Abstract
The relationships between sea levels and single climate indexes have been widely explored. However, sea level is controlled by multiple climate factors simultaneously with differences among places and time scales. Despite this, few studies have addressed the relationships between sea levels and multiple climate indexes. Here, the interrelations between the annual mean sea level (AMSL) and individual climate indexes and combinations of climate indexes were characterized by wavelet coherence (WTC) and multiple wavelet coherence (MWC). The results showed that six climate indexes had a significant correlation with AMSL, among which the Atlantic Multidecadal Oscillation (AMO) had the least significant correlation, but Niño3.4 had the significant correlation. The average significant coherence area (PASC) value for Niño3.4 was 14.03% and its average wavelet coherence (AWC) value was 0.79. By combining climate indexes, the average values of PASC and AWC can be significantly increased. The average PASC and AWC values of the five climate index combinations were the largest, 95.87% and 0.965, respectively, followed by four, three, and two index combinations. A single climate index is not sufficient to explain sea level change in North America. Sea level changes in North America require between three and five climate indexes to explain, depending on the region. By calculating PASC and AWC values, this study provides the possibility to understand the impact of the combined effects of multiple climate indexes on AMSL, is of great help to screen the best predictor of sea level, and provides a new method to reveal the complex mechanism of sea level change.
HIGHLIGHTS
Multiple wavelet coherence is first used to study the relationship between the annual mean sea level and multiple climate indexes.
We determine the optimal climate index combination for the annual mean sea level in North America.
We analyze the multiple time scales and dominant periods of the annual mean sea level in North America.
INTRODUCTION
Sea level is an important indicator of climate change and it changes along broad spatial and temporal scales. To reveal the physical mechanisms underlying sea level change it is necessary to study the relationship between sea level and climate indices on temporal and spatial scales. To date, research on sea level has mostly focused on the estimation of regional and global sea level rise (Hannah & Bell 2012; Wenzel & Schröter 2014; Hamlington & Thompson 2015) and the influence of atmospheric circulation on sea level change (Chafik et al. 2019). There are significant interannual and interdecadal variabilities in sea level, and these variations exhibit the characteristics of multiple periodic signals. Therefore, selecting analytical methods that have a robust periodic analysis ability is crucial to the study of sea level change.
The wavelet coherence (WTC) method is a robust tool for analyzing the physical mechanisms underlying periodic signals. This method was initially proposed to analyze the El Niño-Southern Oscillation (ENSO) time series. Later, Grinsted et al. (2004) applied it to geophysical time series, further demonstrating its capability to analyze the physical mechanisms underlying periodic variability. It has since been widely promoted and applied in various fields. For example, runoff (Tang et al. 2018; Wang et al. 2018) and lake level (Nourani et al. 2019). Over time the cross-wavelet has been greatly developed and used to analyze periodic cycles caused by a variety of physical mechanisms. These have included precipitation and groundwater (Yu & Lin 2015), runoff and climate indexes (Durocher et al. 2016), global mean temperature anomalies and sea level (Kirikkaleli & Sowah 2021), Baltic Sea level and zonal wind (Medvedev & Kulikov 2019), Pacific Decadal Oscillation (PDO) and sea level in the South China Sea (Xi et al. 2020), positive pressure sea level over the tropical Indian Ocean basin and the northern winter Madden-Julian oscillation (Rohith et al. 2019), Atlantic meridional overturning circulation and Mediterranean Sea level (Volkov et al. 2019), ocean signal and sea level change (Jevrejeva et al. 2006), ENSO and the Baltic Sea (Jevrejeva et al. 2003), sea level response of the Indian Ocean and ENSO (Tiwari et al. 2004), and East China Sea level and ENSO events (Liu et al. 2010). Recently, Piecuch et al. (2019) used the cross-wavelet to reveal the internal relationship between sea level off the New England coast and the Atlantic meridional overturning circulation at 26 °N. Little et al. (2021) used cross wavelets to reveal the high power and space–time complexity of the sea level along the east coast of North America on an interdecadal time scale. It can be seen that the cross-wavelet has become prevalent in the analysis of periodic oceanic signals.
WTC can only reflect the relationship between the two variables on the time scale. Previous studies have shown that a single climate index is not sufficient to explain sea level change (Little et al. 2021). In addition, sea level change can be influenced by multiple climate indexes and their interactions simultaneously. Therefore, if we only include a single factor in the analysis, it will be unable to fully characterize the observed changes. However, the disadvantage of existing multivariate methods is that they cannot reflect this relationship on a time scale, multivariate wavelet coherence (MWC) can determine the multivariate relationship between prediction variables and response variables on the time scale. Here, to overcome the shortcomings of wavelet coherence (WTC) and existing multivariate methods, and try to reveal the combined effects of multiple climate indexes on AMSL, we introduced the MWC invented by Hu & Si (2016). For the application of MWC, Gu et al. (2021) used it to evaluate the multivariate relationship of a local groundwater system, Song et al. (2020) used it to explore the potential relationship between extreme precipitation and a large-scale climate model, Nalley et al. (2019) used it to examine the multiscale relationship between precipitation, runoff change, ENSO, North Atlantic Oscillation (NAO), and PDO, and Cheng et al. (2021) used it to determine the relationship between base flow and meteorological factors/large-scale circulation indexes.
Nevertheless, there is still something worth exploring in this field of study. (1) Although cross wavelets have been used to analyze sea level until now no scholars have used MWC to study the relationship between AMSL and multiple climate indexes. (2) We used MWC to determine which four climate indexes are the best combination of the four climate indexes. (3) We determined how many climate indexes are adequate to explain AMSL in North America.
This paper is organized as follows: The principles of the WTC and MWC are introduced in Section 2. The study area and data sources are presented in Section 3. Section 4 relates the results of the WTC and MWC analyses. The discussion section is presented in Section 5. Conclusions are drawn in Section 6.
MATERIALS AND METHODS
Study area
Latitude and longitude information for 82 stations
Station . | Station number . | Latitude . | Longitude . | Station . | Station number . | Latitude . | Longitude . |
---|---|---|---|---|---|---|---|
SAN FRANCISCO | 10 | 37.807 | −122.465 | YAKUTAT | 445 | 59.548 | −139.733 |
NEW YORK (THE BATTERY) | 12 | 40.7 | −74.013 | ADAK SWEEPER COVE | 487 | 51.863 | −176.632 |
FERNANDINA BEACH | 112 | 30.672 | −81.465 | SKAGWAY | 495 | 59.45 | −135.327 |
SEATTLE | 127 | 47.602 | −122.338 | PORT ISABEL | 497 | 26.06 | −97.215 |
PHILADELPHIA (PIER 9N) | 135 | 39.933 | −75.142 | PORT SAN LUIS | 508 | 35.177 | −120.76 |
BALTIMORE | 148 | 39.267 | −76.578 | ST. PETERSBURG | 520 | 27.76 | −82.627 |
HONOLULU | 155 | 21.307 | −157.867 | GRAND ISLE | 526 | 29.263 | −89.957 |
SAN DIEGO (QUARANTINE STATION) | 158 | 32.713 | −117.173 | MONTAUK | 519 | 41.048 | −71.96 |
GALVESTON II, PIER 21, TX | 161 | 29.31 | −94.793 | BAR HARBOR, FRENCHMAN BAY, ME | 525 | 44.392 | −68.205 |
ATLANTIC CITY | 180 | 39.355 | −74.418 | ROCKPORT | 538 | 28.022 | −97.047 |
PORTLAND (MAINE) | 183 | 43.657 | −70.247 | KAHULUI HARBOR, MAUI ISLAND | 521 | 20.895 | −156.477 |
KEY WEST | 188 | 24.555 | −81.807 | KIPTOPEKE BEACH | 636 | 37.165 | −75.988 |
LEWES (BREAKWATER HARBOR) | 224 | 38.782 | −75.12 | NAWILIWILI BAY, KAUAI ISLAND | 756 | 21.953 | −159.355 |
KETCHIKAN | 225 | 55.332 | −131.625 | UNALASKA | 757 | 53.88 | −166.537 |
BOSTON | 235 | 42.353 | −71.053 | MOKUOLOE ISLAND | 823 | 21.432 | −157.79 |
CHARLESTON I | 234 | 32.782 | −79.925 | SEWARD | 266 | 60.12 | −149.427 |
LOS ANGELES | 245 | 33.72 | −118.272 | BRIDGEPORT | 1068 | 41.173 | −73.182 |
PENSACOLA | 246 | 30.403 | −87.21 | NANTUCKET ISLAND | 1111 | 41.285 | −70.097 |
LA JOLLA (SCRIPPS PIER) | 256 | 32.867 | −117.257 | SELDOVIA | 1070 | 59.44 | −151.72 |
ASTORIA (TONGUE POINT), OR | 265 | 46.207 | −123.768 | FORT MYERS | 1106 | 26.647 | −81.87 |
HILO, HAWAII ISLAND | 300 | 19.73 | −155.055 | CAPE MAY | 1153 | 38.968 | −74.96 |
SEWELLS POINT, HAMPTON ROADS | 299 | 36.947 | −76.33 | SOUTH BEACH | 1196 | 44.625 | −124.042 |
ANNAPOLIS (NAVAL ACADEMY) | 311 | 38.983 | −76.48 | APALACHICOLA | 1193 | 29.727 | −84.982 |
EASTPORT | 332 | 44.903 | −66.982 | CORDOVA | 566 | 60.558 | −145.752 |
NEWPORT | 351 | 41.505 | −71.327 | CHARLESTON II | 1269 | 43.345 | −124.322 |
WASHINGTON DC | 360 | 38.873 | −77.022 | CAMBRIDGE II | 1295 | 38.573 | −76.068 |
SANDY HOOK | 366 | 40.467 | −74.008 | LEWISETTA, VA | 2324 | 37.995 | −76.465 |
WOODS HOLE (OCEAN. INST.) | 367 | 41.523 | −70.672 | PORT TOWNSEND | 1325 | 48.112 | −122.757 |
SANTA MONICA (MUNICIPAL PIER) | 377 | 34.008 | −118.5 | BEAUFORT, NC | 2295 | 34.72 | −76.67 |
CRESCENT CITY | 378 | 41.745 | −124.182 | MONTEREY | 1352 | 36.605 | −121.887 |
FRIDAY HARBOR (OCEAN. LABS.) | 384 | 48.547 | −123.01 | VALDEZ | 1353 | 61.125 | −146.362 |
NEAH BAY | 385 | 48.367 | −124.612 | POINT REYES | 1394 | 37.995 | −122.977 |
FORT PULASKI | 395 | 32.033 | −80.902 | PORT ANGELES, WA | 2127 | 48.125 | −123.44 |
WILMINGTON | 396 | 34.227 | −77.953 | PORT CHICAGO, CA | 2330 | 38.055 | −122.04 |
JUNEAU | 405 | 58.298 | −134.412 | TOKE POINT, WILLIPA BAY, WA | 1354 | 46.707 | −123.967 |
SOLOMON'S ISLAND (BIOL. LAB.) | 412 | 38.317 | −76.452 | KODIAK ISLAND, WOMENS BAY | 567 | 57.732 | −152.512 |
SITKA | 426 | 57.052 | −135.342 | SAND POINT, POPOF IS., AK | 1634 | 55.337 | −160.502 |
CEDAR KEY II | 428 | 29.135 | −83.032 | PORT ORFORD | 1640 | 42.738 | −124.498 |
NEW LONDON | 429 | 41.36 | −72.09 | PANAMA CITY, ST.ANDREWS BAY, FL | 1641 | 30.152 | −85.667 |
PROVIDENCE (STATE PIER) | 430 | 41.807 | −71.4 | REEDY POINT | 786 | 39.558 | −75.573 |
ALAMEDA (NAVAL AIR STATION) | 437 | 37.772 | −122.298 | VACA KEY | 1701 | 24.712 | −81.105 |
Station . | Station number . | Latitude . | Longitude . | Station . | Station number . | Latitude . | Longitude . |
---|---|---|---|---|---|---|---|
SAN FRANCISCO | 10 | 37.807 | −122.465 | YAKUTAT | 445 | 59.548 | −139.733 |
NEW YORK (THE BATTERY) | 12 | 40.7 | −74.013 | ADAK SWEEPER COVE | 487 | 51.863 | −176.632 |
FERNANDINA BEACH | 112 | 30.672 | −81.465 | SKAGWAY | 495 | 59.45 | −135.327 |
SEATTLE | 127 | 47.602 | −122.338 | PORT ISABEL | 497 | 26.06 | −97.215 |
PHILADELPHIA (PIER 9N) | 135 | 39.933 | −75.142 | PORT SAN LUIS | 508 | 35.177 | −120.76 |
BALTIMORE | 148 | 39.267 | −76.578 | ST. PETERSBURG | 520 | 27.76 | −82.627 |
HONOLULU | 155 | 21.307 | −157.867 | GRAND ISLE | 526 | 29.263 | −89.957 |
SAN DIEGO (QUARANTINE STATION) | 158 | 32.713 | −117.173 | MONTAUK | 519 | 41.048 | −71.96 |
GALVESTON II, PIER 21, TX | 161 | 29.31 | −94.793 | BAR HARBOR, FRENCHMAN BAY, ME | 525 | 44.392 | −68.205 |
ATLANTIC CITY | 180 | 39.355 | −74.418 | ROCKPORT | 538 | 28.022 | −97.047 |
PORTLAND (MAINE) | 183 | 43.657 | −70.247 | KAHULUI HARBOR, MAUI ISLAND | 521 | 20.895 | −156.477 |
KEY WEST | 188 | 24.555 | −81.807 | KIPTOPEKE BEACH | 636 | 37.165 | −75.988 |
LEWES (BREAKWATER HARBOR) | 224 | 38.782 | −75.12 | NAWILIWILI BAY, KAUAI ISLAND | 756 | 21.953 | −159.355 |
KETCHIKAN | 225 | 55.332 | −131.625 | UNALASKA | 757 | 53.88 | −166.537 |
BOSTON | 235 | 42.353 | −71.053 | MOKUOLOE ISLAND | 823 | 21.432 | −157.79 |
CHARLESTON I | 234 | 32.782 | −79.925 | SEWARD | 266 | 60.12 | −149.427 |
LOS ANGELES | 245 | 33.72 | −118.272 | BRIDGEPORT | 1068 | 41.173 | −73.182 |
PENSACOLA | 246 | 30.403 | −87.21 | NANTUCKET ISLAND | 1111 | 41.285 | −70.097 |
LA JOLLA (SCRIPPS PIER) | 256 | 32.867 | −117.257 | SELDOVIA | 1070 | 59.44 | −151.72 |
ASTORIA (TONGUE POINT), OR | 265 | 46.207 | −123.768 | FORT MYERS | 1106 | 26.647 | −81.87 |
HILO, HAWAII ISLAND | 300 | 19.73 | −155.055 | CAPE MAY | 1153 | 38.968 | −74.96 |
SEWELLS POINT, HAMPTON ROADS | 299 | 36.947 | −76.33 | SOUTH BEACH | 1196 | 44.625 | −124.042 |
ANNAPOLIS (NAVAL ACADEMY) | 311 | 38.983 | −76.48 | APALACHICOLA | 1193 | 29.727 | −84.982 |
EASTPORT | 332 | 44.903 | −66.982 | CORDOVA | 566 | 60.558 | −145.752 |
NEWPORT | 351 | 41.505 | −71.327 | CHARLESTON II | 1269 | 43.345 | −124.322 |
WASHINGTON DC | 360 | 38.873 | −77.022 | CAMBRIDGE II | 1295 | 38.573 | −76.068 |
SANDY HOOK | 366 | 40.467 | −74.008 | LEWISETTA, VA | 2324 | 37.995 | −76.465 |
WOODS HOLE (OCEAN. INST.) | 367 | 41.523 | −70.672 | PORT TOWNSEND | 1325 | 48.112 | −122.757 |
SANTA MONICA (MUNICIPAL PIER) | 377 | 34.008 | −118.5 | BEAUFORT, NC | 2295 | 34.72 | −76.67 |
CRESCENT CITY | 378 | 41.745 | −124.182 | MONTEREY | 1352 | 36.605 | −121.887 |
FRIDAY HARBOR (OCEAN. LABS.) | 384 | 48.547 | −123.01 | VALDEZ | 1353 | 61.125 | −146.362 |
NEAH BAY | 385 | 48.367 | −124.612 | POINT REYES | 1394 | 37.995 | −122.977 |
FORT PULASKI | 395 | 32.033 | −80.902 | PORT ANGELES, WA | 2127 | 48.125 | −123.44 |
WILMINGTON | 396 | 34.227 | −77.953 | PORT CHICAGO, CA | 2330 | 38.055 | −122.04 |
JUNEAU | 405 | 58.298 | −134.412 | TOKE POINT, WILLIPA BAY, WA | 1354 | 46.707 | −123.967 |
SOLOMON'S ISLAND (BIOL. LAB.) | 412 | 38.317 | −76.452 | KODIAK ISLAND, WOMENS BAY | 567 | 57.732 | −152.512 |
SITKA | 426 | 57.052 | −135.342 | SAND POINT, POPOF IS., AK | 1634 | 55.337 | −160.502 |
CEDAR KEY II | 428 | 29.135 | −83.032 | PORT ORFORD | 1640 | 42.738 | −124.498 |
NEW LONDON | 429 | 41.36 | −72.09 | PANAMA CITY, ST.ANDREWS BAY, FL | 1641 | 30.152 | −85.667 |
PROVIDENCE (STATE PIER) | 430 | 41.807 | −71.4 | REEDY POINT | 786 | 39.558 | −75.573 |
ALAMEDA (NAVAL AIR STATION) | 437 | 37.772 | −122.298 | VACA KEY | 1701 | 24.712 | −81.105 |
Change trends (a) and sequence lengths (b) of AMSL at 82 stations. (It is worth noting that the circles in the figure represent stations, for which detailed location information is shown in Table 1.)
Change trends (a) and sequence lengths (b) of AMSL at 82 stations. (It is worth noting that the circles in the figure represent stations, for which detailed location information is shown in Table 1.)
Data sources
According to the research of Zhang & Church (2012), ENSO and PDO can explain 60% of the total sea level change in the era of elevation measurements, so ENSO and PDO are strong influences on the change in AMSL. ENSO indexes use time series of area-averaged SST (e.g., Niño3.4: from 5 °N to 5 °S and from 170 °W to 120 °W) to identify El Niño and La Niña events. The PDO index was identified as the leading principal component of the variability of the North Pacific (poleward of 20 °N) monthly sea surface temperature (SST) (Mantua et al. 1997). Therefore, in this study, we selected the Niño3.4 and PDO indexes. The NAO determines climate change from the East coast of the United States to Siberia, and from the Arctic to the subtropical Atlantic (Hurrell et al. 2003). The Southern Oscillation Index (SOI) is the difference in sea level pressure between Tahiti (148 °05′W, 17 °53′S) or Easter Island (109 °30′W, 29 °00′S) and Darwin (125 °59′E, 12 °20′S). A negative SOI corresponds to an El Niño event, while a positive SOI corresponds to a La Niña event. The Atlantic Multidecadal Oscillation (AMO) index refers to the annual mean of SST anomalies in the region from 75 °W to 7.5 °W and 0 °N to 60 °N. The tripole index (TPI) is based on the difference between the sea surface temperature anomalies (SSTAs) averaged over the central equatorial Pacific and the average of the SSTA in the Northwest and Southwest Pacific (Henley et al. 2015).
The annual mean sea level (AMSL) data for North America were obtained from the Permanent Service for Mean Sea Level (https://www.psmsl.org/data/obtaining/complete.php). The NAO, Niño3.4 index, AMO, and SOI were provided by NOAA/ESRL (data source: https://www.esrl.noaa.gov/psd/gcos_wgsp/Timeseries/). The PDO was acquired from the NOAA National Centers for Environmental Information (https://www.ncdc.noaa.gov/teleconnections/PDO/). The TPI was acquired from the NOAA Physical Sciences Laboratory (https://psl.noaa.gov/data/timeseries/IPOTPI/tpi.timeseries.ersstv5.data). For annual climate data, we took the average of each of the four seasons: spring, summer, autumn, and winter.
Methods
Wavelet coherence






It should be noted that we evaluate the performances of single factors and combined factors using the average wavelet coherence (AWC, the average of the coherence value in the significant coherence region) and the percentage of significant coherence area (PASC, 100 * significant coherence area/(significant coherence area + insignificant coherence area)). The larger the PASC, the more important that single factor or combination of factors is, the same applies to MWC. It should be noted that a larger PASC tends to correspond to a larger AWC, and smaller PASC values do not correspond to larger AWC values. However, whether the variable is the dominant variable should be reflected by PASC value.
Multiple wavelet coherence
WTC can be thought of as the traditional correlation coefficient localized in the scale-location domain (Grinsted et al. 2004). The wavelet correlation coefficient can be extended from two variables to multiple variables (>2) and, in the same way, the wavelet coherence between two variables can be extended to multiple variables. Similar to bivariate wavelet coherence, MWC utilizes a series of auto- and cross-wavelet power spectra at different scales and spatial (or temporal) locations for the response variable and all predictor variables (Hu & Si 2016).









RESULTS
Results of WTC analysis
Wavelet coherence between AMSL and Niño3.4. (Thick contours denote 5% significance levels against red noise, the areas outside the cones represent the areas where edge effects might distort the results, which is an invalid area. The arrow denotes the relative phase relationships (positive correlation, arrows point right; negative correlation, arrows point left). The color denotes the strength of coherence.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
Wavelet coherence between AMSL and Niño3.4. (Thick contours denote 5% significance levels against red noise, the areas outside the cones represent the areas where edge effects might distort the results, which is an invalid area. The arrow denotes the relative phase relationships (positive correlation, arrows point right; negative correlation, arrows point left). The color denotes the strength of coherence.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
We found that AMSL of most stations has significant correlation with Niño3.4 over a period of 4–8 years, which corresponds to the quasi period of ENSO. The coherence between Niño3.4 and AMSL was complex. Some stations had large areas of significant coherence between AMSL and Niño3.4, such as stations 1196–1634, 300–385, 636–823, 537, 508, and 1196–1634. The years when stations exhibited significant coherence generally corresponded to either EI Niño events (i.e., 1939–1941, 1965–1966, 1972–1976, 1982, 1997–1998, 2006–2007, 2009, and 2010) or La Niña events (i.e., 1970, 1984, 1988–1989, 1998–2001, 2007–2008, and 2010–2011) (Haddad et al. 2013). Furthermore, at some stations the AMSL did not have any significant coherence with Niño3.4 (i.e., stations 112, 183, 188, 234, 1641, and 430).
Wavelet coherence between AMSL and PDO. (Thick contours denote 5% significance levels against red noise, the areas outside the cones represent the areas where edge effects might distort the results, which is an invalid area. The arrow denotes the relative phase relationships (positive correlation, arrows point right; negative correlation, arrows point left). The color denotes the strength of coherence.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
Wavelet coherence between AMSL and PDO. (Thick contours denote 5% significance levels against red noise, the areas outside the cones represent the areas where edge effects might distort the results, which is an invalid area. The arrow denotes the relative phase relationships (positive correlation, arrows point right; negative correlation, arrows point left). The color denotes the strength of coherence.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
The coherences between AMSL and PDO were complex. Not all stations had relatively large PASC values (PASC value was above 15%), and some stations even had relatively small (PASC value was less than 15%). For example, the PASC values at stations 12, 112, 161, 224, 246, 351, 360, 395, 396, 299, 311, 332, 1106, 430, 497, and 636 (longitudes ranging from about 70 °W to 95 °W) were relatively small, indicating that PDO had a relatively small effect on the changes in AMSL at these stations. In addition, the PASC of AMSL and PDO at stations 158, 245, 256, 300, 377, 378, 384, 385, 566, 567, 1269, 1325, 1352, 1394, 1634, 1640, 2127, 487, 508, and 757 (ranging approximately from 120 °W to 125 °W) were relatively large, which meant that PDO had a greater influence on the changes in AMSL at these stations. For the stations with larger PASC, we found that the time periods of PDO coherence to AMSL were generally between 4–8 years and 8–16 years, and this periodicity also lasted for a long time. The effects of PDO on sea level changes in North America also appeared localized.
Distributions of PASC of six climate indexes ((a) SOI, (b) PDO, (c) Niño 3.4, (d) TPI, (e) NAO, and (f) AMO) for AMSL.
Distributions of PASC of six climate indexes ((a) SOI, (b) PDO, (c) Niño 3.4, (d) TPI, (e) NAO, and (f) AMO) for AMSL.
Distributions of AWC of six climate indexes ((a) SOI, (b) PDO, (c) Niño 3.4, (d) TPI, (e) NAO, and (f) AMO) for AMSL.
Distributions of AWC of six climate indexes ((a) SOI, (b) PDO, (c) Niño 3.4, (d) TPI, (e) NAO, and (f) AMO) for AMSL.
There were significant differences between the PASC of the six climate indicators for AMSL. The average PASC values of Niño3.4, PDO, TPI, SOI, NAO, and AMO were 14.03, 9.86, 12.92, 10.86, 8.80, and 6.07%, respectively. The average PASC values of Niño3.4, TPI, and SOI were relatively large, while those of NAO and AMO were lower. Niño3.4 had the highest PASC average. This meant that the average influences of NAO and AMO on AMSL were lower than that of Niño3.4, PDO, TPI, and SOI on AMSL. Among the six climate indexes, Niño3.4 had the largest average impact on AMSL, followed by TPI, SOI, PDO, NAO, and AMO, in that order. In addition, the average PASC of Niño3.4, PDO, SOI, and TPI within the 115°–125 °W range was greater than that of the 65°–90 °W range, which meant that the impacts of these indexes on AMSL were greater at 115°–125 °W than at 65°–90 °W.
The differences between the AWC of AMSL for the six climate indexes were not very large, which was also true for the AWC of the different stations. The coherence range (the largest of the 82 stations minus the smallest) was within 0.1. The regions with relatively large climate index AWC were mainly concentrated within 115°–125 °W. PDO and AMSL showed a coherence of about 0.78 along the eastern Pacific coast of North America, PDO and AMSL showed high correlation, which is similar to the conclusion of Hamlington et al. (2014). From the PASC and AWC values, the impact of Niño3.4 on AMSL was greater than SOI, which means that the impact of El Niño on AMSL is greater than that of the southern oscillation. The mean values of PASC and AWC corresponding to Niño 3.4 were the largest among the six climate indexes, which means that ENSO plays a very important role in the change of AMSL in North America.
Results of MWC analysis
Section 3.1 reports on the significant coherence between the individual climatic indexes and AMSL. Next, we will focus on the relationships between AMSL and combinations of multiple climate indexes.
Niño3.4–PDO
MWC between AMSL and Niño3.4–PDO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.)
MWC between AMSL and Niño3.4–PDO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.)
Compared with Niño3.4, under the combined effects of Niño3.4–PDO, the significant coherence area increased significantly, and the average PASC of the 82 stations increased from 14.03 to 49.63%, and the average coherence of the 82 stations also significantly increased from 0.79 to 0.83. This meant that the coherence of Niño3.4–PDO with AMSL was significantly greater than that of Niño3.4 with AMSL. However, the effect of Niño3.4–PDO on AMSL was still localized, and on some time scales there was still no significant coherence.
Distributions of PASC and AWC ((a) PASC and (b) AWC) of Niño3.4–PDO.
It can be seen that the stations with large PASC values (PASC ≥ 85%) were mainly concentrated in the 115 °W–125 °W region, and Niño3.4–PDO had a large impact on the AMSL. In the 70 °W–90 °W region of the east coast of North America, the PASC values were relatively small, and Niño3.4–PDO had a smaller impact on the AMSL. Similar to earlier, the AWC of different stations did not differ much, with a total range of 0.1, and the distribution of AWC was similar to that of PASC, with relatively larger AWC values within the 120 °W–125 °W region.
Moon et al. (2015) studied the response of ENSO–PDO phase relationship changes to sea level changes. He believed that when ENSO and PDO are in phase, the sea level difference is quite large. Our analysis shows that under the action of Niño3.4–PDO, for the east coast of the Pacific Ocean, the high value area of PASC was mainly concentrated in the area of 115–125 °W longitude, and the low value area of PASC was 152–160 °W, this shows that there are regional differences in the influence of Niño3.4–PDO on AMSL. We have a conclusion similar to that of Moon et al. (2015) was reached.
Although we did not discuss the modulation effect of PDO on ENSO in different phases, our analysis showed that: The influence of Niño3.4–PDO on AMSL was greater than that of Niño3.4 and PDO on AMSL, because we can see that compared with the effect of Niño3.4 and PDO under the effect of Niño3.4–PDO, PASC and AWC of 82 stations were significantly increased, relative to the effect of Niño3.4 and PDO.
Niño3.4–PDO–NAO
MWC between AMSL and Niño3.4–PDO–NAO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
MWC between AMSL and Niño3.4–PDO–NAO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
As can be seen from Figure 8, compared with Niño3.4 and Niño3.4–PDO, the combined effects of Niño3.4–PDO–NAO had a much higher significant coherence area, and the average PASC and average coherence of the 82 stations increased to 81% and 0.88, respectively. This was significantly greater than the PASC and AWC of Niño3.4 and Niño3.4–PDO, which meant that Niño3.4–PDO–NAO had a stronger influence on AMSL. Furthermore, the time scale of the effect of Niño3.4–PDO–NAO on AMSL was longer than that of Niño3.4–PDO.
Distributions of PASC and AWC ((a) PASC and (b) AWC) of Niño3.4–PDO–NAO.
It can be seen that the stations with large PASC values (PASC ≥ 95%) were mainly concentrated in the 120 °W–125 °W region and that Niño3.4–PDO–NAO had a large impact on the AMSL. In the 70 °W–90 °W region of the east coast of North America, the PASC values were relatively small, and Niño3.4–PDO–NAO had only a small impact on the change of AMSL. The AWC values of the different stations did not differ much, basically ranging within 0.1. The distribution of AWC was similar to that of PASC, with relatively large AWC values between 120 °W and 125 °W and relatively small AWC values between 70 °W and 90 °W. This was consistent with the significant coherence analysis of Niño3.4–PDO and AMSL.
Niño3.4–PDO–NAO–AMO
MWC between AMSL and Niño3.4–PDO–NAO–AMO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
MWC between AMSL and Niño3.4–PDO–NAO–AMO. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
Distributions of PASC and AWC ((a) PASC and (b) AWC) of Niño3.4–PDO–NAO–AMO.
We found that the PASC values were higher in the 50 °N–60 °N and 150 °W–180 °W regions, and that the PASC values were about 98%. The PASC values were also higher in the 115 °W–125 °W and 30 °N–50 °N regions. In these regions, Niño3.4–PDO–NAO–AMO had the greatest impact on AMSL. Within 120 °W–125 °W, AWC was the largest and at 70 °W–90 °W, AWC was the smallest.
Under the combined effects of Niño3.4–PDO–NAO–AMO, the PASC ranged within 85.35–100%, and the range of the AWC was 0.89–0.98, which were significantly greater than the PASC and AWC under Niño3.4, Niño3.4–PDO, and Niño3.4–PDO–NAO. This meant that Niño3.4–PDO–NAO–AMO had a greater overall impact on AMSL than the other indexes and index combinations.
Niño3.4–PDO–NAO–AMO–TPI
The PASC and AWC of Niño3.4–PDO–NAO–AMO–TPI. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
The PASC and AWC of Niño3.4–PDO–NAO–AMO–TPI. (Thick contours denote 5% significance levels against red noise. The area outside the cone represents the area where edge effects might distort the results.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
Distributions of PASC and AWC ((a) PASC and (b) AWC) of Niño3.4–PDO–NAO–AMO–TPI and AMSL.
Distributions of PASC and AWC ((a) PASC and (b) AWC) of Niño3.4–PDO–NAO–AMO–TPI and AMSL.
Under the combined effects of Niño3.4–PDO–NAO–AMO–TPI, both the PASC and AWC had high values for all stations, with PASC ranging from 95.87 to 100% and AWC ranging from 0.938 to 0.991, which were significantly greater than the PASC and AWC under Niño3.4–PDO–NAO–TPI.
According to Hu & Si (2016), when the addition of a climate index leads to an increase in PASC of more than 5%, the climate index is important. Therefore, with Niño3.4–PDO–NAO–AMO–TPI already returning minimum PASC values of 95.37%, any further increase of 5% in PASC is impossible. With such a high explanatory power already being achieved, there is no need to add additional climate indexes to explain the changes in AMSL.
DISCUSSION
Which four climate indexes are the best combination of the four climate indexes?
Difference between the PASC values of Niño3.4–PDO–NAO–AMO and those of PDO–NAO–AMO–TPI (a) and PDO–NAO–AMO–SOI (b). (Note: The scale of the color bar is more positive, indicating that the PASC of Niño3.4–PDO–NAO–AMO was generally greater than those of PDO–NAO–AMO–TPI or PDO–NAO–AMO–SOI. If the scale of the color bar was negative, it would indicate that the PASC of Niño3.4–PDO–NAO–AMO was smaller than those of PDO–NAO–AMO–TPI or PDO–NAO–AMO–SOI.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
Difference between the PASC values of Niño3.4–PDO–NAO–AMO and those of PDO–NAO–AMO–TPI (a) and PDO–NAO–AMO–SOI (b). (Note: The scale of the color bar is more positive, indicating that the PASC of Niño3.4–PDO–NAO–AMO was generally greater than those of PDO–NAO–AMO–TPI or PDO–NAO–AMO–SOI. If the scale of the color bar was negative, it would indicate that the PASC of Niño3.4–PDO–NAO–AMO was smaller than those of PDO–NAO–AMO–TPI or PDO–NAO–AMO–SOI.) Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/wcc.2022.180.
We found that the PASC of Niño3.4–PDO–NAO–AMO was not very different from PDO–NAO–AMO–TPI or PDO–NAO–AMO–SOI, and the PASC difference at most stations was within plus or minus 3%. The PASC differences between Niño3.4–PDO–NAO–AMO and PDO–NAO–AMO–TPI were greater than or equal to 0 at 59% of stations, and stations 10, 127, 155, 180, and 1068 had the largest PASC differences, at 11.04, −6.91, 14.62, 31.95, and 28.48%, respectively. The PASC differences between of Niño3.4–PDO–NAO–AMO and PDO–NAO–AMO–SOI were greater than or equal to 0 at 40% of the stations, and those of stations 127 and 1068 were largest, −6.17 and 28.48%, respectively.
Furthermore, the average PASC values of Niño3.4–PDO–NAO–AMO, PDO–NAO–AMO–TPI and PDO–NAO–AMO–SOI were 96.00, 95.06, and 95.83%, respectively. The average PASC values of Niño3.4–PDO–NAO–AMO and PDO–NAO–AMO–TPI had a large difference (0.94%), while those of Niño3.4–PDO–NAO–AMO and PDO–NAO–AMO–SOI had a small difference (0.17%). By comparing the PASC of Niño3.4, TPI, and SOI, we found that the PASC of Niño3.4 was higher than those of TPI and SOI at 61 and 57% of the stations, respectively. In addition, the average PASC of Niño3.4 among the 82 stations was 14.03%, higher than the average PASC of TPI (12.92%) and SOI (10.86%), but the differences in AWC were small.
After examining the PASC of single climate indexes and the average PASC values of Niño3.4–PDO–NAO–AMO, PDO–NAO–AMO–TPI, and PDO–NAO–AMO–SOI, we recommend Niño3.4–PDO–NAO–AMO as the optimal combination of four climate indexes.
How many climate indexes are adequate to explain AMSL?
Previous studies (Hamlington et al. 2015; Moon et al. 2015) usually only include a single climate variable, such as ENSO and PDO, to explain sea level change, which may be inadequate in the interpretation of sea level change, because scale dependence and local information were previously ignored (Hu et al. 2017).
It can be seen that the Niño3.4–PDO–NAO combination was best within 122 °W–126 °W, while the best climate index combinations within 150 °W–180 °W and 50 °N–60 °N was Niño3.4–PDO–NAO–AMO. In addition, the region within 65 °W–90 °W corresponded to the Niño3.4–PDO–NAO–AMO combination. The Niño3.4–PDO–NAO–AMP–TPI combination was mainly best within 67 °W–76 °W and 90 °W–98 °W.
In short, to reflect the changes in AMSL in most regions of North America, at least three climate indexes (Niño3.4–PDO–NAO; 122 °W–126 °W) are required, but sometimes four (Niño3.4–PDO–NAO–AMO ) or five (Niño3.4–PDO–NAO–AMO–TPI) climate indexes are best. Although only six climate indexes were examined in this study, they were selected because of their close relationship to AMSL, so the impact of other climate indexes on AMSL were not discussed in depth. Other climate indexes may indirectly affect PDO or ENSO events. However, under various climate index combinations, especially the Niño3.4–PDO–NAO–AMO–TPI combination, AWC and PASC were sufficiently large (see Figure 13). The PASC of Niño3.4–PDO–NAO and Niño3.4–PDO–NAO–AMO were also large enough at some stations, and the PASC values of Niño3.4–PDO–NAO–AMO–TPI were basically large enough across the whole-time domain, which meant that it is sufficient to explain the changes in AMSL on various time scales.
Future prospects
Although MWC cannot reflect the modulation effect of a single climate index on ENSO, nor can it reflect the modulation effect of multiple climate indexes on ENSO, revealing the modulation effect of multiple climate indexes on ENSO (the effect of different phases on ENSO) is an extremely complex and challenging task. The AWC and PASC values of MWC provide the possibility to understand the combined effects of multiple climate indexes on AMSL and ENSO. In addition, wavelet coherence and multivariable wavelet coherence provide the possibility to understand the dominant factors and complex mechanism of sea level change. PASC value and AWC value can be used to screen out the best influencing factors. However, the traditional principal component analysis cannot reflect the influence of dependent variables on independent variables in time scale. In addition, complex factors, such as the rotation of the Earth, affect atmospheric circulation, which in turn influences sea level changes, although these effects are not considered in this paper. The research conducted in this paper also provides the possibility to understand the impact of earth rotation on sea level, which can be qualitatively judged by PASC and AWC values.
In the future, we will further study the influence of multiple climate indexes on AMSL at different phases and the mechanisms of sea level change.
CONCLUSIONS
Using wavelet coherence and MWC to study the time scale relationship between the AMSL and climate indicators at 82 stations in North America, the following conclusions can be drawn: Among the six climate indexes, Niño 3.4 has the strongest significant coherence with AMSL. However, the time scale of the effect of a single climate index on AMSL is localized. Therefore, the joint action of multiple climate indexes was needed to improve the global explanatory power of climate indexes on the time scales of AMSL. Three to five climate indexes (Niño3.4–PDO–NAO, Niño3.4–PDO–NAO–AMO, Niño3.4–PDO–NAO–AMO–TPI) were sufficient to reflect the change of AMSL, depending on the region.
The MWC was used to identify the best combination of independent variables by calculating the PASC and AWC of independent variables. The larger the PASC and AWC, the more important the independent variable, which also provides a new method for screening the best predictor of sea level. This new method provides an effective means to resolve the complex spatial and temporal variability of multiple control factors over multiple temporal scales. The methods used in this study are universal and can serve as a reference for screening the best independent variables for dependent variables in other fields.
It must be noted that we only discussed the impacts of Niño3.4, PDO, NAO, AMO, SOI, and TPI on AMSL, and did not discuss the impacts of other climate indexes or factors (glacier melting, population growth) on AMSL. In future studies, we will further explore the influence of these factors on AMSL in North America.
ACKNOWLEDGEMENTS
Data supporting all figures and tables are freely available upon request from the corresponding author. The codes for calculating wavelet coherence are available from Grinsted et al. (2004) and can be downloaded from the website (http://grinsted.github.io/wavelet-coherence/). The codes for calculating MWC are available from Hu & Si (2016) and can be downloaded from the website (http://www.hydrol-earth-syst-sci.net/20/3183/2016/hess-20-3183-2016-supplement.pdf). This research was financially supported by the National Natural Science Foundation of China (Grant Nos U1911204 and 51861125203), The National Key R&D Program of China (2017YFC0405900), and the Project for Creative Research from the Guangdong Water Resources Department (Grant No. 2018, 2020).
DECLARATION OF COMPETING INTEREST
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.