Detecting significant trends
The detectability of a trend or its significance may be treated from different but coincident points of views. It generally relies on the estimation of the standard deviation of the trend estimate (or the associated interval of confidence), and less usually on the number of observations required to detect a trend among a noise with a given variance. Formally, the statistical assessment of the significance of a trend in a time series of length n resorts to testing either the variable | ̂ |/ ̂ or the variable √ ̂√ −21− ̂² , with r, the coefficient of correlation between the time series and the trend. Both tests are similar [Scherrer, 77] and both variables theoretically follow a student’s T-distribution with n-2 degrees of freedom [Haan, 78; Legendre & Legendre, 79; Scherrer, 77]. Under the considered red noise model assumption, the 90% confidence level is reached for | ̂ |/ ̂ > 1.64 and the 95 % confidence level for | ̂ |/ ̂ > 1.96. In the subsequent, we consider a 95% confidence level, such that we test for | ̂ |/ ̂ > 1.96.
Application to the two-sensor SeaWiFS-MERIS dataset.
Tiao et al.  showed that the existence of a moderate positive value of Ф in the daily measurements is enough to make the trend estimate insensitive to changes in the temporal sampling. Compared to the daily data, the monthly averaged data will lower the length of the time series and the autocorrelation leading to similar trend detection. It implies that geophysical datasets, associated with high autocorrelation levels, may be analyzed using the monthly time series. Two datasets are used here, the global 1998-2010 SeaWiFS monthly chl-a products estimated using the OC4 algorithm [O’Reilly et al., 64 & 80], and the global 2003-2011 MERIS chl-a monthly estimated using the MERIS OC4 algorithm [Morel et al., 81]. Data were projected on a regular 1×1°grid and time series with more than 30% of missing data were withdrawn from the analysis, leaving 31829 for both datasets. For each location of the 1×1° grid, a climatology estimated from the available observations has been subtracted to the original time series to remove the seasonal signal, St. Neither spatial nor temporal interpolations were performed on the dataset. The value of T0 used in Eq.(13), i.e. the starting time of the MERIS time series, is equal to 60 months.
Single-sensor linear trend detection using the SeaWiFS dataset
Figure 1 shows for the period 1998-2010 the estimated model parameters for the single-sensor model Eq.(6) using the SeaWiFS monthly chl-a dataset, namely the long term trend ̂, the noise autocorrelation ϕ̂ and the white noise variance ̂². Overall we detect significant linear trends for 41 % of the 31829 time series (Figure 3). There are several coherent patches with significant trends. The typical magnitude of trends in the chl-a is ~ ± 0.003 mg.m−3 .year-1, with positive peak values of +0.009 mg.m−3.year-1 at the Eastern part of the Argentina, the South of Australia, the Behring Sea and specific coastal areas. Negative peak values of -0.009 mg.m−3.year-1 are reached in the North Atlantic & the Arabian Sea. We observe in the inter-tropical region a majority of negative trends in the chl-a concentration with a mean value of -0.002 mg.m-3.year-1.
Compared to previous trend estimations performed on the SeaWiFS dataset, Gregg  observed globally comparable trends on the period 1998-2003 with the exception of the eastern part of Africa that do not show anymore a positive trend. This difference may be explained either by the absence of a global linear trend for the entire 1998-2010 period, or by the ignorance of noise autocorrelation by Gregg et al. . Henson et al.  showed a similar global distribution of the trend estimates for this period using the same single-sensor model, Eq.(6), and the 1998-2007 SeaWiFS dataset, with nevertheless less positive trends East of South Argentina. This suggests that the data from 2008 to 2010 contributed significantly to the detection of significant positive trends of chl-a in this area.
Single-sensor linear trend detection using the MERIS dataset
We also report the same analysis as above for the 2003-2011 MERIS dataset (Figure 5). Although the MERIS dataset is only a 10-year time series compared to the 13 years of data available for the SeaWiFS dataset, we detect significant linear trends for 50% of the 2003-2011 MERIS time series (resp. 41% for the SeaWiFS data). The Equatorial Pacific shows a linear decrease of -0.002 mg.m-3.year-1 surrounded by a large belt of positive trends, with a mean value of 0.006 mg.m-3.year-1, starting from the East Papua New Guinea and ending in the North in the Mexico and in the South in the North of Chile leading to a wishbone shape of positive trends in the Equatorial Pacific. This region is well known to be strongly influenced by the ENSO signal (http://www.srh.weather.gov/srh/jetstream/tropics/enso.htm). In this region, the difference in terms of detection between SeaWiFS and MERIS dataset is clearly visible. A major bENSO–Niño event occurred during the 1997-1998 followed by a ENSO–Niña period during 1998-2000. Phytoplankton productivity relies on the availability of sunlight, macronutrients (e.g., nitrogen, phosphorous), and micronutrients (e.g., iron), and thus is sensitive to climate-driven changes in the delivery of these resources to the euphotic zone. Turk et al.  showed that the ENSO oscillation strongly impact the chl-a and the primary production in the Equatorial Pacific and one can clearly see Figure 3a that using the SeaWiFS 1998-2010 dataset a limited number of significant trends are detected in this area compared to the MERIS dataset over the period 2003-2011: non-stationary processes such as El-Niño-La-Niña tend to reduce the ability to detect a trend. The problem of estimating and removing ENSO-related variations from climate records has been addressed in many previous studies for the SST using a variety of methods. In this spirit, Compo et al.  used the ENSO pattern filter, EPF [Alexander et al., 85] developed to remove the contribution of ENSO patterns in the Sea Surface Temperature (SST) from 1871–2006. Satellite-derived ocean color time series are nevertheless shorter and show a greater intra-seasonal variability. This alters the ability of filtering the long-term variability caused by such signals. This type of filtering has not yet been implemented for the chl-a time series and its evaluation is in any case beyond the scope of this analysis, as we do not discuss of the quality of the input data.
The estimated noise autocorrelation shows a similar geographical distribution as observed for the SeaWiFS dataset with nevertheless a large band of high auto-correlated noise in the South Pacific. The residual variance is distributed similarly to the one estimated using SeaWiFS with nevertheless a Northward extension of the detected trends.
Two-sensor linear trend detection using both MERIS and SeaWiFS data
Using the two-sensor model, Eq. (13), the joint analysis (Figure 5) of the MERIS and SeaWiFS time series leads to 60% of significant detections of linear trends for the period 1998-2011 (resp. 50% and 41% for MERIS and SeaWiFS data alone). It resorts to much clearer patterns at a global scale for the period 1998-2011 compared to the period 1998-2010 & 2003-2011 considered individually. In Table 2 we summarize at ocean-scale trend estimated statistics. At global scale, the observed median value in the significant trends is 2.83 x 10-4 mg.m−3.year-1. This value is low and opposite to the estimated trend by Boyce et. al.  for the 20th century using this time in-situ data. Indian Ocean shows the largest median value with a decreasing value of -1.40 x 10-3 mg.m−3.year-1 while the Pacific and the Atlantic show similar median positive trends of respectively of 7.27 x 10-4 and 8.27 x 10-4 mg.m−3.year-1. Regarding coastal areas, we detect positive trends especially in the Bering Sea, the Pacific shores of the United States, the Patagonian Shelf (Figure 5a). Regarding the open ocean, Southern regions show as observed in Figure 3a & Figure 4a, a majority of positive trends with local maximum at +0.009 mg.m−3.year-1 in the Eastern part of South Argentina and the South East part of Australia. Although we do not discuss here of the quality of the dataset, we underline nevertheless that both algorithms, SeaWiFS OC4 and MERIS OC4, are calibrated for open ocean waters where the observed radiance is constrained by the water and the chl-a absorption properties. In coastal areas and specific areas, the effect of the suspended matters and the colored dissolved organic matters may alter the observed radiances leading to positive biases in the estimated chl-a retrieval using the OC4 algorithms and possibly affecting the trend estimation in such areas. A full discussion on the estimation of optical properties in coastal areas is available in [IOCCG, 17].
The ‘wishbone’ pattern in the Equatorial Pacific clearly appears Figure 5b with this time some extensions of the positive trends from the Florida to the Mediterranean Sea and from Brazil to South Africa. This Atlantic extension of this structure was not visible using the SeaWifs (Figure 5b) dataset and only partially visible using the MERIS dataset.
Some coastal regions depict negative trends with a minimum of -0.009 mg.m−3.year-1 in the equatorial area, the North Atlantic and the North Pacific. From the joint analysis, the Indian gyre and more generally the Indian Ocean shows a global negative trend except for its Southern area. The decline in the global gyres in the productivity, directly linked to the chl-a, was also observed by Polovina et al. .
In the Atlantic, the inter-tropical zone shows a low decrease of -0.002 mg.m−3.year-1. In the South Atlantic and below 40°S, the trend increases positively. Regarding the North Atlantic, we detect an increase of the chl-a in the North Atlantic Current & the Gulf Stream, and Northwards the Atlantic Western part shows a positive trend conversely to the Eastern part.
Table of contents :
Résumé étendu en Français
0.1 Couleur de l’eau et température de surface de la mer
0.1.1 Les données satellitaires
0.1.2 Principe de la mesure de “la couleur de l’eau”: Introduction à la théorie du transfert radiatif.
0.1.3 Principe de la mesure de “la température de surface”
0.2.1 Régression d’une variable aléatoire géophysique.
0.2.2 Estimation et caractérisation de modes distincts dans des signaux multivariés.
0.2.3 Estimation et caractérisation de régimes physiques distincts
0.3 Principaux résultats
0.4 Conclusions et perspectives
1 Chapter 1: Introduction
2 Chapter II: Detection of linear trends in multi-sensor time series in presence of auto-correlated noise: application to the chlorophyll-a SeaWiFS and MERIS datasets and extrapolation to the incoming Sentinel 3 – OLCI mission.
2.2 Trend estimation
2.3 Statistical modeling
2.3.1 Single-sensor dataset
2.3.2 Multi- sensor dataset
2.4 Detecting significant trends
2.5 Application to the two-sensor SeaWiFS-MERIS dataset.
2.5.1 The dataset
2.5.2 Single-sensor linear trend detection using the SeaWiFS dataset
2.5.3 Single-sensor linear trend detection using the MERIS dataset
2.5.4 Two-sensor linear trend detection using both MERIS and SeaWiFS data
2.6 Optimization of a time-overlap between successive missions for long term monitoring & impact of the incoming ESA Sentinel 3 – OLCI mission.
3 Chapter III: Multi-scale event-based mining in geophysical time series: characterization and distribution of significant time-scales in the Sea Surface Temperature anomalies relative to ENSO periods from 1985 to 2009.
3.2 Event-based analysis of geophysical times series
3.2.1 Wavelet-based extraction of elementary time-scale events
3.2.2 The determination of the elementary time-scale events
3.2.3 Event-based mining of the event database
3.3 Application to the satellite-derived SSTA observed from 1985 to 2009
3.3.1 The pathfinder dataset
3.3.2 The Multivariate Enso Index
3.3.3 Event detection examples in SSTA time series
3.3.4 Characteristic time-scales of SSTA elementary events.
3.3.5 The spatial distribution of the SSTA characteristic scales.
3.4 The density of HF and 1.54-year events with respect to ENSO modes.
3.5 Investigating frequency shifts in the SSTA and the inter-tropical Pacific during the ENSO 1997-2000 event.
3.6 Conclusions and future work
4 Chapter IV: Characterization of time-varying regimes in remote sensing time series: application to the forecasting of satellite-derived suspended matter concentrations.
4.2.1 Markov switching models
4.2.2 Estimation of the model parameters
4.2.3 Forecasting application
4.2.4 Model performance estimation
4.3 The data
4.3.1 The studied variable
4.3.2 Predictors and covariates
4.4.1 Example with the estimation of EC_SPIM1
5 Chapter 5: Ocean Color Atmospheric corrections in coastal complex waters using a Bayesian latent class model and potential for the incoming Sentinel 3 – OLCI mission.
5.2 Review of the standard Ocean Color inversion method
5.2.1 Atmospheric correction principles
5.3 The standard processing atmospheric correction scheme
The Bright Pixel Atmospheric Correction (BPAC)
5.4.1 Spectral representations of the water contributions using Non-Negative Matrix Factorization
5.4.2 Bayesian Formalism
5.4.3 Performance valuation
5.5 The in-situ MERMAID dataset.
5.6.1 Prior distributions of aerosol and water variables
5.6.2 Bayesian ocean-color inversion
5.6.3 Inversion performance for the Mermaid dataset
5.6.4 Example of estimated water reflectance on a very turbid area
5.6.5 Estimated water types associated with the MEETC2 inversion
6 General discussion & perspectives