### 5 − 1. Optimization method

We optimized the hyper-parameter \(\:\alpha\:\), which balances the inversion modelling error and the reference SSP positioning error, by the MCMC method to obtain the vertical positioning error \(\:{\widehat{\sigma\:}}_{\text{Z}}\) defined in Eq. 4. In the MCMC method, \(\:\alpha\:\) is optimized to match the vertical positioning error \(\:{\widehat{\sigma\:}}_{\text{Z}}\) with the deviation from a fitting curve of multiple campaign solutions in a time-series. Assuming that the displacement rate at G02 was constant during the observational period, we applied weighted linear line fitting to the vertical array displacements for the multiple campaign datasets at G02: By defining the vertical array displacement, \(\:\delta\:{p}_{\text{Z}},\) for the \(\:i\)th campaign dataset as \(\:{d}_{i}\), the fitting line can be written as follows:

$$\:{d}_{i}^{\text{c}\text{a}\text{l}}={a}_{\text{Z}}{\bullet\:t}_{i}+{b}_{\text{Z}},$$

7

where, \(\:{d}_{i}^{\text{c}\text{a}\text{l}}\) is the predicted vertical array displacement; \(\:{t}_{i}\) is the time of the \(\:i\)th survey campaign; \(\:{a}_{\text{Z}}\) is the vertical displacement rate; and \(\:{b}_{\text{Z}}\) is the intercept of the fitted line.

Assuming that misfits between \(\:{d}_{i}\) and \(\:{d}_{i}^{\text{c}\text{a}\text{l}}\) follow the normal distribution with the mean of 0 and the standard deviation of \(\:{\widehat{\sigma\:}}_{\text{Z}}\), we approximated the posterior density functions of the hyper-parameter \(\:\alpha\:\) by the random walk Metropolis algorithm (Metropolis et al. 1953) which is one of the basic MCMC algorithms. Following the Bayes’ theorem, the posterior probability density \(\:p\left(\alpha\:|\mathbf{d}\right)\) is calculated as

$$\:p\left(\alpha\:|\mathbf{d}\right)\propto\:p\left(\mathbf{d}|\alpha\:\right)p\left(\alpha\:\right).$$

8

To calculate the marginal likelihood \(\:p\left(\mathbf{d}|\alpha\:\right)\), we need to estimate the linear parameters, \(\:{a}_{\text{Z}}\) and \(\:{b}_{\text{Z}}\), for the weighted line fitting shown in Eq. 7. Defining the linear parameter vector \(\:\mathbf{x}\) as \(\:\mathbf{x}=\left({a}_{\text{Z}},{b}_{\text{Z}}\right)\), we rewrite the observation equation (Eq. 7) as \(\:\mathbf{d}=\mathbf{H}\mathbf{x}\) in matrix form. According to Fukuda & Johnson (2010), the marginal likelihood \(\:p\left(\mathbf{d}|\alpha\:\right)\) is extended using the linear parameters when the observation equation includes linear parameters:

$$\:p\left(\mathbf{d}|\alpha\:\right)=\int\:p\left(\mathbf{d},\mathbf{x}|\alpha\:\right)\:d\mathbf{x}=\int\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)p\left(\mathbf{x}|\alpha\:\right)\:d\mathbf{x}.$$

9

The likelihood \(\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)\) is defined based on the probability density of the normal distribution as:

$$\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}\bullet\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right),$$

10

where, \(\:\mathbf{E}\left(\alpha\:\right)\) is the data variance-covariance matrix and is defined as \(\:\mathbf{E}\left(\alpha\:\right)=\text{d}\text{i}\text{a}\text{g}\left({\widehat{\sigma\:}}_{Z,1}^{2},\cdots\:,{\widehat{\sigma\:}}_{Z,I}^{2}\right)\). \(\:I\) is the total number of campaigns. Note that the vertical positioning error of the \(\:i\)th campaign survey is denoted as \(\:{\widehat{\sigma\:}}_{\text{Z},i}=\sqrt{{\alpha\:}^{2}\bullet\:{\sigma\:}_{\text{Z},i}^{2}+{\sigma\:}_{\text{S}\text{S}\text{P},i}^{2}}\). In this study, we provided no prior information on \(\:p\left(\mathbf{x}|\alpha\:\right)\); thus, \(\:p\left(\mathbf{x}|\alpha\:\right)=1\) and Eq. 9 is rewritten as:

$$\:p\left(\mathbf{d}|\alpha\:\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}\int\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)\:d\mathbf{x}.$$

11

Letting \(\:{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\) as \(\:f\left(\mathbf{x}\right)\), \(\:f\left(\mathbf{x}\right)\) is rewritten as:

$$\:f\left(\mathbf{x}\right)=f\left({\mathbf{x}}^{\varvec{*}}\right)+{\left(\mathbf{x}-{\mathbf{x}}^{\varvec{*}}\right)}^{\text{T}}{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\left(\mathbf{x}-{\mathbf{x}}^{\varvec{*}}\right),$$

12

where, \(\:{\mathbf{x}}^{\varvec{*}}\) is the least squared solution minimizing \(\:f\left(\mathbf{x}\right)\) as:

$$\:{\mathbf{x}}^{\varvec{*}}={\left({\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right)}^{-1}{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{d}.$$

13

According to Fukuda and Johnson (2010), the integral in Eq. 11 is analytically computed as

$$\:\int\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)\:d\mathbf{x}=\text{e}\text{x}\text{p}\left(-\frac{1}{2}f\left({\mathbf{x}}^{\varvec{*}}\right)\right)\times\:2\pi\:{\left|{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right|}^{-\frac{1}{2}}.$$

14

In the present study, we provided a prior information on \(\:\alpha\:\) which restricts \(\:\alpha\:>0\) as:

$$\:p\left(\alpha\:\right)=\left\{\begin{array}{cc}0&\:\text{i}\text{f}\:\alpha\:\le\:0\\\:1&\:\text{i}\text{f}\:\alpha\:>0\end{array}\right.$$

15

.

Thus, when \(\:\alpha\:>0\), the posterior probability density \(\:p\left(\alpha\:|\mathbf{d}\right)\) is rewritten using Equations 8, 11–15 as:

$$\:p\left(\mathbf{d}|\alpha\:\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}{\left|{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right|}^{-\frac{1}{2}}\bullet\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)$$

16

.

In the random-walk Metropolis algorithm, an unknown parameter is iteratively updated in the MCMC chain. Letting \(\:\alpha\:\) in the \(\:j\)th iteration is \(\:{\alpha\:}^{\left(j\right)}\), the unknown parameter in the next iteration \(\:{\alpha\:}^{{\prime\:}}\) is generated as:

$$\:{\alpha\:}^{{\prime\:}}={\alpha\:}^{\left(j\right)}+\delta\:\alpha\:$$

17

,

where, \(\:\delta\:\alpha\:\) is a random value and is generated from a uniform distribution \(\:U\left(-\text{0.5,0.5}\right)\) in this study. The generated unknown parameter is then evaluated using the acceptance probability

$$\:{P}_{\text{a}\text{c}\text{c}\text{e}\text{p}\text{t}}=\text{m}\text{i}\text{n}\left[1,\frac{p\left({\alpha\:}^{{\prime\:}}|\mathbf{d}\right)}{p\left({\alpha\:}^{\left(j\right)}|\mathbf{d}\right)}\right]$$

18

.

If the acceptance probability exceeds a random number generated from a uniform distribution \(\:U\left(\text{0,1}\right)\), the generated unknown parameter is accepted as \(\:{\alpha\:}^{\left(j+1\right)}\). We performed \(\:1.1\times\:{10}^{6}\) iterations and discarded the samples of the first \(\:1\times\:{10}^{5}\) iterations as the burn-in period. We adopted mode of the samples as the optimal value of \(\:\alpha\:\). We then calculated the displacement rate, \(\:{a}_{\text{Z}}\), and its error using a weighted linear least-squares method, considering the optimized positioning errors. Although we can simultaneously estimate posterior distributions of the linear parameters, \(\:{a}_{\text{Z}}\) and \(\:{b}_{\text{Z}}\), as with \(\:\alpha\:\) by the method of Fukuda & Johnson (2010), we employed the displacement rate and its error based on the optimal value of \(\:\alpha\:\) to compare with the solutions from the conventional positioning errors.

### 5 − 2. Optimized positioning errors

Applying the optimization method addressed in Section 5.1 to the positioning results at G02 including \(\:{\sigma\:}_{\text{S}\text{S}\text{P}}\) shown in Table 1. Figure 6 shows a histogram of the sampled \(\:\alpha\:\), and the mode of the samples is 7.8.

Figure 7 shows time-series of the vertical array displacements with three types of the positioning errors: the conventional positioning errors (10\(\:{\sigma\:}_{\text{Z}}\): multiples of the inversion modelling error) in Fig. 7a, the reference SSP positioning errors (\(\:{\sigma\:}_{\text{S}\text{S}\text{P}}\) in Table 1) in Fig. 7b, and the optimized positioning errors (\(\:{\widehat{\sigma\:}}_{\text{Z}}=\sqrt{{7.8}^{2}\bullet\:{\sigma\:}_{\text{Z}}^{2}+{\sigma\:}_{\text{S}\text{S}\text{P}}^{2}}\)) in Fig. 7c. The conventional positioning errors reflected the observation conditions, especially the insufficiency of the moving survey; for example, \(\:{\sigma\:}_{\text{Z}}\) at the campaign number 10 is large (Table 1) because we did not conduct a moving survey at the campaign (Figure S1). However, the deviation from the regression line was occasionally larger than the conventional positioning error when a direct oceanographical measurement of SSP was not conducted (mainly when using WG; squares in Fig. 7a). The reference SSP positioning errors provided large errors to the campaigns without direct oceanographical measurements of the SSP. However, the overall positioning errors were evaluated to be too small to compensate for the deviation of the estimates from the regression line. The optimized positioning errors successfully express the deviation of the estimates from the regression line by including the influence of reference SSP positioning errors, unlike conventional positioning errors.