# A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) at an error level of 2 cm in terms of equivalent water height (EWH) at monthly intervals [19].

## 2. Data and Methods

#### 2.1. GRACE Data

_{20}and C

_{30}terms, which are poorly observed by GRACE and GRACE-FO, in particular, with only a single accelerometer, are replaced with SLR observations by the technical note 14 (TN-14) document recommended by Loomis et al. [25].

#### 2.2. Mascon Modeling

_{max}is 60; ${k}_{l}^{\prime}$ (provided by Wahr et al. [11]) is the load Love number of degree l; P

_{lm}stands for the fully normalized associated Legendre functions; $\mathsf{\Delta}{\mathrm{C}}_{lm}$ and $\mathsf{\Delta}{\mathrm{S}}_{lm}$ are the Stokes coefficients after the mean gravity field is removed.

**y**is an n-vector of pseudo observations of the radial gravitational disturbance calculated by spherical harmonic coefficients;

**A**is an n × t design matrix (n > t) and denotes an overdetermined system of the equation;

**x**is a t-vector of unknown ground mass-points to be estimated (i.e., ${\delta m}_{j}$);

**e**denotes the n-vector of random errors with zero mean and variance of unit weight ${\sigma}_{0}^{2}$. Via the law of error propagation, the weight matrix

**P**is computed with

**P**= (

**BDB**

^{T})

^{−1}, in which

**D**is the covariance matrix from the CSR release 06 products;

**B**is the coefficient matrix of projecting the spherical harmonics to the pseudo observation vector with its ith row elements corresponding to spherical harmonics $\mathsf{\Delta}{\mathrm{C}}_{lm}$ and$\text{}\mathsf{\Delta}{\mathrm{S}}_{lm}\text{}$which are written as [28],

#### 2.3. Combined Use of TSVD and Tikhonov Regularization

**x**, we applied the least-squares adjustment to minimize the square sum of

**e**. The design matrix

**A**(n × t) can be expressed by singular value decomposition (SVD) as

**U**= [

**u**

_{1},

**u**

_{2}, …,

**u**

_{n}] are the left singular vectors of

**A**; the diagonal matrix

**S**= diag[s

_{1}, s

_{2}, …, s

_{t}] is the singular value of

**A**; the columns of

**V**= [

**v**

_{1},

**v**

_{2}, …,

**v**

_{t}] are the right singular vectors of

**A**. The least-squares solution ${\mathsf{x}}_{\mathrm{LS}}$ is,

**A**increases as well and small perturbations in the observations can cause significant perturbations in the solution. Due to the sampling and geometry of pseudo observation, we found a condition number for the design matrix A of 6.9·10

^{17}(with the geodesic grid of 1° × 1°), indicating an ill-conditioned normal equation

**N**=

**A**

^{T}

**PA**. There are two regularization methods presented: TSVD and Tikhonov regularization. By eliminating small singular values, the TSVD solution

**x**

_{k}is in the form of [31],

**A**remains large; however, a small k value leads to losing a large part of the signals. Tikhonov regularization is commonly applied to stabilize ill-posed problems, it is well-known that the solution minimizes the cost function

**R**is the regularization matrix. The solution to Equation (9) is expressed as,

**R**is chosen as an identity matrix

**I**and the design matrix

**A**is truncated to the first k singular values, the solution ${\mathsf{x}}_{\mu k}\text{}$is,

^{−29}is selected by the minimum traced MSE.

^{−33}drop very slowly, and the correspondent eigenvectors form linearly dependent parameters. Truncating them to about a 2° resolution is better than direct parameterizing with 2° grids, since the combinations with larger eigenvalues are left. However, if we truncate more terms to get a stable solution, we may lose the signals corresponding to small eigenvalues. The terms for the eigenvalues larger than 10

^{−33}are corresponding to the spatial resolution of GRACE data; Tikhonov regularization is therefore used to derive a stable solution.

#### 2.4. Leakage Correction

## 3. Results and Discussion

#### 3.1. MSE Roots

#### 3.2. Total Mass Variations

_{2}alias that would remove an error due to incorrect modeling of the S2-tide, as follows,

_{1}, A

_{2}, A

_{3}, ${\theta}_{1}$, ${\theta}_{2}$, ${\theta}_{3}$ stand for the annual, semi-annual, and 161-day amplitudes and phases; t is the time tag in years;

**b**is bias parameter; $\mathsf{\Delta}$ is the residuals. We realize that Equation (14) represents a mathematical model that enables an unbiased trend estimate, rather than a tool for assessing solution errors in the presence of real geophysical signals which do not necessarily follow this model. However, we argue that (1) it is indeed our primary aim to derive the mass trends, and (2) several modeling studies have confirmed that a trend plus (semi-) annual model describes snow accumulation, groundwater, and surface water change in the TP region quite well [37,38].

#### 3.3. Mass Variations Distribution

## 4. Summary

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Glacial isostatic adjustment (GIA) signals of the Tibetan Plateau (TP) in equivalent water height (EWH, mm/year) at 1° resolution.

**Figure 2.**(

**a**) Eigenvalues of the normal equation in January 2004; (

**b**) condition numbers for 172 months; (

**c**) traced mean squared error (MSE) of the combined method varies with regularization parameter in January 2004.

**Figure 5.**Linear trend of the TP in EWH (cm/year): (

**a**) with combined TSVD and Tikhonov regularization; (

**b**) with Tikhonov regularization; (

**c**) with P4M6 + 400 km Gaussian filtering; (

**d**) with TSVD; (

**e**) difference between (

**a**) and (

**c**); (

**f**) difference between (

**b**) and (

**a**).

**Figure 6.**Root mean squared error (RMSE) of mascons in the TP: (

**a**) with combined TSVD and Tikhonov regularization; (

**b**) with Tikhonov regularization; (

**c**) with P4M6 + 400 km Gaussian filtering; (

**d**) with TSVD.

MSE Roots | Maximum | Minimum | Mean |
---|---|---|---|

Tikhonov + TSVD | 6.74 | 1.29 | 3.08 |

TSVD | 8.31 | 1.70 | 4.23 |

Tikhonov | 7.56 | 1.53 | 3.53 |

Method | Trend (Gt/year) | Annual | RMSE (cm) | RMS Ratio | |
---|---|---|---|---|---|

Amplitude(cm) | Phase (°) | ||||

Tikhonov + TSVD | −5.6 ± 4.2 | 2.8 ± 0.5 | 226.8 ± 14.4 | 1.9 | 1.21 |

TSVD | −8.9 ± 5.9 | 2.2 ± 1.9 | 262.6 ± 34.3 | 1.6 | 0.45 |

Tikhonov | −6.8 ± 5.2 | 2.3 ± 0.5 | 220.2 ± 26.4 | 2.2 | 0.78 |

P4M6 + 400 km | −8.6 ± 5.8 | 2.3 ± 0.6 | 223.1 ± 23.5 | 1.8 | 1.16 |

Method | Time Intervals | GRACE Data | Trend (Gt/year) | Trend of Combined Method (Gt/year) |
---|---|---|---|---|

TSVD + Tikhonov | Apr 2002–Apr 2019 | GFZ Release 06 | −5.9 ± 4.3 | −5.6 ± 4.2 |

Jacob et al. [8] | Jan 2003–Dec 2010 | CSR Release 04 | −4 ± 20 | −2.3 ± 5.7 |

Yi and Sun [35] | Jan 2003–Dec 2012 | CSR Release 05 | −7.8 ± 5.7 | −4.6 ± 5.3 |

Zou et al. [37] | Aug 2002–Dec 2016 | CSR Release 05 | −6.2 ± 1.7 | −7.6 ± 3.8 |

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**MDPI and ACS Style**

Chen, T.; Kusche, J.; Shen, Y.; Chen, Q. A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau. *Remote Sens.* **2020**, *12*, 2045.
https://doi.org/10.3390/rs12122045

**AMA Style**

Chen T, Kusche J, Shen Y, Chen Q. A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau. *Remote Sensing*. 2020; 12(12):2045.
https://doi.org/10.3390/rs12122045

**Chicago/Turabian Style**

Chen, Tianyi, Jürgen Kusche, Yunzhong Shen, and Qiujie Chen. 2020. "A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau" *Remote Sensing* 12, no. 12: 2045.
https://doi.org/10.3390/rs12122045