_{2}Spectrum by the TSVD Based Linearized Bregman Iteration

_{2}Spectrum by the TSVD Based Linearized Bregman Iteration

_{2}Spectrum by the TSVD Based Linearized Bregman Iteration

The low-field nuclear magnetic resonance (NMR) technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields. However, the speed and accuracy of the existing numerical inversion methods are still challenging due to the ill-posed nature of the first kind Fredholm integral equation and the contamination of the noises. This paper proposes a novel inversion algorithm to accelerate the convergence and enhance the precision using empirical truncated singular value decompositions (TSVD) and the linearized Bregman iteration. The L1 penalty term is applied to construct the objective function, and then the linearized Bregman iteration is utilized to obtain fast convergence. To reduce the complexity of the computation, empirical TSVD is proposed to compress the kernel matrix and determine the appropriate truncated position. This novel inversion method is validated using numerical simulations. The results indicate that the proposed novel method is significantly efficient and can achieve quick and effective data solutions with low signal-to-noise ratios.

The time domain signal determined by the low field nuclear magnetic resonance (NMR) can be used to detect many features of the porous rock, such as the pore size distribution, pore connectivity, wettability, viscosity, as well as fluid saturation [

The Bregman iteration is considered as an effective method for solving constrained optimization problems. It has been widely applied in many fields such as seismic prospecting, image reconstruction, as well as noise reduction [

Based on the basic principle NMR theory, the longitudinal relaxation time (T_{1}) and the transversal relaxation time (T_{2}) are used to depict the polarizing and decaying behaviors during the nuclear precession process, which are given by [_{1} and T_{2}, determined by the intrinsic properties of the fluids such as viscosity and chemical compositions; _{1} and T_{2}, controlled by the properties of fluid-rock interface such as pore size and surface relaxivity; _{2}, affected by the diffusion coefficient, temperature and pressure; _{1} and T_{2}, respectively [

T_{2} is the most frequently measured parameter in conventional low-field NMR experiments and logging, which are obtained by Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. Therefore, only the inversion problem of T_{2} is examined in this study. The discrete relationship between the magnetization intensity and T_{2} satisfies the following equation:
_{2}; _{2}, which is always predefined as the exponent of 2, such as 32, 64, and 128;

In the conventional inversion algorithms, _{2} with the size of

Since

It is often the case that most elements in the spectrum are zeros, so L1 penalty term is added into the function to enforce the sparsity of the spectrum. Therefore, the cost function is expressed as [

The Bregman iteration is very popular in many optimization problems due to its very nice convergence properties, including monotonic decrease in the residual term, convergence to the original signal, and convergence in terms of Bregman distance to the original signal with noisy data [

Assuming

The Bregman distance has several nice properties that make it an efficient tool for solving L1 regularization problems. Using the Bregman function, the optimization problem is transformed as:

Consequently, the key steps of the Bregman iteration are expressed as follows:

The main function of

Then the problem is transformed to find the optimal values to satisfy

The first order Taylor expansion of

By adding the penalty term of

Substituting

Combining

Hence, the implementation of the linearized Bregman algorithm is as follows: (1) Initialize the values of

The above section gives the full part of the linearized Bregman iteration. However, direct application of this algorithm is impossible since the ill-conditioned kernel matrix

Singular values of

It should be noted that singular values decay quickly to zero, and small singular values are often considered harmful components since they will result in the large error of the solutions. Conventional methods are to set a threshold and drop values smaller than the threshold such as the generalized cross validation (GCV),

Through many times of simulations trials, an empirical equation is established to predict the proper truncated position and can be expressed by:

Therefore, the kernel matrix is expressed as [

Therefore, the iteration function for

The space complexity, time complexity, and computational complexity for

The simulation tests are conducted to verify the proposed novel inversion method using two typical distributions of the T_{2} spectrum, including the unimodal spectrum, the bimodal spectrum. Moreover, numerical data with different SNRs will be used to investigate the noise tolerance of this method. In our simulation, the noise is added using the in-built ‘awgn’ function of the Matlab software. It is typical additive Gaussian white noise. The SNR is specified with dB. All the simulations are conducted on a computer labeled the ‘OptiPlex 7050’ with the Intel(R) Core(TM) i7-7700 CPU @ 3.60 GHz. In the simulation, the echo spacing (T_{E}) and the number of echoes are set as 0.2 ms and 8000, respectively. After many runs of simulations,

The Gaussian’s function is used to establish the forward unimodal distributed T_{2} spectrum with a peak of 10 ms and the decaying signals with SNR of 100, 50, 30, 20, 10, and 5, respectively. It should be noted that for simplicity, the normalized amplitude less than 0.003 is dropped and the spectrum is then renormalized. The simulated echo trains are shown in

Considering the most common case, the bimodally distributed spectrum is constructed. The positions of the two peaks are 10 ms and 60. Similar to the previous case,

Based on the above simulation results, it is concluded that the proposed novel algorithm works very well for the inversion of the T_{2} spectrum. The proposed algorithm can achieve fast convergence and its inversion results are only slightly influenced by the noise when the SNR is higher than 20.

A novel, efficient and accurate algorithm is developed for the inversion of the NMR T_{2} spectrum. In this method, the empirical TSVD and linearized Bregman iteration are used to enhance the speed and accuracy of the numerical inversion of the NMR T_{2} spectrum for the first time. The results of the numerical inversion study show that the linearized Bregman iteration can obtain quick and effective performance in the solution of ill-posed and over-determined problems. Moreover, this novel method can work well for data with the SNR higher than 20.

However, much research should be conducted to generalize this method to field applications and experiments. The effects of the iteration parameters and the acquisition parameters on inversion instability are also very important and can be investigated in the future.

The authors are grateful for the financial support by the

The authors declare that they have no conflicts of interest to report regarding the present study.

_{1}–T

_{2}inversion with double objective functions