Application of Tikhonov regularization method to wind retrieval from scatterometer data Ⅱ: cyclone wind retrieval with consideration of rain

According to the conclusion of the simulation experiments in paper I, the Tikhonov regularization method is applied to cyclone wind retrieval with a rain-effect-considering geophysical model function （called CMF＋Rain）. The CMF＋Rain model which is based on the NASA scatterometer-2 （NSCAT2） GMF is presented to compensate for the effects of rain on cyclone wind retrieval. With the multiple solution scheme （MSS）, the noise of wind retrieval is effectively suppressed, but the influence of the background increases. It will cause a large wind direction error in ambiguity removal when the background error is large. However, this can be mitigated by the new ambiguity removal method of Tikhonov regularization as proved in the simulation experiments. A case study on an extratropical cyclone of hurricane observed with SeaWinds at 25-km resolution shows that the retrieved wind speed for areas with rain is in better agreement with that derived from the best track analysis for the GMF＋Rain model, but the wind direction obtained with the two-dimensional variational （2DVAR） ambiguity removal is incorrect. The new method of Tikhonov regularization effectively improves the performance of wind direction ambiguity removal through choosing appropriate regularization parameters and the retrieved wind speed is almost the same as that obtained from the 2DVAR.

Bathymetry inversion using the modifi ed gravitygeologic method:application of the rectangular prism model and Tikhonov regularization

Bathymetry data are usually obtained via single-beam or multibeam sounding;however,these methods exhibit low efficiency and coverage and are dependent on various parameters,including the condition of the vessel and sea state.To overcome these limitations,we propose a method for marine bathymetry inversion based on the satellite altimetry gravity anomaly data as a modification of the gravity-geologic method(GGM),which is a conventional terrain inversion method based on gravity data.In accordance with its principle,the modified method adopts a rectangular prism model for modeling the short-wavelength gravity anomaly and the Tikhonov regularization method to integrate the geophysical constraints,including the a priori water depth data and characteristics of the sea bottom relief.The a priori water depth data can be obtained based on the measurement data obtained from a ship,borehole information,etc.,and the existing bathymetry/terrain model can be considered as the initial model.Marquardt’s method is used during the inversion process,and the regularization parameter can be adaptively determined.The model test and application to the West Philippine Basin indicate the feasibility and eff ectiveness of the proposed method.The results indicate the capability of the proposed method to improve the overall accuracy of the water depth data.Then,the proposed method can be used to conduct a preliminary study of the ocean depths.Additionally,the results show that in the improved GGM,the density diff erence parameter has lost its original physical meaning,and it will not have a great impact on the inversion process.Based on the boundedness of the study area,the inversion result may exhibit a lower confi dence level near the margin than that near the center.Furthermore,the modifi ed GGM is time-and memory-intensive when compared with the conventional GGM.

Some studies on the Tikhonov regularization method with additional assumptions for noise data

In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were supposed to satisfy some additional monotonic condition. Moreover, with the assumption that the singular values of operator have power form, the improved convergence rates of the regularized solution were worked out.

Representing surface wind stress response to mesoscale SST perturbations in western coast of South America using Tikhonov regularization method

Interaction between mesoscale perturbations of sea surface temperature(SSTmeso)and wind stress(WSmeso)has great influences on the ocean upwelling system and turbulent mixing in the atmospheric boundary layer.Using daily Quik-SCAT wind speed data and AMSR-E SST data,SSTmeso and WSmeso fields in the western coast of South America are extracted by using a locally weighted regression method(LOESS).The spatial patterns of SSTmeso and WSmeso indicate strong mesoscale SST-wind stress coupling in the region.The coupling coefficient between SSTmeso and WSmeso is about 0.0095 N/(m^2·℃)in winter and 0.0082 N/(m^2·℃)in summer.Based on mesoscale coupling relationships,the mesoscale perturbations of wind stress divergence(Div(WSmeso))and curl(Curl(WSmeso))can be obtained from the SST gradient perturbations,which can be further used to derive wind stress vector perturbations using the Tikhonov regularization method.The computational examples are presented in the western coast of South America and the patterns of the reconstructed WS meso are highly consistent with SSTmeso,but the amplitude can be underestimated significantly.By matching the spatially averaged maximum standard deviations of reconstructed WSmeso magnitude and observations,a reasonable magnitude of WSmeso can be obtained when a rescaling factor of 2.2 is used.As current ocean models forced by prescribed wind cannot adequately capture the mesoscale wind stress response,the empirical wind stress perturbation model developed in this study can be used to take into account the feedback effects of the mesoscale wind stress-SST coupling in ocean modeling.Further applications are discussed for taking into account the feedback effects of the mesoscale coupling in largescale climate models and the uncoupled ocean models.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

Application of Tikhonov Regularization Technique to the Kinetic Data of an Autocatalytic Reaction: Pyrolysis of N-Eicosane

A new technique based on Tikhonov regularization, for converting time-concentration data into concentration-reaction rate data, was applied to a novel pyrolysis investigation carried out by Susu and Kunugi [1]. The reaction which involves the thermal decomposition of n-eicosane using synthesis gas for K2CO3-catalyzed shift reaction was reported to be autocatalytic. This result was confirmed by applying Tikhonov regularization to the experimental data (conversion vs. time) presented by Susu and Kunugi [1]. Due to the ill-posed nature of the problem of obtaining reaction rates from experimental data, conventional methods will lead to noise amplification of the experimental data. Hence, Tikhonov regularization is preferably employed because it is entirely independent of reaction rate model and it also manages to keep noise amplification under control, thus, leading to more reliable results. This is shown by the agreement of the kinetic parameters obtained using the resulting conversion-reaction rate profile, with the Ostwald-type process for autocatalysis suggested by Susu and Kunugi [1].

VIBRATING VELOCITY RECONSTRUC-TION USING IBEM AND TIKHONOV REGULARIZATION

The inverse problem to determine the vibrating velocity from known exteriorfield measurement pressure, involves the solution of a discrete ill-posed problem. To facilitate thecomputation of a meaningful approximate solution possible, the indirect boundary element method(IBEM) code for investigating vibration velocity reconstruction and Tikhonov regularization methodby means of singular value decomposition (SVD) are used. The amount of regularization is determinedby a regularization parameter. Its optimal value is given by the L-curve approach. Numerical resultsindicate the reconstructed normal surface velocity is a good approximation to the real source.

Convergence analysis on Browder-Tikhonov regularization for second-order evolution hemivariational inequality

This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality （SOEHVI） with non-coercive operators. With duality mapping, the regularized formulations and a derived first-order evolution hemivariational inequality （FOEHVI） for the problem considered are presented. By applying the Browder-Tikhonov regularization method to the derived FOEHVI, a sequence of regularized solutions to the regularized SOEHVI is constructed, and the strong convergence of the whole sequence of regularized solutions to a solution to the problem is proved.

利用GPS垂直位移反演区域陆地水储量变化的TSVD-Tikhonov正则化方法

利用GPS垂直位移反演区域陆地水储量变化(TWSC)属于典型的病态问题,其关键是如何进行稳定求解并提高反演结果的精度和可靠性.本文引入TSVD-Tikhonov组合正则化方法对利用GPS垂直位移反演区域TWSC的病态问题进行求解,并以四川省TWSC反演为例进行分析与验证.首先,通过数值模拟对TSVD、Tikhonov和TSVD-Tikhonov正则化方法采用不同正则化参数选取策略(RMSE最小准则、GCV法和L-curve法)进行反演,结果显示基于TSVD-Tikhonov正则化反演的TWSC比单独使用TSVD或Tikhonov正则化反演结果的精度和可靠性更高,这三种正则化方法反演2005年1月至12月的TWSC差值的平均STD分别为14.97 mm、7.03 mm和5.04 mm.其次,利用中国地壳运动观测网络(CMONOC)的72个GPS测站的垂直位移数据,基于TSVD-Tikhonov正则化反演了四川省2010年12月至2021年2月的TWSC时间序列,结果表明GPS反演的TWSC与GRACE/GFO Mascon模型(JPL、CSR和GSFC)的空间分布特征及季节性变化符合较好,但其TWSC信号的振幅比GRACE/GFO Mascon模型更强.最后,采用广义三角帽方法(GTCH)融合不同类型的降水、蒸散发和径流数据,并根据水量平衡方程计算的dTWSC/dt序列(PER-dS/dt)对GPS反演的dTWSC/dt序列(GPS-dS/dt)和GRACE/GFO Mascon模型融合的dTWSC/dt序列(GRACE/GFO-dS/dt)进行验证,结果表明这三类dTWSC/dt序列的季节性变化符合较好,平滑后GPS-dS/dt和GRACE/GFO-dS/dt序列与PER-dS/dt序列的相关系数分别为0.78和0.87,但GPS相比GRACE/GFO对降水变化的响应更为敏感.本文研究证明了TSVD-Tikhonov组合正则化方法能够提高GPS垂直位移反演区域TWSC的精度和可靠性,同时也表明GPS观测数据对局部水质量负荷变化更为敏感,可作为GRACE/GFO反演区域TWSC的有益补充.

基于GRACE星间重力位差的Slepian局部地表质量变化反演法

根据Slepian基函数在频域和空域的局部集中特性,建立了利用GRACE星间重力位差(GPD)估计局部地表质量变化的Slepian基函数反演模型及病态问题求解算法,并以亚马逊流域的陆地水储量变化(TWSC)反演为例,评估了反演方法的精度和有效性.首先,通过闭环数值模拟比较了Slepian基函数法(GPD SBF)、基于GPD的mascon方法(GPD Mascon)和球谐系数法(GPD SH)反演2005年亚马逊流域TWSC的性能,结果表明:Slepian基函数法通过限定信号的频域和空域范围,降低了反演过程中向下延拓带来的不确定性,有效削弱了法方程求解的病态性,其解算结果的稳定性明显优于GPD Mascon;GPD SBF的反演精度及可靠性总体上优于GPD Mascon和GPD SH,并且能够更好地恢复边缘区域的信号和减小泄漏误差的影响.其次,利用实测的GRACE GPD数据反演了2004—2015年亚马逊流域的TWSC时间序列,结果显示:GPDSBF相比GPDMascon反演的TWSC与官方mascon模型(CSR、JPL和GSFC RL06 mascon)更为一致,并且GPD SBF反演结果呈现出更多的空间细节和更好的信噪比.最后,利用不同GRACE反演结果计算的(TWSC的一阶导数)和水文气象数据(GPCP降水、ERA5蒸散发)由水量平衡方程估计了亚马逊流域Obidos水文站的月平均径流量,并采用测站的实测径流量进行检验,结果表明:GPD SBF和GPD Mascon估计的径流量与实测值扣除季节性信号后的差值STD分别为12.35 mm和14.54 mm,相关系数分别为0.71和0.69,并且GPD SBF和各种官方mascon模型估计的径流量与实测径流量更为接近.本文研究证明Slepian基函数法可削弱病态问题求解对正则化约束的依赖,其比传统的GPD Mascon解算精度和可靠性更高,为反演高精度、高分辨率的局部地表质量变化提供了一种新的解决方案.

利用点电流源和Tikhonov正则化的潜艇稳恒电场反演方法

为了评估潜艇水下腐蚀相关稳恒电场分布特性,从潜艇水下腐蚀相关稳恒电场产生机理出发,基于等效点电流源建立潜艇水下腐蚀相关稳恒电场正演模型,并利用Tikhonov正则化根据已知电场数据求解等效点电流源电流强度,对潜艇周围海水空间的腐蚀相关稳恒电场进行推算。将某型潜艇的COMSOL软件仿真结果作为模拟试验数据,对所提方法有效性进行验证。结果表明:由水深38 m电场值向水深42.5 m和水深33.5 m进行反演时,相对均方根误差、最大值相对误差、峰峰值相对误差均不超过0.06;由较近平面向较远平面进行反演时,即使推算深度达到45 m,相对均方根误差仍然在0.21以内;噪声标准差为实际电场最大值的0.1倍时,反演误差仍然小于0.1。该算法抗噪声能力强,精度较高,能较好地用于工程实践。

Tikhonov正则化方法在航空γ测量数据处理中的应用

在航空γ测量中,当地形平坦且飞行高度变化不大时,采用传统的高度修正方法可得到良好的结果。实际测量中,地形崎岖不平、飞行高度突变等常见现象会导致高度修正后的结果仍存在偏差甚至错误。基于条带模型构设崎岖地形条件下的反演方程组,应用Tikhonov正则化方法求解可得到地面放射性核素的面活度浓度。Tikhonov正则化方法的应用结果表明:随机噪声较小时,采用广义交叉验证(generalized cross validation,GCV)方法选取正则化参数得到的反演结果较好;随机噪声较大时,采用L曲线方法选取正则化参数得到的反演结果较好。与传统高度修正方法的计算结果相比该方法好,且适用于飞行高度变化较大的情形。

Numerical estimation of choice of the regularization parameter for NMR T2 inversion

Nuclear Magnetic inversion is the basis of NMR Resonance （NMR） T2 logging interpretation. The regularization parameter selection of the penalty term directly influences the NMR T2 inversion result. We implemented both norm smoothing and curvature smoothing methods for NMR T2 inversion, and compared the inversion results with respect to the optimal regular- ization parameters （（Xopt） which were selected by the dis- crepancy principle （DP）, generalized cross-validation （GCV）, S-curve, L-curve, and the slope of L-curve methods, respectively. The numerical results indicate that the DP method can lead to an oscillating or oversmoothed solution which is caused by an inaccurately estimated noise level. The （Xopt selected by the L-curve method is occa- sionally small or large which causes an undersmoothed or oversmoothed T2 distribution. The inversion results from GCV, S-curve and the slope of L-curve methods show satisfying inversion results. The slope of the L-curve method with less computation is more suitable for NMR T2 inversion. The inverted T2 distribution from norm smoothing is better than that from curvature smoothing when the noise level is high.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

用Tikhonov正则化方法同时反演对流扩散方程的对流速度和源函数

在给定两个附加观测数据的条件下,本文基于Tikhonov正则化方法研究了对流扩散方程的对流速度和源函数的同时反演问题.鉴于原问题是一个初始值非零的对流扩散方程,本文通过将初始值转化为源项得到了一个组合源项,首先将原问题转化为一个具有齐次条件的对流扩散问题.由于所得问题是不适定的,本文进而利用Tikhonov正则化方法构建了相应的极小化目标泛函,得到了问题最优解的存在性和应满足的必要条件.最后,对终端时刻较小的特殊情形,本文证明了最优解的唯一性和稳定性.

New Regularization Algorithms for Solving the Deconvolution Problem in Well Test Data Interpretation

Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.

Efficient multiuser detector based on box-constrained dichotomous coordinate descent and regularization

The presented iterative multiuser detection technique was based on joint deregularized and box-constrained solution to quadratic optimization with iterations similar to that used in the nonstationary Tikhonov iterated algorithm.The deregularization maximized the energy of the solution,which was opposite to the Tikhonov regularization where the energy was minimized.However,combined with box-constraints,the deregularization forced the solution to be close to the binary set.It further exploited the box-constrained dichotomous coordinate descent algorithm and adapted it to the nonstationary iterative Tikhonov regularization to present an efficient detector.As a result,the worst-case and average complexity are reduced down as K2.8 and K2.5 floating point operation per second,respectively.The development improves the "efficient frontier" in multiuser detection,which is illustrated by simulation results.In addition,most operations in the detector are additions and bit-shifts.This makes the proposed technique attractive for fixed-point hardware implementation.

基于正则化与恒星日滤波的BDS多路径误差削减方法

在重构的载波相位单差残差基础上,利用分段思想估计北斗卫星的多路径重复时间,分别采用正则化方法和经典小波滤波方法提取载波相位单差残差的多路径信号,得到“干净”的单差残差序列。实验结果表明,利用Tikhonov正则化方法正确提取多路径信号是可行的,多路径信号比原始测量残差更具平滑性,并进一步优化了正则化参数的估计方法。利用优化后的Tikhonov正则化方法与恒星日滤波后,载波相位单差残差平均改进40.5%;坐标残差E、N、U方向分别改进24.8%、26.3%和42.7%。无论是观测值域还是坐标域,优化后的Tikhonov正则化方法较传统的小波滤波方法更具优越性。

Cosmic Dark Energy from ‘t Hooft’s Dimensional Regularization and Witten’s Topological Quantum Field Pure Gravity

We utilize two different theories to prove that cosmic dark energy density is the complimentary Legendre transformation of ordinary energy and vice versa as given by E(dark) = mc2 (21/22) and E(ordinary) = mc2/22. The first theory used is based on G ‘t Hooft’s remarkably simple renormalization procedure in which a neat mathematical maneuver is introduced via the dimensionality of our four dimensional spacetime. Thus, ‘t Hooft used instead of D = 4 and then took at the end of an intricate and subtle computation the limit to obtain the result while avoiding various problems including the pole singularity at D = 4. Here and in contradistinction to the classical form of dimensional and renormalization we set and do not take the limit where and is the theoretically and experimentally well established Hardy’s generic quantum entanglement. At the end we see that the dark energy density is simply the ratio of and the smooth disentangled D = 4, i.e. (dark) = (4 -k)/4 = 3.8196011/4 = 0.9549150275. Consequently where we have ignored the fine structure details by rounding 21 + k to 21 and 22 + k to 22 in a manner not that much different from of the original form of dimensional regularization theory. The result is subsequently validated by another equally ingenious approach due mainly to E. Witten and his school of topological quantum field theory. We notice that in that theory the local degrees of freedom are zero. Therefore, we are dealing essentially with pure gravity where are the degrees of freedom and is the corresponding dimension. The results and the conclusion of the paper are summarized in Figure 1-3, Table 1 and Flow Chart 1.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.