Introduction

According to the optical classification standard given by Morel and Prieur (1977), oceanic waters may be characterized as either Case I, the optical properties of which are dominated by chlorophyll and covarying detrital pigments; or as Case II, in which other substances, which are not only covaried with chlorophyll-a (chla), but are also affected by other optical properties. Such substances include gelbstoff, suspended sediments, coccolithophores, detritus, and bacteria. As a result, pigments retrieved from remote sensing imageries in Case I waters have achieved reasonable results [±20 % for the local best cases (Gordon 1990)], but more research is still required for Case II waters (Moore et al. 2009).

In order to quantify chla in productive turbid waters, a variety of algorithms have been developed, and all were based on the properties of the reflectance peak near 675 nm (Gitelson 1992), including the ratio of the reflectance peak at the NIR (near-infrared) to the reflectance at 670 nm. Gons et al. (2000) used the reflectance ratio at 704 and 672 nm and absorption and backscattering coefficients at these wavelengths to assess chla concentrations ranging from 3 to 185 mg/m3. Ruddick et al. (2000) analyzed the manner in which errors in reflectance measurements affect chla retrievals for an NIR to red reflectance ratio algorithm with a general choice of wavelengths. The authors of the said study found that the effects on chla retrieval depend strongly on the choice of the NIR wavelength if the error is spectrally neutral, suggesting a new type of algorithm, where the NIR wavelength used for retrieval is chosen dynamically for each spectrum to be processed. These approaches rely on a specific spectral feature to biophysical measurements using statistical regression. This type of model is simple and easily implemented (Matthews 2011). However, the method lacks a physical foundation and the relationships within are quite geographically specific and cannot be applied in other areas.

Recently, Dall’Olmo et al. (2003) provided evidence that a three-band reflectance model, originally developed for estimating pigment contents in terrestrial vegetation, could also be used to assess chla concentration in turbid waters (Gitelson et al. 2008, 2007). This semi-analytical algorithm involved three underlying assumptions (Gitelson et al. 2008; Le et al. 2009): (1) the absorption by suspended solids and CDOM (colored dissolved organic matters) at λ 2 was close to that at λ 1; (2) reflectance at λ 3 is minimally affected by the absorption by optically active constituents and could only account for the variability in scattering between samples; and (3) the total backscattering curve is “spectrally flat” in the red and NIR regions. However, the three assumptions for the three-band semi-analytical algorithm may be violated in highly turbid waters, due to the fact that the absorption and scattering of particulate matter at the NIR cannot be ignored in turbid waters (Tassan and Ferrari 2003; Tzitzuiou et al. 2006). Recently, Le et al. (2009) developed a four-band semi-analytical algorithm to improve the performance of the three-band algorithm in turbid waters. The improvement was achieved by subtracting the effects of suspended solids, as well as minimizing the effects of pure water absorption and backscattering in the NIR region using the fourth band. According to the study results carried out by Le et al. (2009), the four-band algorithm showed optimal accuracy in the cases of λ 1 = 663 nm, λ 2 = 693 nm, λ 3 = 705 nm, and λ 4 = 740 nm. However, the reflectance near 704 nm generally corresponded to the reflectance peak owing to the chla scattering, whereas the reflectance near 740 nm was indistinctively influenced by the chla (Gons et al. 2000). As a result, it was difficult for the selected bands of the four-band algorithm to meet the underlying assumption that both (a ssc + a CDOM + a chla ) and b b were spectrally flat in the red-NIR band (Gitelson et al. 2008). Furthermore, the four-band algorithm used the difference R −1rs (λ 3) − R −1rs (λ 4) as the denominator to eliminate the influence of backscattering by all particulate matter b b in the numerator. The four-band algorithm suggested that the λ 3 should be quite close to λ 4 in order to meet the requirement of \( \left[ {a_{{{\text{chl}}a}} \left( {\lambda_{3} } \right) + a_{\text{ssc}} \left( {\lambda_{3} } \right) + a_{\text{CDOM}} \left( {\lambda_{3} } \right)} \right] \sim \left[ {a_{{{\text{chl}}a}} \left( {\lambda_{4} } \right) + a_{\text{ssc}} \left( {\lambda_{4} } \right) + a_{\text{CDOM}} \left( {\lambda_{4} } \right)} \right].\,\)Thus, the difference R −1rs (λ 3) − R −1rs (λ 4) must be a small value variable, which is many times lower than the difference R −1rs (λ 1) − R −1rs (λ 2). The uncertainty in the numerator, R −1rs (λ 1) − R −1rs (λ 2), would be enlarged if a small number was used as the denominator to divide that numerator (Chen et al. 2010a). As a result, the four-band algorithm may possess poor noise tolerance ability in chla concentration estimation in turbid waters, which is another limitation of the four-band algorithm.

The main objective of this study is to validate the performance of the three- and four-band algorithms, and to further improve it for use in the Yellow River estuary, China. The specific goals of this study are as follows: (1) to locate the optimal spectral positions of the three- and four-band algorithms; (2) to evaluate the accuracy and stability of the three- and four-band algorithms in predicting chla in the turbid waters of the Yellow River estuary; (3) to improve the performance of the three-band semi-analytical algorithm by developing an improved semi-analytical algorithm; and (4) to compare the performances of the three- and four-band algorithms with the improved three-band algorithm in terms of estimating chla concentration in the turbid waters of the Yellow River estuary.

Data, methods and techniques

Study area

The Yellow River estuary is a typical turbid water body, located between 117.58°E and 122.25°E and between 37.10°N N and 41.00°N, semi-enclosed by the Bohai Bay and Laizhou Bay of the Bohai Sea, China (Fig. 1). The Yellow River, the second longest river in China, flows into the Yellow River estuary, and is known around the world for its high concentration of sediments. The average run-off is 5.80 × 1010 m3 per year and sediment transport to Bohai Sea is 1.10 × 109 tons per year (Chen et al. 2010b). Consequently, the water column in the Yellow River Estuary is quite turbid, which interferes with the bio-optical properties of the chla signs and creates a challenge for estimating the chla concentration from remote sensing data. The Yellow River estuary represents typical turbid coastal waters which are influenced by river and anthropogenic inputs. Chla concentrations in this area are difficult to estimate accurately as the bio-optical properties of the water bodies are quite complex (Zhang et al. 2010). The signal of the phytoplankton is partially interfered with by the bio-optical properties of SSC and CDOM. As a result, it is challenging to estimate the chla concentration in the Yellow River estuary due to the high level of SSC. This is the reason why the Yellow River estuary was selected as the study area for use in this paper.

Fig. 1
figure 1

Yellow River estuary and sample stations.

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Datasets used

In this study, in order to evaluate the accuracy of the algorithm for estimating chla concentration, two independent data sets, including the spectral optical properties and chla concentration of the water column, were collected from the Yellow River estuary. The first data set was used for model calibration, while the second was used for model performance evaluation. The calibration data set containing 24 samples was collected from the Yellow River estuary on 1 and 4 September, 2009. The validation data set containing eight samples was collected from the Yellow River estuary on 10 September, 2009.

Field measurements

The field measurements were conducted from 10:00 to 14:00 local time in the Yellow River estuary. At each station (Fig. 1), water-leaving reflectance measurements were taken from aboard a boat. The reflectance was measured with a spectroradiometer with 25° fiberoptic, covering the spectral range 350–2,500 nm (Spectral Devices, Boulder, CO, ASD). Although data in the range 350–2500 nm, with a spectral resolution of 3 nm (full-width at half-maximum, FWHM) and a 1.4 nm sampling interval for the 350–1050 nm spectral range (ASD 1999) were collected as well, the data mainly used in this study were those in the range 400–900 nm, which is the most commonly used wavelength for water color remote sensing (Deng and Li 2003; Gons et al. 2008; Ouaidrari and Vermote 1999; Tachiiri 2005; Wang et al. 2009). Following the ocean optics protocols for satellite ocean color sensor validation (Mueller et al. 2003), various measurements were repeated at each station in order to estimate the uncertainty associated with each measurement, and the measurements with <5 % uncertainty were selected for model calibration and validation.

While the measurements were being taken, the tip of the optical fiber was kept ~1 m above the water surface by means of a 3 m long, hand-held black pole. The radiance of both the water surface (L sw(λ)) and a standard gray board (L p(λ)) was measured. Ten curves were acquired for each target. In order to effectively avoid the interference of the ship with the water surface and the influence of direct solar radiation, the instrument was positioned at an angle β of 90–135° with the plane of the incident radiation pointed away from the sun (Le et al. 2009). The view of the water surface, α, was controlled between 30 and 45° with the aplomb direction. In this way, most of the direct sunlight was eliminated while the impact of the ship’s shadow was minimized (Le et al. 2009). Immediately after measuring the water radiance, the spectroradiometer was rotated upwards by 90–120° to measure the skylight. The view azimuth angle in this measurement was kept the same as that when measuring the water radiance (Mueller and Fargion 2002).

Remote sensing reflectance, R rs(λ) was calculated as follows:

$$ R_{\text{rs}} \left( \lambda \right) = \frac{{L_{\text{w}} \left( \lambda \right)}}{{E_{\text{d}} \left( {0^{ + } ,\lambda } \right)}} $$
(1)

where L w(λ)is the water-leaving radiance, and E d(0+,λ) is the total incident radiant flux of the water surface. L w(λ) and E d(0+,λ) in Eq. (1) are further calculated as follows:

$$ L_{\text{w}} \left( \lambda \right) = L_{\text{sw}} \left( \lambda \right) - rL_{\text{sky}} \left( \lambda \right) $$
(2)
$$ E_{\text{d}} \left( {0^{ + } ,\lambda } \right) = \frac{{\pi L_{\text{p}} \left( \lambda \right)}}{{\rho_{\text{p}} \left( \lambda \right)}} $$
(3)

where L sky(λ) represents the diffused radiation of the sky, which contains no information of water properties and therefore must be eliminated; r refers to the reflectance of the skylight at the air–water interface, the value of which depends upon the solar azimuth, measurement geometry, wind speed, and surface roughness; and ρ p(λ) is the reflectance of the gray plate. In this study, r is calculated with assumption of the black water body at wavelengths from 1,000 to 1,020 nm (Hale and Querry 1973) and wavelength-independent (Doxaran et al. 2002). The remote sensing reflectance calculated by Eq. (1) is shown in Fig. 2.

Fig. 2
figure 2

Field measurement datasets a Calibration datasets collected in the Yellow River estuary, China, on 1 and 4 September, 2009. b Validation datasets collected in the Yellow River estuary, China, on 10 September, 2009

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Laboratory measurements

Water samples were collected immediately after the radiance measurements were taken. At each station a standard set of water quality parameters was measured, including chla concentration and transparency. The surface water samples were collected at a depth of 0.5 m below the water–air surface. After sampling, water samples were conserved in bottles at a low temperature and sent for laboratory analysis in the afternoon of the same day.

The laboratory analyses were carried out within 24 h following sample collection. The chla concentrations were extracted and measured with 90 % acetone in accordance with the Ocean Optical Protocols of NASA (Mueller and Fargion 2002), a generally accepted method of quantifying chla concentration in the fields of chemistry and biology (Gilpin and Tett 2001).

Improved semi-analytical algorithm

The R rs(λ) model was given according to the following general equation (Morel and Prieur 1977), which was adapted from Lee et al. (1994):

$$ R_{\text{rs}} \left( \lambda \right) = \frac{{ft^{2} }}{{Q\left( \lambda \right)n^{2} }}\frac{{b_{\text{b}} \left( \lambda \right)}}{{a\left( \lambda \right) + b_{\text{b}} \left( \lambda \right)}} $$
(4)

where f is an empirical factor averaging approximately 0.32–0.33; t is the transmittance of the air–water interface; Q(λ) is the upwelling irradiance-to-radiance ratio E u(λ)/L u(λ); and n is the real part of the index of refraction of water. By making the following two approximations, Eq. (4) may be greatly simplified (Carder et al. 2003):

  1. 1.

    In general, f is a function of the solar zenith angle, θ 0. However, Morel and Gentili (1993) have shown that the ratio f/Q is relatively independent of the θ 0 for sun and satellite viewing angles.

  2. 2.

    t 2/n 2 is approximately equal to 0.54, and although it is capable of changing with sea-state, it is relatively independent of wavelength.

These two approximations lead to a simplified version of Eq. (4):

$$ R_{\text{rs}} \left( \lambda \right) = h\frac{{b_{\text{b}} \left( \lambda \right)}}{{a\left( \lambda \right) + b_{\text{b}} \left( \lambda \right)}} $$
(5a)

where h is unchanging with respect to λ and θ 0. The total absorption coefficient may be expanded as follows:

$$ a\left( \lambda \right) = a_{\text{w}} \left( \lambda \right) + a_{{{\text{chl}}a}} \left( \lambda \right) + a_{\text{CDOM}} \left( \lambda \right) + a_{\text{ssc}} \left( \lambda \right) $$
(5b)

where the subscripts “w”, “chla”, “CDOM”, and “ssc” refer to water, chla, CDOM, and suspended sediment, respectively. According to the underlying assumption of three- and four-band algorithms that the (a ssc + a CDOM) and b b are spectrally flat in the red-NIR region, the R −1rs (λ 1) − R −1rs (λ 2) may be approximated to the following equation:

$$ R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right) \approx h\frac{{a_{{{\text{chl}}a}} \left( {\lambda_{1} } \right) + a_{\text{w}} \left( {\lambda_{1} } \right) - a_{\text{w}} \left( {\lambda_{2} } \right)}}{{b_{\text{b}} }} $$
(6)

Equation (6) is still affected by b b. If backscattering varies between samples, the model output would be different for the same chla. To account for this, the R −1rs (λ 3) is used in the three-band algorithm, and R −1rs (λ 3) − R −1rs (λ 4) is used in the four-band algorithm. Hence, the three- and four-band algorithms may be denoted as follows:

$$ \left[ {{\text{chl}}a} \right] \propto \left[ {R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)} \right]R_{\text{rs}} \left( {\lambda_{3} } \right) $$
(7a)
$$ \left[ {{\text{chl}}a} \right] \propto \frac{{R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)}}{{R_{\text{rs}}^{ - 1} \left( {\lambda_{3} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{4} } \right)}} $$
(7b)

where [chla] is the chla concentration. The structure of Eq. (5a) originates from Morel and Prieur (1977); when they developed a reflectance model, h was given the mean value of 0.33. As shown in Morel and Gentili (1996), the coefficient h is not constant and does vary in an orderly manner with the water optical properties. Thus, it is much more challenging to separate a chla (λ 1) from the remote sensing reflectance using the three-band algorithm provided by Gitelson et al. (2008) or the four-band algorithm developed by Lee et al. (2002), due to the poor approximation of Eq. (5a). If considering the water optical properties dependent γ in Eq. (5a), the three- and four-band algorithms should be written as follows:

$$ \left[ {R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)} \right]R_{\text{rs}} \left( {\lambda_{3} } \right) \approx \frac{{h\left( {\lambda_{3} } \right)}}{{h\left( {\lambda_{1} } \right)}}\frac{{a_{{{\text{chl}}a}} \left( {\lambda_{1} } \right) + a_{\text{w}} \left( {\lambda_{1} } \right) - a_{\text{w}} \left( {\lambda_{2} } \right)}}{{a_{w} \left( {\lambda_{3} } \right)}} $$
(8a)
$$ \frac{{R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)}}{{R_{\text{rs}}^{ - 1} \left( {\lambda_{3} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{4} } \right)}} \approx \frac{{h\left( {\lambda_{3} } \right)}}{{h\left( {\lambda_{1} } \right)}}\frac{{a_{{{\text{chl}}a}} \left( {\lambda_{1} } \right) + a_{\text{w}} \left( {\lambda_{1} } \right) - a_{\text{w}} \left( {\lambda_{2} } \right)}}{{a_{\text{w}} \left( {\lambda_{3} } \right) - a_{\text{w}} \left( {\lambda_{4} } \right)}} $$
(8b)

Due to the fact that h varies in an orderly manner with the water optical properties, the outputs of the three- and four-band algorithms would be different for the same chla concentration. To account for this, Gordon et al. (1988) have carried out extensive computations of R rs(λ) as a function of the optical properties of the water and solar zenith angle θ 0, and have concluded that for θ 0 ≥ 20°, R rs(λ) may be directly related to the inherent optical properties of the water through the following:

$$ r_{\text{rs}} \left( \lambda \right) = \frac{{R_{\text{rs}} }}{{ 0. 5 2 { + 1} . 7R_{\text{rs}} }} $$
(9a)
$$ r_{\text{rs}} \left( \lambda \right) = \frac{s\left( \lambda \right)}{s\left( \lambda \right) + 1}\left[ {g_{0} + \frac{{g_{1} s\left( \lambda \right)}}{s\left( \lambda \right) + 1}} \right] $$
(9b)
$$ s\left( \lambda \right) = \frac{{ - g_{0} + \sqrt {g_{0}^{2} + 4g_{1} r_{\text{rs}} \left( \lambda \right)} }}{{2g_{1} + g_{0} - \sqrt {g_{0}^{2} + 4g_{1} r_{\text{rs}} \left( \lambda \right)} }} $$
(9c)

where s(λ) is the ratio of total backscattering to total absorption; and r rs is remote sensing reflectance spectral measured just below water surface. For nadir-viewed r rs, Gordon et al. (1988) found that g 0 ≈ 0.0949 and g 1 ≈ 0.0794 for oceanic Case I waters. Recently, Lee et al. (1999) suggested that g 0 ≈ 0.084 and g 1 ≈ 0.17 are more ideal for higher-scattering coastal waters.

The first spectral band must be maximally sensitive to chla. This means that λ 1 must be restricted to within the range 660–690 nm (Gitelson et al. 2008). In order to minimize the influences of absorption by CDOM and suspended sediment at the λ 1, the second spectral band is used. The second band must meet the requirements of a CDOM(λ 1) ~ a CDOM(λ 2) and a ssc(λ 1) ~ a ssc(λ 2). The underlying assumption within is that a CDOM(λ) and a ssc(λ) are spectrally flat in the red-NIR region (Gitelson et al. 2008). It is worth noting that the improved three-band algorithm does not require λ 2 to be minimum sensitive to absorption by chla, because [a chla (λ 1) − a chla (λ 2)] is still quite well related to chla concentration, provided that both a chla (λ 1) and a chla (λ 2) are very well related to chla concentration. This means that the λ 2 should be set to from 660 to 730 nm, which is beyond the expected wavelength range suggested by Dall’Olmo and Gitelson (2006). The difference s −1(λ 1) − s −1(λ 2) is still affected by b b(λ 1), i.e. if backscattering varies between samples, the model output would be different for the same chla concentration. To account for this, a third spectral band, λ 3, has been adopted. According to the study results carried out by Mobley (1994), Babin and Stranski (2004), and Binding et al. (2008), a w > 2.8 m−1 and a ssc < 0.052 m−1 in highly turbid waters (based on a ssc(λ) = 0.04 × [SSC] × exp[−0.011(λ − 440)], where [SSC] refers to SSC concentration), while λ > 750 nm, and SSC concentration is <40 kg/m3. This means that the λ 3 should be set to >750 nm in order to minimize the absorption by suspended sediments in highly turbid waters (a w ≫ a ssc). Additionally, in order to minimize the influences of absorption by chla at the λ 3 in the samples with high chla concentration (Lee and Carder 2004), the a chla (λ 3) in made non-negligible at the NIR bands in the improved three-band algorithm. Thus, the three-band algorithm may be improved as follows:

$$ \left[ {s^{ - 1} \left( {\lambda_{1} } \right) - s^{ - 1} \left( {\lambda_{2} } \right)} \right]s\left( {\lambda_{3} } \right) \approx \frac{{a_{{{\text{chl}}a}} \left( {\lambda_{1} } \right) - a_{{{\text{chl}}a}} \left( {\lambda_{2} } \right) + a_{\text{w}} \left( {\lambda_{1} } \right) - a_{\text{w}} \left( {\lambda_{2} } \right)}}{{a_{{{\text{chl}}a}} \left( {\lambda_{3} } \right) + a_{\text{w}} \left( {\lambda_{3} } \right)}} $$
(10)

Linear regression of a chla (λ) versus chla concentration, [chla], yielded an equation of this form (Carder et al. 2003; Dall’Olmo and Gitelson 2006; Gitelson et al. 2008):

$$ a_{{{\text{chl}}a}} \left( \lambda \right) = K_{0} \left( \lambda \right)\left[ {{\text{chl}}a} \right] + K_{1} \left( \lambda \right) $$
(11)

Substituting Eq. (11) into Eq. (10) and performing simplification then yielded the following improved three-band algorithm:

$$ [{\text{chl}}a] = \frac{1}{{P_{0} \left[ {s^{ - 1} \left( {\lambda_{1} } \right) - s^{ - 1} \left( {\lambda_{2} } \right)} \right]s\left( {\lambda_{3} } \right) + P_{1} }} + P_{2} $$
(12)

In a similar way, the linear regression of the three- and four-band algorithms may be written as follows:

$$ \left[ {{\text{chl}}a} \right] = x_{0} \left[ {R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)} \right]R_{\text{rs}} \left( {\lambda_{3} } \right) + x_{1} $$
(13a)
$$ \left[ {{\text{chl}}a} \right] = y_{0} \frac{{R_{\text{rs}}^{ - 1} \left( {\lambda_{1} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{2} } \right)}}{{R_{\text{rs}}^{ - 1} \left( {\lambda_{3} } \right) - R_{\text{rs}}^{ - 1} \left( {\lambda_{4} } \right)}} + y_{1} $$
(13b)

where x 0, x 1, y 0, y 1, P 0, P 1, and P 2 are the empirical coefficients determined by the non-linear iterative method suggested by Chen and Quan (2012).

Noise tolerance ability of the chla estimation algorithm

In this study, the three-band, four-band, and improved three-band algorithms were calibrated and validated against bio-optical datasets collected from in situ measurements. It may be considered reasonable to ignore the noise of field measurements provided that the bio-optical measurements strictly follow the standards of bio-optical experiments (Mueller and Fargion 2002). However, the final goal of oceanic remote sensing is not only to determine the optimal estimation algorithms from field measurements, but also to accurately estimate the spatial distribution of the chla concentration from satellite images. In fact, the satellite images are contaminated by atmospheric path scattering and absorption, resulting in a residual uncertainty of 5–9 %, even with an accurate atmospheric correction (Chen et al. 2011a; Gordon and Voss 1999; Hu and Carder 2002). Thus, it is necessary to consider the noise tolerance ability of the remote sensing algorithm while applying it to estimating the chla concentration from satellite data.

According to Chen et al. (2010a), the uncertainty caused by a small data noise may be expressed as follows:

$$ \Updelta \left[ {{\text{chl}}a} \right]_{\text{D}} = \frac{{\sum\limits_{i = 1}^{n} {\frac{{\partial \left[ {chla} \right]}}{{\partial R_{\text{rs}} \left( {\lambda_{i} } \right)}}\Updelta R_{\text{rs}} \left( {\lambda_{i} } \right)} }}{{\left[ {chla} \right]}} $$
(14)

where ΔR rs(λ i) is a data noise at λ i, Δ[chla]D is the uncertainty in chla estimation caused by ΔR rs(λ i), and n is the number of bands. The principle of Cauchy–Buniakowsky–Schwarz (Chen et al. 2003) suggested that Eq. (14) must meet the following requirement:

$$ \Updelta \left[ {{\text{chl}}a} \right]_{\text{D}} \le \frac{{\sqrt {2\sum\limits_{i = 1}^{n} {\left( {\frac{{\partial \left[ {{\text{chl}}a} \right]\Updelta R_{\text{rs}} \left( {\lambda_{i} } \right)}}{{\partial R_{\text{rs}} \left( {\lambda_{i} } \right)}}} \right)^{2} } } }}{{\left[ {{\text{chl}}a} \right]}} $$
(15a)

A small noise is generally defined as the ratio of the noise to the measurement value. For the sake of simplicity, in this study it is assumed that the remote sensing reflectance at all bands contains the same residual uncertainty, e.g. the 5 % residual uncertainty in the atmospheric correction of the MODIS and SeaWiFS data obtained during NASA’s mission. This assumption may be unrealistic, but it avoids controversy concerning the exact definition of uncertainty in remote sensing reflectance at each band and its respective method of measurement:

$$ \Updelta \left[ {{\text{chl}}a} \right]_{\text{D}} \le k\frac{{\sqrt {2\sum\limits_{i = 1}^{n} {\left( {\frac{{\partial \left[ {{\text{chl}}a} \right]}}{{\partial R_{\text{rs}} \left( {\lambda_{i} } \right)}}} \right)^{2} \left[ {R_{\text{rs}} \left( {\lambda_{i} } \right)} \right]^{2} } } }}{{\left[ {{\text{chl}}a} \right]}} $$
(15b)

where k, in this study, is defined as a small data noise. In brief, only the case associated with k = 1 % is discussed here, since any other cases may be linearly denoted using this one. In order to compare different models, noise tolerance ability was normalized by k, denoted as follows:

$$ \Updelta \left[ {{\text{chl}}a} \right]_{\text{N}} = \frac{{\sqrt {2\sum\limits_{i = 1}^{n} {\left( {\frac{{\partial \left[ {{\text{chl}}a} \right]}}{{\partial R_{\text{rs}} \left( {\lambda_{i} } \right)}}} \right)^{2} \left[ {R_{\text{rs}} \left( {\lambda_{i} } \right)} \right]^{2} } } }}{{\left[ {{\text{chl}}a} \right]}} $$
(16)

where Δ[chla]N refers to the normalized Δ[chla]D, that is, the noise tolerance ability defined as the uncertainty in chla estimation caused by 1 % data noise.

Statistical criteria

In order to evaluate algorithm performance, the RMS (root-mean-square) statistic was used in this study. The RMS statistic described here was based on the ratio of root-mean-square error to measured values (O’Reilly et al. 1998). This statistic was described by (Carder et al. 2003):

$$ {\text{RE}} = \left| {\frac{{\left[ {{\text{chl}}a} \right]_{\text{obs}} - \left[ {{\text{chl}}a} \right]_{\bmod } }}{{\left[ {{\text{chl}}a} \right]_{\text{obs}} }}} \right| \times 100\% $$
(17a)
$$ {\text{RMS}} = \sqrt {\frac{{\sum\limits_{i = 1}^{m} {{\text{RE}}_{i}^{2} } }}{m}} $$
(17b)

where, [chla]mod,i is the modeled value of the ith element, RE is the relative error, REi is the relative error of the ith element, [chla]obs,i is the observed (or in situ measured) value of the ith element, and m is the number of elements.

Results

Chla concentration

The data sets used to calibrate and validate the performance of the three-band, four-band, and improved three-band algorithms contained 32 samples of chla concentration. The chla concentration ranged from 1.57 to 13.64 μg/l for the calibration data set, and varied from 2.35 to 6.76 μg/l for the validation dataset (Table 1), indicating that the Yellow River estuary has been slightly eutrophied. The poor correlation (Fig. 3) between chla concentration and transparency emphasizes the fact that chla and the covarying pigment are not the only contributors of water optical properties. Qiao et al. (2009) indicated that the average SSC in the Yellow River estuary may reach as high as >10 kg/m3, showing that water of the Yellow River estuary is SSC-rich. Thus, the water of the Yellow River estuary is categorized as extremely turbid Case II.

Table 1 Descriptive statistics of the optical water quality parameters measured: chla concentration, and transparency (STD refers to standard deviation)
Full size table
Fig. 3
figure 3

Chla concentration plotted against transparency.

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“Spectrally flat” characteristics at red-NIR regions

According to the results of the study performed by Gitelson et al. (2008), the underlying assumption of the three-band algorithm was that a CDOM(λ 1) ~ a CDOM(λ 2), a ssc(λ 1) ~ a ssc(λ 2), and b b (λ 1) ~ b b (λ 2) at the red-NIR regions, which also causes the remote sensing reflectance to be “spectrally flat” within those ranges. Thus, the spectral characteristic of the remote sensing reflectance may be used to check the validity of the underlying assumption of the three-band, four-band, and improved three-band algorithms at red-NIR regions. Figure 4 shows the spectral characteristics within the ranges 660–690 nm and 710–730 nm, revealing that the remote sensing reflectance was spectrally flat within the ranges 660–690 nm, but linearly decreased in the range 710–730 nm. Therefore, at least for this data set, the wavelength range 660–690 nm for improved three-band algorithm may be more reasonable than that of 710–730 nm; therefore in this study, the λ 2 of the improved three-band algorithm should be set to the range 660–690 nm.

Fig. 4
figure 4

Spectral characteristics within the ranges 660–690 nm and 710–730 nm, respectively. a Spectral characteristics within the range 660–690 nm. b Spectral characteristics within the range 710–730 nm

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Algorithm calibration

The optimal positions of the three-band, four-band, and improved three-band algorithms were calculated by the band tuning method, e.g., the λ 1 of the three-band algorithm must be restricted to within the range 660–690 nm; the λ 2 of the three- and four-band algorithms must be restricted to within the range 710–730 nm, while the λ 2 of the improved three-band algorithm must be in the range 660–690 nm; the λ 3 of the three-band algorithm and the λ 3 and λ 4 of the four-band algorithm must be restricted to within ranges from the red to NIR wavelengths, while the λ 3 of the improved three-band algorithm must be larger than 750 nm. The non-linear iterative method suggested by Chen and Quan (2012) was used to determine the most ideal functions of the three-band, four-band, and improved three-band algorithms.

Figure 5 showed the optimal three-band, four-band, and improved three-band algorithms, indicating that the three-band algorithm possesses the optimal accuracy in the cases of λ 1 = 668 nm, λ 2 = 726 nm, and λ 3 = 765 nm; the four-band algorithm possesses the optimal accuracy in the cases of λ 1 = 688 nm, λ 2 = 710 nm, λ 3 = 671 nm, and λ 4 = 710 nm; and the improved three-band algorithm possesses the optimal accuracy in the cases of λ 1 = 671 nm, λ 2 = 685 nm, and λ 3 = 917 nm. Therefore, the improved three-band algorithm (R 2 = 0.9168) produces superior performance in comparison to both the three-band (R 2 = 0.6963) and four-band algorithms (R 2 = 0.8325).

Fig. 5
figure 5

Optimal a three-band, b four-band, and c improved three-band algorithms in the Yellow River estuary, Chinam

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Algorithm validation

The accuracy and stability in the chla concentration predicted by the three-band, four-band, and improved three-band algorithms were assessed by examining their respective RMS against the validation dataset. Figure 6 shows the accuracy of the three-band, four-band, and improved three-band algorithms assessed by the validation data set, indicating that the improved three-band algorithm produced a superior performance in comparison to both the three- and four-band algorithms, which is consistent with the algorithm calibration results shown in Sect. Algorithm calibration. Using the improved three-band algorithm in estimating the chla concentration in the Yellow River estuary produced 23.86 % RMS, which decreased 24.93 % RMS from the four-band algorithm and 38.62 % RMS from the three-band algorithm, a significant improvement.

Fig. 6
figure 6

Performance of three chla estimation algorithm assessed by validation dataset

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Discussion

Generally speaking, the stability of chla concentration estimation algorithm is greatly dependent on the equivalent degrees of freedom of the chla estimators (Walden 1990), so that although the four-band algorithm produces a superior performance in estimating chla concentration from turbid waters in comparison to the three-band algorithm, its stability may in fact be inferior to the three-band algorithm, due to the fact that the addition of the fourth band also increased the equivalent degrees of freedom of model. In order to illuminate this problem, the calibration data set was used for the noise tolerance ability analysis of these three chla concentration estimation algorithms. Figure 7 shows the noise tolerance ability of three-band, four-band, and improved band algorithms computed by Eq. (16), indicating that the improved three-band algorithm possesses a superior noise tolerance ability in comparison to both the three- and four-band algorithms. It is generally considered reasonable to ignore the residual uncertainty of field measurements provided that the bio-optical measurements strictly follow the standards of bio-optical experiments (Mueller and Fargion 2002). However, the final goal of oceanic remote sensing is not only to construct optimal estimation algorithms from field measurements, but also to accurately estimate the chla concentration from satellite images. Noise tolerance ability is of importance to algorithms for estimating chla concentration in turbid waters from satellite data. The results of this study suggest that an optimal algorithm should not only have a good chla concentration prediction ability, but it should also have an excellent noise tolerance ability for noise of remote sensing imagery. Accordingly, from the perspectives of both noise tolerance ability and chla prediction ability, the improved three-band algorithm is the most ideal of the three algorithms.

Fig. 7
figure 7

Anti-disturbance ability of three-band algorithm, four-band algorithm and improved three-band algorithm

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Due to the poor approximation of Eq. (4) developed by Morel and Prieur (1977), the absorption by chla cannot be very well isolated by either the three- or four-band algorithms. A more accurate remote sensing reflectance model developed by Gordon et al. (1988) is used to construct the improved three-band algorithm. Moreover, compared to the three-band algorithm, the improved three-band algorithm removes the backscattering of suspended solids over the NIR region more effectively, and when compared to the four-band algorithm, the improved three-band algorithm is more capable of reducing the equivalent degrees of freedom. The calibration and validation results indicate that all three of the algorithms produce good performance in terms of estimating chla concentration in the Yellow River estuary, but the improved three-band algorithm provides a performance which is superior to both the three- and four-band algorithm, not only in terms of chla prediction ability, but also in terms of noise tolerance ability.

The improved three-band algorithm presented in this study may be applied to NASA HYPERION and the Compact High Resolution Imaging Spectrometer (CHRIS). As an example, the HYPERION channels at 681 nm and 915 nm is closed to the position of λ 2 = 685 and λ 3 = 917 nm, respectively, and the 32nd channel of HYPERION is located at the position of λ 1 = 671 nm. A comparison of the measured and predicted estimates of the chla by using the improved three-band algorithm with HYPERION spectral bands is presented in Fig. 8, and the results indicate that the improved three-band algorithm with HYPERION spectral bands produces good performance in estimating the chla concentration in the Yellow River estuary, the regression coefficient of which is 0.8997, and the corresponding RMS is 27.23 %. These findings imply that, provided that an atmospheric correction scheme for visible and near-infrared bands is available, the improved three-band algorithm may be used for the quantitative monitoring of chlorophyll-a concentration from the HYPERION sensor in the Yellow River estuary.

Fig. 8
figure 8

Chla concentrations measured in Yellow River estuary reservoirs in October, 2009 plotted versus chla concentration predicted by the improved three-band algorithm using HYPERION spectral bands

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One limitation of this study is that the calibration and validation datasets contained only a narrow range of optical properties of natural turbid waters, being collected only from the Yellow River estuary. The results are insufficient to completely validate the accuracy of the algorithms in other waters with different bio-optical properties. Thus, it is concluded that the improved algorithm should be used for estimating chla in highly turbid coastal waters, although it may be necessary to accordingly reposition the wavelengths of the three bands for the given aquatic bio-optical conditions. The researchers also suggest calibration and validation of the algorithms based on more in situ measurements of waters with different optical properties.

Summary

In this study, an improved three-band algorithm was developed and constructed. The results suggest that the optimal wavelengths of the improved three-band semi-analytical algorithm must meet the requirements of the λ 1 and λ 2 being restricted to within the range 660–690 nm, and the λ 3 must be longer than 750 nm. The respective performances of the three-band, four-band, and improved three-band algorithms were validated and assessed using bio-optical datasets collected from the turbid waters of the Yellow River estuary, China. By comparison, using the improved three-band algorithm in estimating chla concentration from the Yellow River estuary, uncertainty decreased by 24.93 % compared to the four-band algorithm, and 38.62 % compared to the three-band algorithm. The noise tolerance ability is of importance to algorithms for estimating chla concentration in turbid waters. The researchers advise that an optimal algorithm must not only have a good chla concentration prediction ability, but it must also possess an excellent noise tolerance ability in order to decrease the impacts of residual data uncertainty of satellite imagery when performing the chla concentration estimation. Due to its good performance in noise tolerance ability and chla concentration estimation accuracy, the improved three-band algorithm is superior to both the three- and four-band algorithms, and may be used for retrieving chlorophyll-a concentration from extremely turbid waters with similar bio-optical properties, although it may be necessary to reposition the optimal band positions of the algorithm using local bio-optical information.