Evaluating Modeling Approaches for Phytoplankton Productivity in Estuaries

Abstract

Phytoplankton comprise the base of the food web in estuaries and their biomass and rates of growth (productivity) exert a bottom-up control in pelagic ecosystems. Reliable means to quantify biomass and productivity are crucial for managing estuarine ecosystems. In many estuaries, direct productivity measurements are rare and instead are estimated with biomass-based models. A seminal example of this is a light utilization model (LUM) used to predict productivity in the San Francisco Estuary and Delta (SFED) from long timeseries data using an efficiency factor, ψ. Applications of the LUM in the SFED, Chesapeake Bay, and the Dutch Scheldt Estuary highlight significant interannual and regional variability, indicating the model must be recalibrated often. The objectives of this study are to revisit the LUM approach in the SFED and assess a chlorophyll-a to carbon model (CCM) that produces a tuning parameter, Ω. To assess the estimates of primary productivity resulting from the models, productivity was directly measured with a ¹³C-tracer at nine locations during 22 surveys using field-derived phytoplankton incubations between March and November of 2023. For this study, ψ was determined to be 0.42 ± 0.02 (r² = 0.89, p < 0.001, CI₉₅ = 319). Modeling productivity using an alternative CCM approach (Ω = 3.47 × 10⁴ ± 1.7 × 10³, r² = 0.84, p < 0.001, CI₉₅ = 375) compared well to the LUM approach, expanding the toolbox for estuarine researchers to cross-examine productivity models. One practical application of this study is that it confirms an observed decline in ψ, suggesting a decline in light utilization by phytoplankton in the SFED. This highlights the importance of occasionally recalibrating productivity models in estuaries and leveraging multiple modeling approaches to validate estimations before application in ecological management decision making.

1. Introduction

Phytoplankton productivity is important in supporting food webs and fishes of aquatic ecosystems, including wetlands. Worldwide and throughout history, wetlands have been claimed by humanity, chiefly for agriculture, freshwater resources, and inland shipping ports [1]. The associated pelagic ecosystems, which once relied on phytoplankton carbon from productive wetlands as a source of primary production, are now primarily dependent on phytoplankton carbon produced within less productive pelagic systems to support the base of estuarine food webs. This phenomenon is exemplified by the San Francisco Estuary and Delta (SFED, see

Table 1. List of acronyms and symbols with units if applicable.

Symbol	Parameter	Units
SFED	San Francisco Estuary and Delta
LUM	light utilization model
CCM	chlorophyll-a to carbon model
ρC	transport (uptake) rate of carbon	mg-C m−3 d−1
PNpd	depth-integrated daily net phytoplankton productivity	mg-C m−2 d−1
PNpy	depth-integrated annual net phytoplankton productivity	g-C m−2 yr−1
B	chlorophyll-a concentration	mg-Chl m−3
Eo	ambient photosynthetically active radiation	Einsteins m−2 d−1
Zp	photic zone depth	m
Cp	phytoplankton carbon concentration	mg-C m−3
BEoZp	composite parameter	Einsteins mg-Chl m−4 d−1
B:Cp	chlorophyll-a to phytoplankton carbon ratio	mg-Chl mg-C−1
ψ	light utilization model regression slope	mg-C m2 mg-Chl−1 Einsteins−1
Ω	chlorophyll-a to carbon model regression slope	mgC2 mgChl−1 m−2 d−1
MAE	mean absolute error

for acronyms) in California where between 77 and 85% of the wetlands have been reclaimed and irreversibly altered due to the allocation of land and water resources for agriculture [2,3]. Abundances of pelagic organisms in the SFED sharply declined after 2000, raising concerns from natural resource agencies that the food web was degrading [4]. This observation spurred decades of research into potential causes for the decline, one being chronically depressed phytoplankton biomass [5].

Enhanced phytoplankton biomass is one criterion by which ecosystem restoration can be assessed for success [6]. As a result, phytoplankton biomass monitoring is a staple among natural resource agencies and researchers. Typically, the monitoring of chlorophyll-a fluorescence, a common proxy for phytoplankton biomass, is conducted continuously by telemetered fluorometers (in vivo) and discretely through grab samples (in vitro). Chlorophyll-a fluorometry is cost effective; however, the metric is unable to explain if the biomass is growing or simply accumulating due to hydrodynamics [7]. For this reason, complementary direct measurements of phytoplankton primary productivity, the growth that supports the increase in biomass, are necessary to establish baseline data for interannual or regional comparisons and assess restoration success, ecosystem function, and food web resources. These rates can be measured from the uptake and incorporation of isotopic tracers (e.g., ¹³C, ¹⁴C) by phytoplankton or changes in oxygen during an incubation period [8,9]. Due to cost, technical, and time barriers for many agencies and researchers, these rates are rarely measured with direct approaches and instead are usually estimated from simple linear regression models utilizing phytoplankton biomass and other easily monitored parameters. Here, we use direct measurement techniques with isotopic tracers to evaluate two simple linear regression models for productivity in the SFED.

One such model that is widely used in the SFED is a linear regression known as the light utilization model (LUM) [5,10]. The LUM estimates photic zone daily net phytoplankton productivity (PN_pd, also referred to as ∫P and ΣP in some publications) from chlorophyll-a and light availability. PN_pd (mg-C m⁻² d⁻¹) is a measure of the uptake of carbon over a cross-section of a water column and is relevant to larger coupled hydrodynamic-ecological models [6]. This model was initially calibrated with direct measurements of phytoplankton productivity in the major saline embayments of the SFED in the 1980s [10]. Later calibrations were focused in the freshwater Delta [5] and ‘low salinity zone’ [11,12], which extends from San Pablo Bay to the Delta (Figure 1) depending on annual freshwater outflow. These studies each conducted a linear regression between measured PN_pd and a composite parameter (BE₀Z_p) made from the product of chlorophyll-a concentration (B, mg-Chl m⁻³), ambient photosynthetically active radiation (E₀, E m⁻² d⁻¹), and photic zone depth (Z_p, m). The slope of the regression line—the efficiency factor (ψ)—is then used to predict PN_pd from long running historical records of BE₀Z_p going forward (e.g., [3,5,13,14]). Another study [12] made its own determination of ψ and recalculated prior determinations of ψ in the SFED. The authors concluded that (1) the observed interannual variability in ψ detracts from its reliability as a general modeling approach on timeseries greater than one year and (2) continued effort should be aimed at making direct measurements of phytoplankton productivity in estuarine systems. Studies from other estuaries present mixed results but emphasize the first conclusion [15,16]. Still, the value of a reliable and simplistic model for phytoplankton productivity cannot be understated.

Another modeling approach, which, to our knowledge, is untested in estuaries, uses chlorophyll to phytoplankton carbon ratios (B:C_p) to predict PN_pd. This modeling approach is referred to here as a chlorophyll-to-carbon model (CCM). The inspiration for this approach stems from the following observations. First, off of the coast of British Columbia, the lowest values of C_p:B (highest values of B:C_p) were constrained to the highly productive upwelling zone, and the highest values of C_p:B (lowest values of B:C_p) were constrained just seaward of the continental shelf, which are classically less productive [17]. Additionally, B:C_p ratios have been used to estimate the growth rate (µ) of phytoplankton assemblages and exhibit a dynamic relationship, with µ depending on light and nutrient resources in laboratory studies [18]. Intrigued, this research has the following aims: (1) to revisit the LUM and address recent conclusions about its general use in estuaries [12] and (2) evaluate the performance of the LUM in comparison to the performance of the CCM at retrieving observed phytoplankton primary productivity (PN_pd) for estuaries. One novelty is to provide an alternative model for estimating estuarine productivity in areas where direct measurements are lacking.

2. Materials and Methods

2.1. Study Region Description

The SFED (Figure 1) is an urban estuary in central California, serving as the drain for 40% of the state’s acreage. The brackish northern SFED includes Suisun Bay, while further inland is the Delta, beginning at the confluence of the Sacramento and San Joaquin Rivers and roughly bounded northward by the Yolo Bypass (YB) floodplain. The majority of freshwater flow into the SFED originates during the winter and early spring in northern California and enters the SFED through the Sacramento River and Yolo Bypass floodplain. Freshwater flow into the SFED is mostly managed by upstream reservoirs. This research was conducted on a hydrological transect (Figure 1) consisting of nine monitoring stations. Five stations (I-80TD, LIS, STTD, BL5, and LIB) were along a slough known as the Toe Drain in the YB, and four stations (US655, US649, US5, DWR-D7) were within the northern San Francisco Estuary (nSFE), which encompasses Suisun Bay and the seaward portion of the Delta.

2.2. Sampling and Laboratory Methods

This research was conducted between 31 March and 2 November 2023, with the transect sampled during 22 ~biweekly surveys. When data were aggregated by months, 31 March was lumped with April dates to better calculate confidence intervals. Water samples from the nSFE stations US655, US649, US5, and DWR-D7 were collected aboard the R/V Questuary by deploying a Sea Bird Electronics SB-32 rosette fitted with four 3-L Niskin bottles and a Seabird SBE-19 plus CTD. This allowed for the collection of water samples at the near surface (<1 m) while also taking real-time vertical profiles of PAR. Secchi depths were measured in parallel with vertical PAR profiles. Water samples from the YB stations I-80TD, LIS, STTD, BL5, and LIB were collected at the near-surface by the California Department of Water Resources personnel from a boat using a Van Dorn Bottle or from land with a dip stick. Secchi depths were measured at all stations along the YB.

Water samples for the analysis of dissolved inorganic carbon (DIC) concentrations, necessary for calculating phytoplankton carbon uptake, were collected and analyzed using an MBARI clone DIC analyzer with acid sparging and non-dispersive infrared analysis [19]. For chlorophyll and nutrient concentration analyses, sample water was filtered through 25-mm Whatman GF/F filters, and the filter and filtrate were immediately put on ice during transit and then frozen. Filters were analyzed with a Turner Designs Model 10-AU fluorometer for chlorophyll-a concentrations with phaeophytin-a correction [20]. Filtrates were analyzed for ammonium, nitrate, nitrite, silicate, and phosphate. Ammonium samples were analyzed using spectrophotometry (10 cm path-length cell) [21]. Other nutrients were analyzed with a Bran and Luebbe AutoAnalyzer II with the MT-19 manifold chemistry module [22,23,24].

To measure phytoplankton carbon uptake, 150-mL clear polycarbonate incubation bottles were filled with sample water from each station and spiked with a trace amount of NaH¹³CO₃ (99 at%) equivalent to ~10% of the ambient DIC concentration. Incubation bottles were placed into wire-screened, neutral density tubes or bags with a nominal shading percentage determined with a LI-COR Quantum Sensor (a dark bottle, 1, 3, 10, 25, and 70% of ambient irradiance) [11,25]. Dark bottles were spray-painted black and incubated in a capped, opaque PVC pipe. All incubations were corrected for dark bottle carbon uptake. All incubations were conducted in outdoor, flow-through incubation tables over a 24-h photoperiod under natural light and near in situ temperatures at the Estuary and Ocean Science (EOS) Center seawall located in Central SF Bay.

Incubations were terminated by filtration onto a pre-combusted (450 °C for 4 h) 25 mm Whatman GF/F filter and immediately frozen. Filters were analyzed for isotopic enrichment and particulate organic carbon concentrations (POC, mg-C m⁻³) in a PDZ Europa 20/20 gas chromatography mass spectrometer [8,25,26]. Uptake (transport) rates, ρC, (mg-C m⁻³ day⁻¹) were calculated following Dugdale and Wilkerson [9] and Legendre and Gosselin [27]. POC was converted to phytoplankton carbon concentrations (C_p in mg-C m⁻³), assuming that one third of POC was phytoplanktonic [28,29]. More accurate ways to measure C_p exist (i.e., [17]); however, the required instrumentation was not readily available for this study.

Photosynthetically active radiation (PAR as µE m⁻² s⁻¹) was measured with a LI-COR Quantum sensor at the EOS Center seawall. Where this PAR timeseries had data gaps, it was complemented with a weather station 12 km north in San Pablo Bay Point San Pedro (CIMIS station ID #157) reporting total solar radiation (W m⁻²). Total solar radiation was converted to PAR (2.06 µE s⁻¹ W⁻¹), assuming PAR represents 45% of total solar radiation [11,30]. Irradiance was then averaged over the daytime portion of the incubation period. With these two timeseries, E_o was determined for each incubation period.

2.3. Modeling Approaches

Values of ρC were used to determine PN_pd through trapezoidal integration under a light attenuation curve. Z_p is defined as the depth at which 99% of surface PAR is attenuated [10,11], equivalent to 1% of surface PAR. To measure Z_p, a LI-COR Quantum PAR sensor was fit to the sampling rosette and cast from surface to two meters above the channel bottom. The depth where only 1% of surface PAR was detected was recorded as Z_p. For stations surveyed in the YB, Z_p could not be measured in this way; therefore, it was estimated from the Secchi depth and mathematically derived from a light attenuation coefficient (k, in m⁻¹), which was determined using a linear regression (Equation (1)). We performed a linear regression of Z_p and Secchi depths at stations in the nSFE where both parameters were measured in parallel, following the approach of Kimmerer et al. [11]. Equation (1) fits within the 95% confidence intervals (CI₉₅) of other studies that have carried out similar procedures (e.g., [7,11]).

$$k={\displaystyle \frac{0.80\pm 0.08}{secchi\left(m\right)}}; {\mathrm{r}}^2=0.69,p-\mathrm{value}\le 0.001$$

(1)

Using the statistical package ‘lmtest’ in R Version 4.3.3 [31,32], a linear regression was performed between BE_oZ_p and measured PN_pd using Equation (2) to calibrate the LUM by deriving ψ for the YB and nSFE. Similarly, a linear regression was performed to calibrate the CCM productivity model between B:C_p and measured PN_pd, with the tuning parameter Ω derived from the regression slope (Equation (3)).

$${PN}_{pd}=\psi B{E}_0{Z}_p=\psi B{E}_0{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(0.01\right)}{k}}$$

(2)

$${PN}_{pd}=\Omega {\displaystyle \frac{B}{C_p}}$$

(3)

CI₉₅ were calculated for ψ, and r² values and the mean absolute error (MAE) were calculated for each fit of Equations (2) and (3). The MAE can be used as a measure of uncertainty between paired predictions and observations [3,33]. Depth-integrated annual net phytoplankton production (PN_py as g-C m⁻² yr⁻¹) was calculated from observed and estimated PN_pd by applying the study period average of PN_pd to a full year (Equation (4)). This is consistent with the application of ψ and Ω, which are the results of aggregating months, as opposed to discriminating between them.

$${PN}_{py}={\displaystyle \frac{365\times \overline{{PN}_{pd}}}{1000}}$$

(4)

3. Results

The YB and nSFE regions (Figure 1) showed different trends in nutrients and DIC during the study period. As regional averages, silicate (SiO₃) was abundant (>>150 µM) throughout the study, with the lowest concentration appearing on the first sampling date in the YB (8 April), where the average SiO₃ was 60 µM (Figure 2A). Phosphate (PO₄) was relatively constant in the nSFE throughout the study (<1–2 µM) and more variable in the YB, where average PO₄ was ~1 µM in April and rose to 8 µM in the summer before falling to 5–6 µM in the fall (Figure 2B). Regional averages of nitrate (NO₃) were relatively low in the YB throughout the study (<5 µM) and more variable in the nSFE, where NO₃ was >25 µM in the early spring and <10 µM in the summer (Figure 2C). Ammonium (NH₄) was variable throughout the year in both the YB and nSFE; however, averaged regional values remained below 3 µM (Figure 2D). The concentrations of dissolved inorganic carbon (DIC) were between 2 and 6 times higher in the YB (1700–3200 µM) than in the nSFE (<500–1300 µM) (Figure 2E).

Regionally averaged phytoplankton biomass, B, was relatively high in April and May in the YB (~20 mg-Chl m⁻³) and decreased to ~6 mg-Chl m⁻³ in the summer and fall months (Figure 3A). The elevated B in the spring may have been driven by a diatom bloom as the increase in B was accompanied by a decline in SiO₃, PO₄, and NO₃ (Figure 2A–C). Limited visualization of the phytoplankton community using a FlowCam qualitatively revealed that much of this B was composed of pennate diatoms. B in the nSFE was variable throughout the year, reaching a peak in May (~7 mg-Chl m⁻³) and a low in the fall (~1 mg-Chl m⁻³). Regionally averaged C_p was variable throughout the year but was consistently higher in the YB (~560–870 mg-C m⁻³) than in the nSFE (~260–570 mg-C m⁻³) throughout the year (Figure 3B). In both the YB and the nSFE, as expected, E_o (daily PAR) primarily decreased as a function of the number of Julian days away from the summer solstice (June 21st), although cloudy or foggy weather interrupted this signal in the nSFE (e.g., May and August in Figure 3C). Z_p was shallow in the spring (<1 m) and deepened to ~3.5–4 m by the fall in the YB (Figure 3D). Sharp fluctuations in Z_p occurred in nSFE during the spring (between <1 m and ~7 m) and fall (~4 m and ~7 m) but Z_p was less variable in the summer (~4–5 m). Water temperature increased in both regions between March (~12–14 °C) and August (~21–24 °C) and then decreased into October and November (~16–19 °C) (Figure 3E).

Depth-integrated daily primary productivity, PN_pd, decreased gradually over the year in the YB, ranging from 1000 mg-C m⁻² d⁻¹ in April to 250 mg-C m⁻² d⁻¹ in October (Figure 4A). PN_pd was more variable over the year in the nSFE, ranging between <200 and ~820 mg-C m⁻² d⁻¹ (Figure 4A). Overall, PN_py for 2023 was 137 ± 24 g-C m⁻² yr⁻¹ (

Table 2. Comparison of LUM and CCM model estimates for PN_py to observed values of PN_py from ¹³C incubations.

Estimate Source	Regression Slope	PNpy Estimate (g-C m−2 yr−1)	Within CI95	MAE	Percent Error
Observed	-	137	±24	-	-
LUM 2023	Ψ = 0.42	125	Yes	31%	−9%
LUM 1997	Ψ = 0.73	217	No	61%	58%
CCM 2023	Ω = 3.47 × 104	131	Yes	36%	−4%

). Estimating PN_py by integrating monthly averages of PN_pd over a full year and filling in missing winter months with the smallest monthly average resulted in 111 ± 54 g-C m⁻² yr⁻¹. This approach is not significantly different from the approach in Equation (4); therefore, it was not repeated for the models. The composite parameter used in the LUM, BE₀Z_p, ranged between 250 and 2500 E mg-Chl m⁻⁴ d⁻¹ (Figure 4B) and that used in the CCM, B:C_p, ranged between 5 × 10⁻³ and 3 × 10⁻² mg-Chl mg-C⁻¹ (Figure 4C).

There were strong linear relationships between BE₀Z_p and measured PN_pd (r² = 0.89, p < 0.001, n = 76) and between B:C_p and measured PN_pd (r² = 0.85, p < 0.001, n = 76) (Figure 5A,B). The linear regression between BE₀Z_p and measured PN_pd in this study (i.e., LUM 2023) resulted in ψ = 0.42 (Figure 5A). Based on the MAE, estimates of PN_pd—produced using the product of ψ = 0.42 and measured BE₀Z_p from this study—are accompanied with 31% uncertainty (Table 2). Using ψ = 0.42 to estimate monthly averages of PN_pd produces overestimates of measured PN_pd for April and May and underestimates for the remainder of the months in the year (Figure 6). Overall, the estimate of PN_py produced using ψ = 0.42 was 125 g-C m⁻² yr⁻¹, or a 9% underestimate of measured PN_py (Table 2). All but one monthly average PN_pd estimate (August) and the PN_py estimate fell within the observed CI₉₅ (Figure 6 and Table 2). Finally, the linear regression between B:C_p and measured PN_pd resulted in Ω = 3.47× 10⁴ (i.e., CCM 2023) (Figure 5B) and is accompanied with 36% uncertainty according to the MAE (Table 2). PN_py estimated using Ω = 3.47 × 10⁴ is 131 g-C m⁻² yr⁻¹, which is 4% smaller than the observed PN_py. All monthly average PN_pd estimates produced using Ω = 3.47 × 10⁴ fell within the observed CI₉₅ (Figure 6), a fact that is also true of the PN_py estimate (Table 2).

4. Discussion

This study set out to directly measure net estuarine phytoplankton productivity (PN_pd) using the stable isotope ¹³C tracer along with measures of phytoplankton biomass (B and C_p) and light availability (E_o and Z_p), such that modeling approaches for net phytoplankton productivity could be compared. The results show that daily rates of PN_pd and B were quite variable within and between distinct regions of the SFED throughout 2023 (Figure 3A and Figure 4A), even though E_o and Z_p were relatively similar in pattern and magnitude. A comparison of estimated values of PN_pd and PN_py produced from the LUM calibrated in this study and in prior studies [5] shows that the LUM is not reliable for estimating carbon uptake beyond the dataset it was calibrated with. In this study, applying an LUM calibrated in 1997 [5] to data collected in 2023 resulted in a 58% overestimate of PN_py, which did not fall within the CI₉₅ of observed PN_py. A CCM approach was shown to be a successful alternative to the LUM approach for estimating PN_pd and PN_py and could be run simultaneously to improve model validation and predictability.

Although ¹⁴C is more sensitive, the use of ¹³C was particularly advantageous for this study as it did not require strict safety protocols for using radioactive ¹⁴C in the field [34]. PN_py estimated in this study (137 ± 24 g-C m⁻² yr⁻¹) compares well with prior determinations in the SFED, which have ranged from 90 ± 13 g-C m⁻² yr⁻¹ in the Delta [5] to 120 g-C m⁻² yr⁻¹ in San Francisco Bay [10,35]. PN_py was also estimated by integrating monthly averages of PN_pd over a full year, filling in missing winter months with the smallest monthly average measured. This approach yielded an estimate of PN_py that was smaller (111 ± 54 g-C m⁻² yr⁻¹) but not significantly different from the estimate produced from Equation (4). As such, the SFED continues to rank in the lower half of estuaries where PN_py has been measured [36]. The SFED also continues to straddle the line between an oligotrophic (<100 g-C m⁻² yr⁻¹) and mesotrophic estuary (100–300 g-C m⁻² yr⁻¹) as defined in Nixon [37], even though there are abundant nutrients (Figure 2). As such, the SFED can continue to be described as a high-nutrient, low-chlorophyll (growth) estuary [38,39,40].

A leading explanation for why the SFED classifies as an oligo-mesotrophic estuary, despite substantial nutrient loading, is that phytoplankton are chronically light-limited [5,10]. The Dutch Scheldt Estuary is another example of this phenomenon [41] in addition to the Bristol Channel, U.K. [42], Roskeeda Bay, U.K. [43], and the Colne Estuary, U.K. [44]. It is indisputable that phytoplankton require light energy to fix ambient inorganic carbon and that these are in some way proportional. As such, the LUM approach for modeling PN_pd using ψ is well iterated in estuarine systems, none more than the SFED (e.g., [5,10,12,35,45,46]). For datasets limited to ~1–2 years, the LUM estimates PN_pd well, based on a single calibration of ψ [10,47]. However, when a single ψ is applied to longer duration datasets, the predictive potential of the LUM is considerably reduced—an issue all phytoplankton productivity models suffer from [15,48,49]. This ‘long dataset’ issue is made particularly evident for the LUM when studies determining ψ in different years are compiled [12] (

Table 3. Determination of the linear regression slope (ψ) reported here and refit or calculated from other published studies.

Year(s)	Location(s)	Season	Photoperiod	ψ	Reference
2023	nSFE + YB	March–November	24 h	0.42 ± 0.02	This study
	nSFE		data	0.38 ± 0.02
	YB		data	0.45 ± 0.02
1980	South SF Bay	Annual	24 h	0.99 ± 0.09	[10] Cole & Cloern (1984), refit data
	San Pablo Bay		data	1.22 ± 0.15
1988	Suisun Bay	May–October	24 h	0.81 ± 0.10	[45] Alpine & Cloern (1992), calculated from data in Table 2
1997	Delta	May–November	30 m	0.73 ± 0.05	[5] Jassby et al. (2002), as reported
2004–2005	Central SF Bay	Annual	6 h	0.37 ± 0.04	[46] Lorenzi 2006, as reported
2006	San Pablo—Suisun Bay	March–August	6 h	0.29 ± 0.03	[12] Parker et al. (2012), as reported
2007	Suisun Bay		24 h	0.56 ± 0.04

). Each study determining ψ in the SFED, including this study, shows that ψ varies interannually and has generally decreased since the turn of the millennium (Table 3), confirming recent conclusions about the general use of the LUM in estuaries [12]. Some of the differences in values of ψ are certainly due to methodological differences (e.g., incubation length, the isotopic tracer used, and instrumentation), though much of the differences must also be due to limitations of the model [50]. Additionally, the implications of differences in methodology have been discussed previously and are mostly thought to be inconsequential [5,8,12].

For 2023, ψ was determined to be 0.42 ± 0.02—a significant deviation from all prior determinations save one study in the Central SF Bay [46] (Table 3). The LUM and ψ for this study (referred to as LUM 2023) are highlighted by a strong linear relationship (r² = 0.89, p < 0.001) between PN_pd and BE_oZ_p but come with a large CI₉₅ for estimates (319 mg-C m⁻² d⁻¹) (Figure 5). CI₉₅ were not reported in prior studies. The PN_py estimate using LUM 2023 fell within the observed CI₉₅ and is overall a 9% underestimate of measured PN_py. Furthermore, as a model, LUM 2023 is ~31% uncertain based on the MAE (Table 2).

To emphasize the ‘long dataset’ issue discussed earlier, a commonly used value of ψ (0.73) in the SFED [5] was applied to BE_oZ_p collected in this study (referred to as LUM 1997) to assess its retrieval of measured PN_pd (Figure 6). Monthly averaged PN_pd estimates derived from LUM 1997 are consistent overestimates of measured PN_pd. Notably, during spring months (April and May) when a putative diatom bloom occurred in the YB (see Figure 2A–C and Figure 3A), monthly averaged PN_pd estimates from LUM 1997 fell outside of the observed CI₉₅. The PN_py estimate from LUM 1997 also fell outside of the observed CI₉₅ and, overall, there was a 58% overestimate accompanied by 61% uncertainty (Table 2). We can, therefore, echo conclusions that the LUM model is not fit for general use on long estuarine timeseries [12,15]—at least as a solitary approach. In this study, determinations of ψ were also significantly different for the nSFE and YB regions, a finding similar to those in the Scheldt Estuary, where ψ varied greatly station to station [16]. For the Scheldt estuary, this was partly due to the variability in Z_p over a tidal cycle; however, other explanations may be related to differences in residence time between the nSFE and YB in this study as well [51].

The CCM performed here between B:C_p and PN_pd (CCM 2023) had a strong linear relationship (r² = 0.85, p < 0.001) and CI₉₅ (366 mg-C m⁻² d⁻¹) and uncertainty (36%) that were similar to values seen for LUM 2023 (Figure 6). Overall, monthly averaged PN_pd and the annual PN_py estimate (131 g-C m⁻¹ yr⁻¹) from CCM 2023 all fell within the observed CI₉₅ (Figure 6 and Table 2). In comparing the CCM and LUM, these results suggest that neither is better than the other. Specifically, the LUM had superior model statistics (i.e., r² and CI₉₅ width); however, the CCM produced a more precise annual estimate (4% vs. 9% underestimate). Thus, these models performed similarly, produced agreeable results, and could be used together prospectively—or retrospectively—to validate PN_pd estimates in estuarine systems. Still, it continues to be important to recognize the limitations of these simplistic models that arise from underlying assumptions. For example, the assumption in this study that a fixed proportion of POC was phytoplanktonic is complicated by the potential variability of detrital matter that was not assessed. Additionally, the assumption that ambient PAR and photic zone depths are the absolute measures to assess light availability to phytoplankton denies the consideration of how rates of vertical mixing affect access to the photic zone and what the physiological readiness of phytoplankton to utilize available light is. Finally, this study has shown that it is a poor assumption to consider the values of ψ and Ω to be constants and that the utilization of both the LUM and CCM require occasional recalibration to assess variability in these models. As such, these models and underlying assumptions could be better tested together in estuaries by utilizing pre-existing continuous monitoring networks, like the one present in the SFED to complement discrete studies such as this one. This could be accomplished by leveraging optical instrumentation that is or can be installed on pilings and piers to measure chlorophyll-a fluorescence and turbidity (EXO2 sondes), ambient PAR (LI COR sensors), and estimate C_p (e.g., WET Labs ECO-BB3 [17]). For estimating C_p from optical instrumentation, special care would need to be taken to consider different ambient conditions in estuaries, since previous applications were conducted in oceanic settings mostly free of detrital carbon [17,52].

On their own, simple linear models, such as the LUM and CCM, fail to capture the reality of stochastic fluid systems—something estuaries exemplify. In its a priori, the LUM does not consider the many abiotic and biotic factors (initial conditions) driving PN_pd that may be significantly different over small or large timescales [10]. The CCM begins to inherently incorporate other factors, like nutrient conditions [18 and references therein], into its estimation of PN_pd. For example, in a scenario where there was a temporal or regional change to nutrient loading, such that a limiting nutrient emerged, estimations of ψ from the LUM would not necessarily capture this effect on productivity, leading to variability in ψ, whereas estimations of Ω from the CCM would likely capture this effect, leading to more consistency in Ω. By pairing simple linear models that capture a wider range of initial conditions, we can cross-examine and validate results and predict reality with more confidence. The simultaneous application of multiple models is a common practice in forecasting natural scenarios [53]. The same practice can be applied to modeling PN_pd, where using an ensemble of simplistic models would improve our confidence in outputs. By packaging simplistic models together, we can assess model agreement, identify patterns of cases where models disagree, and investigate the roots of these disagreements.

5. Conclusions

Here, we suggest a simple approach of packaging the LUM with a CCM for future efforts with an aim to model PN_pd. The choice to complement the LUM with a CCM is based on success with CCMs in oceanography [28], correlative behavior observed between B:C_p, the growth rate µ, and net phytoplankton productivity [17,18] and because B:C_p is simple to measure alongside the LUM methodology and can be measured simply using shipboard sensors or satellites [17,28,54]. The results of this research show that (1) the SFED continues to be an oligo-mesotrophic estuary despite nutrient loading, (2) the ‘long dataset’ issue persists with the LUM approach to modeling PN_pd, reinforcing the importance of periodic recalibration, and (3) we demonstrate how expanding the toolbox available for modeling may lead to more precise estimates of net phytoplankton productivity through an ensemble modeling approach.