Multi-Parameter Algorithms of Remote Sensing Reflectance, Absorption and Backscattering for Coastal Waters of the Southern Baltic Sea Applied to Pomeranian Lakes

Abstract

The Pomeranian lakes in Northern Poland and the nearby coastal waters of the Baltic Sea belong to optically complex water bodies characterised by high eutrophication levels. These water types require a local approach when developing bio-optical algorithms that combine the inherent and the apparent properties of seawater. Well-established local algorithms are of great value for understanding and addressing rapid changes in water quality related mostly to human activities in coastal and near-shore zones, as well as in optically similar lakes. Our research analyses the possibility of using the multi-parameter algorithms of absorption a(λ), backscattering b_b(λ) and remote sensing reflectance R_rs(λ), originally developed for the coastal waters of the Southern Baltic Sea, for three selected Pomeranian lakes. Our multi-parameter algorithms are based on the input concentrations of the biogeochemical components measured in the lake waters, i.e., chlorophyll a (Chl a), suspended particulate matter (SPM), inorganic suspended particulate matter (SPM_inorg), the sum of the surface concentrations of accessory pigments (ΣC) and coloured dissolved organic matter with a wavelength of 400 nm (a_CDOM(400)). R_rs(λ) and a(λ) output values were compared with independent measurements of these parameters conducted in the lake waters at 20 sampling stations. Our algorithm output values of b_b(λ) were compared to the values obtained based on the algorithm provided by Ficek, previously developed and validated for Pomeranian lakes, at the same stations. The statistical analyses conducted afterwards showed that the multi-parameter algorithms of R_rs(λ) and a(λ) for the Southern Baltic Sea are sufficient to be used for the stations investigated in the aforementioned three lakes. Specifically, the correlations between the b_b(λ) values obtained based on the Ficek algorithm and the b_b(λ) values obtained using our multi-parameter algorithm reveal a statistical error rate of less than 20%.

1. Introduction

In terms of optics, the division of water bodies was introduced in 1977 by Morel and Prieur [1], who identified “Case 1 waters” and “Case 2 waters”. In subsequent years, Gordon and Morel refined this classification [2,3], and thus, Case 1 waters [4] were found to constitute nearly 98% of the volume of all ocean waters. Case 1 waters (mostly open ocean waters) are areas where their optical properties are primarily determined by autogenous (locally produced) components of seawater, mainly phytoplankton and suspended particulate matter (SPM). Moreover, the concentrations of these components (phytoplankton, organic detritus and organic suspended particulate matter (SPM_org)) correlate statistically with the concentration of chlorophyll a (Chl a) [5].

Whenever the optical properties of water indicate the presence of substances of allogeneic origin (coming from the outside, e.g., from river waters in the form of suspended particles, sediments, coloured dissolved organic matter (CDOM) and river phytoplankton, as well as from the precipitation of atmospheric dust of urban, industrial or volcanic origin, or coming from dredged material dumping or pollutants, etc.), these are classified as Case 2 waters (mostly coastal waters, bays, semi-closed seas and lakes) [6,7]. This type of water includes, among others, the Baltic Sea and Pomeranian lakes [5,6,7,8,9]. Water-quality-related research is especially important in near-shore basins which are under the direct influence of pollution linked to human activities [10,11]. Similarly, monitoring water quality in inland water bodies is of growing importance in the context of the carbon cycle and regional climate change [12,13]. Recently, numerous optics-related research projects have been conducted in lakes [14,15], and successful attempts have been made to monitor key optical parameters from space on a regular basis [16,17].

Understanding the mutual relations between the inherent optical properties (IOPs) and the apparent optical properties (AOPs) of a water body and changes in the solar light transfer in the sea as a result of the appearance of water admixtures is an important issue in the optics of the sea, especially when developing satellite methods for monitoring marine and lake environments [18,19,20]. For example, the Chl a concentration can be used as a proxy for a lake’s trophic state [20]. Moreover, the knowledge of these relationships allows for the estimation of optical values that are more difficult to determine empirically, e.g., the backscattering coefficient (b_b). This parameter plays a significant role in the assessment of the so-called “ocean colour”, i.e., the basic optical parameter calculated as a weighted ratio of remote sensing reflectance (R_rs) values for different wavelengths, measured by sensors placed on satellites. The value of R_rs is proportional to the value of b_b [21]. Therefore, measurements provided on this scale and the development of semi-empirical models to understand the biogeochemical background will greatly benefit from the remote monitoring of Case 2 waters, especially lakes.

Over the past decades, many bio-optical algorithms for Case 2 waters have been developed. They are usually based on equations describing the physical relationships between IOPs and AOPs as well as statistical correlations between IOPs and the concentrations of biogeochemical components of seawater [22,23,24,25,26,27].

In 2013, Ficek showed similarities between the eutrophic waters of the Baltic Sea and the Pomeranian, Estonian and Scandinavian lakes in terms of optical properties and optically significant water components [14]. On the basis of relevant research and analyses, he developed a semi-empirical model of light absorption by SPM in Pomeranian lakes, which allowed for determining the absorption spectra for phytoplankton and particles other than algae, taking into account the known concentrations of Chl a or SPM. In addition, Ficek derived a spectral model for the waters of Pomeranian lakes, which allows for calculating the value of the backscattering coefficient (b_b(λ)) for any wavelength in the VIS range based on the known concentrations of SPM or Chl a. Ficek also showed that some semi-empirical models and algorithms developed for waters in other regions can be adapted to Pomeranian lakes. He stated that these algorithms should be modified by determining and introducing into the equations other values of the coefficients that would be characteristic of Pomeranian lakes. Since then, more publications have appeared that address the possibility of using marine algorithms for lake waters, e.g., [16,28,29]. In 2016, Meler et al. [28] presented models for estimating the absorptive properties of coloured dissolved organic matter (CDOM) for the Baltic Sea and Pomeranian lakes. They developed two empirical mathematical models that enabled them to design the absorption coefficients of coloured dissolved organic matter (a_CDOM) dynamics in natural waters and reconstruct the complete a_CDOM spectrum in the UV and visible spectral domains. The input variable in the first model was Chl a, and in the second one, it was the surface absorption coefficient of CDOM at 400 nm (a_CDOM(400)). The comparison of these approximations with other models of light absorption by CDOM demonstrated that their parameterisations were superior for estimating CDOM absorption in the optically complex waters of the Baltic Sea and Pomeranian lakes [28].

The utility and various applications of algorithms have been widely presented in the existing literature. Some of the articles address algorithms specifically developed and implemented to obtain coefficients that serve as an indicator of water quality, and some are used to study lake ecosystems [30,31,32,33]. Other algorithms help in exploring the relationship between lakes and rivers. For example, in 2021, Jiang et al. proposed an algorithm to monitor rivers’ backflow into lakes using satellite images of Lake Poyang (the largest freshwater lake in China) [34]. The proposed solution allowed for independence from hydrological stations, which up until that point had been needed for lake monitoring.

The Baltic Sea is characterised by a low salinity compared to the waters of the oceans and most seas, and thus it is referred to as a brackish sea. This is mainly due to the predominance of the inflow of river waters and rainwater over the inflow of salty waters from the Kattegat. Our multi-parameter algorithms of a(λ_i), b_b(λ_i) and R_rs(λ_i) developed for the coastal waters of the southern Baltic Sea [22] were based on in situ optical measurements carried out in the coastal zone (see Figure 1), where freshwater inflows occurred. In some of the studied sites, the salinity was as low as 4 PSU. On the other hand, the salinity of Łebsko Lake may even reach 3 PSU. Therefore, we decided to investigate whether the bio-optical algorithms developed for the coastal waters of the Baltic Sea can be applied to Pomeranian lakes. Based on this premise, the authors decided to investigate the possibility of using the multi-parameter algorithms of a(λ_i), b_b(λ_i) and R_rs(λ_i), developed for the coastal waters of the southern Baltic Sea [22], for three selected Pomeranian lakes (Łebsko, Chotkowskie and Obłęskie).

2. Materials and Methods

2.1. Multi-Parameter Algorithms of R_rs(λ_i), a(λ_i) and b_b(λ_i)

For the analysis, the authors used five-parameter algorithms of R_rs(λ_i) and a(λ_i), and a two-parameter algorithm of b_b(λ_i) developed for the coastal waters of the southern Baltic Sea and described in [22], where λ_i = 420, 488, 555 and 620 nm. The R_rs(λ_i) model is a mathematical description of the relationships between the concentrations and properties of natural water components (Chl a, SPM, SPM_inorg, ΣC and a_CDOM(400)) and the following optical properties: the absorption coefficient of phytoplankton a_ph(λ_i), the absorption coefficient of non-algal particles a_d(λ_i), the absorption coefficient of CDOM a_CDOM(λ_i) and the backscattering coefficient b_b(λ_i). The model was based on the following equation [21]:

$$R_{rs}\left({\lambda }_i\right)={\left[\frac{f}{Q}\right]}_{{\lambda }_i}\frac{b_b\left({\lambda }_i\right)}{a\left({\lambda }_i\right)+b_b\left({\lambda }_i\right)}$$

(1)

In the above equation, the backscattering coefficient b_b(λ_i) was estimated as follows [5]:

b_b(λ_i) = b_bp(λ_i) + b_bw(λ_i),

(2)

where b_bp(λ_i) is the backscattering coefficient of particles and b_bw(λ_i) is the backscattering coefficient of water molecules given by Morel [35]. The absorption coefficient a(λ_i) was calculated as the sum of the following [5]:

a(λ_i) = a_CDOM(λ_i) + a_ph(λ_i) + a_d(λ_i) + a_w(λ_i),

(3)

where a_w(λ_i) is the absorption coefficient of seawater molecules [36].

The values of [f/Q]_λi parameter were calculated for the chosen areas of the southern Baltic Sea based on a large dataset of measurements of L_u(λ_i), E_d(λ_i), b_b(λ_i) and a(λ_i) (

Table 1. The constants of the five-parameter model of a(λ_i), the two-parameter model of b_b(λ_i) and the five-parameter model of R_rs(λ_i) (Equations (4)–(7)).

λi	420	488	555	620
B1	0.009	0.006	0.005	0.004
B2	0.911	0.891	0.935	0.881
B3	0.337	0.827	0.977	1.230
O1	0.077	0.624	1.037	1.488
O2	1.006	1.077	1.072	1.136
O3	0.132	0.485	0.689	0.794
D1	0.057	0.035	0.022	0.015
D2	0.807	0.762	0.646	0.592
D3	0.750	0.903	1.157	1.542
P1	0.041	0.022	0.011	0.007
P2	0.827	0.820	0.815	0.926
P3	0.493	0.824	0.257	0.261
bbw	0.0023	0.0012	0.0007	0.0004
aw	0.0045	0.0147	0.0596	0.2755
f/Q	0.07	0.10	0.12	0.13

). Having assumed the above, it was possible to determine R_rs(λ_i) for four wavelengths in the visible light range using the knowledge of the concentration of admixture components in the coastal waters of the southern Baltic Sea based on Equations (4)–(7) [22].

$$b_{bp}{\left({\lambda }_i\right)}_{cal}={\mathrm{B}}_1\left({\lambda }_i\right){SPM}^{{\mathrm{B}}_2\left({\lambda }_i\right)}{e}^{{\mathrm{B}}_3\left({\lambda }_i\right)\frac{{SPM}_{inorg}}{SPM}}$$

(4)

$$a_{CDOM}{\left({\lambda }_i\right)}_{cal}={10}^{\left[{\mathrm{O}}_1\left({\lambda }_i\right){\left(a_{CDOM}\left(400\right)\right)}^2+{\mathrm{O}}_2\left({\lambda }_i\right)a_{CDOM}\left(400\right)-{\mathrm{O}}_3\left({\lambda }_i\right)\right]}$$

(5)

$$a_d{\left({\lambda }_i\right)}_{cal}={\mathrm{D}}_1\left({\lambda }_i\right){SPM}^{{\mathrm{D}}_2\left({\lambda }_i\right)}{e}^{{\mathrm{D}}_3\left({\lambda }_i\right)\frac{{SPM}_{inorg}}{SPM}}$$

(6)

$$a_{ph}{\left({\lambda }_i\right)}_{cal}={\mathrm{P}}_1\left({\lambda }_i\right){Chl a}^{{\mathrm{P}}_2\left({\lambda }_i\right)}{e}^{{\mathrm{P}}_3\left({\lambda }_i\right)\frac{\mathsf{\Sigma }\mathrm{C}}{Chl a}}$$

(7)

These multi-parameter algorithms were further used to calculate R_rs(λ_i), a(λ_i) and b_b(λ_i) for the three chosen lakes. The first stage was to examine the possibility of using the five-parameter total absorption algorithm to calculate a(λ_i) for the selected lakes. For this purpose, we used Equations (3) and (5)–(7) as well as relevant constants (Table 1). Next, based on Equations (2) and (4) and Table 1, b_b(λ_i) was calculated. The last step was to calculate R_rs(λ_i) based on Equation (1) and previously determined a(λ_i) and b_b(λ_i). The values of [f/Q]_λi parameters used for calculations of R_rs(λ_i) of lake waters, presented in Table 1, come from [22].

2.2. Study Area

The multi-parameter marine algorithms were previously developed on the basis of measurements carried out in the area of the southern Baltic Sea, i.e., the Gulf of Gdańsk, Szczecin Lagoon and the Southern Baltic Sea coastal zone (Figure 1). In order to examine the possibility of using these algorithms in lake waters, we used empirical materials collected during several field trips to the lakes of the Pomeranian Lake District: Łebsko, Chotkowskie and Obłęskie (Figure 1). Łebsko Lake is the third largest lake in Poland, with an area of 71 km² and a maximum depth of 6.3 m. It is a coastal lake separated from the sea by a sandy spit. The basin is a remnant of the former sea bay, the Łeba River is its tributary and it is also connected by canals with Gardno and Sarbsko Lakes [37]. Chotkowskie Lake is a glacial lake with a total area of 0.56 km² and a maximum depth of 10.3 m. The third of the analysed lakes, Obłęskie Lake, is a flow-through lake, connected by a small stream with the Wieprza River basin. The maximum depth of the lake is 8.9 m, and it covers an area of 0.62 km² [14]. These lakes represent three optical lake types in Northern Poland according to [38].

2.3. Data Acquisition and Processing

The acquisition and processing of the data used to develop marine multi-parameter algorithms are described in [22]. In situ R_rs spectra in lakes were measured with the HyperOCR (Satlantic Inc., Halifax, NS, Canada) instrument and processed according to the standard procedure described in [22]. At the same sampling stations, separate water samples were also collected for laboratory measurements of the following parameters: the concentration of chlorophyll Chl a (absorption spectrum of the pigment ethanol extract) [39] and the mass concentration of the total suspension SPM and its inorganic fraction SPM_inorg (standard gravimetric method) [40]. The composition of accessory pigments ΣC (the sum of chlorophyll b, chlorophyll c, and photosynthetic and photoprotective pigments) was determined using HPLC method (high-performance liquid chromatography) [41]. The absorption coefficient of CDOM at 400 nm (a_CDOM(400)) was obtained using spectrophotometric analysis followed by two-step filtration [42].

All parameters were tested in a manner resembling the test methods used previously for the Southern Baltic Sea. Finally, the analysed materials from the three Pomeranian lakes included 20 sets of measurement data for the surface water layer (Chl a, ΣC, SPM, SPM_inorg, a_CDOM(400), L_u(0⁻) and E_d(0⁻)).

To analyse the data, we used arithmetic and logarithmic statistical methods. Lake waters are areas of high diversity in terms of water physics and chemistry. The concentrations of biogeochemical constituents for the three lakes varied up to three orders of magnitude. Therefore, we applied logarithmic statistics in addition to arithmetic statistics, for more accurate evaluation of the relationship between the measured values and the calculated ones.

3. Results

3.1. Remote Sensing Reflectance R_rs Estimated for the Lakes of the Pomeranian Lake District

The empirical materials collected during the spring and summer months (from April to July) from the Łebsko, Chotkowskie and Obłęskie Lakes are characterised by high concentrations of Chl a (from 8.6 mg·m⁻³ up to 81.6 mg·m⁻³), SPM_inorg (from 1.2 g·m⁻³ to 21.1 g·m⁻³) and SPM_org (from 2.6 g·m⁻³ to 51.3 g·m⁻³). The data for the Southern Baltic Sea waters that were used to develop the algorithms of R_rs(λ_i), a(λ_i) and b_b(λ_i) were collected both in the spring and during the autumn–winter months and are characterised by significantly lower concentrations of Chl a, SPM_inorg and SPM_org (the average concentration of Chl a was 9.8 mg·m⁻³, the average concentration of SPM_inorg was 1.1 g·m⁻³ and the average concentration of SPM_org was 3.2 g·m⁻³). All the measured R_rs spectra for each studied lake are shown in Figure 2 below. There are differences both the spectral shape and range of R_rs values for each water basin. Łebsko Lake usually had high R_rs values and a high peak at 705 nm, which is characteristic for CDOM-rich eutrophic waters. In the case of Chotkowskie Lake, the R_rs values were much lower with more diverse shapes, while Obłęskie Lake had the widest range of recorded R_rs values and the highest peak in the central part of the visible spectrum.

Figure 3 below presents the graphs of the R_rs values determined using the HyperOCR meter R_rs(λ_i)_m, in relation to the values calculated on the basis of the five-parameter model of R_rs for the Southern Baltic Sea R_rs(λ_i)_cal and the probability density distributions of the calculated R_rs to the measured R_rs ratio for four wavelengths.

The obtained estimation errors of R_rs(λ_i) are shown in

Table 2. Definitions and values of relative errors of the R_rs(λ_i) coefficients for three lakes in the Pomeranian Lake District (Łebsko, Chotkowskie and Obłęskie Lakes) (for the wavelengths: 420, 488, 555 and 620 nm), calculated on the basis of the five-parameter R_rs for chosen stations in the Southern Baltic Sea [22].

Arithmetic Statistic of Rrs(λ) [%]		Rrs(420)	Rrs(488)	Rrs(555)	Rrs(620)
Relative mean error $⟨\epsilon ⟩=\frac{1}{N}{\displaystyle \sum _i}{\epsilon }_i,$		45.41	54.96	32.74	17.42
RMSE—root mean square error of ε ${\sigma }_{\epsilon }=\sqrt{\frac{1}{N}\left({\displaystyle \sum _i}{\left({\epsilon }_i-⟨\epsilon ⟩\right)}^2\right)}$		71.34	41.84	14.77	38.72
Logarithmic Statistic of Rrs(λ) [%]		Rrs(420)	Rrs(488)	Rrs(555)	Rrs(620)
Mean logarithmic error ${⟨\epsilon ⟩}_g={10}^{g_i}-1$		31.58	49.86	31.92	12.61
Standard error factor $x={10}^{{\sigma }_{log}},$		1.56	1.30	1.12	1.33
Statistical logarithmic errors	${\sigma }_{+}=x-1$	56.25	30.27	12.25	33.21
	${\sigma }_{-}=x^{-1}-1$	−36.00	−23.24	−10.91	−24.93
where: Xi,m—measured values of Rrs(λi); Xi,cal—estimated values of Rrs(λi), ${\epsilon }_i=\left(\frac{X_{i,cal}-X_{i,m}}{X_{i,m}}\right),$ $g_i=log\left(\frac{X_{i,cal}}{X_{i,m}}\right),$ ${\sigma }_{log}=\sqrt{\frac{1}{N}\left({\sum }_i{\left(g_i-⟨g⟩\right)}^2\right)},$ $⟨g⟩=\frac{1}{N}{\sum }_i{g}_i$

. Comparing the values of R_rs(λ_i)_m measured in lakes with the HyperOCR radiometer and those calculated on the basis of the five-parameter R_rs(λ_i)_cal for the Southern Baltic coastal waters, it is clearly visible that the R_rs(λ_i)_cal coefficients calculated using the model take higher values than the R_rs(λ_i)_m obtained by way of measurements, especially for the wavelengths of 488 nm and 555 nm (see the histograms in Figure 3). Therefore, the systematic errors have positive values (Table 2).

The strongest correlation (R² > 0.9) was achieved for 555 nm and 620 nm and the weakest (R² = 0.53) for 420 nm. The highest consistency between the measured and the calculated R_rs was obtained for 620 nm, which is the minimum for remote sensing reflectance corresponding to the maximum for water absorption. On the other hand, the most linear correlation obtained for 550 nm corresponds to the maximum for R_rs which is the minimum for absorption. The statistical logarithmic errors vary from −36.00% to +56.25%.

The values of statistical errors (Table 2) affecting the R_rs calculated for lakes on the basis of the five-parameter R_rs model exceed 50% only for a wavelength of 420 nm. For other wavelengths, the statistical error rate is much lower, and for 620 nm, it is a small percentage. Moreover, the greatest differences in the values of R_rs(λ_i)_cal and R_rs(λ_i)_m are observed for a wavelength of 420 nm, for which the standard error factor has the highest value x = 1.56 (Table 2, Figure 3). This corresponds to the spectral region of significant influence of a_CDOM(λ_i) on a(λ_i).

3.2. Absorption Coefficients Estimated for the Lakes of the Pomeranian Lake District

The total absorption values measured in the tested Pomeranian lakes at the four tested wavelengths are presented in Figure 4. The highest values were obtained at 420 nm in the region of strong CDOM absorption. Samples no. 1–5 characterised by the highest absorption values and range were collected in Łebsko Lake. Samples no. 6–13 were collected in Chotkowskie Lake and samples no. 14–20 in Obłęskie Lake.

Figure 5 below presents the comparison of the measured and modelled absorption coefficients for lake waters. The values of absorption coefficients measured in lakes a(λ_i)_m for the three wavelengths (420, 488 and 555 nm) in most cases are higher than the values of the absorption coefficients calculated on the basis of the five-parameter model a(λ_i)_cal. Only for the wavelength of 620 nm, the values of the absorption coefficients obtained on the basis of the calculations are mostly higher than the measured ones.

The strongest correlation (R² > 0.9) was obtained for 420 nm. The statistical error rates of the light absorption coefficients that were calculated using the five-parameter model of absorption range from 10% to 70% (

Table 3. Relative errors in the total absorption coefficients a(λ) for the wavelengths: 420, 488, 555 and 620 nm for three lakes in the Pomeranian Lake District (Łebsko, Chotkowskie and Obłęskie Lakes), calculated on the basis of the total absorption model for selected waters of the Southern Baltic Sea [22]. The measurements were completed using 20 sets of data for the surface layer.

Arithmetic Statistic of a(λ) [%]		a(420)	a(488)	a(555)	a(620)
Relative mean error $⟨\epsilon ⟩=\frac{1}{N}{\displaystyle \sum _i}{\epsilon }_i,$		−7.49	−10.35	−3.66	65.97
RMSE—root mean square error of ε ${\sigma }_{\epsilon }=\sqrt{\frac{1}{N}\left({\displaystyle \sum _i}{\left({\epsilon }_i-⟨\epsilon ⟩\right)}^2\right)}$		10.65	23.92	35.63	11.07
Logarithmic Statistic of a(λ) [%]		a(420)	a(488)	a(555)	a(620)
Mean logarithmic error ${⟨\epsilon ⟩}_g={10}^{g_i}-1$		−8.05	−13.14	−10.08	42.82
Standard error factor $x={10}^{{\sigma }_{log}},$		1.12	1.29	1.48	1.69
Statistical logarithmic errors	${\sigma }_{+}=x-1,$	29.19	47.51	69.26	33.21
	${\sigma }_{-}=x^{-1}-1,$	−22.59	−32.21	−40.92	−24.93
where: Xi,m—measured values of a(λi); Xi,cal—estimated values of a(λi), ${\epsilon }_i=\left(\frac{X_{i,cal}-X_{i,m}}{X_{i,m}}\right),$$g_i=log\left(\frac{X_{i,cal}}{X_{i,m}}\right),$  ${\sigma }_{log}=\sqrt{\frac{1}{N}\left({\sum }_i{\left(g_i-⟨g⟩\right)}^2\right)},$  $⟨g⟩=\frac{1}{N}{\sum }_i{g}_i$

). This indicates that the diversity of the chemical composition of the suspended and dissolved matter greatly influences the optical absorption capacity. Relatively small values of the standard error factors presented in Table 3 (except for the 620 nm wavelength, where x = 1.69) show good agreement between the results obtained with the use of the five-parameter absorption model and the measurements in the lake waters. The statistical logarithmic errors range from −40.92% to +69.26%.

3.3. Backscatttering Coefficient Comparison between Two Algorithms

In the case of the backscattering coefficient b_b(λ_i), we applied a semi-empirical method based on the regression equation between b_b(λ_i) and SPM for Pomeranian lakes in 2009–2010 developed by Ficek [12]. None of the wavelengths used in the multi-parameter model coincided with the wavelength used by Ficek, i.e., 532 nm. Therefore, the b_b at 532 nm was estimated by linearly approximating the data from the two nearby wavelengths: 488 nm and 550 nm. The difference in the values of b_b(532) calculated according to Ficek’s algorithm [14] and those calculated on the basis of Lednicka and Kubacka [22] is small (Figure 6). The b_b(532) calculated according to Ficek [14] clearly is slightly higher than the values calculated using the algorithm by Lednicka and Kubacka [22]. This tendency can be noted for all twenty measurements taken with regard to the lakes of the Pomeranian Lake District (see Figure 6).

The relative differences between b_b(532) for the three lakes of the Pomeranian Lake District (Łebsko, Chotkowskie and Obłęskie), calculated on the basis of the b_b model for the Southern Baltic Sea [22] and based on Ficek’s algorithm [12], are as follows:

• A relative mean error $⟨\epsilon ⟩$ of 19.89%;
• A root mean square error ${\sigma }_{\epsilon }$ of 16.76%;
• A mean logarithmic error ${⟨\epsilon ⟩}_g$ of 18.74%;
• A standard error factor x of 1.15;
• A statistical logarithmic error ${\sigma }_{+}$ of 15.45% and ${\sigma }_{-}$ of 13.38%.

A high correlation between b_b(532)_F and b_b(532)_L&K is confirmed by the rate of statistical errors that affect the b_b values, which range from 13% to 17%.

4. Discussion

4.1. Performance of the Rrs Model in Lakes

The remote sensing of lakes is probably just as important (because of the need to control water quality and climate change) as it is challenging (because of the high spatial and temporal variability in lake water components) [9,16]. Similarly to that of near-shore waters and river estuaries, it requires a strictly local approach, and there are many advantages of applying existing marine bio-optical models for inland waters [16]. The presented study aims to analyse the applicability of a set of multi-parameter algorithms for the Southern Baltic Sea as described in [22] to Pomeranian lakes.

The performance of multi-parameter models was evaluated using several statistical measures. The most commonly used one is the root mean squared error (RMSE), which shows differences between values measured in situ and values predicted by a model. The highest RMSE for the remote sensing reflectance was obtained at 420 nm, where the range of the measured values was the greatest. This is also the spectral region with the highest absorption by CDOM of all tested wavelengths [43] and the region with the lowest R_rs signal. The studied Pomeranian lakes are CDOM-dominated waters, and high CDOM absorption reduces the R_rs. The five-parameter R_rs model worked less efficiently for low R_rs values (RMSEs of over 70% at 420 nm) and was the most efficient for high R_rs values (RMSEs of less than 15% at 555 nm). Ficek et al. [38] divided Pomeranian lakes into three groups according to the shape of the remote sensing reflectance spectra, and this study covers a representative of each group: waters with high CDOM (Chotkowskie Lake), waters with low CDOM but high Chl a (Obłęskie Lake) and waters with moderate concentrations of all optically significant constituents (Łebsko Lake)—see Figure 1 in [38]. The structure of the tested multi-parameter models was based on a set of constants calculated on the basis of a large dataset of ship cruises spanning several years. The f/Q constant depends on the irradiance conditions and geometry between the sun and the sensor’s position and is usually assumed to be wavelength independent, although by definition, it depends on λ [44] and is considered seasonally dependent [45]. In our five-parameter R_rs model, the wavelength-dependent f/Q was applied as the most accurate for the Southern Baltic Sea. Although the 20 measurements taken in the Pomeranian lakes are not enough to draw substantial conclusions, we believe that the performance of our multi-parameter model is encouraging enough for such tests to be continued in inland waters. After gathering a larger dataset covering inland waters, it will be possible to calculate specific f/Q factors for these waters and improve the model’s accuracy.

4.2. Performance of the Absorption Model in Lakes

Water absorption properties result from all optically significant components in a given body of water and strongly depend on their concentrations and chemical composition. Among organic particles, there are over twenty chemical types differing in terms of absorption properties. For mineral particles, there are at least a dozen types of these minerals with different chemical compositions [46]. In coastal zones, especially in river mouths, a_CDOM(λ_i) can be very high and determine the entire light absorption spectrum [47]. Figure 5 presents graphs for four wavelengths: 420, 488, 555 and 620 nm, regarding the relative share of a_CDOM (λ_i), a_ph(λ_i) and a_d(λ_i) for selected stations in the Southern Baltic Sea and the studied lakes in the Pomeranian Lake District. A significant difference can be observed in the chemical composition of sea and lake waters. The first three graphs show a strong advantage of a_CDOM for both the lakes and coastal waters of the Southern Baltic Sea (in the case of the seawater, the average percentage share for these three wavelengths is well above 50%). In the case of the lakes, there is also a large contribution of a_d(λ_i), the average percentage of which for all wavelengths is high, and for 555 nm, it is even 37%. In the case of both the lake waters and the Southern Baltic Sea waters at a wavelength of 420 nm, the a_CDOM(λ_i) has the greatest contribution to the spectrum of the entire absorption, and its average percentage share is 58% for the lakes and 68% for the Baltic coastal waters (the average percentage share of other absorption coefficients for the lakes is a_ph(λ_i)—20% and a_d(λ_i)—22% and for the selected reservoirs of the Southern Baltic Sea is a_ph(λ_i)—20% and a_d(λ_i)—12%, respectively). The average percentage share of a_CDOM (λ_i), a_ph(λ_i) and a_d(λ_i) in the absorption process for all wavelengths of the analysed waters is shown in

Table 4. Average percentage share of CDOM, SPM_org and SPM_inorg in the absorption process.

Lakes	λ = 420 nm	λ = 488 nm	λ = 555 nm	λ = 620 nm
aph(λ) [%]	20	22	19	35
ad(λ) [%]	22	31	37	33
aCDOM(λ) [%]	58	47	44	32
Coastal Waters	λ = 420 nm	λ = 488 nm	λ = 555 nm	λ = 620 nm
aph(λ) [%]	20	27	18	19
ad(λ) [%]	12	15	17	17
aCDOM(λ) [%]	68	58	65	64

.

The graphs presented in Figure 7 clearly show a significant difference in the share of various absorption components between the lake waters and the Southern Baltic Sea coastal zone. This differentiation is particularly visible for the longwave part of the spectrum. Therefore, the possibility of using the five-parameter model to calculate the a(λ_i) coefficient in the selected lakes of the Pomeranian Lake District was tested. A very good agreement was obtained at 420 nm and a relatively good agreement at other wavelengths, which enables model testing to be continued after gathering a larger dataset. The correlation plots in Figure 3 indicate that a shift on the y-axis could provide a better linear correlation between the modelled data and the measured data. However, such a calibration of the model will most probably be seasonally and locally variable and wavelength dependent, which is why it will require a much larger dataset.

In 2019, Xue et al. [48] used local data to build a model of IOPs based on satellite R_rs(λ) data in turbid inland waters. They measured the R_rs(λ) and absorption coefficients from lakes in the Yangtze River and Huai River basin, China. Their iterative IOP inversion model was developed to retrieve, among others, the total absorption coefficients in turbid lakes. They established that the total absorption coefficients derived from their algorithm have a mean RMSE of 39.45%, ranging from 400 to 720 nm. The RMSEs obtained in our study were, respectively: for 420 nm—10.65%, 488 nm—23.92%, 555 nm—35.63% and 620 nm—11.07%. Here, we again observe better model performance for high absorption values at 420 nm and 620 nm.

4.3. Performance of the Backscattering Model in Lakes

The backscattering coefficient is a fundamental quantity in regulating the optical signal available for remote sensing and is of particularly great importance for optically complex coastal and inland waters [49]. In this study, the correlation between two semi-empirical models was tested: the model developed by Ficek for the Pomeranian lakes and our multi-parameter model developed by Lednicka and Kubacka for the Southern Baltic Sea. Both models were based on different sets of measured data. The backscattering coefficient in the Baltic Sea and optically similar inland waters is strongly spectral dependent [50], and its values can vary by two orders of magnitude [51]. The RMSEs between the two models and the R² were almost 17% and 0.995, respectively, which is a very satisfactory result. The data measured in the lake waters were mostly characterised by high concentrations of Chl a. However, for low concentrations of Chl a, the values of b_b(λ_i) measured in the lake waters are very similar to those represented by the turbid coastal waters of the Baltic Sea [14].

5. Conclusions

The study was focused on analysing the possibility of using the multi-parameter algorithms of a(λ_i), b_b(λ_i) and R_rs(λ_i), developed for the coastal waters of the Southern Baltic Sea [22], for three Pomeranian lakes. The conducted tests show that the five-parameter model of R_rs(λ_i) and the five-parameter model of a(λ_i) developed for the Southern Baltic Sea can be applied with satisfactory accuracy to the Pomeranian lakes. The RMSE obtained for R_rs(λ) exceeds 50% only for a wavelength of 420 nm. In the case of a(λ_i), the statistical error rates do not exceed 40% for all tested wavelengths.

Moreover, the difference in the b_b(532)_F values obtained on the basis of the Ficek algorithm for the lakes of the Pomeranian Lake District and the b_b(532)_L&K values obtained on the basis of the Lednicka and Kubacka algorithm for the selected areas of the Southern Baltic Sea reveals systematic errors with a rate lower than 20% and statistical errors with a rate below 17%. Although the number of measurements available for testing was relatively small, their range was representative for all optical types occurring in the Pomeranian Lake District. The results obtained encourage us to derive the optical parameters of inland waters using the multi-parameter algorithms developed for the Southern Baltic Sea.

It is commonly known that bio-optical algorithms are designed to associate AOPs (especially R_rs(λ)) with IOPs and seawater component concentrations. However, the great variety in optically significant constituents of Case 2 waters limits the use of global bio-optical algorithms. This is why we believe that our contribution of the presented results will allow for an expansion of our knowledge about the optical properties of Case 2 waters represented by the lakes of the Pomeranian Lake District and encourage testing, applications and further calibrations of the existing models in inland waters.