Comparative Analysis of Jet Detection Methods on the Basis of Satellite Altimetry Data by Example of the Antarctic Circumpolar Current Sector to the South of Africa

Abstract

For the period of satellite altimetry observation 1993–2018, a comparison was made of estimates of the linear meridional shifts of the jet structure and variations in current intensity in the sector of the Antarctic Circumpolar Current (ACC) south of Africa (10° E–25° W), obtained on the basis of the parameters derived from the module of the absolute dynamic topography (ADT) gradient $\left| {\nabla \zeta } \right|$: directly $\left| {\nabla \zeta } \right|$, module of the velocity of geostrophic current |u| on the ocean surface, half of square of the ADT gradient $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, and specific kinetic energy $\frac{1}{2}{{u}^{2}}$. The analysis was carried out by a method developed earlier and refined in this paper using linear regression analysis. It is shown that qualitatively the characters of the meridional shifts (displacement of the curves of these parameters averaged over latitude and for each year) in latitude and the corresponding current intensity variations (i.e., variations in the parameters themselves) over 26-year observations, when calculated through each of the four parameters, turned out to be similar. The same similarity was obtained from the calculations with respect to the ADT scale. Meanwhile, the quantitative differences between calculations for different pairs of parameters can be significant even in the $\left| u \right|$ and $\left| {\nabla \zeta } \right|$ pair. In quantitative terms, the absolute values of shifts of the jet structure and variations in the intensity of currents in the ACC band in the studied sector of the Southern Ocean generally increase from those linearly dependent on the ADT gradient module to quadratically dependent ones.

INTRODUCTION

The Antarctic Circumpolar Current (ACC) encircles the Antarctic continent from west to east and is divided into jets [2, 15], i.e., zones characterized by increased current velocity on the ocean surface. These zones correspond to an increased slope of the absolute dynamic topography (ADT, ζ) on the ocean surface and to an increased slope of isopycnal surfaces deep in the ocean. ADT isolines (isohypses) are streamlines of the geostrophic current on the ocean surface. In the Southern Hemisphere, higher ADT values remain to the left of the current direction, and, in the Northern Hemisphere, to the right. Accordingly, the compactions of isohypses (zones with increased ADT gradients, ∇ζ) on the ADT maps correspond to jets.

In accordance with the ideas proposed in [15], considered classical today, three jets are distinguished in the ACC zone throughout the entire circumpolar circle (terms are proposed by [15]): the Subantarctic Front, the Polar Front, and the Southern Front of the ACC. Following [3], we will call these jets the Subantarctic (SAC), South Polar (SPC), and South Antarctic (SthAC) currents, respectively. More recent studies based on the satellite altimetry data consider a larger number of jets: up to 9 in the entire circumpolar circle [17] and up to 12 in regional studies [5–7]. According to the conclusions by [18], the ACC jets are tied to the same isohypses throughout the circumpolar circle and in time; under the observed ocean level rise, it would mean a meridional shift of the ACC jets to the south. Meanwhile, the most recent studies do not reveal a systematic long-term meridional displacement of the ACC jets (reviews in [11] and [19]). In addition, to date, a number of studies [10, 13, 20] do not indicate that any one front maintains a continuous structure throughout the ACC.

The long-term linear meridional shift of the ACC jet structure was studied in the Southern Ocean sector to the south of Africa (from 10° W to 25° E) (Fig. 1) for the period of 1993–2018 by the original author’s method based on linear regression analysis [19]. The jet structure of currents is understood as alternating zones with increased and decreased ADT gradient module $\left| {\nabla \zeta } \right|$ in some selected (meridional in the case of the ACC) direction. Its shift, thus, is a meridional displacement of the averaged (over latitude and for each year over the specified sector of the Southern Ocean) curves $\left| {\nabla \zeta } \right|$ depending on latitude or ADT values for the entire observation period. It should be noted that the calculation of the meridional shift is due to the fact that the ACC jets are quasi-zonal in this sector (Fig. 1). In the mentioned work, in particular, it was shown that the entire ACC band (57°–42° S) shifts northward by 0.05° which, however, does not exceed the calculation error. Alongside with that, in some zones of the ACC band, the shift turns out to be significant, reaching 0.4° to the south inside the SAC and 1.5° to the north inside the SPC. The ACC jet structure is displaced with respect to the ADT scale (in the band from –130 to 20 cm on the ADT scale adjusted to the middle of the 1993–2002 interval) by 8.3 ± 1.0 cm in the above-mentioned period. These data agree well with the average ocean level increase estimated at 7.5 cm for 22 years (1993–2014) of satellite altimetry observations [8]. It should be noted that the estimates given in [19] refer, strictly speaking, to the period from mid-1993 to mid-2018, i.e., a 25-year interval. The method was modified and refined in [4] to analyze seasonal harmonic oscillations of the meridional shift of the ACC jet structure.

Fig. 1.

Mean dynamic topography (isolines and color shading) in the region south of Africa. Thick isolines –130 and 20 cm show the approximate boundaries of the ACC. Acronym AC is the Agulhas Current, in which ADT values are also fall within the range characteristic of the ACC. The shaded areas outlined by white lines correspond to ocean areas with depths of less than 3000 m. Oblique lines show the main tracks of T/P, Jason-1, -2, -3 satellites. Dashed line at 39° S shows the north limit for calculating h versus ζ curves.

Jet cores and interjet gaps, based on the ADT data, can be related not only to $\left| {\nabla \zeta } \right|$ maxima and minima, as, e.g., in [4, 6, 7, 10, 11, 19], but also to the corresponding extrema of the parameters derived from $\left| {\nabla \zeta } \right|$: geostrophic velocity module $\left| u \right| = \left| {\frac{g}{f}\nabla \zeta } \right|$ (g is a gravity acceleration and f is the Coriolis parameter) (e.g., [1]), specific kinetic energy, $\frac{1}{2}{{u}^{2}}$ (e.g., [9]), and also half of square of the ADT gradient module, $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$. All these parameters are different physical values, and calculating the meridional shift of the jet structure based on them will obviously yield different quantitative results. Our study objective is to evaluate such shifts and to compare them with each other using the example of the same ACC sector to the south of Africa as in [4, 19]. The jet structure shift in the ACC is studied not only in the meridional direction, but also with respect to the ADT scale. Section 2 describes the data used to carry out the analysis; section 3 describes the data analysis method; section 4 discusses the calculation results in detail; section 5 contains a discussion of the study results and basic conclusions; and Appendices 1 and 2 describe the calculation error assessment procedure.

DATA

As in [19], we used the daily ADT data with a ${\raise0.5ex\hbox{$\scriptstyle {1^\circ }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} \times {\raise0.5ex\hbox{$\scriptstyle {1^\circ }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}$ grid from the SEALEVEL_GLO_PHY_ L4_REP_OBSERVATIONS_008_047 product developed and distributed by the Copernicus Marine and Environmental Monitoring Service (CMEMS) (http://marine.copernicus.eu). An example of an ADT map constructed using these data is shown in Fig. 1. The synoptic (instantaneous, in particular, at a certain point in time) ADT is the sum of the mean (over time) ADT and the instantaneous sea level anomaly (SLA) determined by satellite altimeters. The above product uses a version of the mean ADT (MDT CNES-CLS18 [14]) calculated from various measurements of hydrophysical parameters in the ocean, satellite altimetry observations, geoid and mean sea surface models, and wind reanalysis data, while the SLA data are interpolated from satellite tracks drawn on the Earth’s surface onto a regular grid for each day. Figure 1 shows a location of the so-called main tracks of the TOPEX/Poseidon (T/P) and Jason-1, -2, -3 satellites with altimeters to the south of Africa, which are repeated with a period of τ ≈ 10 days during the entire era of satellite altimetry observation.

METHODS

Assessing the long-term linear meridional shift of the jet structure is based on the linear regression analysis. Following the algorithm described in [19], first, the curves of the dependence of each studied physical value ($\left| {\nabla \zeta } \right|$, $\left| u \right|$, $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, $\frac{1}{2}{{u}^{2}}$) on the parameter a (latitude, φ or ADT, ζ), averaged over the study area and for the each year, are calculated. The curves obtained in this way are denoted h(a). In the region located south of Africa, the h(a) calculation, as in [19], is bounded from the north to 39° S in order to cut off the northern periphery of the Agulhas Current and its cyclonic eddies; the ζ values which occur inside these eddies are characteristic of the ACC (Fig. 1). It should be noted that when calculating the dependence h(φ), the h minima and maxima can be blurred or even disappear due to the non-zonality of these jets and their meandering; in particular, the maxima of $\left| {\nabla \zeta } \right|$ values averaged along isohypses and over a certain period of time, as shown for the sector to the south of Africa in [6], are tied at time intervals of up to a year to the same relatively narrow ADT ranges. The h(ζ) calculation is less prone to such blurring. The curve calculation methods are described more thoroughly in [19]. Hence, a set of 26 average annual curves is obtained for each physical value.

Based on the set of curves calculated above, the derivative of the $\bar {h}$ (average for the entire observation time) distribution with respect to a:

$${{x}_{i}} = \frac{{{{X}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{X}_{{i{\kern 1pt} - {\kern 1pt} 1}}}}}{{2\Delta a}},$$

(1a)

where

$${{X}_{i}} = \frac{1}{L}\mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{h}_{{i,l}}}.$$

(1b)

In this case, h_i,l is the value h for the ith value of argument $a$ in the l-th curve; the summation is carried out by year; and L is the number of years. In the formula (1а), for i = 1, it is necessary to replace i – 1 by 1, and for i = N, i + 1 is replaced by N, where N is the total number of the argument values a, and 2∆a is replaced by ∆a. Linear variations in the h values over the entire observation period are also calculated for each ${{a}_{i}}$ value; the magnitude of this variation, y_i, is calculated through the linear regression coefficient k of h_i,l dependence on time reduced to a unit interval:

$${{y}_{i}} = k\left( {t,{{h}_{i}}} \right) = \frac{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {\left( {{{h}_{{i,l}}} - \overline {{{h}_{i}}} } \right)} \left( {{{t}_{l}} - \bar {t}} \right)}}{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{l}} - \bar {t}} \right)}}^{2}}} }},$$

(2a)

where ${{t}_{l}} = \frac{l}{L}$. A free term of the linear regression is also calculated:

$$b\left( {t,{{h}_{i}}} \right) = \overline {{{h}_{i}}} - k\left( {t,{{h}_{i}}} \right)\bar {t}.$$

(2b)

Formula (2a) was modified in relation to [19] to calculate y_i for the full 1993–2018 interval. The corresponding changes were made to the error calculation formulas in Appendix 1.

A pair of x($a$) and y($a$) distributions can be represented on the same plane with a parametric dependence on a. A linear regression between x and y is estimated on this plane for some a range, e.g., corresponding to the ACC band. In this case, the coefficient k of the linear regression with the opposite sign represents the linear shift of the ACC jet structure for the entire observation period:

$$k\left( {x,y} \right) = \frac{{\sum\limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {\left( {{{y}_{i}} - \tilde {y}} \right)} \left( {{{x}_{i}} - \tilde {x}} \right){{w}_{i}}}}{{\sum\limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{i}} - \tilde {x}} \right)}}^{2}}} {{w}_{i}}}},$$

(3a)

and the free regression term is the variation in h in this range over the same period

$$b\left( {x,y} \right) = \tilde {y} - k\left( {x,y} \right)\tilde {x}.$$

(3b)

The upper tilde sign means averaging over index i; ${{w}_{i}}$ is a weight coefficient, and $\sum\nolimits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{{w}_{i}}} = 1$. In this study, we take ${{w}_{i}} = {1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-0em} N}$ for $a$ corresponding to latitude and ${{w}_{i}} = {{{{s}_{i}}} \mathord{\left/ {\vphantom {{{{s}_{i}}} {\sum\nolimits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{s}_{j}}} }}} \right. \kern-0em} {\sum\nolimits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{s}_{j}}} }}$ for $a$ corresponding to ζ; in this case, ${{s}_{i}}$ is an estimate of the area of map zones limited by isohypses ζ_i ± ∆ζ/2. The difference between edge values of a range will be further understood as the calculation scale, and the central value ${{a}_{0}}$ will be understood as the median latitude or median ADT value depending on the choice of parameter a.

According to [19], it is essential that the shifts in the jet structure of currents are less than the calculation scale with respect to a to provide a correct application of the method proposed (in its actual form). This wording should be adjusted by adding one more necessary condition. The method is based on the assumption that the estimated shifts of the jet structure are smaller than the calculation scale with respect to a and smaller than the characteristic scale of this structure (e.g., the distance between the h maximum points in the h(a) distribution). In the cases when this assumption is unfair, the method needs to be refined.

In this study, the difference in linear shifts is estimated to compare calculations in terms of different h parameters. The difference in shifts calculated from a pair of parameters h1 and h2 (hereinafter, a parameter whose shift is subtracted is indicated the second) is determined as follows:

$$\Delta k = - \left( {k1\left( {x1,y1} \right) - k2\left( {x2,y2} \right)} \right).$$

(4a)

To compare linear variations in physical parameters h1 and h2, they are reduced to one dimension, a velocity, by dividing by the dimensional coefficient β corresponding to the median value ${{a}_{0}}$ for the calculated a range. The parameter β is indicated for each physical value in Table 1. Then the difference between the reduced values is calculated:

$$\Delta b = \frac{1}{{\beta {\text{1}}}}b1\left( {x1,y1} \right) - \frac{1}{{\beta {\text{2}}}}b2\left( {x2,y2} \right).$$

(4b)

Table 1.   Coefficients for reducing parameters derived from the ADT gradient module to the velocity dimension

The procedure for assessing the error in calculating linear shifts and variations in parameters is described in Appendix 1. It should be noted that we changed the method of taking into account the mean ADT error in [4] compared to [19]. This method has been revised once again in this paper, and the procedure for taking into account the sea level anomaly error was also changed. These changes reduced the estimated total error by approximately two to five times. The fields of not only the ADT gradient module, but also of the module of geostrophic velocity, specific kinetic energy, and half of square of the ADT gradient are analyzed in this work. Based on this, the error assessment procedure is given in full. Appendix 2 describes the procedure for assessing the error in calculating differences in linear shifts and variations in parameters calculated from different physical values.

CALCULATION RESULTS

Calculations with Respect to the Latitude Scale

Figure 2 shows 26-year series of average annual h distributions (i.e., $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\left\langle {\left| u \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle ,$ and $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$; hereinafter, angle brackets denote averaging in direction, in particular, along latitude or isohypses and over an annual period of time) depending on φ, estimated linear shifts of the jet structure of currents and linear variations in indicated physical parameters over 26-year observations, depending on the calculation scale and median latitude. The calculation was carried out with a step of 0.5° latitude along the scale axis and 0.25° latitude along the median latitude axis. In Figs. 2b, 2c, 2e, 2f, 2h, 2i, 2k, and 2l, each point corresponds to one calculation performed according to the method described in section 3. In this case, the resulting value of the calculated parameter is assigned to the median latitude. For example, the northern and southern boundaries are conventionally assumed to be 42° S and 57° S for the ACC. Thus, the calculation for the ACC band in the indicated figures corresponds to 15° latitude along the scale axis and 49.5° S along the median latitude axis. Dark and light shading in these figures highlight the areas with absolute values of the estimated parameters which are less than the standard calculation error and the 95% probability level (according to the Student’s t-test). Figure 3 shows estimates of the standard error of calculation in terms $\left\langle {\left| u \right|} \right\rangle$. Calculation errors for other parameters are not given due to the quantitative similarity of the shift calculation errors to those shown in Fig. 3a.

Fig. 2.

To calculate the shift of the ACC jet structure with respect to latitude: (a, d, g, j) 26-year series of average annual distributions of $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\left\langle {\left| u \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$, and $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ depending on latitude φ for the ACC band to the south of Africa; (b, e, h, k) corresponding linear shifts (° latitude) of the jet structure of currents depending on the median latitude and calculation scale; dark shading corresponds to areas of calculated points with a shift estimate less than the standard calculation error, while a light shading corresponds to less than 95% of the probability level (according to Student’s t-test); (c, f, i, l) corresponding variations in current intensity; (c) cm/km × 10^–3, (f) cm/s, (i) cm²/km² × 10^–3, and (l) cm²/s². Dashed and dash–dotted horizontal lines are the conventional boundaries of the ACC and the boundaries between zones of increased ADT gradient within the ACC. Crosses in (b, c, e, f, h, i, k, l) indicate the points corresponding to the SAC, SPC, and SthAC calculation bands, as well as the ACC as a whole; the calculation results for these points are given in Table 2. Acronyms in (a, d, g, j): SAC is the Subantarctic Current, SPC—the South Polar Current, SthAC—the South Antarctic Current, and WC—the Weddell Current.

Fig. 3.

Standard calculation errors in terms of geostrophic velocity depending on the median latitude and calculation scale: (a) errors in calculating the linear shift (° latitude) of the jet structure of currents; (b) errors in calculating the current velocity variations (cm/s). The rest is as in Fig. 2.

In each of Figs. 2a, 2d, 2g, and 2j, in the ACC band, three zones of increased h can be distinguished at 45°–46° S, 48°–50° S, and 55°–57° S. In accordance with [19], the latitudinal bands covering these zones will be conventionally called the Subantarctic (SAC), South Polar (SPC), and South Antarctic (SthAC) currents, and the zones with maximum values of these parameters will be termed as cores of these currents. The zone of increased h values, observed to the south of the ACC band at approximately 57°–58° S, will also be conventionally called the Weddell Current (WC), and the zone to the north at 37°–41° S as the Agulhas Current (AC).

The calculations of the linear meridional shift of the jet current structure carried out using different physical parameters yield a qualitatively identical result: a shift is southward in the SAC band and northward in the SPC band (Figs. 2b, 2e, 2h, 2k). The ACC as a whole shifts slightly northward by 0.04°–0.08° latitude (Table 2). It is possible, however, to observe slightly higher values of the shift in quadratic parameters, in particular, in terms of the squared ADT gradient and specific kinetic energy, especially in the SPC zone (Table 2). The patterns of linear variations in the parameters are also qualitatively similar: the SAC band is marked by a decrease in current intensity, while the SPC and SthAC are marked by an increase. The characteristic absolute values of variations in these parameters are 5–10% of the average values over 26 years with extreme values of up to 20%.

Table 2.   Estimates of the shift of the meridional jet structure of the ACC and variations in $\left| {\nabla \zeta } \right|$, $\left| u \right|$, $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, and ${{{{u}^{2}}} \mathord{\left/ {\vphantom {{{{u}^{2}}} 2}} \right. \kern-0em} 2}$ in the ACC band for 26-year observations

For a more detailed comparison of calculations for different calculated physical parameters, Fig. 4 shows for three different pairs ($\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$, $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$) the difference in linear shifts in the structure of currents with respect to median latitude and corresponding variations in the parameters reduced to one dimension, a velocity. The differences obtained within each pair are due to the fact that one of the parameters differs from the other by a coefficient which changes the weight of the points when calculating the linear regression with a parametric dependence on latitude. It should be emphasized that this coefficient varies with latitude. This effect is illustrated in Fig. 5 which shows latitudinal averages for the studied meridional band and for the entire observation period of $\left| {\nabla \zeta } \right|$, $\left| u \right|$, $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, and $\frac{1}{2}{{u}^{2}}$ distribution. For instance, in the $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair, when calculating the linear regression for $\left\langle {\left| u \right|} \right\rangle$, parameter $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ is preceded by the factor of $\left| {\frac{g}{f}} \right|$ which increases the weight of low-latitude points. In the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ pair, the factor of ${{\left( {\frac{g}{f}} \right)}^{2}}$ increases the weight of low-latitude points even more. In the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair, the factor of $\frac{1}{2}{{\left( {{g \mathord{\left/ {\vphantom {g f}} \right. \kern-0em} f}} \right)}^{2}}\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ increases the weight not only of low-latitude points, but also of points with higher $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$. A comparison within this pair illustrates the influence of the nonlinear (quadratic) relationship between the parameters on the resulting estimated difference. Their absolute value is the highest exactly in this pair. As for other pairs with such relationship, qualitative and quantitative differences in calculation results from the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair are generally negligible.

Fig. 4.

To estimate the difference in linear shifts of the jet structure of currents with respect to median latitude, calculated using different parameters: (a) difference (° latitude) for the pair of parameters $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ (the parameter whose shift is subtracted is indicated second) depending on the median latitude and calculation scale; (b) difference in linear variations in current intensity reduced to the velocity dimension (cm/s) for the same pair of parameters; (c, d) the same calculations for the pair $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$; (e, f) the same for $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$. The rest is as in Fig. 2.

Fig. 5.

Distributions of the ADT gradient module, geostrophic velocity module, half of square of the ADT gradient module, and specific kinetic energy averaged over the studied sector of the Southern Ocean along latitude and over the entire period of observations.

In the $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair, the obtained difference in the shift of the jet structure of the currents is generally small (no more than 0.03° latitude). The difference values are reliable and comparable with the shift in the zone between the SPC and SAC cores, where the difference is negative (in other words, the northward shift in terms of $\left\langle {\left| u \right|} \right\rangle$ is less than in terms of $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, and the absolute shift value generally does not exceed the standard calculation error (Fig. 3a). The area of reliable positive difference in Fig. 4a approximately corresponds to the SPC core, and the difference is much less than the shift. The difference in shifts in the pair of $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ (Fig. 4c) is qualitatively the same as in Fig. 4a, but with an increase in values of up to 0.05° latitude. In the $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair, the region of a reliable positive shift difference corresponds to the SPC core at calculation scales of 3°–5° latitude and the southern periphery of the SPC zone on larger scales (Fig. 4d). In this case, the difference reaches 0.15° latitude. Such values are only 2.5–3 times less than the shift estimate, but greater than the standard calculation error (Fig. 3a); in other words, they are reliable even with respect to the shift. Reliable negative difference values (up to –0.05° latitude) are noted in the SAC southern periphery at small calculation scales.

The difference in variations of $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ parameters reduced to the velocity dimension (Fig. 4b), i.e., in fact, the difference in current intensity variations, is negative in the SAC zone and positive in the SPC, thus being coincident with the signs of changes in these parameters. Hence, variations in terms of $\left\langle {\left| u \right|} \right\rangle$ turn out to be more pronounced in general than in terms of $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$. The reliable negative difference values are also observed in the area of small positive variations in parameters in the boundary zone between the SAC and the SPC on a scale of 10° latitude and more. The obtained absolute difference values are 5–10% of the variations under considerations. These values are much less than the standard error when calculating the variations in these parameters reduced to the velocity dimension (Fig. 3b). The difference in intensity variations in the pair of $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ (Fig. 4d) is qualitatively the same as in Fig. 4b, but with an increase in absolute values from 2 to 5 times. In this case, they can already exceed the standard error when calculating the intensity variations, especially in the southern part of the SPC zone and in the ACC southern periphery on larger scales. The difference in variations of the current intensity calculated in the pair of $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ is positive almost everywhere, except for the northern periphery of the SAC. Absolute values of this difference can locally not only exceed the standard error when calculating the $\left\langle {\left| u \right|} \right\rangle$ variations (Fig. 4b), but also be comparable to the intensity variations calculated in terms of $\left\langle {\left| u \right|} \right\rangle$, especially in the southern part of the SPC zone (Fig. 2e). Extreme values of up to 0.5 and 1 cm/s are observed approximately in the SPC core on a scale of 5° latitude and in the southern periphery of the current on a scale of about 10° latitude, respectively.

Calculations with Respect to the ADT Scale

When constructing the series of h relationships to ζ, we took into account the shift of the ACC jet structure as a whole by 8.3 cm, obtained in [19] in terms $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, in the period of satellite altimetry observations from mid-1993 to mid-2018. These series were adjusted to the ζ ' scale corresponding to the middle of the 1993–2012 interval, in turn corresponding to the averaging interval of the mean dynamic topography MDT CNES-CLS18. Figure 6 shows 26-year series of average annual h relationships to ζ ', the estimated linear shift of the jet structure, and linear variations in the indicated physical parameters for 26-year observations depending on the calculation scale and the median ζ ' value. The calculation was carried out with a step of 0.5 cm along the scale axis and 0.2 cm along the axis of median ζ ' values. All four series of dependences of h on ζ ' obviously show the presence of three well-defined h maximum zones at –25 to –5 cm, –75 to ‒45 cm, and –130 to –115 cm, corresponding to the SAC, SPC, and SthAC. In addition, all time series are marked by one more slight h maximum at –95 to –85 cm. Figure 7 shows the standard calculation error estimated in terms of $\left\langle {\left| u \right|} \right\rangle$.

Fig. 6.

To calculate the shift of the ACC jet structure with respect to ADT: (a, d, g, j) 26-year series of the average annual distributions of $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\left\langle {\left| u \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$, and $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ depending on the ADT ζ' adjusted to the middle of the 1993–2012 interval, for the ACC band to the south of Africa; (b, e, h, k) corresponding linear shifts (cm) of the jet structure of currents depending on the median value of ζ' and calculation scale; dark shading corresponds to areas of calculated points with a shift estimate less than the standard calculation error, light shading corresponds to less than 95% of the probability level (according to Student’s t-test); (c, f, i, l) corresponding variations in current intensity; (c) 10^–3 cm/km, (f) cm/s, (i) 10^–3 cm²/km², and (l) cm²/s². The calculation results for points marked with crosses are given in Table 3.

Fig. 7.

Standard calculation errors in terms of geostrophic velocity depending on the median values of ADT, ζ' and calculation scale: (a) calculation errors of the linear shift (cm) of the jet structure of currents; (b) calculation errors of current velocity variations (cm/s). The rest is as in Fig. 6.

The calculations of the shift in the jet structure of currents, carried out using different physical parameters, yield a qualitatively similar result: the SAC band and the northern part of the SPC are marked by a positive shift, while the southern part of the SPC and the SthAC band, by a southward shift (Figs. 6a, 6d, 6g, 6j; Table 3). Extreme values of the positive shift reach 9 cm and correspond to a zone of low h values between the SAC and SPC cores on a small calculation scale. Meanwhile, quantitative differences in the shift estimates are noted. In particular, the shifts reached 0.0 ± 0.3, 0.1 ± 0.3, 0.9 ± 0.5 and 1.1 ± 0.7 cm for $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\left\langle {\left| u \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle ,$ and $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$, respectively, in the ACC as a whole (Table 3). A smaller negative shift in quadratic parameters is also evident in the southern part of the SPC and in the SthAC band. The patterns of linear h variations are also qualitatively similar: a decrease, on the verge of calculation accuracy, in the intensity of current in the SAC band and an increase in the SPC and SthAC bands with extreme values in the SPC core. As in the case of the dependence on latitude, the intensity variations is 5–10% of its average values over 26 years.

Table 3.   Estimates of the shift of the jet structure of the ACC with respect to the ADT adjusted to middle of the interval 1993–2012 and variations in $\left| {\nabla \zeta } \right|$, $\left| u \right|$, $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, and ${{{{u}^{2}}} \mathord{\left/ {\vphantom {{{{u}^{2}}} 2}} \right. \kern-0em} 2}$ in the ACC band for 26-year observations

For a more detailed comparison of the calculations, Fig. 8 shows the differences in shifts of the jet structure of currents with respect to the median ζ ' values calculated using pairs of different physical parameters. The reliable difference in shifts in the $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair is observed mainly in the SPC and ACC zones as a whole on calculation scales of more than 50 cm (Fig. 8a). In this case, the difference values are positive and reach 0.15 cm, indicating a larger shift when calculated using $\left\langle {\left| u \right|} \right\rangle$. These differences and estimated absolute values of the shifts calculated in terms of these two parameters (Figs. 6b, 6e) are generally significantly less than the shift calculation accuracy (Fig. 7a). Figure 8a shows the area (scales 140–180 cm, median ADT from –30 to –60 cm) where the absolute shift difference exceeds the estimated shift error. However, the calculations in this area partially include zones located outside the ACC. In terms of quality, the patterns of the differences in shifts calculated from pairs of $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ (Fig. 8c), $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ (Fig. 8e) are the same, but with some expansion of the range of reliable values and with an increase in absolute values. In the first case, the increase is approximately twofold, due to which these values become comparable to the standard shift error (Fig. 7a), although smaller than it. In the second case, the increase is more than 10 times, with extreme values of about 1.8 cm in the SPC band, respectively. Such values substantially exceed the standard shift error (Fig. 7a).

Fig. 8.

To estimate the difference in linear shifts of the jet structure of currents with respect to the median ADT ζ' value, calculated using different parameters: (a) difference (cm) for the pair of parameters $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ depending on the median ζ' values and calculation scale; (b) difference in linear variations in current intensity, reduced to the velocity dimension (cm/s), for the same pair of parameters; (c, d) the same calculations for $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$; (e, f) the same for $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$. The rest is as in Fig. 6.

The patterns of differences in linear variations in current intensity calculated from $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ (Fig. 8b), $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ pairs (Fig. 8d) and reduced to the dimension of velocity are basically similar to each other in qualitative terms. Positive, on the verge of calculation accuracy, difference values are observed in the zone between the SPC and SAC cores, and reliable (up to 0.06 cm/s) values are observed approximately in the SPC core on calculation scale of up to 100 cm. Negative reliable difference values correspond to the southern periphery of the SPC and the SthAC zone. When calculating in the $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair (Fig. 8b), there is another region of negative reliable values, approximately corresponding to the ACC band as a whole. The absolute values of the difference are approximately 2 times greater when calculated using the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$ pair. In both calculations, the difference estimates are much smaller than the standard error in calculating the intensity variations (Fig. 7b). The difference in linear intensity variations calculated in the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair (Fig. 8f) is qualitatively and in many ways similar to the previous two cases, but with considerably higher difference values (up to 0.4 cm/s and up to 30% of the absolute values of variation on small scales of calculation). In this case, only reliable positive values are noted: they correspond to the southern part of the SAC and the entire SPC band in a wide range of calculation scales. These values can significantly exceed the standard error when calculating the variations itself (Fig. 7b).

DISCUSSION AND CONCLUSIONS

The linear meridional shift of the jet structure and current intensity variations in the ACC sector to the south of Africa (10° E–25° W) were estimated and compared in the satellite altimetry observation period of 1993–2018. The estimates were obtained based on the parameters derived from the ADT gradient module: the ADT gradient module itself $\left| {\nabla \zeta } \right|$, module of the velocity of geostrophic current $\left| u \right|$ on the ocean surface, half of square of the ADT gradient $\frac{1}{2}{{\left| {\nabla \zeta } \right|}^{2}}$, and specific kinetic energy $\frac{1}{2}{{u}^{2}}$. The comparison was based on the method described in [19] and applied therein to the ADT gradient module field. The method was improved in the course of the present study in terms of calculating error estimates of the parameters derived from the ADT gradient module and with respect to taking into account the estimated data errors. In addition, the estimation interval was changed from mid-1993–mid-2018 to the entire interval of 1993–2018.

The 26-year time series of average annual values of parameters $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\left\langle {\left| u \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$, and $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ averaged in the ACC sector under study along latitude, are divided into three zones with higher values of these parameters conventionally corresponding to the SAC, SPC, and SthAC bands. When averaging along isohypses, four such zones are identified. An additional fourth zone is noted in the southern part of the SPC band. The meridional shift of the jet structure of currents (displacements of the annual latitude-averaged curves of the listed parameters) along latitude and the corresponding variations in the parameters (actually current intensity variations) over 26-year observations, when calculated using each of the four parameters, are similar in nature: the SAC band is marked by a southward shift and a decrease in intensity, while the SPC and SthAC are characterized by a northward shift and an increase in intensity; as for the ACC as a whole, shift to the north and an intensity increase are slight (on the verge of assessing accuracy). When calculating with respect to the ADT scale adjusted to the middle of the 1993–2012 interval, the patterns of shift and intensity variations are also similar for each of the four parameters. A positive shift is noted in the zone between the SAC and SPC cores, as well as in the northern part of the SPC, whereas a negative shift is observed in the southern part of the SPC and in the SthAC. The intensity variations are positive in the SPC and SthAC and slight, negative in the SAC. The above-mentioned similarity is regular for $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ pairs, whereas such a similarity is not a priori obvious for pairs of parameters that depend linearly and quadratically on the ADT gradient module. Time variation in the linear parameters does not necessarily lead to a variation in the same sign of the quadratic ones.

In quantitative terms, the absolute shift values of the jet structure of currents in the ACC band and current intensity variations in the studied sector of the Southern Ocean generally increase from those linearly dependent on the ACC gradient module to quadratically dependent ones. Shifts in the ACC jet structure over 26 years with respect to the initial unadjusted ADT scale look very indicative: 8.6 ± 0.3 cm when calculated through $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$, 8.8 ± 0.3 cm through $\left\langle {\left| u \right|} \right\rangle$, 9.7 ± 0.5 cm through $~\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$, and 9.9 ± 0.7 cm through $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$. While the first two values are close up to almost tenths of a centimeter to the available estimates of the World Ocean level growth over the same period, the other two significantly exceed it.

The quantitative differences between the calculations in terms of $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ are generally much smaller than the calculation errors of each of these parameters, although the differences in shifts of the jet structure and current intensity variations can be reliable. Thus, a known quantitative equivalence of calculations can be suggested directly for the ACC band in the sector to the south of Africa in terms of $\left\langle {\left| u \right|} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$. However, when the areas located outside ACC are included in the calculation, such equivalence may no longer be observed. The quantitative differences already for the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\frac{1}{2}\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle$ pair in the ACC band, although less than the calculation errors for each of these parameters, can locally be relatively comparable with them. The differences for the $\frac{1}{2}\left\langle {{{u}^{2}}} \right\rangle$ and $\left\langle {\left| {\nabla \zeta } \right|} \right\rangle$ pair can already be very significant, many times greater than the calculation error of these parameters, and even comparable to the shifts and current intensity variations. Hence, the calculations in these pairs are not quantitatively equivalent.

Appendices

APPENDIX 1

Estimated Calculation Errors of the Linear Shift of the ADT Gradient Structure

The total calculation error consists of the error in the calculation procedure error and data errors. To derive formulas for estimating the total calculation errors of the coefficient k(x, y) and the free term b(x, y), they are considered as N × L functions of independent variables ${{h}_{{i,l}}}$ which can be understood as ${{\left\langle {\left| {\nabla \zeta } \right|} \right\rangle }_{{i,l}}}$, ${{\left\langle {\left| u \right|} \right\rangle }_{{i,l}}}$, $\frac{1}{2}{{\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle }_{{i,l}}}$, and $\frac{1}{2}{{\left\langle {{{u}^{2}}} \right\rangle }_{{i,l}}}$, where $u = {{g\nabla \zeta } \mathord{\left/ {\vphantom {{g\nabla \zeta } {2\Omega \sin \varphi }}} \right. \kern-0em} {2\Omega \sin \varphi }}$ is the geostrophic current velocity on the ocean surface. The squared total calculation error can be represented as several terms corresponding to the calculation stages of x and y distributions based on the h data, the final k and b calculation stage based on x and y distributions, as well as the contribution of data errors and h_i,l calculations based on the initial daily data and the mean dynamic topography. The total squared calculation error of the coefficient k(x, y) has the following form:

$${{\delta }^{2}}\left( {k\left( {x,y} \right)} \right) = \delta _{k}^{2} + \delta _{{{{k}_{x}}}}^{2} + \delta _{{{{k}_{y}}}}^{2} + \delta _{{{{k}_{e}}}}^{2} + \delta _{{{{k}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} + \delta _{{{{k}_{{\hat {\zeta }}}}}}^{2},$$

(A1.1)

where

$$\begin{gathered} \delta _{k}^{2} = \frac{1}{{L\frac{{\Delta T}}{\tau }N\frac{{\Delta a}}{A} - 2}} \\ \times \,\,\frac{{\sum\limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{y}_{i}} - k\left( {x,y} \right){{x}_{i}} - b\left( {x,y} \right)} \right)}}^{2}}{{w}_{i}}} }}{{\sum\limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{i}} - \tilde {x}} \right)}}^{2}}{{w}_{i}}} }}, \\ \end{gathered} $$

(A1.2)

$$\delta _{{{{k}_{x}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N k_{{{{x}_{i}}}}^{2}\delta _{{{{x}_{i}}}}^{2},$$

(A1.3)

$$\delta _{{{{k}_{y}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N k_{{{{y}_{i}}}}^{2}\delta _{{{{y}_{i}}}}^{2},$$

(A1.4)

$$\delta _{{{{k}_{e}}}}^{2} = \frac{A}{{\Delta a}}\frac{\tau }{{\Delta T}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L k_{{{{h}_{{{\kern 1pt} i,l}}}}}^{2}\delta _{{{{e}_{{i,l}}}}}^{2},$$

(A1.5)

$$\delta _{{{{k}_{{\zeta {\kern 1pt} '}}}}}^{2} = \frac{1}{{R_{0}^{2}}}{{\left( {\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{k}_{{h{{{\kern 1pt} }_{{i,l}}}}}}{{\gamma }_{{i,l}}}} \right)}^{2}}{{\delta }^{2}}\zeta {\kern 1pt}^{ '},$$

(A1.6)

$$\delta _{{{{k}_{{\hat {\zeta }}}}}}^{2} = \frac{1}{{R_{0}^{2}}}{{\left( {\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{k}_{{{{h}_{{{\kern 1pt} i,l}}}}}}{{\gamma }_{{i,l}}}} \right)}^{2}}{{\delta }^{2}}\hat {\zeta }.$$

(A1.7)

In formula (A1.2), ${{\delta }_{k}}$ is the standard error of calculating k from x and y distributions, which is based on the estimated residual variance; the tilde means averaging over index i; ${{w}_{i}}$ are the weighting coefficients, as indicated in the comments to formula (3) in the main text of this paper. In formulas (A1.3) and (A1.4), $\delta _{{{{k}_{x}}}}^{2}$ and $\delta _{{{{k}_{y}}}}^{2}$ are contributions of x and y calculation errors based on the h data to the squared total calculation error; ${{k}_{{{{x}_{i}}}}}$ and ${{k}_{{{{y}_{i}}}}}$ are partial derivatives in terms of x_i and y_i; $\delta _{{{{x}_{i}}}}^{2}$ and $\delta _{{{{y}_{i}}}}^{2}$ are estimates (for each i-th value of a) of the squared standard errors of average (over the full time observation interval) h values and linear variations in h, respectively. In formula (A1.5), $\delta _{{{{k}_{e}}}}^{2}$ is the contribution of h_i,l calculation error based on the initial daily data; ${{\delta }_{{{{e}_{{i,l}}}}}}$ is the root-mean-square h deviation per year for each i, l pair; ${{k}_{{{{h}_{{{\kern 1pt} i,l}}}}}}$ is the partial derivative in terms of h_i,l. In formula (A1.6), $\delta _{{{{k}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2}$ is the data error contribution in terms of the sea level anomaly; $\delta \zeta {\kern 1pt}^ {{'}}$ is the value of this error, by which we mean the root-mean-square value of the formal mapping error for the entire observation period and over the entire calculation band. In formula (A1.7), $\delta _{{{{k}_{{\hat {\zeta }}}}}}^{2}$ is the contribution of the mean dynamic topography data error; $\delta \hat {\zeta }$ is the estimate of this error, understood as its root-mean-square value over the entire calculation band. In formulas (A1.6) and (A1.7), ${{\gamma }_{{i,l}}}$ depends on which variable h is used (Table A.1). The A and τ scales correspond to single measurements along the argument axis and in time, ∆a is the step along the argument axis, ΔT = 365.25 is the number of days equivalent to the step along the time axis. On the map, a single measurement area corresponds to a “square” formed by adjacent satellite tracks (Fig. 1 in the main text of this paper). In the studied sector of the Southern Ocean, the linear size R of such “square” is estimated at 167 km. If J is the number of “squares” which occur in the current band on the map, then, when averaging in a specific direction (along latitude or isohypse), the scale of a single measurement A along the argument axis is estimated as the ratio of the band width to J. Hence, $L\frac{{\Delta T}}{\tau }N\frac{{\Delta a}}{A}$ is the total number of single measurements in the current band. In the studied sector of the Southern Ocean, according to [19], the step is ∆$a =$ 0.25° latitude and the scale is A = 0.1° latitude for $a = \varphi$; the step is ∆$a = 0.2$ cm and the scale is A = 1 cm for $a =$ ζ. In formulas (A1.6) and (A1.7), ${{R}_{0}}$ is a smaller linear size of the calculated current band on the map. In particular, in the studied sector of the Southern Ocean with a width of 35° longitude, the smaller size for $a = \varphi$ corresponds to the width of the band between marginal latitudes, and for $a =$ ζ, to the width of the band between marginal isohypses, based on the fact that 1 cm of ADT is on average equivalent to 0.1° latitude.

Table A.1.  Correspondence between a variable h and γ in formulas (A1.6) and (A1.7)

${{h}_{{i,l}}}$	${{\gamma }_{{i,l}}}$
Module of the sea level gradient, ${{\left\langle {\left| {\nabla \zeta } \right|} \right\rangle }_{{i,l}}}$	1
module of geostrophic velocity, ${{\left\langle {\left| u \right|} \right\rangle }_{{i,l}}}$	$\frac{g}{{2\Omega \sin {{\varphi }_{i}}}}$
Half of square of sea level gradient, $\frac{1}{2}{{\left\langle {{{{\left| {\nabla \zeta } \right|}}^{2}}} \right\rangle }_{{i,l}}}$	$\sqrt {2{{h}_{{i,l}}}}$
Specific kinetic energy, $\frac{1}{2}{{\left\langle {{{u}^{2}}} \right\rangle }_{{i,l}}}$	$\frac{g}{{2\Omega \sin {{\varphi }_{i}}}}\sqrt {2{{h}_{{i,l}}}}$

1. g is the acceleration of gravity, $2\Omega \sin {{\varphi }_{i}}$ is the Coriolis parameter for latitude φ_i. Angle brackets indicate averaging over a selected direction on the ocean surface (along latitude or isohypses) and over a year.

To calculate the total error of the free term b(x, y), formulas (A1.1, A1.3–A1.7) are given with complete similarity to k replaced by b, and formula (A1.2) is replaced by:

$$\delta _{b}^{2} = \delta _{k}^{2}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N x_{i}^{2}{{w}_{i}}.$$

(A1.8)

In formulas (A1.3 and A1.4), for the coefficient k(x, y), the derivatives with respect to x_i and y_i are as follows:

$$\begin{gathered} {{k}_{{{{x}_{i}}}}} = \frac{{{{w}_{i}}}}{{\sum\limits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{j}} - \tilde {x}} \right)}}^{2}}{{w}_{j}}} }} \\ \times \,\,\left( {\left( {{{y}_{i}} - \tilde {y}} \right) - 2k\left( {x,y} \right)\left( {{{x}_{i}} - \tilde {x}} \right)} \right), \\ \end{gathered} $$

(A1.9)

$${{k}_{{{{y}_{i}}}}} = \frac{{{{w}_{i}}}}{{\sum\limits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{j}} - \tilde {x}} \right)}}^{2}}{{w}_{j}}} }}\left( {{{x}_{i}} - \tilde {x}} \right).$$

(A1.10)

For free terms b(x, y),

$${{b}_{{{{x}_{i}}}}} = - {{k}_{{{{x}_{i}}}}}\tilde {x} - k\left( {x,y} \right){{w}_{i}},$$

(A1.11)

$${{b}_{{{{y}_{i}}}}} = {{w}_{i}} - {{k}_{{{{y}_{i}}}}}\tilde {x}.$$

(A1.12)

The squared error $\delta _{{{{x}_{i}}}}^{2}$ has the following form:

$$\begin{gathered} \delta _{{{{x}_{i}}}}^{2} = \frac{1}{{{{{\left( {2L\Delta a} \right)}}^{2}}}}\frac{\tau }{{\Delta T}} \\ \times \,\,\mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{\left( {\left( {{{h}_{{i{\kern 1pt} + {\kern 1pt} 1,l}}} - {{h}_{{i{\kern 1pt} - {\kern 1pt} 1,l}}}} \right) - \left( {{{{\bar {h}}}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{{\bar {h}}}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right)} \right)}^{2}}. \\ \end{gathered} $$

(A1.13)

In this formula, for i = 1, it is necessary to replace i – 1 by 1, and for i = N, i + 1 is replaced by N, where N is the total number of the argument values a, and 2∆a is replaced by ∆a. The same should be done for the rest of the formulas below, when the index i goes to edges of the set. The overbar in formula (A1.13) means averaging over index l. The squared error $\delta _{{{{y}_{i}}}}^{2}$ is based on the estimated residual variance:

$$\delta _{{{{y}_{i}}}}^{2} = \frac{1}{{L\frac{{\Delta T}}{\tau } - 2}}\frac{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{h}_{{i,l}}} - k\left( {t,{{h}_{i}}} \right){{t}_{l}} - b\left( {t,{{h}_{i}}} \right)} \right)}}^{2}}} }}{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{l}} - \bar {t}} \right)}}^{2}}} }},$$

(A1.14)

where ${{t}_{l}} = \frac{l}{L}$, $L\frac{{\Delta T}}{\tau }$ is the total number of single measurements over 26 years for argument a. To derive the formula for calculating the total data error contribution, the total differential of the coefficient k is written in terms of total differentials of variables x_i and y_i:

$$dk\left( {x,y} \right) = \mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \left( {{{k}_{{{{x}_{i}}}}}d{{x}_{i}} + {{k}_{{{{y}_{i}}}}}d{{y}_{i}}} \right).$$

(A1.15)

Differentials $d{{x}_{{i,l}}}$ and $d{{y}_{{i,l}}}~$ can be written in terms of those of a single ADT measurement:

$$d{{x}_{i}} = \frac{1}{{2L\Delta a}}\mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L \left( {d{{h}_{{i{\kern 1pt} + {\kern 1pt} 1,l}}} - d{{h}_{{i{\kern 1pt} - {\kern 1pt} 1,l}}}} \right),$$

(A1.16)

$$d{{y}_{i}} = dk\left( {t,{{h}_{i}}} \right) = \left( {1 - \frac{1}{L}} \right)\frac{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {\left( {{{t}_{l}} - \bar {t}} \right)d{{h}_{{i,l}}}} }}{{\sum\limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{l}} - \bar {t}} \right)}}^{2}}} }}.$$

(A1.17)

Then

$$\begin{gathered} dk\left( {x,y} \right) = \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L \mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \left( {\frac{{{{k}_{{{{x}_{i}}}}}}}{{2L\Delta a}}{{{\left( {d{{h}_{{i{\kern 1pt} + {\kern 1pt} 1,l}}} - d{{h}_{{i{\kern 1pt} - {\kern 1pt} 1,l}}}} \right)}}^{{^{{^{{^{{^{{^{{^{{^{{}}}}}}}}}}}}}}}}}} \right. \\ \left. { + \,\,{{k}_{{{{y}_{i}}}}}\left( {1 - \frac{1}{L}} \right)\frac{{\left( {{{t}_{l}} - \bar {t}} \right)d{{h}_{{i,l}}}}}{{\sum\limits_{m{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{m}} - \bar {t}} \right)}}^{2}}} }}} \right). \\ \end{gathered} $$

(A1.18)

The equation under the sum signs in A1.16 for each i depends on dh at three successive steps i – 1, i, and i + 1. To adjust A1.16 to a form that depends only on $d{{h}_{{i,l}}}$ at each step i, we take into account the fact that the following equality is valid for arbitrary variables U and V in a discrete representation: $\sum\nolimits_{i{\kern 1pt} = {\kern 1pt} - \infty }^{ + \infty } {{{U}_{i}}} \left( {{{V}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{V}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right) \equiv$ $- \sum\nolimits_{i{\kern 1pt} = {\kern 1pt} - \infty }^{ + \infty } {\left( {{{U}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{U}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right){{V}_{i}}}$. Then, neglecting the discrepancy at edges of the set and the difference between weighting coefficients w at three successive steps i – 1, i, and i + 1, we can write the finite set as follows:

$$dk\left( {x,y} \right) = \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L \mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{k}_{{{{h}_{{{\kern 1pt} i,l}}}}}}d{{h}_{{i,l}}},$$

(A1.19)

where

$$\begin{gathered} {{k}_{{{{h}_{{{\kern 1pt} i,l}}}}}} = \frac{{{{w}_{i}}}}{{\sum\limits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{j}} - \tilde {x}} \right)}}^{2}}{{w}_{j}}} }} \\ \times \,\,\left( {\frac{1}{{2L\Delta a}}{{{\left( {2k\left( {x,y} \right)\left( {{{x}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{x}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right) - \left( {{{y}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{y}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right)} \right)}}^{{^{{^{{^{{^{{^{{^{{}}}}}}}}}}}}}}}} \right. \\ \left. { + \,\,\left( {1 - \frac{1}{L}} \right)\left( {{{x}_{i}} - \tilde {x}} \right)\frac{{\left( {{{t}_{l}} - \bar {t}} \right)}}{{\sum\limits_{m{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{m}} - \bar {t}} \right)}}^{2}}} }}} \right). \\ \end{gathered} $$

(A1.20)

It should be taken into account that variables ${{h}_{{i,l}}}$ are dependent in the part determined by the time-independent mean ADT. The total differential $d{{h}_{{i,l}}}~$ corresponds to the error, the square of which can be written as the sum of three terms:

$$\delta _{{{{h}_{{{\kern 1pt} i,l}}}}}^{2} = \delta _{{{{e}_{{i,l}}}}}^{2} + \delta _{{h_{{{\kern 1pt} i,l}}^{{{'}}}}}^{2} + \delta _{{{{{\hat {h}}}_{{{\kern 1pt} i,l}}}}}^{2},$$

(A1.21)

where ${{\delta }_{{{{e}_{{i,l}}}}}}$ is the error in the procedure for calculating ${{h}_{{i,l}}}$ in terms of initial daily data, ${{\delta }_{{h_{{{\kern 1pt} i,l}}^{{{'}}}}}}$ and ${{\delta }_{{{{{\hat {h}}}_{{{\kern 1pt} i,l}}}}}}$ are data errors of the sea level anomaly gradients and the mean ADT. Thus, the formula (A1.5) is obtained.

As for the contribution of the sea level anomaly error to the error in calculating the free term $b\left( {x,y} \right)$, the formula is analogous to (A1.5) with ${{k}_{{{{h}_{{{\kern 1pt} i,l}}}}}}$ replaced by ${{b}_{{{{h}_{{{\kern 1pt} i,l}}}}}}$, where

$$\begin{gathered} {{b}_{{{{h}_{{{\kern 1pt} i,l}}}}}} = \frac{{{{w}_{i}}}}{{\sum\limits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {{{x}_{j}} - \tilde {x}} \right)}}^{2}}{{w}_{j}}} }} \\ \times \,\,\left( {\frac{{\tilde {x}}}{{2L\Delta a}}{{{\left( {2k\left( {x,y} \right)\left( {{{x}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{x}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right) - \left( {{{y}_{{i{\kern 1pt} + {\kern 1pt} 1}}} - {{y}_{{i{\kern 1pt} - {\kern 1pt} 1}}}} \right)} \right)}}^{{^{{^{{^{{^{{^{{}}}}}}}}}}}}}} \right. \\ + \,\,\left( {1 - \frac{1}{L}} \right)\left( {\left( {{{x}_{i}} - \tilde {x}} \right)\tilde {x} - \mathop \sum \limits_{j{\kern 1pt} = {\kern 1pt} 1}^N {{{\left( {{{x}_{j}} - \tilde {x}} \right)}}^{2}}{{w}_{j}}} \right) \\ \left. { + \,\,\frac{{\left( {{{t}_{l}} - \bar {t}} \right)}}{{\sum\limits_{m{\kern 1pt} = {\kern 1pt} 1}^L {{{{\left( {{{t}_{m}} - \bar {t}} \right)}}^{2}}} }}} \right). \\ \end{gathered} $$

(A1.22)

The data error of the mean ADT gradient for each variable ${{h}_{{i,l}}}~$ is as follows:

$${{\delta }_{{{{{\hat {h}}}_{{{\kern 1pt} i,l}}}}}} = {{\gamma }_{{i,l}}}\frac{{\delta \hat {\zeta }}}{{{{R}_{0}}}}.$$

(A1.23)

This expression of the error is based on the fact that it should not depend on the scale of a single measurement of the sea level anomaly on the map, because when constructing the mean ADT, different data unrelated to satellite altimetry measurements were used. In this case, the linear scale of the calculation band becomes the unit scale. On the maps, the calculation band for $a = \varphi$ corresponds to a trapezoid. Its linear scales are determined by the distance along the meridian between extreme latitudes, and along the latitude by the length of the midline. Choosing the smaller of these two scales, i.e., ${{R}_{0}}$, gives an estimate of the ADT gradient error “from above”. Taking into account the quasi-zonality of isohypses in the studied ocean area, the calculation band for $a =$ ζ is also approximated by a trapezoid. In the described approach, the data error of the mean ADT gradient within the calculation band depends on index i. Taking into account also the time independence of this error, equivalent to its dependence on the index l for the entire observation period, we obtain formula (A1.7).

As for the data error of the sea level anomaly gradient, the formula is similar to (A1.23):

$${{\delta }_{{h_{{{\kern 1pt} i,l}}^{{{'}}}}}} = {{\gamma }_{{i,l}}}\frac{{\delta \zeta {\kern 1pt}^{ '}}}{{{{R}_{0}}}}.$$

(A1.24)

In this case, we assume that the formal mapping error which we interpret as the data error of the sea level anomaly, as well as the mean ADT error, is a continuous function of coordinates in a discrete representation. In addition, it is also a continuous time function. With this approach, it seems possible to consider it dependent on both indices i and l over the entire calculation band and for the entire observation period. Thus, the formula (A1.6) is obtained. It should be noted that the data error of the sea level anomaly gradient can be considered through the scale R of a single measurement of the sea level anomaly on the map and the sea level anomaly error $\delta \zeta _{{{\kern 1pt} i,l}}^{{{'}}}$ for each i, l pair, as we did in [4]. Then, the contribution of the data error of the sea level anomaly, instead of formula (A1.6), is written as

$$\delta _{{{{k}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} = \frac{A}{{{{R}^{2}}}}\frac{1}{{\Delta a}}\frac{\tau }{{\Delta T}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L k_{{{{h}_{{{\kern 1pt} i,l}}}}}^{2}\gamma _{{i,l}}^{2}{{\delta }^{2}}\zeta _{{{\kern 1pt} i,l}}^{{{'}}}.$$

(A1.25)

In this case, the error does not depend on the absolute value of R, but on the ratio A/R ² (unit measurement along the a-axis to the squared unit measurement of R on the map) based solely on the configuration of the analyzed area on the maps and the choice of a.

According to [12], the formal mapping error δζ ' is less than 10% of the signal variance (<2 cm for the Southern Ocean). It is this estimate that we take to calculate ${{\delta }_{{{{k}_{\zeta }}}}}$, while assuming the constancy of δζ ' in time and over the ocean surface. For the CNES-CLS09 version of the mean ADT, $\delta \hat {\zeta }$ is estimated to be 1–2 cm in the Southern Ocean (Fig. 13 in [16]). Based on these data and assuming the $\delta \hat {\zeta }$ constancy over the ocean surface, we can take $\delta \hat {\zeta }$ to be on average 1.5 cm.

The formulas derived above are valid if the step along the a axis is no less than that along the latitude of the initial geographic grid. Nevertheless, these formulas may also be relatively applicable for a smaller step along the a axis if we minimize the additional variance of disturbances in the distributions ${{h}_{{i,l}}}~$ that arises when interpolating to a finer grid. This can be done, e.g., by small-scale smoothing of ${{h}_{{i,l}}}~$ distributions, although this approach leads to some overestimations of the error. In particular, in this study, the distributions of the dependences of h on ζ were smoothed along the ζ axis using a sliding cosine filter (Tukey filter) with a base of 2.5 cm. In our case, this scale is approximately equivalent to a latitude step of 0.25° of the initial geographic grid.

APPENDIX 2

Estimated Error of the Difference in Linear Shifts of the ADT Gradient Structure Obtained on the Basis of Different Physical Values

The total squared error of the difference in estimated shifts Δk = –(k1(x1, y1) – k2(x2, y2)) obtained from different physical values of h1 and h2, taking into account the mutual dependence of these values, is represented as

$${{\delta }^{2}}\left( {\Delta k} \right) = \delta _{{\Delta k}}^{2} + \delta _{{\Delta {{k}_{x}}}}^{2} + \delta _{{\Delta {{k}_{y}}}}^{2} + \delta _{{\Delta {{k}_{e}}}}^{2} + \delta _{{\Delta {{k}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} + \delta _{{\Delta {{k}_{{\hat {\zeta }}}}}}^{2},$$

(A2.1)

where

$${{\delta }_{{\Delta k}}} = {{\delta }_{{k1}}} - {{\delta }_{{k2}}},$$

(A2.2)

$$\delta _{{\Delta {{k}_{x}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{\left( {k{{1}_{{x{{1}_{i}}}}}{{\delta }_{{x{{1}_{i}}}}} - k{{2}_{{x{{2}_{i}}}}}{{\delta }_{{x{{2}_{i}}}}}} \right)}^{2}},$$

(A2.3)

$$\delta _{{\Delta {{k}_{y}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{\left( {k{{1}_{{y{{1}_{i}}}}}{{\delta }_{{y{{1}_{i}}}}} - k{{2}_{{y{{2}_{i}}}}}{{\delta }_{{y{{2}_{i}}}}}} \right)}^{2}},$$

(A2.4)

$$\delta _{{\Delta {{k}_{e}}}}^{2} = \frac{A}{{\Delta a}}\frac{\tau }{{\Delta T}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{\left( {k{{1}_{{h{{1}_{{i,l}}}}}}{{\delta }_{{e{{1}_{{i,l}}}}}} - k{{2}_{{h{{2}_{{{\kern 1pt} i,l}}}}}}{{\delta }_{{e{{2}_{{{\kern 1pt} i,l}}}}}}} \right)}^{2}},$$

(A2.5)

$$\delta _{{\Delta {{k}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} = \frac{1}{{R_{0}^{2}}}{{\left( {\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L k{{1}_{{h{{1}_{{{\kern 1pt} i,l}}}}}}\gamma {{{\text{1}}}_{{{\kern 1pt} i,l}}} - k{{2}_{{h{{2}_{{{\kern 1pt} i,l}}}}}}\gamma {{{\text{2}}}_{{{\kern 1pt} i,l}}}} \right)}^{2}}{{\delta }^{2}}\zeta {\kern 1pt}^{ '},$$

(A2.6)

$$\delta _{{\Delta {{k}_{{\hat {\zeta }}}}}}^{2} = \frac{1}{{R_{0}^{2}}}{{\left( {\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L k{{1}_{{h{{1}_{{i,l}}}}}}\gamma {{{\text{1}}}_{{i,l}}} - k{{2}_{{h{{2}_{{{\kern 1pt} i,l}}}}}}\gamma {{{\text{2}}}_{{{\kern 1pt} i,l}}}} \right)}^{2}}{{\delta }^{2}}\hat {\zeta }.$$

(A2.7)

The dimension and values of b(x, y) are different for each physical h value. To compare them, they are reduced to one dimension, e.g., to the geostrophic velocity module, by dividing by some dimensional parameter β corresponding to the median ${{a}_{0}}$ value for the calculated a range. For each physical value, the parameter β is indicated in Table 1 in the main text of the paper. Already for the reduced values, we can write the total squared error of the difference $\Delta b = \frac{{b1\left( {x1,y1} \right)}}{{\beta 1}} - \frac{{b2\left( {x2,y2} \right)}}{{\beta 2}}$:

$${{\delta }^{2}}\left( {\Delta b} \right) = \delta _{{\Delta b}}^{2} + \delta _{{\Delta {{b}_{x}}}}^{2} + \delta _{{\Delta {{b}_{y}}}}^{2} + \delta _{{\Delta {{b}_{e}}}}^{2} + \delta _{{\Delta {{b}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} + \delta _{{\Delta {{b}_{{\hat {\zeta }}}}}}^{2},$$

(A2.8)

where

$${{\delta }_{{\Delta b}}} = \frac{{{{\delta }_{{b1}}}}}{{\beta 1}} - \frac{{{{\delta }_{{b2}}}}}{{\beta 2}},$$

(A2.9)

$$\delta _{{\Delta {{b}_{x}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{\left( {\frac{1}{{\beta 1}}b{{1}_{{x{{1}_{i}}}}}{{\delta }_{{x{{1}_{i}}}}} - \frac{1}{{\beta 2}}b{{2}_{{x{{2}_{i}}}}}{{\delta }_{{x{{2}_{i}}}}}} \right)}^{2}},$$

(A2.10)

$$\delta _{{\Delta {{b}_{y}}}}^{2} = \frac{A}{{\Delta a}}\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{\left( {\frac{1}{{\beta 1}}b{{1}_{{y{{1}_{i}}}}}{{\delta }_{{y{{1}_{i}}}}} - \frac{1}{{\beta 2}}b{{2}_{{y{{2}_{i}}}}}{{\delta }_{{y{{2}_{i}}}}}} \right)}^{2}},$$

(A2.11)

$$\begin{gathered} \delta _{{\Delta {{b}_{e}}}}^{2} = \frac{A}{{\Delta a}}\frac{\tau }{{\Delta T}} \\ \times \,\,\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{\left( {\frac{1}{{\beta 1}}b{{1}_{{h{{1}_{{i,l}}}}}}{{\delta }_{{e{{1}_{{i,l}}}}}} - \frac{1}{{\beta 2}}b{{2}_{{h{{2}_{{i,l}}}}}}{{\delta }_{{e{{2}_{{i,l}}}}}}} \right)}^{2}}, \\ \end{gathered} $$

(A2.12)

$$\begin{gathered} \delta _{{\Delta {{b}_{{\zeta {\kern 1pt}^{ '}}}}}}^{2} = \frac{A}{{{{R}^{2}}}}\frac{1}{{\Delta a}}\frac{\tau }{{\Delta T}} \\ \times \,\,\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N \mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L {{\left( {\frac{1}{{\beta 1}}b{{1}_{{h{{1}_{{i,l}}}}}}\gamma {{{\text{1}}}_{{i,l}}} - \frac{1}{{\beta 2}}b{{2}_{{h{{2}_{{i,l}}}}}}\gamma {{{\text{2}}}_{{i,l}}}} \right)}^{2}}{{\delta }^{2}}\zeta _{{i,l}}^{{{'}}}, \\ \end{gathered} $$

(A2.13)

$$\begin{gathered} \delta _{{\Delta {{b}_{{\hat {\zeta }}}}}}^{2} = \frac{A}{{{{R}^{2}}}}\frac{1}{{\Delta a}} \\ \times \,\,\mathop \sum \limits_{i{\kern 1pt} = {\kern 1pt} 1}^N {{\left( {\mathop \sum \limits_{l{\kern 1pt} = {\kern 1pt} 1}^L \left( {\frac{1}{{\beta 1}}b{{1}_{{h{{1}_{{i,l}}}}}}\gamma {{{\text{1}}}_{{i,l}}} - \frac{1}{{\beta 2}}b{{2}_{{h{{2}_{{i,l}}}}}}\gamma {{{\text{2}}}_{{i,l}}}} \right)} \right)}^{2}}{{\delta }^{2}}{{{\hat {\zeta }}}_{i}}. \\ \end{gathered} $$

(A2.14)