# Quantifying the Impact of Linear Regression Model in Deriving Bio-Optical Relationships: The Implications on Ocean Carbon Estimations

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## Abstract

**:**

_{bp}), and 19 years of monthly TChla and b

_{bp}ocean colour data. Results of the regression analysis have been used to calculate phytoplankton carbon biomass (C

_{phyto}) and POC from: i) BGC-Argo float observations; ii) oceanographic cruises, and iii) satellite data. These applications enable highlighting the differences in C

_{phyto}and POC estimates relative to the choice of the method. An analysis of the statistical properties of the dataset and a detailed description of the hypothesis of the work drive the selection of the linear regression method.

## 1. Introduction

_{bp}) is in situ measured or derived from ocean colour imagery. It is at the base of the estimation of the particulate organic carbon (POC) [6,7,8,9,10] and the phytoplankton carbon biomass (C

_{phyto}) [11,12,13], both of which are fundamental variables used to constrain and understand the total carbon budget in the ocean [14,15]. Even though there are many works in which regression methods are correctly used and clearly mentioned in the text giving the opportunity to understand and reproduce the work [16,17,18,19,20], there are several cases where no information is provided about the linear regression method used [12,21,22,23,24,25], thus, preventing an evaluation of the impact of the methodology on the derived parameters. The lack of such information is crucial as the use of one method over another can return significantly different estimates on parameters. Indeed, differences due to the application of a sub-optimal regression method, instead, might be considered as errors and propagate if the wrong parameters are then used as inputs for modelling (e.g., empirical algorithms of ocean parameters). For this reason, as McArdle (2003) [26] pointed out “…if the slope, the intercept, or both parameters of the line are important, then care must be taken that the scientific conclusions follow from the data”. In other words, the scientific conclusions must be based on the appropriate methodology, i.e., a methodology adapted to the statistical properties of the data set to be analysed.

_{bp}and b

_{bp}-POC relationships from discrete samples. Afterwards, we assessed the impact of the derived linear models with to the estimation of C

_{phyto}and POC base on the time series of b

_{bp}vertical profiles from the BGC-Argo floats and by applying either type-I or type-II regression method. Finally, a similar analysis was also conducted relying on satellite observations, namely C

_{phyto}was evaluated, by using 19-years of monthly TChla and b

_{bp}.

## 2. Data and Methods

#### 2.1. Theoretical Background

#### 2.2. Field and Satellite Measurements

#### 2.2.1. Cruise Data

_{bp}collected from October 2011 to December 2013, whose measurement protocols are summarized below:

_{bp}(λ), is obtained following [33], with few differences: (1) one dark 0–50 m β(140, λ) profile was measured, averaged and subtracted from all profiles within each cruise; (2) data were binned around ±0.5 m of each nominal depth (1 m resolution); (3) the total absorption and beam attenuation coefficients used for the σ correction [34] were measured respectively with an a-Sphere absorption meter (HOBI Labs) and a with Gamma-4 transmissometer (Hobi Labs).

#### 2.2.2. BGC-Argo Floats Data

_{bp}(700) profiles were calibrated, quality-controlled and additionally corrected by removing positive spikes greater than twice the 90th quantiles of the residual signal calculated as the difference between the profile and a median filter (window of 5 dots).

#### 2.2.3. Ocean Colour Data

^{−3}) and b

_{bp}(m

^{−1}; 443nm) data time-series at 4 km resolution for the period 1997–2015 over the global ocean was downloaded from the ESA-CCI website (http://www.esa-oceancolour-cci.org/). ESA-CCI products are the results of the merging between SeaWiFS, MERIS, MODIS-Aqua, and VIIRS time-series [25,36,37,38]. TChla was estimated with a blending of the OCI (as implemented by NASA, itself a combination of CI and OC4), the OC5 (NASA, 2010) and the OC3 algorithms (http://www.esa-oceancolour-cci.org/?q=webfm_send/684). The Quasi-Analytical Algorithm (QAA) was used to compute b

_{bp}[39,40]. The accuracy of the QAA algorithm was demonstrated in several recent studies [12,13,19,23,24,25,41]. Both datasets were remapped at 100 km resolution, enough to resolve the broader oceanographic scales of variability. In such a context, monthly TChla and b

_{bp}data were selected for the specific area of the northwestern Mediterranean basin to maintain the same domain of the analysis performed by using field measurements (see Figure 1).

#### 2.3. Statistics

_{phyto}and POC):

- (i)
- “Anomalies” here defined as the difference between parameters established with OLS and SMA linear regression methods.
- (ii)
- The relative percentage differences (RPD) between parameters computed by the application of the unsuitable and suitable method.

## 3. Results and Discussion

_{phyto}as a function of the TChla -b

_{bp}relationship, and POC as a function of b

_{bp}as examples to highlight the impact of using a regression method not adapted to the data set on typical bio-optical oceanographic problems.

#### 3.1. Total Chlorophyll-a versus Optical Backscattering

_{phyto}based on the relationship between TChla and b

_{bp}(443) and applied their model to SeaWiFS ocean color data on a global scale. Bellacicco et al. (2016) [12] revisited the model for regional tuning respective to the Mediterranean Sea and used the 555 nm band instead of 443 nm for b

_{bp}. Recently, Bellacicco et al. (2018) [13] generalized this approach on a global scale by using b

_{bp}(443). The equation for the computation of C

_{phyto}is:

_{bp}at 700 nm for compatibility also with BGC-Argo float measurements. The b

^{k}

_{bp}(700) is the backscattering coefficient, at 700 nm, of the background fraction of non-algal particles that does not covary with TChla (e.g., heterotrophic bacteria and viruses) [11]. This value corresponds to the b

_{bp}(700) when TChla is zero: it is the intercept of the linear fit between the two variables. SF is the scaling factor chosen to give satellite Chl:C values (average value of 0.010) consistent with laboratory results, and also for the average contribution of phytoplankton to total particulate organic carbon (±30%) to be consistent with field estimates. In the original work, SF is equal to 13,000 mg C m

^{−2}[11]. Here, taking into account the change of wavelength for b

_{bp}(700 nm instead of 443 nm) and to remain consistent with [11], we computed, according to in situ data, a SF of 16,455 mg C m

^{−2}, 26% more with respect to the value of the original work. About the b

^{k}

_{bp}(700), Bellacicco et al. (2016) [12] demonstrated that b

^{k}

_{bp}(555) varies both in space and time. However, for sake of simplicity, we considered it to be a constant as in the original work of Behrenfeld et al. (2005) [11]. The main assumption of the model is the good relationship between TChla and b

_{bp}[12,13,23]. The first order co-variability between TChla and b

_{bp}is expected because phytoplankton cells contain TChla and also act as light backscatterers [13,42,43,44]. This co-variability also indicates that particles population abundance covaries with phytoplankton biomass, whereas the physiological photoacclimation process plays a secondary role in determining the chlorophyll variations [11,44]. In such a specific context, the underlying hypothesis is that TChla is the independent variable while b

_{bp}is the dependent one. There is no likelihood of interchanging the variables for the evaluation of the b

^{k}

_{bp}. Indeed, b

^{k}

_{bp}is defined as the intercept of the linear regression fit between TChla and b

_{bp}, and it is the b

_{bp}when TChla is equal to 0. The choice of the most appropriate regression method is founded upon which is the dependent variable and which is the independent one. In this case, the main goal is the estimation of a parameter (b

^{k}

_{bp}), allowing for the definition of another parameter (C

_{phyto}), thus OLS is the preferable method to be applied [11,12,13].

_{phyto}biomass estimates that we can compute using in situ data. Furthermore, the intercept of the linear fit (i.e., b

^{k}

_{bp}coefficient), in fact, has a biological meaning, as being the background contribution of the non-algal particles to the total b

_{bp}[11]. Figure 2 shows the TChla-b

_{bp}relationship with indicated both slopes and intercepts as computed by applying the two different regression methods. Here, the intercept varies from 5.8 (±0.5) × 10

^{−4}to 4.5 (±0.3) × 10

^{−4}m

^{−1}when calculated with OLS and SMA, respectively (Figure 2). These values are lower than those reported for the same region (though only surface measurements were used) [12]. This is consistent with a theoretically higher carbon (and its proxy b

_{bp}) to TChla ratio in more illuminated waters [14].

_{bp}, the goal being their comparison, the SMA (or RMA) has to be applied because it is statistically more robust in the context of the analysis of field measurements as explained in Section 2.1.

_{phyto}from Equation (2) and by using the b

_{bp}(700) 0–400 m profiles collected during the BOUSSOLE cruises. To each profile, we applied the b

^{k}

_{bp}(700) values (i.e., intercepts) obtained after the application of both OLS and SMA methods (Figure 2). The RPD on C

_{phyto}estimation is equal to 23.5% (overestimation of total C

_{phyto}using SMA instead of the appropriate OLS method).

_{phyto}changes when using either methods, we applied the relationships reported in Figure 2 to the time series of b

_{bp}(700) vertical profiles from the BGC-Argo dataset. When assessing the integral of C

_{phyto}over depth and time, the RPD between C

_{phyto,SMA}and C

_{phyto,OLS}is 28.7%. In this example, the use of SMA (the less adapted method) leads to an overestimation of C

_{phyto}.

_{bp}(443) for the period 1997–2015 as shown in Figure 3. As described earlier, the good relationship between TChla and b

_{bp}enables defining the b

^{k}

_{bp}coefficient, a fundamental parameter for the C

_{phyto}computation. Figure 3a shows a moderate correlation between TChla and b

_{bp}(r

^{2}equal to 0.56) in the northwestern Mediterranean Sea. The correlation implies the reliable estimation of b

^{k}

_{bp}coefficient by using all the pixels for the period 1997–2015 [12,13]. In such a context, the b

^{k}

_{bp}is 8.5 (±0.2) × 10

^{−4}m

^{−1}(with the OLS method), a value consistent with the order of magnitude found by a recent work always based on ocean colour data [13]. With the SMA, the b

^{k}

_{bp}becomes lower with respect to the computation performed by OLS: 6.3 (±0.3) × 10

^{−4}m

^{−1}. Figure 3b shows the subsequent crucial application of this coefficient on the satellite averaged b

_{bp}time series for the C

_{phyto}computation by using Equation (2) (443 nm instead of 700 nm as a wavelength of reference). Figure 3b shows how both the obtained C

_{phyto}time series follow a similar temporal pattern but with different values. Regarding the entire time series, the mean difference between C

_{phyto,SMA}(with SMA-based b

^{k}

_{bp}) and C

_{phyto,OLS}(with OLS-based b

^{k}

_{bp}) is 2.56 mg C m

^{−3}, that is the 28% of the mean C

_{phyto,OLS}: C

_{phyto,SMA}overestimates C

_{phyto,OLS}. Therefore, in this specific context, there is a general overestimation of C

_{phyto}if one uses the SMA method instead of OLS. This critical point has to be taken into account because of its potential impact in the case of phytoplankton carbon studies on regional and global scales, mostly in ocean carbon budget studies.

#### 3.2. Optical Backscattering vs. Particulate Organic Carbon

_{bp}[6,7,8,9,10,23] as follows:

_{bp}in oceanic water determining, therefore causing a strong correlation with POC.

_{bp}. In such a context, it is possible to interchange POC and b

_{bp}for a simple comparison aimed at establishing a relationship between them, i.e., not for optimizing slope and intercept in view of a further application.

_{bp}from both satellite and in situ data (Table 2 in Thomalla et al., 2017 [10]). Figure 4 shows the established linear relationship between POC and b

_{bp}(700) using both SMA and OLS methods. Our estimates of the slope and the intercept, computed using both methods, are consistent with previous results from the Mediterranean Sea [6], Atlantic and Pacific Oceans [7], Southern Ocean [10] and North Atlantic Ocean [23].

^{2}, i.e., the variance explained by the linear model) as well as the correlation coefficient (r, i.e., a measure of the linear correlation between two variables) are not dependent on the regression method.

_{bp}(700) vertical profiles acquired from BGC-Argo floats in the same area sampled to establish the linear models. Figure 4b shows the anomalies of POC as the difference between POC estimated using linear models based on OLS and SMA methods (POC

_{OLS}and POC

_{SMA}respectively). The anomalies, in general, are weak; however, several areas of large differences have impacted the computation of the POC budget over the time series. In detail, at the end of spring 2014 the largest anomalies are between 5.0 and 20.0 mg m

^{−3}in surface waters. In other periods, the anomalies are between −10.0 and +5.0 mg m

^{−3}. When evaluating the integral of POC along the water column and over time, the RPD between POC

_{OLS}and POC

_{SMA}is 13.3%, showing the importance of selecting the correct regression method which avoids an incorrect estimate of the POC budget. In this example, the use of OLS (the unsuitable method) causes an overestimation of POC.

## 4. Conclusions

- The phytoplankton carbon biomass based on the TChla-b
_{bp}relationship needs to be computed using the OLS method due to the asymmetry assumption between the two variables. In such a context, the intercept of the linear fit between TChla and b_{bp}, which is necessary to compute the C_{phyto}, represents the fraction of b_{bp}that does not co-vary with TChla, confirming that the dependent and independent parameters cannot be interchanged from a theoretical perspective. Only in this specific case, the application of the SMA is unsuitable, as it assumes symmetry of the parameters. Its application always determines an overestimation of phytoplankton carbon biomass. - For all linear regression analysis in which the main aim is to compare two parameters (e.g., b
_{bp}-POC or TChla-b_{bp}), the most appropriate method is SMA due to its theoretical symmetry, and because of the uncertainties that affect both variables. It is thus possible to interchange the x and y axes without any impact on the interpretation of the results.

_{phyto}and POC retrievals. The introduction of sizeable errors is a key factor in the carbon budget estimates when linear models are used on a global scale. Indeed, the total C

_{phyto}:POC ratio utilizing the time series of b

_{bp}(700) vertical profiles give an RPD of 13.6% overestimation using C

_{phyto,SMA}to POC

_{OLS}(the ratio computed using both unsuitable approaches) with respect to the ratio C

_{phyto,OLS}to POC

_{SMA}(the appropriate methods to be used). It has to be kept in mind that two single relationships are applied to the full time-series of the BGC-Argo floats in the example shown here. It is understood however that spatio-temporal variations of the two relationships exist and could have an impact on the budget estimates. In this work, we thus highlighted the importance of the selection and use of the correct regression method. The choice of the model, and hence the method, has to be done a priori relative to any computation based on the data set properties. Given that, it cannot be overlooked that a fraction of the variation of the data around a linear regression fit can also be due to biogeochemical variability rather than error measurements. This type of error represents that portion of variability unresolved by the fitting function adopted, especially in case of ocean color data, where retrieval models are ofter oversimplified. Furthermore, in case of high correlation between variables, both slope and intercept estimations computed by type-I and type-II regression methods do not show large differences between. Therefore, for a correct application of linear regression methods in optical and satellite oceanography, a deeper study of the relationship between the two variables from a theoretical point of view needs to be performed. In fact, as demonstrated, the influence of the unsuitable method in cases of carbon estimations can be considerable and potentially impactful in the context of global carbon budget studies from space or by using field measurements.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Details

**Figure A1.**For an OLS line, the error is defined as the vertical dispersion of a point from the straight line (distance 1 to 2) and the quantity minimized is the sum of squares of these linear distances. In case of SMA, on the other hand, the error is defined as the area of the triangle 3-4-5 and the quantity minimized is the sum of these area (redrawn from Smith et al., 2009 [30]) (

**a**). Scatter plot and linear fits calculated with OLS (blue) and SMA (red) methods by using a syntehtical datasets with a normal distributed error added to both X and Y variables (

**b**).

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**Figure 1.**The northwestern Mediterranean Sea showing the southern coast of France, the island of Corsica, and the location of the BOUSSOLE buoy in the Ligurian Sea (black star) redrawn from [22]. Black dots are the locations where the float surfaced, while the float trajectory is overlaid in the plot with dashed black line.

**Figure 2.**Scatter-plot and linear fit (continuous lines) calculated with ordinary least square (OLS) (blue) and standard major axis (SMA) (red) methods in the TChla-b

_{bp}relationship at the BOUSSOLE site. For both the coefficients, intercepts (A) and slopes (B), the standard errors are also indicated.

**Figure 3.**Scatter plot between TChla and b

_{bp}from ocean colour data in the northwestern Mediterranean Sea with linear fits (continuous lines) calculated with OLS (blue) and SMA (red) methods (

**a**). For both the coefficients, intercepts (A) and slopes (B), the standard errors are also indicated. Time series of C

_{phyto}(

**b**) based on the b

^{k}

_{bp}computed by OLS (in blue) and SMA (in red) methods.

**Figure 4.**Scatter-plot and linear fits calculated with OLS (blue) and SMA (red) methods in the b

_{bp}-POC relationship at the BOUSSOLE site (

**a**). For both the coefficients, intercepts (A) and slopes (B), the standar errors are also indicated. Time series anomalies of particulate organic carbon (POC) derived from BGC-Argo b

_{bp}vertical profiles (0–250 m) using OLS and SMA and relationships (

**b**).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bellacicco, M.; Vellucci, V.; Scardi, M.; Barbieux, M.; Marullo, S.; D’Ortenzio, F. Quantifying the Impact of Linear Regression Model in Deriving Bio-Optical Relationships: The Implications on Ocean Carbon Estimations. *Sensors* **2019**, *19*, 3032.
https://doi.org/10.3390/s19133032

**AMA Style**

Bellacicco M, Vellucci V, Scardi M, Barbieux M, Marullo S, D’Ortenzio F. Quantifying the Impact of Linear Regression Model in Deriving Bio-Optical Relationships: The Implications on Ocean Carbon Estimations. *Sensors*. 2019; 19(13):3032.
https://doi.org/10.3390/s19133032

**Chicago/Turabian Style**

Bellacicco, Marco, Vincenzo Vellucci, Michele Scardi, Marie Barbieux, Salvatore Marullo, and Fabrizio D’Ortenzio. 2019. "Quantifying the Impact of Linear Regression Model in Deriving Bio-Optical Relationships: The Implications on Ocean Carbon Estimations" *Sensors* 19, no. 13: 3032.
https://doi.org/10.3390/s19133032