Traditional Inversion Methods Research Articles

Fast Inversion of Subsurface Target Electromagnetic Induction Response With Deep Learning

With the development of electromagnetic detection technology, the inversion of electromagnetic induction (EMI) response based on a 3-D orthogonal dipole model can provide an estimation of parameters of a high-conductivity target, such as the target’s position, orientation, and shape. However, the traditional inversion methods suffer from several limitations including relying on the initial value, easy to fall into the local optimal solution, and high computational complexity. To overcome these disadvantages, in this letter, we propose a deep learning (DL) inversion method of subsurface target EMI response based on deep neural network (DNN) architecture, which is combined with adaptive moment estimation (Adam) optimization algorithm and learning rate attenuation strategy to improve the model accuracy. Datasets are obtained from forward modeling in different target parameters. By limiting the range of target parameters, errors caused by the nonuniqueness of inversion results are avoided. Through simulation and field experiments, we verify the performance of this method. The experimental results show that compared with the traditional inversion algorithms, the inversion accuracy is higher and the inversion speed is three to four orders of magnitude faster.

Double-scale supervised inversion with a data-driven forward model for low-frequency impedance recovery

Low-frequency information is important in reducing the nonuniqueness of absolute impedance inversion and for quantitative seismic interpretation. In traditional model-driven impedance inversion methods, the low-frequency impedance background is from an initial model and is almost unchanged during the inversion process. Moreover, the inversion results are limited by the quality of the modeled seismic data and the extracted wavelet. To alleviate these issues, we have investigated a double-scale supervised impedance inversion method based on the gated recurrent encoder-decoder network (GREDN). We first train the decoder network of GREDN called the forward operator, which can map impedance to seismic data. We then implement the well-trained decoder as a constraint to train the encoder network of GREDN called the inverse operator. Besides matching the output of the encoder with broadband pseudowell impedance labels, data generated by inputting the encoder output into the known decoder match the observed narrowband seismic data. The broadband impedance information and the already-trained decoder largely limit the solution space of the encoder. Finally, after training, only the derived optimal encoder is applied to unseen seismic traces to yield broadband impedance volumes. Our approach is fully data driven and does not involve the initial model, seismic wavelet, and model-driven operator. Tests on the Marmousi model illustrate that our double-scale supervised impedance inversion method can effectively recover low-frequency components of the impedance model, and we determine that low frequencies of the predicted impedance originate from well logs. Furthermore, we apply the strategy of combining the double-scale supervised impedance inversion method with a model-driven impedance inversion method to process field seismic data. Tests on a field data set indicate that the predicted impedance results not only reveal a classic tectonic sedimentation history but also match the corresponding results measured at the locations of two wells.

Biot's equations-based reservoir parameter inversion using deep neural networks

AbstractReservoir parameter inversion from seismic data is an important issue in rock physics. The traditional optimisation-based inversion method requires high computational expense, and the process exhibits subjectivity due to the nonuniqueness of generated solutions. This study proposes a deep neural network (DNN)-based approach as a new means to analyse the sensitivity of seismic attributes to basic rock-physics parameters and then realise fast parameter inversion. First, synthetic data of inputs (reservoir properties) and outputs (seismic attributes) are generated using Biot's equations. Then, a forward DNN model is trained to carry out a sensitivity analysis. One can in turn investigate the influence of each rock-physics parameter on the seismic attributes calculated by Biot's equations, and the method can also be used to estimate and evaluate the accuracy of parameter inversion. Finally, DNNs are applied to parameter inversion. Different scenarios are designed to study the inversion accuracy of porosity, bulk and shear moduli of a rock matrix considering that the input quantities are different. It is found that the inversion of porosity is relatively easy and accurate, while more information is needed to make the inversion more accurate for bulk and shear moduli. From the presented results, the new approach makes it possible to realise accurate and pointwise inverse modelling with high efficiency for actual data interpretation and analysis.

A Decoupled Calibration Method Based on the Multi-Output Support Vector Regression Algorithm for Three-Dimensional Electric-Field Sensors.

Aiming at the problem that the measured accuracy of the electric field intensity which is affected by the coupling interference by sensor output signal from the component of a three dimensional electric field, the causes of the coupling error was analyzed, and a decoupled calibration method based on support vector regression algorithm for three-dimensional electric field sensor is proposed. The solution of the decoupled calibration matrix was regarded as a multi-objective optimization process, and the optimal decoupling calibration matrix was obtained by the ν-SVR algorithm. The complex inverse calculation of the matrix was avoided, and the calculation error was reduced. A rotary calibration device was designed to accurately measure the angle between the induction electrode of the sensor and the electric-field vector, and an accurate calculation model of the theoretical electric field was established. The experimental results showed that the error between the calculated and theoretical values of the electric-field components obtained by the proposed method were smaller than those obtained by the traditional inverse matrix calibration method, the accuracy of the calibration was improved, the rationality of the calibration method was proven, and the accuracy of the three-dimensional electric-field intensity measurements was further improved.

A new method of mineral inversion based on error analysis and static response equation error: A case study of a shale gas reservoir in the Wufeng-Longmaxi Formation

The accurate quantitative calculation of mineral components is very important and basic work in formation evaluation. Using well-log data to estimate mineralogy and porosity is a mainstream method with core measurements often used. However, in shale reservoirs, there are many mineral components, such as organic matter and pyrite. In addition, the pore structure is complex, and gas exists in the pores as free state, adsorbed state, and dissolved state. These factors make the logging response characteristics more complex and thus the estimation of the mineral components more difficult. To address this problem, we have adopted a mineral inversion method based on error analysis and response equation error. Based on the error analysis of the mineral inversion method, we first establish a technique to obtain interpretation parameters and the function of the response equation error combined with the core data. Then, based on the weighted total least-squares method, we construct the objective function, and we use the improved krill herd algorithm to solve the problem. Finally, we estimate the mineral component volume. The calculated results indicate that the method can accurately determine the clay, quartz + feldspar, carbonate contents, and porosity by conventional logging data. Compared to the traditional mineral inversion method, the average relative error is reduced by 11.1%. The method has high applicability to shale reservoirs and can supply the basic parameters for formation evaluation.

Machine learning based inverse modeling of full-field strain distribution for mechanical characterization of a linear elastic and heterogeneous membrane

Heterogeneous membranes or films are thin and soft structures with spatial variations in material property and thickness. Mechanical behavior of heterogeneous membranes is not well understood, mainly due to the difficulty in obtaining accurate and reliable material property data. To understand the mechanical behavior of these materials, accurate and efficient characterization methods for heterogeneous membranes are needed. In this paper, an inverse method based on machine learning is developed to efficiently extract mechanical properties from full-field strain distributions. This approach is demonstrated on a flat heterogeneous membrane with uniform thickness formed by up to four linear elastic synthetic materials in a grid arrangement, and deforming in a moderate strain range (true strain ∼10%). The results show that the machine learning method achieves accuracy comparable to the traditional inverse finite element method, and is 6 orders of magnitude faster in the demonstrated case studies.

An inverse design method of the acoustic lens

The traditional forward and inverse diffraction iterative lens design method that replaces the amplitude according to the desired sound field distribution can achieve most sound field distributions under unitary conditions. However, when the sound field distribution we desired contains a sharp plane with abrupt changes, the traditional method will ignore the severe energy loss of the truncated plane. The designed lens will produce a ringing effect, which reduces the quality of the lens. Inspired by the Yang Gu algorithm that can be applied to non-unitary conditions, this paper proposes an acoustic lens design method based on a simulated annealing particle swarm algorithm to overcome the shortcomings of traditional forward and inverse iterative algorithms. The feasibility of this method is verified by designing a lens with a flat-top sound pressure. This paper provides a more widely applicable method in an acoustic lens design.

New Inverse Method for Determining Uniaxial Flow Properties by Spherical Indentation Test

The spherical indentation test has been successfully applied to inversely derive the tensile properties of small regions in a non-destructive way. Current inverse methods mainly rely on extensive iterative calculations, which yield a considerable computational costs. In this paper, a database method is proposed to determine tensile flow properties from a single indentation force-depth curves to avoid iterative simulations. Firstly, a database that contain numerous indentation force-depth curves is established by inputting varied Ludwic material parameters into the indentation finite elements model. Secondly, for a given experimental indentation curve, a mean square error (MSE) is designated to evaluate the deviation between the experimental curve and each curve in the database. Finally, the true stresses at a series of plastic strain can be acquired by analyzing these deviations. To validate this new method, three different steels, i.e. A508, 2.25Cr1Mo and 316L are selected. Both simulated indentation curves and experimental indentation curves are used as inputs of the database to inversely acquire the flow properties. The result indicates that the proposed approach provides impressive accuracy when simulated indentation curves are used, but is less accurate when experimental curves are used. This new method can derive tensile properties in a much higher efficiency compared with traditional inverse method and are therefore more adaptive to engineering application.

Evaluation of the benefits of improved permeability estimation on high-resolution characterization of DNAPL distribution in aquifers with low-permeability lenses

High-resolution characterization of dense non-aqueous phase liquid (DNAPL) source zone architecture (SZA) is crucial for developing remediation strategies. Inverse modeling approaches are widely used to characterize DNAPL SZA. However, unknown parameters (permeability and DNAPL saturation) are usually assumed as Gaussian distribution in traditional geostatistical inverse methods (e.g., ensemble smoother with multiple data assimilation, ESMDA). This assumption may not be true in channels of fluvial deposits. The presence of clay lenses will bring significant challenges in reconstructing the DNAPL SZA using geostatistical inverse methods based on the Gaussian assumption. Moreover, the DNAPL characterization also suffers from a limited number of borehole data in the field. In this study, we use ESMDA's variant, ensemble smoother with multiple data assimilation-direct sampling (ESMDA-DS), to improve the estimation of non-Gaussian permeability field and evaluate the benefits of improved permeability estimation on the reconstruction of DNAPL source zone. The inversion technique is based on the iterative inversion framework to incorporate the clay content information into the DNAPL characterization. The performance of our proposed framework is examined and discussed through a numerical synthetic case with challenging pool-dominated SZA. To overcome data scarcity, we integrated hydrogeological and geophysical (electrical resistance tomography) datasets. Results show that the permeability is characterized well using ESMDA-DS method compared with ESMDA method. With the improved permeability estimation, the DNAPL source zone can be characterized in higher resolution. Compared with ESMDA method, the root mean square error (RMSE) of permeability is reduced by 38.5% using the ESMDA-DS method. Then the estimation RMSE of DNAPL saturation is reduced by 24.9% by improved permeability estimation. We also demonstrated the significant benefits of improved permeability estimation on DNAPL depletion behavior and plume evolution. Numerical simulations demonstrate with the improved permeability estimation, the DNAPL depletion prediction is highly improved, the prediction RMSE of the DNAPL dissolved plume is reduced by 34.5% after 50 years.

Inverted Algorithm of Terrestrial Water-Storage Anomalies Based on Machine Learning Combined with Load Model and Its Application in Southwest China

Dense Global Position System (GPS) arrays can be used to invert the terrestrial water-storage anomaly (TWSA) with higher accuracy. However, the uneven distribution of GPS stations greatly limits the application of GPS to derive the TWSA. Aiming to solve this problem, we grid the GPS array using regression to raise the reliability of TWSA inversion. First, the study uses the random forest (RF) model to simulate crustal deformation in unobserved grids. Meanwhile, the new Machine-Learning Loading-Inverted Method (MLLIM) is constructed based on the traditional GPS derived method to raise the truthfulness of TWSA inversion. Second, this research selects southwest China as the study region, the MLLIM and traditional GPS inversion methods are used to derive the TWSA, and the inverted results are contrasted with datasets of the Gravity Recovery and Climate Experiment (GRACE) Mascon and the Global Land Data Assimilation System (GLDAS) model. The comparison shows that values of Pearson Correlation Coefficient (PCC) between the MLLIM and GRACE and GRACE Follow-On (GRACE-FO) are equal to 0.91 and 0.88, respectively; and the values of R-squared (R2) are equal to 0.76 and 0.65, respectively; the values of PCC and R2 between MLLIM and GLDAS solutions are equal to 0.79 and 0.65. Compared with the traditional GPS inversion, the MLLIM improves PCC and R2 by 8.85% and 7.99% on average, which indicates that the MLLIM can improve the accuracy of TWSA inversion more than the traditional GPS method. Third, this study applies the MLLIM to invert the TWSA in each province of southwest China and combines the precipitation to analyze the change of TWSA in each province. The results are as follows: (1) The spatial distribution of TWSA and precipitation is coincident, which is highlighted in southwest Yunnan and southeast Guangxi; (2) this study compares TWSA of MLLIM with GRACE and GLDAS solutions in each province, which indicates that the maximum value of PCC is as high as 0.86 and 0.94, respectively, which indicates the MLLIM can be used to invert the TWSA in the regions with sparse GPS stations. The TWSA based on the MLLIM can be used to fill the vacancy between GRACE and GRACE-FO.

An improved dual-baseline PolInSAR method for forest height inversion

P-band PolInSAR technique has been validated to be a useful tool for forest height inversion. Aiming at the residual ground scattering contribution and temporal decorrelation in repeat-pass interferometry that affect the inversion accuracy, we proposed a dual-baseline method based on the RVoG + VTD model. This method is firstly applied to E-SAR P-band data acquired over the Krycklan Catchment, Sweden with mixed forests, and the influence of ground scattering contribution and temporal decorrelation related to vertical wavenumber kz and height hv are theoretically and experimentally explored by comparing the traditional three-stage inversion method, the fixed extinction method and the proposed method. The experimental results show that the traditional three-stage inversion method has a overall serious overestimation for the whole study area, resulting in a R2 of ca. 0.4 and RMSE of ca. 7 m. Compared with the three-stage inversion method, the fixed extinction method improves the overestimation in the near range but has little effect in the far range. After introducing the RVoG + VTD model, both the overestimation in the near and far range are improved. In general, the study area is affected by both the influence of ground scattering contribution and temporal decorrelation. The overestimation in the near range (with kz larger than ca. 0.1 rad/m) is mainly caused by ground scattering contribution while for the far range (with kz smaller than ca. 0.06 rad/m) temporal decorrelation is the dominated reason. The simulation experiment further reveals the relationship between the influence of ground scattering contribution, temporal decorrelation and kz. Simulative results also show that ground scattering contribution has a greater influence for a large kz while temporal decorrelation generally brings overestimation, and has a greater impact for a small kz inversely. The proposed method simultaneously reduces the effects of ground scattering contribution and temporal decorrelation, and produces the highest accuracy with R2 reaching to ca. 0.6 and RMSE reducing to ca. 3.48 m, further demonstrating its validity for forest height inversion.

An entirely new approach based on remote sensing data to calculate the nitrogen nutrition index of winter wheat

The nitrogen nutrition index (NNI) is a reliable indicator for diagnosing crop nitrogen (N) status. However, there is currently no specific vegetation index for the NNI inversion across multiple growth periods. To overcome the limitations of the traditional direct NNI inversion method (NNIT1) of the vegetation index and traditional indirect NNI inversion method (NNIT2) by inverting intermediate variables including the aboveground dry biomass (AGB) and plant N concentration (PNC), this study proposed a new NNI remote sensing index (NNIRS). A remote-sensing-based critical N dilution curve (Nc_RS) was set up directly from two vegetation indices and then used to calculate NNIRS. Field data including AGB, PNC, and canopy hyperspectral data were collected over four growing seasons (2012–2013 (Exp.1), 2013–2014 (Exp. 2), 2014–2015 (Exp. 3), 2015–2016 (Exp. 4)) in Beijing, China. All experimental datasets were cross-validated to each of the NNI models (NNIT1, NNIT2 and NNIRS). The results showed that: (1) the NNIRS models were represented by the standardized leaf area index determining index (sLAIDI) and the red-edge chlorophyll index (CIrededge) in the form of NNIRS=CIrededge/(a×sLAIDIb), where “a” equals 2.06, 2.10, 2.08 and 2.02 and “b” equals 0.66, 0.73, 0.67 and 0.62 when the modeling set data came from Exp.1/2/4, Exp.1/2/3, Exp.1/3/4, and Exp.2/3/4, respectively; (2) the NNIRS models achieved better performance than the other two NNI revised methods, and the ranges of R2 and RMSE were 0.50–0.82 and 0.12–0.14, respectively; (3) when the remaining data were used for verification, the NNIRS models also showed good stability, with RMSE values of 0.09, 0.18, 0.13 and 0.10, respectively. Therefore, it is concluded that the NNIRS method is promising for the remote assessment of crop N status.

An adaptive stratified joint PP and PS AVA inversion using accurate Jacobian matrix

Amplitude variation with incidence angle (AVA) analysis is an essential tool for discriminating lithology in hydrocarbon reservoirs. Compared to traditional AVA inversion using only compressional wave (P-wave) information, joint AVA inversion using PP and PS seismic data provides better estimation of rock properties (e.g., density, P- and shear wave [S-wave] velocities). Currently, the most used AVA inversions depend on the approximations of the Zoeppritz equations (e.g., the Shuey and Aki-Richards approximations), which are not suitable for formations with strong contrast interfaces and seismic data with large incidence angles. Based on the previous derivation of the accurate Jacobian matrix, we have found that the sign of each partial derivative of reflection coefficient with respect to the P-, S-wave velocities and density changes across the interface represents a good indicator for the reflection interfaces. Accordingly, we adopt an adaptive stratified joint PP and PS AVA inversion using the accurate Jacobian matrix that can automatically obtain the layer information and can be further used as a constraint in the inversion of in-layer rock properties (density and P- and S-wave velocities). Due to the use of the exact Zoeppritz equations and accurate Jacobian matrix, this proposed inversion method is more accurate than traditional AVA inversion methods, has higher computational efficiency, and can be applied to seismic wide-angle reflection data or seismic data acquired for formations with strong contrast interfaces. The model study shows that this proposed inversion method works better than the classic Shuey and Aki-Richards approximations at estimating reflection interfaces and in-layer rock properties. It also works well in handling a part of the complex Marmousi 2 model and real seismic data.

Ant Colony Optimization Inversion Using the L1 Norm in Advanced Tunnel Detection

During the construction of the tunnel, there may be water-bearing anomalous structures such as fault fracture zone. In order to ensure the safety of the tunnel, it is necessary to carry out advanced tunnel detection. The traditional linear inversion method is highly dependent on the initial model in the tunnel resistivity inversion, which makes the inversion results falling into the local optimal optimum rather than the global one. Therefore, an inversion method for tunnel resistivity advanced detection based on ant colony algorithm is proposed in this paper. In order to improve the accuracy of tunnel advanced detection of deep anomalous bodies, an ant colony optimization (ACO) inversion is used by integrating depth weighting into the inversion function. At the same time, in view of the high efficiency and low cost of one-dimension inversion and the advantages of L1 norm in boundary characterization, a one-dimensional ant colony algorithm is adopted in this paper. In order to evaluate the performance of the algorithm, two sets of numerical simulations were carried out. Finally, the application of the actual tunnel water-bearing anomalous structure was carried out in a real example to evaluate the application effect, and it was verified by excavation exposure.

Deep learning audio magnetotellurics inversion using residual-based deep convolution neural network

In this study, we developed a novel 18-layers residual full convolutional neural network (18RFCN) for audio magnetotellurics (AMT) data inversion. Different from traditional inversion methods, the 18RFCN-based inversion algorithm is based on big-data training rather than prior-information assumptions, and the prediction is almost no time-consuming after network training. The problem of gradient disappearance was solved by adding shortcut connections when training deep neural networks, and we proposed two novel methods to quickly generate millions of samples for network training to improve the network generalization. We first tested on four synthetic data sets with Gaussian noises, which are entirely different from the data of sample sets, and the inversion results showed that the deep learning algorithm could accurately recover the underground conductivity structure almost no time-consuming compared to the Gauss Newton (GN) algorithm. We further tested the inversion algorithm on a field AMT data set acquired in Qinghai province, China. The inversion results of the deep learning algorithm agree well with the GN algorithm and the drilling data.

A fast robust Kalman filter for initial alignment of strapdown inertial navigation system

This paper investigates the non-Gaussian initial alignment for the strapdown inertial navigation system (SINS) with main focus on reducing the computational burden, meanwhile, ensuring the robustness of alignment filter. Conventional Kalman filter (KF) is the commonly used alignment filter, which brings a large computational burden during the filtering process, and assumes the statistic characteristics of the system noise is Gaussian in advance. When there are non-Gaussian characteristics in practice, it is difficult to improve the alignment filter speed without sacrificing the filter accuracy. In view of this problem, a fast robust Kalman filter (FRKF) is proposed. The approximated matrix inversion based on Neumann series expansion is used in the update step of the maximum correntropy criterion KF (MCC-KF). The computational complexity of the KF/MCC-KF/FRKF is analyzed, the approximated inversion method requires much less computational cost than the traditional inversion method, such that the FRKF is attractive from a complexity point of view. The alignment results show that the FRKF can obtain the state estimation accuracy as well as MCC-KF, while the computational burden is reduced remarkably.

Semi-Supervised Learning for Seismic Impedance Inversion Using Generative Adversarial Networks

Seismic impedance inversion is essential to characterize hydrocarbon reservoir and detect fluids in field of geophysics. However, it is nonlinear and ill-posed due to unknown seismic wavelet, observed data band limitation and noise, but it also requires a forward operator that characterizes physical relation between measured data and model parameters. Deep learning methods have been successfully applied to solve geophysical inversion problems recently. It can obtain results with higher resolution compared to traditional inversion methods, but its performance often not fully explored for the lack of adequate labeled data (i.e., well logs) in training process. To alleviate this problem, we propose a semi-supervised learning workflow based on generative adversarial network (GAN) for acoustic impedance inversion. The workflow contains three networks: a generator, a discriminator and a forward model. The training of the generator and discriminator are guided by well logs and constrained by unlabeled data via the forward model. The benchmark models Marmousi2, SEAM and a field data are used to demonstrate the performance of our method. Results show that impedance predicted by the presented method, due to making use of both labeled and unlabeled data, are better consistent with ground truth than that of conventional deep learning methods.

Inversion of magnetic data using deep neural networks

As a novel tool, deep learning is used to solve complex problems in the real world and has been successfully applied to invert for seismic and electromagnetic data. In this study, we propose to use deep neural networks (DNNs) to recover the distribution of the physical properties of buried magnetic orebodies from the surface and airborne magnetic anomaly data. This approach is based on data training instead of prior-knowledge assumptions used in traditional inversion methods. Once our generalized network is established, the computing time to predict new magnetic data using this method can be significantly reduced. By implementing the forward modeling of different types of synthetic physical property models, we obtained enough datasets to train a DNN model so that the network can establish a nonlinear mapping directly from magnetic anomaly data to physical properties. The pre-trained network can be used to estimate the distribution of magnetization intensity from new input magnetic data. Two DNN structures were employed to test the feasibility and generalization of the proposed method by implementing Experiments in serval two-dimensional (2D) synthetic examples. Compared with the conventional method, the predicted distribution of magnetization intensity obtained by using our method is more concentrated and has better resolution to determine the boundary of the magnetic body. In a field example of Galinge iron ore deposits in China, the magnetization distribution of concealed iron orebodies inverted by the proposed approach was in good agreement with that detected by borehole data. DNNs have good nonlinear inversion capability and exhibit some excellent merits of strong learning ability, wide coverage and strong adaptability, which is a considerable application prospect in geophysical exploration.

Modulus Inversion Layer by Layer of Different Asphalt Pavement Structures

In order to improve the accuracy of modulus inversion of the pavement structure layer, a layer‐by‐layer inversion method was proposed to be compared with the traditional inversion method by inverting the modulus of each structural layer of the inverted asphalt pavement and semirigid asphalt pavement. The results show that the influence of cushion modulus on the modulus of inverted subgrade and modulus of cement‐stabilized crushed stone is restricted by the cushion modulus and pavement structure characteristics, and the thicker cement‐stabilized crushed stone layer is beneficial for improving inverted modulus of subgrade; besides, for the inverted asphalt pavement, the modulus of the graded crushed stone transition layer has a significant influence on the modulus inversion of cement‐stabilized crushed stone. The modulus of the graded gravel transition layer inverted by these two methods is underestimated, the modulus of cement‐stabilized gravel is overestimated using the traditional inversion method, and the inversion result of the inverted asphalt pavement is more significantly affected by the inversion method than the semirigid base asphalt pavement. Moreover, the modulus of the pavement structural layer is determined by the material and structural characteristics, and its recommended empirical value or the value in the indoor test does not conform to the actual value of the site; by contrast, the inversion modulus obtained using the layer‐by‐layer inversion method is closer to the actual value, which can be used in the design of similar pavement structures to accumulate data for determining the material modulus or the pavement structure adjustment coefficient in the pavement structure.

A Valid Model for Prediction Tibiofemoral Force for Healthy People Based on Static Optimization

The musculoskeletal model plays an important role in the investigation of human lower limb diseases. Although different methods are used for musculoskeletal models, the prediction of the tibiofemoral and muscle forces still needs more improvements. This paper introduces a model for the lower limb; 3-DOF hip, 1-DOF knee, and 3-DOF ankle. The model estimates the tibiofemoral and muscle forces based on static optimization. The shank and the foot are considered as one element to avoid the high-cost computation of the traditional inverse dynamic method. The direction of the tibiofemoral force is estimated based on the analytical method to tune the weight factors of the predicted force. Two subjects (A and B) performed walking at 1m/s for about 5 gait cycles with recording the kinematics, ground reaction force (GRF), and foot center of pressure (CoP) and electromyographic (EMG) signals. A static optimization technique was performed to predict the tibiofemoral and 6 lower limb muscles forces based on the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> -Newton law in three-dimensional (3D). Muscles moment arms were verified by compared to measured data in the literature. The validity of the model was guaranteed by testing the model on a subject of total knee replacement (TKR) and compared the predicted tibiofemoral force with the measured one. The predicted tibiofemoral forces had the root mean square error (RMSE) of 0.19, 0.56, and 0.43 BW for TKR subject, subject A and subject B, respectively. The predicted muscles force was validated by comparing it to EMG signals. The analysis and the validation of the model showed that it could be used for different activities like walking and running and assessment rehabilitation devices such as knee brace.