Static array positioning with gradients

As discussed in various studies (e.g., Kido (2007), Honsho et al. (2019)), horizontal array displacements has trade-off relationship with horizontal heterogenity of a sound speed structure. Recent GNSS-A studies has considered a horizontally sloping sound speed structure (Yokota et al. (2018), Yasuda et al. (2017), Honsho et al. (2019)). Here, a static array positioning method considering a horizontally sloping sound speed structure (Tomita & Kido, 2022) is simply introduced.

Effect of a horizontally sloping sound speed structure on a travel-time is expressed by two component: shallow and deep gradients (e.g., Yokota et al., 2018).

with

$\mathbf{g}_{\mathrm{s}}$ is the shallow gradient, and $\mathbf{g}_{\mathrm{d}}$ is the deep gradient; they individually have EW and NS components. $\xi$ is a shot angle, and $\phi$ is an azimuth. They originally have time flcuatuation, but it is assumed o be constant during a campaign for simplicity. $C_0$ is NTD corresponding to the average sound speed fluctuation. The other terms are same with the previous pages (static array positioning.

As long as using single sea-surface platform, the NTD ($C_0$) and the effect of the shallow gradient ($\mathbf{g}_{\mathrm{s}}\mathbf{u}^{\mathrm{hor}}\left(t_{n}\right)$) are strongly trade-off, This means that the effect of the shallow gradient does not strongly affect horizontal array displacements by the single sea-surface platform case. Considering this, we simultaneously model the NTD and the effect of the shallow gradient by the superposition of the 3d B-spline functions:

< Observation Equation [0] >

According to the observation equation [0], we can model the sound speed gradient straucture by estimating EW and NS components of the deep gradient ($\mathbf{g}_{\mathrm{d}}$) in addition to the array position and the 3d B-splined NTDs. You can perform this optimization through Gauss-Newton method by static_array_grad().

However, this optimization is quite unstable except acoustic data obtained from spatially well-distributed sea-surface platform (for example, the data of Japan Coast Guard: Watanabe et al., 2020) or obtained at a site with a bunch of seafloor transponders (for example, multi-angled transponders: Tomita et al., 2019).

To stabilize this optimization, Honsho et al. (2019) provided a further constaint: the sound speed gradient uniformly exsits from the sea-surface to a certain depth (gradient depth $D$). This assumption menas that the deep gradient has a linear relationship with the shallow gradient, and its linear coeffient is the gradient depth as:

Thus, if you somehow obtain the shallow depth, you can estimate the deep gradient by optimizing only the gradient depth:

< Observation Equation [1] >

Meanwhile, because of the trade-off relationship between the temporal fluctuation of NTD and the shallow gradient, it is difficult to grasp the shallow gradient. Separation of them can be roughly conducted by modeling long-term component of the temporal fluctuation of NTD (e.g., Honsho et al., 2019). Here, we model the long-term NTD as the 4th polynomial functions:

< Observation Equation [2] >

Iterative optimization of the observation equations [1] and [2], note that the red terms are the unknown parameters for each equation, can conduct relatively stabler analysis under the sound speed gradient than the observation equation [0] in Tomita & Kido (2022). Moreover, the gradient depth is limited from the sea-surface (zero) to the seafloor. To flexibly inculude the limitation, we employ a MCMC technique to optimize the observation equations (1) and (2) static_array_mcmcgrad().

To perform static_array_mcmcgrad(), we initially obtain solutions by the static array positioning with horizontally stratified sound speed structure static_array(). Using the solutions of static_array() as the initial values, static_array_mcmcgrad() optimize the observation equations [1] and [2] for odd and even MCMC loops, respectively. Note that we also optimize a scaling factor for each observation equations; the scaling factors express the observational error of each equation and balance the weights of the observation equations (e.g., Tomita et al., 2021).

Assuming that $\alpha_1$ and $\alpha_2$ express the scaling factors for observation equations [1] and [2], respectively, the variances of the observation equations are expressed as $\left(10^{\alpha_1}\right)^2$ and $\left(10^{\alpha_2}\right)^2$. Thus, if $\alpha_1=-4.0$, the variance of the observation equation [1] is expressed as 1.e-8 (i.e., the standard deviation of 1.e-4 [sec]).

The details of this MCMC optimization are shown in Tomita & Kido (2022).

Optionally, SeaGap includes a developing function static_array_mcmcgradc() which add some constraints to static_array_mcmcgrad() to obtain solutions more stably. See Tutorials in detail for this developing function.