Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series

Abstract

Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths $N=2048$, 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.

Appendices

Discrete wavelet transform (DWT)

This section recalls the basics of discrete wavelet transform (DWT). For further details about wavelet theory, Percival and Walden (2000) can be referred. Let $\varvec{x} \in \mathbb {R}^{N \times 1}$ be a signal with a sample size given by $N=2^J$. With $\varvec{V}_0\equiv \varvec{x}$, the jth level DWT wavelet and scaling coefficients are $N_j=N/2^j$ dimensional vectors $\varvec{W}_j$ and $\varvec{V}_j$ whose elements are given by:

$$\begin{aligned} {W}_{j,n}&=\sum _ {l=0} ^ {L-1} {h}_{l}\varvec{V}_{j-1 , (2n+1-l) ~\mathrm {mod}~ N_{j-1}}; \nonumber \\&\quad n=0,\ldots ,N_j-1\nonumber \\ {V}_{j,n}&=\sum _ {l=0} ^ {L -1} {g}_{l}\varvec{V}_{j-1 , (2n+1-l)~ \mathrm {mod}~ N_{j-1}}, \end{aligned}$$

(24)

where $\mathrm {mod}$ denotes modulo operator. The vectors $\varvec{h}=[ h_0 ~h_1~ \ldots ~h_{L-1}]^\mathrm {T} \in \mathbb {R}^{L \times 1}$ and $\varvec{g}=[ g_0 ~g_1~ \ldots ~g_{L-1}]^\mathrm {T}\in \mathbb {R}^{L \times 1}$ are wavelet (high-pass) and scaling (low-pass) filters at the unit scale, respectively. If, however, the sample size N be an integer multiple of $2^{J_0 }$ ($J_0 \le J$), the algorithm can be stopped after $J_0$ repetitions which is known as partial DWT to level $J_0$ . These coefficients also may be obtained directly from the signal $\varvec{x}$ via:

$$\begin{aligned} {W}_{j,n}&=\sum _ {l=0} ^ {L_j -1} {h}_{j,l}x_{(2^j (n+1)-1-l)~\mathrm {mod}~N},\nonumber \\ {V}_{j,n}&=\sum _ {l=0} ^ {L_j -1} {g}_{j,l}x_{(2^j (n+1)-1-l)~\mathrm {mod}~N}, \end{aligned}$$

(25)

where $\varvec{h}_j=[ h_{j,0} ~h_{j,1}~ \ldots ~h_{j,L_j-1}]^\mathrm {T}\in \mathbb {R}^{L_j \times 1}$ and $\varvec{g}_j=[ g_{j,0} ~g_{j,1}~ \ldots ~g_{j,L_j-1}]^\mathrm {T}\in \mathbb {R}^{L_j \times 1}$ are wavelet and scaling filters, relating $\varvec{W}_{j}$ and $\varvec{V}_{j}$ to $\varvec{x}$, respectively, and $L_j=(2^j -1)(L-1)+1$. jth level wavelet filter $\varvec{h}_j$ is formed by convolving together the following j filters:

$$\begin{aligned} \text {filter~1}&: g_0,g_1,\ldots , g_{L-2},g_{L-1}\nonumber \\ \text {filter~2}&: g_0,0,g_1,0,\ldots , g_{L-2},0,g_{L-1}\nonumber \\&\qquad \qquad \qquad \qquad \vdots \nonumber \\ \text {filter~j-1}&: g_0,\underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},g_1,\underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},\ldots , g_{L-2}, \nonumber \\&\quad \underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},g_{L-1}\nonumber \\ \text {filter~j}&: h_0,\underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},h_1,\underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},\ldots , h_{L-2}, \nonumber \\&\quad \underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},h_{L-1}. \end{aligned}$$

(26)

In a similar manner, jth level scaling filter $\varvec{g}_j$ is formed by convolving together the following j filters:

$$\begin{aligned} \text {filter~1}&: g_0,g_1,\ldots , g_{L-2},g_{L-1}\nonumber \\ \text {filter~2}&: g_0,0,g_1,0,\ldots , g_{L-2},0,g_{L-1}\nonumber \\&\qquad \qquad \qquad \qquad \vdots \nonumber \\ \text {filter~j-1}&: g_0,\underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},g_1,\underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},\ldots , g_{L-2}, \nonumber \\&\quad \underbrace{0,\ldots ,0}_{2^{j-2}-1~\text {zeros}},g_{L-1}\nonumber \\ \text {filter~j}&: g_0,\underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},g_1,\underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},\ldots , g_{L-2},\nonumber \\&\quad \underbrace{0,\ldots ,0}_{2^{j-1}-1~\text {zeros}},g_{L-1}. \end{aligned}$$

(27)

In matrix notation, Eq.  (25) could be rewritten as:

$$\begin{aligned} {\varvec{W}}_j={\mathcal {W}} _j \varvec{x};\quad {\varvec{V}}_j={\mathcal {V}} _j \varvec{x}, \end{aligned}$$

(28)

where the rows of ${\mathcal {W}} _j\in \mathbb {R}^{N_j \times N}$ and ${\mathcal {V}} _j\in \mathbb {R}^{N_j \times N}$ contain circularly shifted versions of $\varvec{h}^\circ _j$ and $\varvec{g}^{\circ }_j$, respectively, which by definition are $\varvec{h}_j$ and $\varvec{g}_j$ periodized to the length N. The first row of ${\mathcal {W}} _j$ is given by:

$$\begin{aligned}&[h^{\circ }_{j,2^j-1},h^{\circ }_{j,2^j-2},\ldots ,h^{\circ }_{j,1},h^{\circ }_{j,0},h^{\circ }_{j,N-1},h^{\circ }_{j,N-2},\ldots , \nonumber \\&\quad h^{\circ }_{j,2^j+1},h^{\circ }_{j,2^j}] \end{aligned}$$

(29)

subsequent rows are formed by circularly shifting the above equation to the right by $k2^{j}$ with $k=1,\ldots ,N_{j}-1$. The final row is:

$$\begin{aligned}{}[h^{\circ }_{j,N-1},h^{\circ }_{j,N-2},\ldots ,h^{\circ }_{j,1},h^{\circ }_{j,0}] \end{aligned}$$

(30)

A similar construction holds for ${\mathcal {V}} _j$ with each $h^{\circ }_{j,l}$ replaced by $g^{\circ }_{j,l}$. Consequently, the partial DWT of level $J_0$ of $\varvec{x}$ as an orthonormal transform is given by:

$$\begin{aligned} \varvec{W}=\mathcal {W}\varvec{x}, \end{aligned}$$

(31)

where $\varvec{W}\in \mathbb {R}^{N \times 1}$ and $\mathcal {W}\in \mathbb {R}^{N \times N}$ are partitioned such that:

$$\begin{aligned} \varvec{W}=\begin{bmatrix} \varvec{W}_1\\ \varvec{W}_2\\ \vdots \\ \varvec{W}_{J_0}\\ \varvec{V}_{J_0}\\ \end{bmatrix} ; \quad \mathcal {W}=\begin{bmatrix} \mathcal {W}_1\\ \mathcal {W}_2\\ \vdots \\ \mathcal {W}_{J_0}\\ \mathcal {V}_{J_0}\\ \end{bmatrix}, \end{aligned}$$

(32)

Furthermore, the jth level DWT detail and smooth are defined by:

$$\begin{aligned} {D}_j={\mathcal {W}} ^{\mathrm {T}}_j {\varvec{W}}_j;\quad {S}_j={\mathcal {V}}^{\mathrm {T}} _j {\varvec{V}}_j, \end{aligned}$$

(33)

in terms of which, the DWT-based additive decomposition (i.e., MRA) can be expressed as:

$$\begin{aligned} \varvec{x}=\mathcal {W}^\mathrm {T}\varvec{W}=\sum _{j=1}^{J_0}{D}_j+{S}_{J_0}. \end{aligned}$$

(34)

1.1 An illustrative example for Haar wavelet and scaling filters

The Haar wavelet and scaling filters at unit scale are defined by, respectively, $\varvec{h}=\varvec{h}_1=[\dfrac{-1}{\sqrt{2}} ~\dfrac{1}{\sqrt{2}}]^\mathrm {T}$ and $\varvec{g}=\varvec{g}_1=[\dfrac{1}{\sqrt{2}} ~\dfrac{1}{\sqrt{2}}]^\mathrm {T}$. By using Eq. (26), one can obtain wavelet filters $\varvec{h}_j$ as:

$$\begin{aligned} \varvec{h}_2&=\begin{bmatrix} \dfrac{-1}{2}&\quad \dfrac{-1}{2}&\quad \dfrac{1}{2}&\quad \dfrac{1}{2} \end{bmatrix}\nonumber \\ \varvec{h}_3&=\begin{bmatrix} \dfrac{-1}{2\sqrt{2}}&\quad \dfrac{-1}{2\sqrt{2}}&\quad \dfrac{-1}{2\sqrt{2}}&\quad \dfrac{-1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}} \end{bmatrix}\\ \varvec{h}_4&=\bigg [\begin{array}{cccccccccccccccc} \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{-1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4} \end{array}\bigg ]\nonumber \\ \vdots \nonumber \end{aligned}$$

(35)

similarly by using Eq. (27) scaling filters $\varvec{g}_j$ can be obtained as:

$$\begin{aligned} \varvec{g}_2&=\begin{bmatrix} \dfrac{1}{2}&\dfrac{1}{2}&\dfrac{1}{2}&\dfrac{1}{2} \end{bmatrix}\nonumber \\ \varvec{g}_3&=\begin{bmatrix} \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}}&\quad \dfrac{1}{2\sqrt{2}} \end{bmatrix}\\ \varvec{g}_4&=\bigg [\begin{array}{cccccccccccccccc} \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4}&\quad \dfrac{1}{4} \end{array}\bigg ]\nonumber \\ \vdots \nonumber \end{aligned}$$

(36)

By periodizing the above filters (defined in Eqs. (35) and (36)) to length N one can obtain, respectively, $\varvec{h}^\circ _j$ and $\varvec{g}^\circ _j$. Now by using Eqs. (29) and (30) matrix ${\mathcal {W}} _j$ (and similarly ${\mathcal {V}} _j$) can be obtained. As an example, matrix $\mathcal {W}$ associated with Haar filters for $N=16$ is:

(37)

Derivation of Eq. (19)

In matrix $Q'_f$, on average, the absolute values of the main diagonal elements are about 12, 54 and 125 times bigger than the elements of first, second and third diagonal, respectively. Furthermore, in matrix $Q'_{rw}$, the main diagonal elements are at least $10^{12}$, $10^{12}$ and $7\times 10^{11}$ times bigger than the elements of first, second and third diagonal, respectively. Therefore, the cofactor matrices $Q'_f$ and $Q'_{rw}$ are almost diagonal matrices that indicate the decorrelation property of DWT for the wavelet coefficients. Hence, by ignoring off-diagonal coefficients, cofactor matrices $Q'_f$ and $Q'_{rw}$ can be approximated by diagonal matrices as:

$$\begin{aligned}&Q'_f= {\mathcal {W}} Q_f{\mathcal {W}} ^\mathrm {T}\nonumber \\&\quad \approx \begin{bmatrix} {\mathcal {W}_1} Q_f{\mathcal {W}_1} ^\mathrm {T} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad {\mathcal {W}_2} Q_f{\mathcal {W}_2} ^\mathrm {T} &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad {\mathcal {W}_J} Q_f{\mathcal {W}_J}^\mathrm {T} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad {\mathcal {V}_J} Q_f{\mathcal {V}_J}^\mathrm {T} \\ \end{bmatrix} \nonumber \\&\quad \approx \begin{bmatrix} c_1I_{N_1} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad c_2I_{N_2} &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad c_J &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad c^*_J \\ \end{bmatrix}, \end{aligned}$$

(38)

$$\begin{aligned}&Q'_{rw}= {\mathcal {W}} Q_{rw}{\mathcal {W}} ^\mathrm {T}\nonumber \\&\quad \approx \begin{bmatrix} {\mathcal {W}_1} Q_{rw}{\mathcal {W}_1} ^\mathrm {T} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad {\mathcal {W}_2} Q_{rw}{\mathcal {W}_2} ^\mathrm {T} &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad {\mathcal {W}_J} Q_{rw}{\mathcal {W}_J}^\mathrm {T} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad {\mathcal {V}_J} Q_{rw}{\mathcal {V}_J}^\mathrm {T} \\ \end{bmatrix}\nonumber \\&\quad \approx \begin{bmatrix} d_1I_{N_1} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad d_2I_{N_2} &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad d_J &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad d^*_J \\ \end{bmatrix}, \end{aligned}$$

(39)

where $c_j$’s and $d_j$’s ($j=1,2,\ldots ,J$) are the constants along the diagonal elements of ${\mathcal {W}_j} Q_f{\mathcal {W}_j} ^\mathrm {T}$’s and ${\mathcal {W}_j} Q_{rw}{\mathcal {W}_j} ^\mathrm {T}$’s, respectively, and do not depend on the time series length. Furthermore, $c^*_J$ and $d^*_J$ are equal to ${\mathcal {V}_J} Q_f{\mathcal {V}_J}^\mathrm {T}$ and ${\mathcal {V}_J} Q_{rw}{\mathcal {V}_J}^\mathrm {T}$, respectively. (The detailed explanations on the construction of matrices ${\mathcal {W}_j}$ and $\mathcal {V}_J$ are presented in “Appendix A”). It should be noted that these constants depend upon scaling factor $\Delta T ^{-\kappa /4}$ defined in Eq. (6). As an example, these constants associated with Haar wavelet up to level $J_0=5$ are shown in Table 5. By substituting Eqs. (38) and (39) in Eq. (18), we can derive Eq. (19).

Table 5 Constants $c_j$’s and $d_j$’s associated with Haar wavelet up to level $J_0=5$

Derivation of Eqs. (22) and (23)

Let us start with Eq. (21). According to Eqs. (11–13) and (20), the coefficients of matrix $M'$ and vector $q'$ could be obtained as:

$$\begin{aligned}&M'_{11}= \mathrm {tr}(U' U');\quad M'_{12}= \mathrm {tr}(U' U'Q'_f);\nonumber \\&M'_{13}= \mathrm {tr}(U' U'Q'_{rw})\nonumber \\&M'_{21}=M'_{12};\quad M'_{22}= \mathrm {tr}(U'Q'_f U'Q'_f);\nonumber \\&M'_{23}= \mathrm {tr}(U'Q'_fU'Q'_{rw})\nonumber \\&M'_{31}= M'_{13};\quad M'_{32}= M'_{23};\quad M'_{33}= \mathrm {tr}(U'Q'_{rw} U'Q'_{rw}), \end{aligned}$$

(40)

and:

$$\begin{aligned}&q'_{1}= \varvec{W}^{\mathrm {T}}U'U'\varvec{W};\quad q'_{2}= \varvec{W}^{\mathrm {T}}U'Q'_fU'\varvec{W};\nonumber \\&q'_{3}= \varvec{W}^{\mathrm {T}}U'Q'_{rw}U'\varvec{W}, \end{aligned}$$

(41)

where:

$$\begin{aligned} U'=Q_{\varvec{W}}^{-1}-Q_{\varvec{W}}^{-1}A'(A'^{\mathrm {T}}Q_{\varvec{W}}^{-1}A')^{-1}A'^{\mathrm {T}}Q_{\varvec{W}}^{-1} . \end{aligned}$$

(42)

Substituting Eq. (19) in Eq. (42) yielded as:

$$\begin{aligned} U'=\begin{bmatrix} k_1^{-1}I_{N_1} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad k_2^{-1}I_{N_2} &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad k_j^{-1} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 \\ \end{bmatrix}. \end{aligned}$$

(43)

By replacing Eqs. (38), (39) and (43) in Eqs. (40) and (41), the simplified forms for coefficients of matrix $M'$ and vector $\varvec{q}'$ could be obtained as Eqs. (22) and (23).