Symmetrical and Asymmetrical Rectiﬁcations Employed for Deeper Ocean Extrapolations of In Situ CTD Data and Subsequent Sound Speed Proﬁles

: The multinational Argo program, which was initiated in 1999, has completed its global requirement of 3000 ﬂoats deployed by 2007. This program has revolutionized ocean observations with the provision of varying data in the upper half of the ocean. However, various studies have reiterated the requirement for deep ocean coverage, since the ocean below 2000 meters (m) is warming. In this regard, full-depth studies are mandatory in order to estimate the rising sea level due to thermal expansion and analyze critical parameters of deep ocean circulation sub 2000 m; further, data below 2000 m are mandatory for multifarious model simulations. As a landmark initiative, in mid-2015, the “Deep Argo Implementation Workshop” was held in Hobart. An array comprising 1228 ﬂoats was suggested by G. C. Johnson, rendering coverage of 5 ◦ latitude × 5 ◦ longitude × 15-day cycles. This was conclusively agreed to be an a ﬀ ordable solution for varying scientiﬁc needs for assessing data in abyssal oceans. Thence, Deep New proﬁlINg ﬂoat of JApan (NINJA) and Deep Arvor ﬂoats were developed by Japan and France, respectively, to cover depths of 0–4000 m. Similarly, Deep Autonomous Proﬁling Explorer (APEX) and Deep Sounding Oceanographic Lagrangian Observer (SOLO) by the United States were designed to cover 0–6000 m. The data o ﬀ ered by this underdeveloped deep pilot array are scarce on both temporal and spatial scales. This particular study o ﬀ ers an ingenious and novel approach to extrapolating conductivity–temperature–depth (CTD) proﬁles, as well as sound speed proﬁles (SSPs), in abyssal oceans below 2000 m. The primitive results of this method exhibited certain discrepancies which were subsequently rectiﬁed by modifying the aforementioned method both symmetrically and asymmetrically in an innovative way. The ﬁnal outcomes of this method are almost identical to the in situ values obtained from Deep Argo ﬂoats, and in this way, o ﬀ er a way to compute deep ocean calculations both spatially and temporally since Deep Argo ﬂoats are aimed at relatively sparse deployments and require a longer duration to provide data (5 ◦ latitude × 5 ◦ longitude × 15-day cycles) as compared to Core Argo data (3 ◦ latitude × 3 ◦ longitude × 10-day cycles). The SSP computations were conducted by employing multiple equations such as Chen and Millero, Del Grosso, and UNESCO (United Nations Educational, Scientiﬁc, and Cultural Organization) algorithms. The study concludes by o ﬀ ering transmission loss rectiﬁcations by employing the aforementioned method as a future course of action.


Introduction
The international Argo program, which was initiated in 1999, began deployments of floats globally and achieved its landmark success of 3000 floats by the year 2007. The in situ data of around 2000 m method are in situ data from Deep Argo floats for the sake of extrapolation which takes course below 2000 m until the end of the available data. The study proceeds by conducting experimentation on varying cycles of a Deep Argo float to offer wider spatiotemporal coverage, since Deep Argo floats are aimed at sparse deployments (5 • latitude × 5 • longitude × 15-day cycles) in comparison to Core Argo data (3 • latitude × 3 • longitude × 10-day cycles) [13,15]. Furthermore, symmetric and asymmetric extrapolations are applied on three Argo floats to illustrate the outcomes of extrapolation which are analogous to in situ values of floats. Similarly, the study concludes with transmission loss rectifications based on these outcomes as a future course of action. The SSPs for this study are computed using three equations, i.e., the Chen and Millero equation (1977) [16], the UNESCO (United Nations Educational, Scientific, and Cultural Organization) equation [17,18], and the Del Grosso equation (1974) [17,18]. The method shows promise in offering wider spatiotemporal coverage with the help of the most appropriate extrapolated values by employing Deep Argo as well as Core Argo floats' in situ data. In this way, the deeper oceans are measured more accurately on both temporal and spatial scales. The method initially relies on Deep Argo floats in the proximity and can then be employed for any time and almost any place, depending solely on the presence of conventional Core Argo floats with profiling depths of 2000 m in that particular locality for subsequent symmetric and/or asymmetric (whichever generates accurate values) extrapolations to the depths based on the particular Deep Argo float.

The Proposed Extrapolation Methodology
The aforementioned method to rectify erroneous extrapolated values is presented mathematically in this particular section. This proposed method is based on the least squares method [14]. The data are presented in Equation (1) below, where x represents the salinities, temperatures, and SSPs in psu,°C , and m/s, respectively. Similarly, y represents the depth in meters, and i = 1, 2, 3, . . . , n.
Employing polynomials for curve fitting, we have The desired design parameters are k = (k 0 , k 1 , . . . , k m ). Then, we have The classical least square fitting for the most appropriate fitting is applied [19]. By taking the differential on both sides, we have By substituting Equation (2) into Equation (4), we have which is of the form . , m and i = 1, . . . , n. The values m and n are free, and a particular range is set for them in order to obtain extrapolations for the salinities, temperatures, and SSPs which are almost identical to the in situ values of the Deep Argo floats. Initially, m was set to 1 for extrapolations against varying values of n which are mentioned in the ensuing results section, and the resulting first-order extrapolation presented anomalies relative to the in situ values of Argo floats. Then, m was set to 2, and the outcomes were improved with this second-order implementation. This activity was conducted for three Deep Argo floats of 4000 m, as well as on a couple of floats with profiling capability of 6000 m depth; the results are graphically presented in Sections 3.1 and 3.2, respectively. Furthermore, a couple of activities were conducted as detailed in Section 3.3 by employing extrapolations on three Deep Argo floats offering data up to depths of~5000 m in the Pacific Ocean. One such activity dealt with the illustration of wider spatiotemporal coverage by conducting extrapolations on multiple cycles of one of these three floats. Similarly, the SSPs for these three floats were computed and extrapolated to exhibit the improved extrapolations asymmetrically as compared to the symmetrical ones. This was done by manipulating the ranges of both m and n.
The in situ vertical profiles of salinity, temperature, and subsequent SSPs employed for this entire activity are available online [20]. The database offers Argo floats for the Atlantic, Indian, and Pacific Oceans by searching any of them using the World Meteorological Organization identification (WMO ID) number. After selection, the website offers multifarious details regarding that particular float. The desired vertical profiles are available both in Netcdf and Ascii formats. The sound speed profiles were computed by employing the vertical profiles of salinity, temperature, and pressure obtained from these data using the equations below: • The Chen and Millero Equation (1977) [16]; • The UNESCO Equation [17,18]; and • The Del Grosso Equation (1974) [17,18].
These equations are discussed in the ensuing Section 2.2.

Sound Speed Profile (SSP) Computations
The sound speed computations for this study were conducted by employing three equations: the Chen and Millero (1977) [16], Del Grosso (1974) [17,18], and UNESCO (United Nations Educational, Scientific, & Cultural Organization) algorithms [17,18]. In this study, we compared the extrapolations for all of these equations; the Del Grosso equation [17,18] was employed for the improved extrapolated rectifications, as will be illustrated in the figures of ensuing sections.

SSP Computations Using the Chen and Millero Equation (1977)
This sound speed equation was numerically computed for specimens of seawater, which were either diluted by employing pure water or concentrated with the help of evaporation by Millero and Chen in 1977 [16], and it is given as follows.

SSP Computations Using the UNESCO Algorithm
The UNESCO equation, also called the UNESCO equation: Chen and Millero, is based on Chen and Millero (1977) [17]. The original UNESCO paper was presented in a 1983 publication by Fofonoff and Millard [16]. In 1995, Wong and Zhu reworked the coefficients in this particular algorithm by considering the International Temperature Scale of 1990, and their version of the UNESCO equation is presented below [18]: The speed of sound in seawater is already presented in Equation (9) above. In that particular notation, C w is the speed of sound in pure water, whereas A, B, and D are terms related to salinity.
T is the temperature represented in degrees Celsius, S is the salinity represented in Practical Salinity Unit parts per thousand (ppt), and P represents pressure, computed in bar kPa. The range of validity for temperature varies from 0 to 40°C, that for salinity varies from 0 to 40 parts per thousand (ppt), and that for pressure is from 0 to 1000 bar.

SSP Computations Using the Del Grosso Equation (1974)
There is another equation regarded as a substitute for the UNESCO algorithm, called the Del Grosso Equation (1974). This equation was also reworked by Wong and Zhu (1995) for the new International Temperature Scale of 1990, and it is given as follows [17,18].
∆C S (S) = C S1 S + C S2 S 2 (17) T represents the temperature in degrees Celsius, S exhibits the salinity in Practical Salinity Units, and P illustrates the pressure in kg/cm 2 . The range of validity for temperature is 0 to 30°C, whereas that for salinity is 30 to 40 parts per thousand, and that for pressure is 0 to 1000 kg/cm 2 , where 100 kPa = 1.019716 kg/cm 2 .

Results
This section presents outcomes which are categorized initially for a depth of 4000 m, followed by 6000 m, displaying the outcomes of basic first-order extrapolations as well as the improved second-order ones. As a special case scenario, three floats (WMO5905738, WMO5905739, WMO5905740) with profiling depths of up to~5000 m were selected in the Pacific Ocean for two particular activities. The first activity was to illustrate the wider spatiotemporal coverage by employing extrapolation for one of these three floats i.e., WMO5905738 (considering 10, 11, and 18 cycles). Furthermore, these three floats were employed for asymmetric extrapolations, and the outcomes proved to be almost identical to the in situ values of floats.

Extrapolations and Their Subsequent Rectifications for 4000 m Floats
The details of the Argo floats employed for computations in the category of 4000 m are detailed in Table 1, which lists the relative float identities, cycle numbers, date, and latitude and longitude, along with the locations of these floats. Initially, the salinities for these floats are illustrated as in situ measured and first-order extrapolations by computing values for m = 1, along with comparative analysis of improved second-order extrapolations produced using m = 2 in (a,b), respectively, of Figures 1-3. The temperatures for floats exhibited in Table 1 are illustrated in a similar manner in (c,d) of Figures 1-3. Likewise, the sound speed profiles for such floats were computed and extrapolated in the aforementioned ways and illustrated as their first-order extrapolations using Chen and Miller (1977) [14] and the UNESCO algorithm [15,16] in (a,b), respectively, of Figures 4-6. However, Del Gross (1974) is exhibited for both in situ measured and first-order extrapolations, as well as with improved second-order extrapolations, in (c,d), respectively, in Figures 4-6. It is pertinent to mention here that the Del Grosso equation [15,16] was considered since it offers a relatively better SSP estimation for acoustically returning patterns in comparison to Chen and Millero (1977) [21]. The values of m are discussed above, whereas the values of n for WMO2902510, WMO2902971, and WMO19274 were 32, 78, and 14, respectively.   (a,b) for the salinities where a difference of~0.2 psu was reduced to a negligible level after improved second-order rectification and, similarly, (c,d) for temperatures where a difference of~1.5°C due to basic first-order extrapolation was reduced to an infinitesimal amount by second-order extrapolation.

Figure 2.
The in situ measured and extrapolated salinities and temperature both by first-order and by improved second-order extrapolation (a,b) for the salinities, where a difference of~0.3 psu due to basic first order was reduced to a negligible amount after improved second-order rectification and, similarly, (c,d) for temperatures, where a difference of~1.3°C due to basic first-order extrapolation was reduced to an infinitesimal amount by second-order extrapolation. Figure 3. The in situ measured and extrapolated salinities and temperature both by first-order and by improved second-order extrapolation (a,b) for the salinities, where a difference of~0.3 psu due to basic first order was reduced to a negligible amount by improved second-order rectification and, similarly, (c,d) for temperatures, where a difference of~1.3°C due to basic first-order extrapolation was reduced to an infinitesimal amount by second-order extrapolation.

Extrapolations and Their Subsequent Rectifications for 6000 m Floats
The data from two Argo floats presented in Table 2 were utilized for computations in this particular section to offer profiling floats of CTD for abyssal depths of 6000 m. The scheme of the illustrations follows that of the aforementioned, i.e., 4000 m floats in Section 3.1. The salinities and temperatures for the aforementioned Argo floats were compared for their measured in situ values using basic first-order extrapolations with m equal to 1, as well as their rectifications with second-order extrapolations by computing with m = 2, with (a,b) presenting salinities and (c,d) presenting temperatures in Figures 7  and 8. The SSPs for the aforementioned Argo floats are also depicted, just like in the 4000 m case detailed in Section 3.1. The basic extrapolations were compared by employing the Chen and Millero equation [16] as well as the UNESCO algorithm [17,18], presented in (a,b) of Figures 9 and 10. Similarly, the SSP given by Del Grosso equation [17,18] extrapolation is compared initially with in situ measured data and the first-order basic equation in (c) and with improved second-order extrapolation in (d) of Figures 9 and 10. Similarly, n for the extrapolations of WMO4902322 and WMO5902521 was chosen to be 49 and 35, respectively.  Figure 7. The in situ measured and extrapolated salinities and temperature both by first-order and by improved second-order extrapolation (a,b) for the salinities, where a difference of~0.1 psu due to basic first-order extrapolation was reduced to a negligible amount after improved second-order rectification and, similarly, (c,d) for temperatures, where a difference of~4°C due to basic first-order extrapolation was reduced to an infinitesimal amount by second-order extrapolation at 6000 m depth. However, a deviation was observed at 3000-5000 m depth with a maximum of~2 • C.

Experimentation in the Pacific Ocean for Spatiotemporal Coverage and Asymmetric Rectification
This section focuses on three floats, i.e., WMO5905738, WMO5905739, and WMO5905740, located in the midst of the Pacific Ocean and with profiling depths of up to~5000 m. One of these floats, WMO5905738, was employed to extrapolate its varying cycles with the aim to demonstrate wider spatiotemporal coverage in terms of salinities and SSPs. Similarly, the aforementioned three floats were employed to exhibit rectifications of discrepancies by asymmetric extrapolations as compared to symmetric ones.

Spatiotemporal Coverage through Extrapolations
As discussed above, varying cycles of an Argo float (WMO5905738) were extrapolated in order to illustrate wider spatiotemporal coverage by employing the proposed method. The details of the cycles, their latitude and longitude, and the date and time are presented in Table 3. The m and n values varied for varying cycles of this particular float, i.e., cycle 10 (m = 3 and n = 112), with temperature and SSP shown in Figure 11a,b, cycle 11 (m = 3 and n = 52), with temperature and SSP illustrated in Figure 11c,d, and cycle 18 (m = 2 and n = 38), with temperature and SSP depicted in Figure 11e,f. It is worth mentioning that the extrapolations produced in this particular activity showed remarkable outcomes, especially to the abyssal depth of 4000 m. This particular activity illustrates the application of extrapolations on varying cycles of the floats to acquire broader spatiotemporal coverage of the abyssal depths of the ocean in certain regions. A deep in situ reference in the vicinity is required, as well as Core Argo buoys with depth 2000 m. The analysis in this particular section is focused on temperature and the subsequently computed SSPs since the role of salinity is minimal in deducing SSPs, as observed from the aforementioned computations and the literature [22,23].

Symmetric and Asymmetric SSP Rectifications
This particular section compares the extrapolation rectifications of SSPs, both symmetrically and asymmetrically. The floats for this typical activity, along with their cycles, latitude and longitude, and time are illustrated in Table 4. In this activity, the same floats were initially rectified using fixed values of both m and n, which were 2 and 40, respectively; this is illustrated in Figure 12a,c,e. However, both m and n were varied for the various floats, i.e., for WMO5905738 (m = 2 and n = 46), for WMO5905739 (m = 3 and n = 68), and for WMO5905740 (m = 2 and n = 37); this is depicted in Figure 12b,d,f. It is pertinent to mention that by varying the values of both m and n, i.e., by rectifying asymmetrically, the achieved rectified values were almost analogous to the in situ measured values of the floats.

Discussion
The study illustrates promising outcomes, and the presented method can be employed for optimum extrapolations in global abyssal oceans to offer better approximation of the vertical profiles of salinity, temperature, and SSPs. The limitations of this particular method are the availability of abyssal reference inputs at the desired depths to ascertain the extrapolations and the reliance on varying Core Argo floats at 2000 m depth for further computations. The Deep Argo array is in the process of deployment and even after its deployment will remain sparsely populated (5 • latitude × 5 • longitude × 15-day cycles) compared to the Core Argo array (3 • latitude × 3 • longitude × 10-day cycles) [13,15]. This method, in turn, is capable of filling the gaps in both the temporal and spatial scales by extrapolating to optimum levels, as illustrated in the outcomes detailed herein for varying Argo floats.
Transmission loss rectifications are displayed for a source depth of 300 m and frequency of 50 Hz in Figure 13   . For the transmission loss computation, accordingly, (a) illustrates the measured, basic first-order extrapolated, and improved second-order extrapolated SSPs. Similarly, the transmission loss based on in situ measured values is exhibited in (b). The transmission loss after employing the first-order extrapolation is presented in (c). Finally, the transmission loss with improved second-order extrapolation is illustrated in (d).

Conclusions
Deep ocean observations are of utmost importance for both scientific and seasonal requirements. In this context, in this study, we took advantage of both the Deep Argo and Core Argo programs to cover vertical CTD profiles and SSPs of the abyssal oceans using as accurate extrapolations as possible. The study was initiated with a novel approach of extrapolating the aforementioned parameters by employing the least squares method in deeper oceans and then rectifying the remaining discrepancies via symmetric and asymmetric extrapolations. We rectified discrepancies of salinity on the order of 0.3 psu to negligible values for depths of 4000 m and from~0.6 psu to minute values for 6000 m. Likewise, the temperatures were rectified on the order of~1.5°C at maximum to negligible for 4000 m and from~4°C at maximum to almost perfect accuracy for 6000 m depths. It is pertinent to mention that the discrepancies in the SSPs rectified for the 4000 m Argo floats were~8.5 m/s and~16 m/s for the 6000 m floats. The asymmetric extrapolations for the SSP were improved from~7 m/s to negligible values as compared to the symmetric extrapolations. Similarly, we conducted experiments on varying cycles of a float to prove the wider spatiotemporal extrapolation capability of this particular method in certain regions based on the availability of abyssal ocean in situ data and the Core Argo profiles in that particular vicinity. Finally, we concluded with an example of transmission loss rectification capability, which can be addressed as a future course of action based on this particular research. The evolutionary process of extrapolations conducted in this specific study ingeniously conjoins the Core Argo (3 • latitude × 3 • longitude × 10-day cycles) and Deep Argo programs (5 • latitude × 5 • longitude × 15-day cycles) to offer broader coverage on both the spatial and temporal scales in global abyssal oceans. Similarly, transmission loss computations based on these extrapolations are presented as a future course of action.