1. Introduction
The large amount of studies aiming at measuring and understanding climate change shows the importance of having long and accurate temperature records. The outstanding Argo program provides measurements from the top 2000 m of the global ocean since 2006 (Roemmich et al. 2015; Riser et al. 2016). A global average estimate shows warming, over the last decade, of 50 m°C decade−1 for waters between the surface and 500 m and of 20 m°C decade−1 from 500 to 2000 m (Roemmich et al. 2015). Measuring programs, like Argo, together with historical and quality controlled hydrographic data are the prime sources of empirical information for ocean climate change studies. However, a particular constraint on these studies is the limited amount of high-quality measurements in the abyssal region, in waters deeper than ~2000 m (e.g., Wunsch, 2016; Johnson et al. 2015, to mention a few). Notably, the recent study by Desbruyères et al. (2016) offered a warming estimate of 20 ± 12 m°C decade−1 for the Atlantic Ocean at ~2000 m for the time period 2000–16 (see their Fig. 2b). Kouketsu et al. (2011, see their Fig. 3) estimated global ocean warming, for depth levels between 3000 and 4000 m of the same magnitude as the rates presented here for the deep Gulf of Mexico (GOM), but with large uncertainties. Recent estimates of global surface rate of warming derived from Argo floats is ~0.15–0.2°C decade−1, but decreases rapidly with depth and is typically 10–30 m°C decade−1 from 400 to 2000 m (e.g., Wijffels et al. 2016, Johnson et al. 2019). For the western North Atlantic, a 73-yr hydrographic record at Bermuda indicates that the layer between 1500 and 2500 dbar shows a secular warming trend of 0.5°C century−1 (Joyce and Robbins, 1996). Also, comparing hydrographic surveys taken 43 years apart in the western North Atlantic, a latitudinal average between 20° and 35°N gives a warming trend of 30 ± 10 m°C decade−1 at 2000-m depth (Joyce et al. 1999, see their Fig. 7). Clearly there is a wide range of warming trends in the available literature for the 2000-m layer in the North Atlantic.
In this study, we use moored measurements at four sites just ~10 m above the bottom at ~3500 m, within the western GOM taken between 2007 and 2018, that show much smaller uncertainty than those of Kouketsu et al. (2011) and Desbruyères et al. (2016). Our CTD measurements indicate a very similar warming trend from the bottom up to ~2000-m depth. The CTD measurements also show the properties of the densest waters that enter the gulf just above the Yucatan Channel sill, which is located between the Caribbean Sea and the GOM. These waters reach the Yucatan Sill via flows from the Windward Passage (Smith 2010).
The following section presents the data and results. The third section elaborates on results from this and other studies, in particular for the Caribbean Sea (Joyce et al. 1999; MacCready et al. 1999). The fourth section summarizes the analysis and results. Two appendixes deal with instrumental drift, the warming trend estimation, and the calculation of confidence intervals in the warming trends reported here from moored observations.
2. Data and results
One source of data for this study comes from moored thermometers at ~10 m above the bottom at depths of ~3500 m at four GOM sites, with time series extending close to 10 years. The moored instruments (Sea-Bird Electronics Inc., SBE37SM) collected data during consecutive periods, each lasting ~1–2 years from deployment to recovery (Fig. 1). The moored time series gaps, which mostly correspond to the time encompassing from recovery to next deployment, are typically shorter than 3 days, except for one measurement period in one site which was completely lost. In addition to the moored time series, CTD profiles from a variety of oceanographic cruises complement the data (Fig. 2 and Table 1). These profiles are relevant for the analysis and interpretation of the warming of GOM’s deep waters.
The map shows the locations of moored thermometers as filled magenta circles at the western side of the Gulf of Mexico, and CTD profiles in other symbols: black diamonds and dots for the western Gulf of Mexico (WGM), red pentagrams for the sill region in the Yucatan Channel (SILL), inverted green triangles for the Yucatan Basin (YUB), and blue squares for the Venezuela Basin (VEB). The Yucatan Channel sill depth is 2040 m. The 2000-m isobath surrounds five basins: WGM, YUB, Cayman Basin (CAY), Colombia Basin (COL), and VEB. The two deep connections between the Caribbean Sea and the Atlantic Ocean are the Windward Passage (WP) and the Anegada–Jungfern Passage complex (AJP). The blue contours between Florida and the Bahamas indicate the 725-m isobath. Black dots within the WGM differentiate the 19 shallower profiles from the 18 profiles deeper than 3700 dbar, in diamonds.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
Following the same color distinction as in Fig. 1, the distribution of depth vs (a) σ0, (b) Absolute Salinity SA, and (c) Conservative Temperature Θ; (d) the corresponding Θ–SA diagram with σ0 contours. The same coloring in profiles is used as the symbol for its corresponding location in Fig. 1: black diamonds (WGM), red pentagrams (SILL), inverted green triangles (YUB), and blue squares (VEB). These abbreviations, of geographical region, are used in Table 1. The rectangle in (c) is shown expanded in Fig. 3.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
This table presents the locations, maximum pressure, and dates of CTD profiles plotted or used in the analysis. Figures 2 and 3 do not include the last 19 profiles listed, to avoid crowded figures. All 37 WGM profiles listed are used in calculations of warming trends.
a. CTD profiles
The selected CTD profiles allow a direct comparison of properties between the Caribbean Sea and the GOM, in particular, just north of the Yucatan Channel sill (red symbols in Fig. 1, denoted by SILL in Table 1). We use TEOS-10 for seawater properties unless otherwise noted (IOC et al. 2010; McDougall and Barker 2011). The set of 37 CTD profiles, well within the deep western side of the GOM (WGM in Table 1), also permits an estimate of the warming from the bottom to ~2000 m below the surface. The 37 WGM profiles, as labeled in Table 1 and shown in black symbols in Fig. 1, come from eight cruises that span from August 2003 to June 2019 and reach at least 3000 m.
Distributions of Conservative Temperature, salinity and density against pressure, and of Conservative Temperature against salinity highlight the hydrographic differences among the basins (Fig. 2). The very deep (pressures exceeding 4500 dbar) fractions of the profiles in the Venezuela Basin, identified as VEB in Table 1, are out of range in the distributions against pressure of Fig. 2 (blue traces).
The temperature profiles in the western GOM (Fig. 3) show clearly a progressive increase of deep temperatures. The temperature increase is quite uniform from ~2000 m down to the deepest common level (i.e., 3640 m at 3700 dbar). The trend is shown in Fig. 4, where a conventional least squares fit provides an estimate of the warming rate and its confidence interval. A warming of 18 m°C decade−1, at such depths, translates in an increase of energy of 2.5 × 105 W km−3. The volume below 2000 m within the GOM is ~7 × 105 km3, hence the rate of energy increase in such volume is ~1.7 × 1011 W (or ~5.4 × 1018 J yr−1).
The two frames show the deep end of 18 available profiles within the GOM that reach 3700 dbar: (left) the in situ temperature (ITS-90) and (right) the Conservative Temperature. The set of profiles in the right frame are the same as the ones shown within the black rectangle of Fig. 2c. The corresponding year of cruises is listed.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
(left) In situ temperatures anomalies at four different depths, from the 37 profiles listed as WGM in Table 1, are plotted along a conventional least squares fit. The individual slopes (m°C decade−1), with corresponding confidence intervals at 95% confidence level, are depicted in the text as well as the corresponding depth. Offsets in the anomalies, for 3500, 3000, and 2000 m are −0.10°, −0.05°, and 0.05°C, respectively. The 2500-m depth anomalies are without offset. (right) The profile of slopes as a function of depth, and the band associated with the confidence interval.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
b. Moored thermometer data
Further observations for this study come from moored thermometers at four locations, called from north to south PER, LMP, ARE and LNK (magenta circles in Fig. 1, Table 2). Table S1 (in the online supplemental material) shows the locations, dates, and instrument’s serial number. The raw data have sampling intervals of 2, 5, or 15 min. These data are filtered to avoid aliasing when resampling to 1-h interval. The set of in situ (ITS-90) temperature time series show sharp discontinuities, expected due to different instrument deployment depths and offsets from thermometer calibration (Fig. 5). Instrumental calibration drift magnitudes are relatively small compared to the observed temperature trends (appendix A).
Locations of four moorings, with number of deployments and records lost.
The four frames show the temperature time series for locations shown in magenta circles in Fig. 1. From north to south these are PER, LMP, ARE, and LNK. Subplot titles show estimated temperature rate of change and its 90% confidence interval. The superimposed green “reconstruction” that avoids the discontinuities, with an offset of 5 m°C, shows the trend in black. Deviations of the individual record means from the reconstruction are shown as the black horizontal traces centered at the ordinate of 4.42°C (see also Fig. B1).
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
There is an extensive processing of meteorological data to account for the effects and to eliminate “mean shifts” that are produced by changes in instrument location (Venema et al. 2018). In our case, the knowledge of precisely when such discontinuities occur facilitates a great deal the data processing.
To concatenate continuous time series consistently, the mean temperature of each fraction of the time series, as defined from deployment to the recovery of the mooring, is modified for an updated mean, without changing the full series mean. The set of individual means plus the two parameters of the standard linear fit become the free parameters of the least squares fit. Details of the fitting algorithm are given in appendix B. The individual temperature records as well as the reconstructed time series for each mooring indicate that typical deviations of such means are on the order of 1 m°C (Fig. 5), which in terms of the general in situ temperature gradient (see Fig. 3), represents a vertical displacement of 8 m. The largest mean shift of individual means is for the fourth record in PER with an anomaly of 3.3 m°C, which corresponds to a vertical displacement of 26 m. To compute confidence intervals, of the slope (or temperature trend), and in general of the free parameters, a Monte Carlo computation was performed, as shown in appendix B. The temperature rates of change range between 14.6 and 17.5 m°C decade−1 with confidence intervals around 1–2 m°C decade−1 (Fig. 5).
3. Discussions
The bathymetry in use is from Smith and Sandwell (1997). For this study the Cayman Basin and Yucatan Basin can be considered as a single basin, since the deepest waters that have direct connection between them are at ~3900 m, and the deepest waters that enter the GOM are at ~2000 m. A similar situation holds for the Colombia Basin and the Venezuela Basin: the deepest waters that freely connect them are at ~4030 m (through the Aruba gap), while the sill between the Colombia Basin and Cayman Basin is found in the Jamaica Rise (i.e., JRS in Fig. 1) at ~1520 m.
The densest waters entering the GOM at the Yucatan Channel (see flow fields shown in Ochoa et al. 2001; Sheinbaum et al. 2002; Candela et al. 2019) are denser (see Fig. 2a and left frame of Fig. 6) than the waters in the deepest regions within the GOM. The densest waters at the deepest CTD casts within the western GOM (Fig. 2a) have σ0 that appears at ~1800 m in the Yucatan Sill vicinity.
Mean vertical profiles within GOM (black traces) and fit of the analytical solution [Eq. (2)] below 2000 dbar in magenta trace. See text for the choice of w ~ 1.6 cm day−1 used in the right panel. The red traces correspond to the mean profile at the Yucatan Sill.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
Surface cooling events within the GOM cannot produce waters as cold and dense as those found below 2000 m; the only known source of such waters are the deep inflows through the Yucatan Channel, as proposed by Sturges (2005). The moored temperature series shown are in the GOM’s interior bottom boundary layer (BBL), a layer that is supplied by the deep inflow at Yucatan Channel, likely in a cyclonic circulation (DeHaan and Sturges 2005; Pérez-Brunius et al. 2018). There is no evidence of significant outflow of deep GOM’s interior BBL waters through the Yucatan Channel, as it was described by Banyte et al. (2018a) for the Panama Basin.
The depth extension of the warming, as evident from CTD profiles, is from the bottom up to ~2000-m depth (Figs. 2–4) and is nearly uniform. It is remarkable that the vertical temperature gradient over such a thick layer shows no significant time dependence.
The geothermal bottom heat flux in the GOM, which is comparable to Mesozoic ocean basins in general, is ~40–47 mW m−2 (Nagihara et al. 1996). In contrast, there is an average geothermal heating 4 times larger in the Panama Basin in the Pacific Ocean, which is considered the primary driver of its abyssal circulation (Banyte et al. 2018b). The geothermal heat flux in the GOM, if distributed uniformly in a 200-m-thick stagnant layer, implies a warming of ~15–18 m°C decade−1. The observed warming is similar but consistent over a much thicker layer. Considering the volume below 2000 m and the bottom, and the observed 18 m°C decade−1 warming (see Fig. 4) results in a heat gain that is ~6 times larger than that expected by the geothermal flux over a stagnant layer.
Besides the geothermal heating, the role of the sloping BBL, given the differential intensity of mixing relative to the same isopycnal levels in the interior, is of paramount importance (Kunze et al. 2012; Ferrari et al. 2016; Holmes et al. 2018). In Munk and Wunsch (1998), the role of lateral advection was introduced. This allows for nonuniform distribution of upwelling velocities as well as eddy coefficients. This perspective is consistent with the study of Holmes et al. (2018), in which interleaving exchanges occur (mainly along isopycnals) between interior and BBL waters (along sloping bottoms). Within this scheme or perspective, an expected feature for the GOM is the interleaving of the spilling waters that cross the sill and mix during their downhill descent.
The Windward Passage has a sill depth of 1680 m, whereas the Anegada–Jungfern Passage complex has a maximum depth of 1815 m (Smith 2010). The Venezuela Basin, for waters below the Anegada–Jungfern passage sill, show no temperature trend before 1970, followed by a ~10 m°C decade−1 warming from 1970 to 2003 (Johnson and Purkey 2009). Hydrographic data (Fig. 2b) suggest that the deep waters that flow into the GOM through the Yucatan Channel come from the Windward Passage, transiting the Cayman Sea to the Yucatan Basin. Notice that the deep waters near the Anegada–Jungfern Passage complex have lower salinities relative to the deepest waters at the entrance to the GOM. Bulgakov et al. (2003) and Smith (2010) have measured the flow of deep waters into the Cayman Basin through the Windward Passage. Although snapshots of the flow in the Windward Passage show inflows and outflows below ~1200 m, the persistent flow at its sill is from the Atlantic Ocean into the Caribbean Sea (see Fig. 14 of Smith 2010).
A warming trend of 30 m°C decade−1 at the levels around 2000 m has been reported from CTD sections in the western subtropical North Atlantic Ocean (1950s–1990s; Joyce et al. 1999) and 50 m°C decade−1 in the Bermuda station “S” for the layer 1500–2500 m (1922–95; Joyce and Robbins 1996). Such a warming trend of inflowing waters through the Anegada–Jungfern Passage contributes to the warming inside the Venezuela Basin (Joyce et al. 1999). For the GOM, the incoming deep flows are not so intermittent as in the Venezuela Basin, and there is evidence of vigorous mixing during their downslope transit (see Fig. 5a of Pérez-Brunius et al. 2018). Extensive mixing under similar conditions is also suggested by the conceptual model of Bryden and Nurser (2003).
An approximate volume of the BBL over the GOM’s interior, limited by the 2000-m isobath is the horizontal area comprised within such isobath, which is 6.2 × 105 km2, times a thickness of 100 m above bottom, i.e., 6.2 × 104 km3. If we consider a certain inflow below ~1800 m through the Yucatan Channel [see Fig. 3a of Candela et al. 2019 for a cross section of the mean flow, and Table S3] that feeds the deep GOM’s BBL, then the same volume has to be removed from the deep GOM by upwelling and flows along the bottom, and eventually be exported as lighter waters. For this inflow transport to replace the estimated BBL volume, a time lapse of 20 ± 14 or 25 ± 20 years is required, depending on if the transport is calculated from below 1790 or 1810 m, respectively. The confidence limits are at 80% in these estimates, and the considerable variability relative to the mean in deep transport estimates implies large uncertainties. Longer estimates of residence time are obtained using the same transports, but assuming a thicker layer than the BBL, for example, by replacing the waters below the isopycnal at ~1800 m in the Yucatan Channel. The density at ~1800 m at the Yucatan Channel (σ0 ≈ 27.7618 is found at ~2700 m in the GOM interior. The area extending horizontally within the Gulf’s 2700-m isobath is 4.7 × 105 km2 enclosing a volume of 3.1 × 105 km3 and the transports below ~1800 m at Yucatan Channel are 0.10 ± 0.07 Sv of inflow and 0.01 ± 0.02 Sv (1 Sv ≡ 106 m3 s−1) of outflow. This estimate comes from Candela et al. (2019, their Fig. 3a), and was obtained by best matching the total 4-yr mean transport at the Yucatan Channel and the Florida Strait sections. The residence time, or time that it takes for the inflow to refill such volume, is on the order of 100 ± 70 years, and, accordingly, an average vertical velocity of 1.6 ± 1.7 cm day−1. Another valid estimate of the GOM’s residence time can be calculated from Candela et al. (2019, their Fig. 3c), which was obtained by considering the mean from all the measured current series, within the observed 4-yr period, that had at least 1-yr-long record. In this case the mean inflow below 1800 m gives 0.22 Sv with a corresponding deep water (>2700 m) residence time of 45 years. Both the inflow and the residence time are within the estimated uncertainty limits mentioned before. These are crude, but databased, estimates of the residence time, which agree with radiocarbon based estimates for the GOM (Amon et al. 2020, manuscript submitted to Nat. Commun.).
Given the limited knowledge on the velocity and mixing within the gulf, only crude estimates of the mechanisms that cause the ~18 m°C decade−1 from the bottom to ~2000 m below the surface are possible; some assumptions that, in our view are deemed reasonable, cannot be verified with present-day knowledge. Nonetheless, the evidence for warming is quite similar to the study of MacCready et al. (1999), which shows, for the Caribbean Sea, a warming of the deep Venezuela Basin waters consistent with dense episodic spills into the basin through the Anegada–Jungfern passage. These episodes rarely involve waters as dense as those found in the deepest bottom of the basin, but remain the only plausible source of the deep waters. In the GOM, the inflow of dense waters through Yucatan Channel is more continuous and the only plausible source of the densest waters found in the GOM’s interior. In contrast, in the Venezuela Basin the interleaving of a thick intermediate layer seems to be the dominant mechanism for the warming (MacCready et al. 1999).
Schematic of the model. Notice the isopycnals deepening, the high intensity of mixing on the downslope transit of the waters, the upwelling along other sloping areas of the BBL, and the interleaving communicating the BBL with the interior.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
Time derivative of solution (6),
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
The time derivative of this solution suggests that, for the GOM deep waters to attain their present 16 m°C decade−1 warming trend, the waters of the western North Atlantic at the entrance to Windward Passage should have started warming about 100 years ago (Fig. 8, green curve). Given the crude approximations used to obtain this number, these 100 years should be taken as the lower-limit estimate of when the Atlantic Ocean started warming. Finally, we note that the derivative of the above solution will be equal to the derivative of the solution of (4) if
What this analysis implies is that the inflow of warmer deep waters and their entrainment during mixing are key ingredients for the observed warming in the deep GOM. However, it could be argued that the warming of the source waters does not necessarily have to be directly reflected in the overall warming pattern in the abyssal gulf. For example, changes in the circulation could result in changes in the depths of isotherms near the sill, or changes in the T–S relation could also impact the temperature of the source waters. Nevertheless, given the evidence of the long-term warming of the deep waters in the Atlantic (e.g., Joyce and Robbins 1996; Joyce et al. 1999), we consider it the most important contributor to observed temperature trend in the deep gulf.
The slow warming of deep waters and the faster warming above 1000 m imply that the GOM large-scale stratification is increasing, similar to observations for global oceanic conditions. Vertical diffusion of heat, from warmer waters above is not supported by our observations as a significant factor in the warming (see Fig. 6 and requirement of excessive diffusivity in Munk’s model fit). In fact, the increase in stratification is more likely to reduce the vertical turbulent exchanges.
4. Conclusions
Two independent time series observations, CTD profiles taken from 2003 to 2019 and moored thermometers, deployed from 2008 to 2019 in the western GOM, show that the overall temperature trend in the abyssal depths of the Gulf of Mexico is ~18 m°C decade−1. The low uncertainty of ±~2 m°C decade−1, at 90% confidence level, is due to the low variability in temperature measurements at such depths and, for the moored data, the appreciable length of the available measurements. This heating is, as shown by the CTD profiles, uniform from ~2000 m to the bottom at ~3700 m in the abyssal plains. In particular, for the moored observations, the time series show a steady increase in the last decade without any sign of a hiatus or deviations from a linear trend.
The geothermal heat flux is an essential contributing factor in the observed temperature distribution; it induces convection at the base of the BBL and therefore is a factor for the renewal of waters. Nonetheless, it cannot explain by itself the warming trend, as such flux has existed for millenniums and is too weak to account for the warming trend in a layer as thick as observed in the deep GOM (>2000 m).
The evidence presented in this paper emphasizes the critical role of mixing and entrainment of overflow waters that ultimately fills the deep regions of the GOM and contributes to a relatively homogeneous warming of the GOM below 2000 m.
An important consideration is the fact that the warming trend measured by the deep thermometers in the GOM is a statistically reliable and solid measurement that adds key information for understanding the warming behavior of the intermediate waters in the Atlantic Ocean. This points to the importance of long-term measurements in marginal seas that, in combination with simple models, can be used to investigate the warming history of the broad oceans connected to them.
Acknowledgments
Funding for this study comes from (i) PEMEX Exploración y Producción under contracts SAP nos.428217896, 428218855, and 428229851, (ii) the National Council of Science and Technology of Mexico, Mexican Ministry of Energy, Hydrocarbon Trust, project 201441, as contribution of the Gulf of Mexico Research Consortium (CIGoM), and (iii) CICESE’s internal funds. We acknowledge PEMEX’s specific request to the Hydrocarbon Fund to address the environmental effects of oil spills in the Gulf of Mexico. Comments and clarifications were provided by Helmut Maske, Myrl Hendershott, Ryan Smith, and Paul Sortiriadis. We thank the crew of the Research Vessel B/O Justo Sierra for their beyond duty support at sea. The technical staff of the “CANEK Measuring Program” has been indispensable for the measurements. The processing of raw CTD data has been carried by Joaquín García. CTD data for the Venezuela Basin was provided by Clivar 2012. (https://doi.org/10.3334/cdiac/otg.clivar_a22_2012). We thank the Instituto de Oceanología de Cuba for their collaboration. We thank the reviewers for their quite thorough and constructive reviews. We manifest credit and special gratitude to one of the reviewers who proposed Eqs. (4) and (5), opening up a nice and important contribution to this work.
We would like to express our immense grief for the loss of our two great friends and colleagues José Ochoa and Vicente Ferreira-Bartrina. This work belongs to them; both were excellent scientists and their motivation will be with us forever.
APPENDIX A
Instrumental Drift
(a) The relationship from instrument output (i.e., counts) to temperatures as determined by the 19 May 2008 calibration. (b),(c) Considering the post-cruise calibration performed on 17 Oct 2017, the distribution of δT0 as a function of the instrument output and temperature, respectively.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
A set of seven SBE37 with calibration and post-cruise calibration done by the manufacturer (Sea-Bird Scientific) showing the instrumental drift in temperatures. The table shows the serial number (SN) the instrumental drift at the reference temperature of 4.5°C, the dates of calibrations, and the time span between such calibrations (years).
These estimates are larger than those of Uchida et al. (2008), which report mean drifts of ~0.1 m°C decade−1 in same instrument model. In our measurements, the drifts remain tolerable, as without drift corrections the statistical uncertainty of the warming trend is of the same order.
APPENDIX B
Least Squares Fit and Confidence Intervals via Monte Carlo Simulations
The ad hoc least squares fit used to estimate the temperature trend is straightforward in the sense that it minimizes the sum of squares of the set of differences between an analytical function and the data. The analytical function has some free parameters, and for any choice of such parameters the sum of squares can be computed. The common and practical procedure is to build a function that, given the set of free parameters, computes the residuals (i.e., the differences between the data and the analytical function) and their sum of squares. Hence, minimizing such a sum of squares, via iterations as the parameters vary, allows finding the least squares result sought. We use the function “fminsearch” from MATLAB to solve for this minimum, which is based in the Nelder–Mead simplex method as described for the “amoeba algorithm” in Press et al. (2007).
The analytical function of choice makes the least squares fit ad hoc; it consists of the addition of two functions, one that is the addition of rectangular functions that modifies the mean temperature of each “deployment to recovery period” (i.e., continuously measured by one instrument) plus a linear contribution. The slope of the linear contribution is the main interest of this study. The fit does not require “regularization” because the discontinuous function, defined by the set of partial means, is restricted to conserve the overall series mean. This discontinuous contribution depends on n − 1 free parameters where n is the number of available deployment to recovery periods. This choice does not eliminate the overall series mean as one of the free parameters, which, in addition to the slope of the trend, makes the least squares problem of n + 1 free parameters.
The least squares problem is defined as the minimization of
The common algorithm for estimating confidence intervals for the slope of the linear contribution does not apply. In this ad hoc fitting, (i) there is the presence of other nonstandard parameters and (ii) the residuals do show significant deviation from a Gaussian distribution. Figure B1 shows the residuals for the data shown in Fig. 5 LMP frame.
The graph shows the fitted analytical function (i.e., the discontinuous function with piecewise linear segments of identical slope) and residuals in a different perspective than Fig. 5 frame LMP. The dashed line along the fitted function is the linear contribution once the discontinuities have been removed. The lower irregular trace is the series of residuals with the zero as a dashed horizontal line surrounded by plus and minus the standard deviation of the residuals.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
For Monte Carlo simulations, it is fundamental to build series with the same statistics as the residuals, referred as synthetic residuals. It is possible to add synthetic residuals to the fitted function from the data and compute the resulting free parameters with the same algorithm. These last parameters differ from the estimated data-derived parameters, precisely because the residuals are not the same. The synthetic residuals are assumed to be a plausible outcome as implied by the data themselves. A sufficiently large set of statistically independent realizations with synthetic residuals, allows for the calculation of confidence intervals for any of the free parameters.
The two functions that are basic in the statistics of the residuals are the probability density function (pdf) and the power spectral density (psd). Alternative equivalences of pdf and psd are the cumulative probability function (cpf) and the lagged autocovariance function. Further statistical characteristics will not be pursued to be estimated from the data or reproduced by the synthetic residuals. We follow Sotiriadi (2014) procedure to build synthetic residuals series with the psd and pdf dictated by the residuals. The procedure, which is based on Price’s theorem (Price 1958), is only an approximation in this case because the residuals have considerable skewness and Sotiriadi’s procedure is formally for pdfs without skewness. The skewness of the simulations is matched completely to those implied by the data residuals, but the psd is distorted.
Figure B2 shows several examples of synthetic residuals, and Table B1 shows some of the set of fitted parameters in those examples. The relevant quantity to figure out via 500 Monte Carlo simulations is the expected error (under the same residuals statistics) for the temperature trend, as shown in the example in Table B1 for location LMP and just a few parameters. In 500 simulations, sorting the fitted parameters, the 90% confidence interval follows from the 25 and 475 fitted values. The 90% confidence limits in the estimate of the temperature trend at each location are shown in labels in Fig. 5.
The graph shows the empirical residuals in the uppermost trace for the location LMP, as is shown in Fig. B1, and with a uniform offset of 5 m°C for four simulations of residual series for the Monte Carlo simulations.
Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-19-0295.1
The list presents, as an example, the fitting of a few parameters for the Monte Carlo simulations. The first three columns show the fitted mean temperature minus the observed mean for deployments from first to third (see text about the series discontinuities and free parameters in the ad hoc least squares fit and Fig. B1). The first row (highlighted in bold) shows the values whose confidence interval are to be found, which are shown at the 90% limit in the last row (also in bold), as deduced from 500 simulations.
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