Correction to: Acta Geophysica https://doi.org/10.1007/s11600-023-01279-y

The publication of this article unfortunately contained mistakes. Equation 6 was not correct. The corrected equation is given below.

Incorrect Eq. 6:

$$\begin{aligned} \varvec{Am}=\varvec{b}, \end{aligned}$$
(6)

where for \(\alpha = x,y,z\), \({\varvec{A}} = {[{\varvec{A}^1},...,{\varvec{A}^i},...,{\varvec{A}^{{n_w}}}]^\mathrm{{T}}}\), \({n_w} = {w_x} \times {w_y}\), \({{\varvec{A}}^i} = \left[ {{{\left( {{{\partial {g_\alpha }} / {\partial x}}} \right) }^i},{{\left( {{{\partial {g_\alpha }} / {\partial y}}} \right) }^i},{{\left( {{{\partial {g_\alpha }}/{\partial z}}} \right) }^i},{{\left( {{g_\alpha } - {B_\alpha }} \right) }^i}} \right]\) \({\varvec{b}} = {\left[ {{{\varvec{b}}^1},...,{{\varvec{b}}^i},...,{{\varvec{b}}^{{n_w}}}} \right] ^\mathrm{{T}}}\), \({{\varvec{b}}^i} = {\left( {x{{\partial {g_\alpha }} / {\partial x}}} \right) ^i} + {\left( {y{{\partial {g_\alpha }}/ {\partial y}}} \right) ^i}\), \({{\varvec{b}}^i} = {\left( {x{{\partial {g_\alpha }} / {\partial x}}} \right) ^i} + {\left( {y{{\partial {g_\alpha }}/ {\partial y}}} \right) ^i} + {\left( {z{{\partial {g_\alpha }}/ {\partial z}}} \right) ^i}\), \(\left\{ {{\varvec{m}}} \right\}\) is the Euler solution set, in which the \(i^{th}\) solution can be expressed as \({{\varvec{m}}^{i}} = {[{x^i_o},{y^i_o},{z^i_o},{N^i_o}]^\mathrm{{T}}}.\)

Correct Eq. 6:

$$\begin{aligned} \varvec{Am}=\varvec{b}, \end{aligned}$$
(6)

where for \(\alpha = x,y,z\), \({\varvec{A}} = {[{\varvec{A}^1},...,{\varvec{A}^i},...,{\varvec{A}^{{n_w}}}]^\mathrm{{T}}}\), \({n_w} = {w_x} \times {w_y}\), \({{\varvec{A}}^i} = \left[ {{{\left( {{{\partial {g_\alpha }} / {\partial x}}} \right) }^i},{{\left( {{{\partial {g_\alpha }} / {\partial y}}} \right) }^i},{{\left( {{{\partial {g_\alpha }}/{\partial z}}} \right) }^i},{{\left( {{g_\alpha } - {B_\alpha }} \right) }^i}} \right]\) \({\varvec{b}} = {\left[ {{{\varvec{b}}^1},...,{{\varvec{b}}^i},...,{{\varvec{b}}^{{n_w}}}} \right] ^\mathrm{{T}}}\), \({{\varvec{b}}^i} = {\left( {x{{\partial {g_\alpha }} / {\partial x}}} \right) ^i} + {\left( {y{{\partial {g_\alpha }}/ {\partial y}}} \right) ^i} + {\left( {z{{\partial {g_\alpha }}/ {\partial z}}} \right) ^i}\), \(\left\{ {{\varvec{m}}} \right\}\) is the Euler solution set, in which the \(i^{th}\) solution can be expressed as \({{\varvec{m}}^{i}} = {[{x^i_o},{y^i_o},{z^i_o},{N^i_o}]^\mathrm{{T}}}.\)

The original article has been corrected.