ℓ-distribution method. The technique is founded on two components: 1/ an original uniform method to tabulate band averaged transmissivities of gaseous paths and 2/ its extension to non-uniform paths, based on the Godson-Weinreb-Neuendorffer’s (GWN) approximation that consists of the definition of effective scaling factors to relate gaseous spectra in distinct states. As the GWN method is known to provide results dependent on the ordering of the gas layers, two path reordering strategies are introduced and compared. One of them is founded on results from statistical theory and requires the introduction of a new coefficient (Kendall’s Ke). This coefficient together with its role on the path reordering strategy is introduced, detailed and analyzed. The two schemes are then assessed against Line-By-Line calculations in line-of-sight geometries. Two configurations representative of radiative transfer in the O2 A-band for scattering atmospheres are then studied: the first one only considers molecular (Rayleigh) scattering and the second one involves various cloud configurations (single or bi-layer clouds located at various altitudes). For these calculations, the ℓ-distribution methods based on the two path reordering schemes were implemented in the 3D Monte Carlo code 3DMCPOL. Results of the ℓ-distribution method are compared with solutions provided by the Correlated k-distribution method and assessed against Line-By-Line calculations. The ℓ-distribution approach combined with the Kendall’s reordering strategy is shown to provide results more accurate than k-distributions at smaller calculation costs. The ℓ-distribution method is shown to be a relevant candidate for radiative transfer in the atmosphere, both for remote sensing applications but also for weather forecasting or radiative budget studies, with or without scattering phenomena.

Tomographic retrieval algorithm of OH concentration profiles using double spatial heterodyne spectrometers

Abstract. The hydroxyl radical (OH) determines the capability of atmospheric self-cleansing and is one of the significant oxidants in atmospheric photochemistry reactions. Global OH has been monitored by satellites with the traditional limb mode in the past decades. This observed mode can achieve the acquisition of high-resolution vertical OH data but cannot obtain enough horizontal OH data for inverting high-precision OH concentrations because OH has a high reactivity that makes OH concentrations extremely low and distributions complicated. The double spatial heterodyne spectrometer (DSHS) is designed to obtain higher-resolution and more detailed OH data. This sensor can measure OH by the three-dimensional limb mode to obtain comprehensive OH data in the atmosphere. Here we propose a new tomographic retrieval algorithm based on the simulated observation data because the DSHS will work officially on the orbit in the future. We build up an accurate forward model. The main part of it is the SCIATRAN radiative transfer model which is modified according to the radiation transmission theory. The error in results obtained by the forward model is ±44.30 % in the lower atmosphere such as at a 21 km height and decreases gradually until the limit of observation altitude. We also construct the tomographic retrieval algorithm of which the core is a lookup table method. A tomographic-observation database is built up through the atmospheric model, the spatial information (the position of the target area and satellite position), the date parameters, the observation geometries, OH concentrations, and simulated observation data. The OH concentrations can be found from it directly. If there are no corresponding query conditions in the tomographic-observation database, the cubic spline interpolation is used to obtain the OH concentrations. The inversion results are given, and the errors in them increase as the altitudes rise until about a 41 km height then start to decrease. The errors in the inversion results reach the maximum of about ±25.03 % at the 41 km height and decrease to ±8.09 % at the limited observation height. They are also small in the lower atmosphere at ±12.96 % at 21 km. In summary, the tomographic retrieval algorithm can obtain more accurate OH concentrations even in the lower atmosphere where the OH data are not high quality and avoids the setting of initial guess values for solving the iteration problems. Our research not only provides support for the scientific theory of the construction of the DSHS but also gives a new retrieval algorithm idea for other radicals.

Dynamic simulation of FWD tests on flexible transversely isotropic pavements with imperfect interfaces

Developing an accurate and efficient forward model of the dynamic response of pavement structure is of great significance for obtaining the design parameters of the layered pavement structure through an inverse calculation based on the measured pavement data. However, the existing algorithms rarely consider the interlayer contact. Based on the traditional spectral element method (SEM), this study proposes a new forward model to simulate the falling weight deflectometer (FWD) dynamic tests for the transversely isotropic layered pavement structure. The interfaces between the pavement layers can be imperfect, and the displacement boundary conditions are discontinuous. Moreover, the developed model considerably enhances the efficiency of the model in solving the dynamic response of transversely isotropic layered media by avoiding the infinite integration in integral transformation. Numerical examples verify the accuracy and efficiency of the model developed in this study. At the same time, an extensive parameter analysis is conducted on the interlayer contact of the pavement structure and inhomogeneity of the layered soil, which provides a reliable, numerical, theoretical basis for the design of pavement structure.