Combining Artificial Neural Networks with Causal Inference for Total Phosphorus Concentration Estimation and Sensitive Spectral Bands Exploration Using MODIS

Abstract

The total phosphorus (TP) concentration is a key water quality parameter for water monitoring and a major indicator of the state of eutrophication in inland lakes. Using remote-sensing to estimate TP concentration is useful, as it provides a synoptic view of the entire water region; however, the weak optical characteristics of TP lead to difficulty in accurately estimating TP concentration. The differences in water characteristics and components between lakes mean that most TP estimation methods are not applicable to all lakes. An artificial neural network (ANN) model was created to represent the correlation between TP concentration and the spectral bands of Moderate Resolution Imaging Spectroradiometer (MODIS) images in different research areas. We investigated the causal inference under the potential outcome framework to analyze the sensitivity of each band with regard to the TP concentration of different lakes for the research of water characteristics. Our results show that the accuracy of the ANN-based TP concentration estimation, with R² > 0.73, root mean squared error (RMSE) < 0.037 mg/L in Lake Okeechobee and R² > 0.73, RMSE < 4.1 μg/L in Lake Erie, respectively, is much higher than traditional empirical methods, e.g., linear regression. We found that the sensitive bands of TP concentration in Lake Erie are blue bands, whereas the sensitive bands in Lake Okeechobee are green bands. Various TP concentration maps were drawn to indicate the distribution of TP concentration and its tendency to change. The maps show that the distribution of TP concentration closely corresponds to the shore land-use, and a high TP concentration corresponds to the latest algal blooms breakout. Our proposed approach shows good potential for the remote-sensing estimation of TP concentration for inland lakes. Identifying the sensitive bands not only help characterize the lakes, but will also help the researchers to further observe the TP concentration of specific lakes in an efficient way.

1. Introduction

Water is a vital resource for humanity and is associated with all aspects of our lives [1,2,3]. Inland lakes are vulnerable to pollution from industry, agriculture, transportation, and other activities. Monitoring and managing the water quality of lakes is important for environmental protection and the sustainable development of ecosystems. Total phosphorus (TP) concentration is a key water quality parameter for the monitoring and assessment of water supplies. Phosphorus is a major indicator of trophic states and an essential element for plants to grow [4,5,6,7]. TP is closely associated with optically active substances, such as chlorophyll-a (Chl-a), colored dissolved organic matter (CDOM), and total suspended matter (TSM) [7,8]. TP concentration is also the main index of water quality [9]. Domestic, industrial, and agricultural wastewater contribute to the increase of TP concentration in inland lakes [10,11,12,13]. Accurate long-term monitoring of TP concentration is essential for the protection of water resources and prevention of lake eutrophication.

Remote sensing is widely used in the assessment of water quality [14,15,16,17] due to its ability to provide a synoptic view of the entire water region [18,19,20]. Visible light and infrared bands of remote-sensing images are useful for the estimation of water quality parameters, such as TP [4,5]; however, no unified models have been created for the estimation of TP concentration in different lakes [21,22]. Differences in water characteristics and components between lakes lead to the low transferability of TP concentration estimation methods. Some authors fail to provide an explanation of their selection of spectral bands for the estimation of water quality parameters [23]. Finding the sensitive spectral bands for the estimation of TP concentration is an important step forward for the exploration of lake characteristics, and is beneficial for establishing an interpretable and accurate connection between remote-sensing imagery and TP concentration. In this paper, we present our model with an aim to establish a connection between the remote-sensing imagery and in situ measurement data for the accurate estimation of TP concentration, and explore sensitive spectral bands concerning TP concentration for the study of lake characteristics.

Research on the estimation of TP concentration based on remote sensing has been presented in recent years. As a non-optical active substance, the concentration of TP is often hard to accurately estimate from remote-sensing imagery due to weak optical characteristics and low signal-to-noise ratio [5,9,24,25]. TP concentration estimation methods based on remote sensing can be divided into semi-analytical approaches and empirical approaches. Semi-analytical approaches [26,27,28] construct a relationship between the reflectance of surface water and the inherent optical characteristics of lakes to inverse TP concentration. The semi-analytical approaches can decouple the contribution of water substances, which lends to the approaches’ explainability and provides a physical underpinning [26]. Because the inherent optical characteristics of lakes are often different, the semi-analytical methods lack generality [8,29]. Empirical approaches [7,30], such as indirect and direct methods [8], can be easily operated. These approaches are based on the connection between remote-sensing reflectance and optically active substances. As indirect methods, the concentration of optically active substances (e.g., Chl-a) is estimated first because an optically active substance often has a higher signal-to-noise ratio in remote-sensing imagery, then the TP concentration is computed in terms of the relationship between the TP concentration and the optically active substance [31]. A time lag may exist between the change of the TP concentration and that of the optically active substance [5]. The unstable and inaccurate correlation function results in a low accuracy of the indirect methods [7]. Direct methods use spectral bands, band transformations, or band combinations to construct statistical models to directly estimate the TP concentration. Empirical approaches have high estimation accuracy at specific lakes; however, they are site-specific [32]. The interpretability of these statistical methods needs additional validation [22,23]. New methods should be introduced to estimate TP concentration to improve accuracy, adaptability, and interpretability.

Neural networks [26,33,34,35] have been proven to have higher accuracies in TP concentration estimation. Their results show that the correlation between remote-sensing imagery and TP concentration can be modeled by complex neural networks. Artificial neural networks (ANNs) that are built with deep hierarchical architectures and activation units can learn hierarchical and discriminate features. The hierarchical structure and powerful learning abilities improve the applicability and performance of ANN methods. Applying an ANN for the assessment of water quality and the estimation of TP concentration has been implemented in recent years. The estimation results of ANNs have been demonstrated to be better than that of band combination methods [36,37]. Compared with statistical regression, ANNs have no need to calculate the correlation coefficient between TP concentration and remote-sensing reflectance, and the training process makes an ANN easy to apply in different study areas. However, providing an explanation for the connection between remote-sensing imagery and TP concentration found by the ANN model is difficult. Exploring the sensitive band with regard to TP concentration is useful for the study of water characteristics, and can also improve the interpretability of ANN models.

Causal inference was used to explore sensitive spectral bands for the assessment of TP concentration in this paper. Causal inference [38,39,40] refers to the process of seeking a causal relationship between a cause and its effect. It is a useful tool for explanatory analysis and has been introduced into machine learning to confirm the correlation between variables and outcomes [41,42]. Under the potential outcome framework [40], the changing variable is referred to as ”treatment” and the corresponding response as ”outcome”. To explore the spectral bands that are sensitive to the TP concentration, the data of a Moderate Resolution Imaging Spectroradiometer (MODIS) were used, as MODIS captures data in a high spectral density, with 36 spectral bands ranging in a wavelength from 0.4 $\mathsf{\mu }$m to 14.4 $\mathsf{\mu }$m and at varying spatial resolutions—2 bands at 250 m, 5 bands at 500 m, and 29 bands at 1 km. Each of the MODIS band-related data were considered as an individual treatment. The individual treatment effect (ITE), in terms of the errors in the estimation of the TP concentration through an ANN, was used to find the sensitive bands.

We used an ANN to establish the correlation between the in situ TP concentration of inland lakes and the remote-sensing reflectance of MODIS due to its powerful learning ability. Although water quality parameters vary in a few hours under the conditions of wind and rain [21,26,34], MODIS has high temporal resolution and can follow the daily change of TP concentration in inland lakes [43]. MODIS has a broad spectrum, spanning visible, infrared spectra, and thermal infrared. Not only the data of visible and infrared bands, but also their exponential, logarithmic, and power transformations were inputted to our ANN model with the aim of increasing the estimation accuracy. Obtaining long-term in situ measurement data enabled us to build the ANN model and test the estimation accuracy.

This paper has three main contributions. (1) A hierarchical ANN model was constructed to model the correlation between TP concentration and remote-sensing reflectance. The results demonstrate that our approach is appropriate for the remote-sensing estimation of the TP concentration of different inland lakes. (2) Causal inference under the potential outcome framework was introduced to analyze the sensitivity of each band to the TP concentration of different lakes. Causal inference analysis improves the interpretability of the ANN model and provides explanations of estimation results. (3) Spatial–temporal TP concentration maps were drawn to investigate the distribution and change tendency of the TP concentration in the study areas. Our work provides an efficient and effective method to monitor the TP concentration of inland lakes.

The paper is organized as follows: Section 1 is the introduction. Materials are introduced in the Section 2. Methods are presented in Section 3. Results and discussions are shown in Section 4. Conclusions are given in Section 5.

2. Materials

To accurately estimate the TP concentration in the long-term and explore sensitive spectral bands with regard to TP concentration, more than 20 years of observation images from MODIS were utilized. Considering the low spatial resolution of MODIS [44], lakes with large areas were selected as research cases. We selected two inland lakes, Lake Okeechobee and Lake Erie. The in situ TP measurement data of the two lakes since 2000 were collected as references for the long-term remote-sensing estimation of TP concentration.

2.1. Study Areas

Lake Okeechobee and Lake Erie were chosen as study areas and are shown in Figure 1. Lake Okeechobee (26.66${}^{\circ }$ N–27.23${}^{\circ }$ N, 80.59${}^{\circ }$ W–81.15${}^{\circ }$ W) is the largest freshwater lake in Florida, USA. Lake Okeechobee is a shallow lake. The surface area of the water is about 1900 km${}^2$, and the average water depth is about 2.7 m. The Kissimmee River in the northern part is its main inflow. The lake contains high concentrations of phosphorus. A large area of algal bloom broke out in 2016 [45]. The monitoring of the TP concentration of Lake Okeechobee is important for the control of eutrophication.

Lake Erie (41.34${}^{\circ }$ N–42.96${}^{\circ }$ N, 78.79${}^{\circ }$ W–83.56${}^{\circ }$ W) is located in North America. It is one of the Great Lakes, with a water surface area of 25,700 km${}^2$, an average depth of 19 m, and a maximum depth of 64 m. The water depth of western Lake Erie is shallower than the depth of the eastern side. The water turbidity of Lake Erie is the highest among the Great Lakes [46]. Cyanobacteria blooms and eutrophication in Lake Erie have become a concern among a wide variety of people in recent years. The water safety of Lake Erie is closely related to agriculture, tourism, shipping traffic, and the ecological environment around the lake. Monitoring water quality and the TP concentration of Lake Erie is beneficial for pollution prevention and water management.

2.2. In-Situ Data

The in situ TP data of Lake Okeechobee were collected from an environmental database named DBHYDRO of the South Florida Water Management District (https://apps.sfwmd.gov/WAB/EnvironmentalMonitoring/index.html). The South Florida Water Management District is the largest water management government agency in Florida. The main responsibility involves improving water quality, preventing floods, and protecting water resources. Historical and up-to-date data of hydrological, meteorological, and water quality are stored in DBHYDRO. The spatial locations of 21 monitoring stations are shown in Figure 1. The year-round in situ TP concentration data from 2000 to 2019 were used for the estimation of TP.

In-situ data of Lake Erie were collected from the Environment and Climate Change Canada Data (http://data.ec.gc.ca/data/substances/monitor). The database offers a large number of data, including air, climate, water, and soil for research. The in situ TP data from 2000 to 2018 were collected from this official website. The spatial locations of monitoring stations are also shown in Figure 1.

2.3. Satellite Data

MODIS was launched by NASA on board the Terra satellite in 1999 and on board the Aqua satellite in 2002. In its 36 spectral bands, 29 bands have 1 km spatial resolutions. MODIS images the entire Earth every 1 or 2 days. Due to its high temporal resolution and spectral density, MODIS is widely used in water quality monitoring. The MODIS Level-1B Calibrated Radiances data products (MOD021KM) of the MODIS/Terra sensor from 2000 to 2019 were collected for the estimation of TP concentration. The images are available from the website of the Level 1 Atmosphere Archive and Distribution System (LAADS, https://ladsweb.modaps.eosdis.nasa.gov/search/).

3. Methods

The downloaded MODIS images were influenced by illumination and cloud. To acquire an accurate reflectance of surface water, the multi-spectral images must first be preprocessed. The correlation between remote-sensing reflectance and TP concentration was modeled by an ANN. Both the MODIS data and the nonlinear transformations of reflectance were input into the ANN model to achieve better results. Causal inference was introduced for the exploration of the sensitivity and importance of each band to TP concentration.

3.1. MODIS Imagery Preprocessing

To shorten the temporal gap between imaging and in situ sampling time, according to the date of in situ TP monitoring, the remote-sensing images were downloaded on the same date. The images influenced by the cloud must be removed first. Bands 1 to 19 of the MOD021KM product belong to the visible and near-infrared bands. These bands were chosen for TP estimation and sensitivity exploration. All selected bands were scaled to 1 km spatial resolution. Reflectance of water leaving was acquired by:

$$\rho =\pi \times R_{\lambda }/(ESU{N}_{\lambda }\times cos\theta ),$$

(1)

where $\rho$ is the reflectance of surface water, $\lambda$ is the wavelength of each band, $R_{\lambda }$ is the radiance at top of the atmosphere, ESUN is the extraterrestrial solar irradiance, and $\theta$ is the solar zenith angle. The pixels were masked if they were not within 0–1 or the solar zenith was larger than 75${}^{\circ }$ [26,43]. A three-by-three mean average filter was applied to get the mean reflectance of each monitoring station and remove high-frequency noise [34,47]. The in situ TP concentration was matched with the reflectance of leaving water in MODIS images based on the minimum distance between the spatial location of each monitoring station and the coordinates of remote sensing pixels. Finally, 338 match-ups in Lake Okeechobee from 2000 to 2019 and 265 match-ups in Lake Erie from 2000 to 2018 were acquired for TP estimation.

3.2. TP Concentration Estimation Based on an ANN

ANNs are advanced methods in the monitoring of water quality and the estimation of TP concentration. With a deep structure and activation layers, ANNs have a powerful learning ability and are capable of modeling complex relationships between water quality parameters and large amounts of remote-sensing reflectance [23]. Given the inputs and outputs, an ANN can automatically learn hierarchical and nonlinear features. In the processing of training, redundant and unimportant features are given little weight. Abundant characteristics make ANN approaches adaptive to different research areas and outperform traditional empirical methods. Because nonlinear components were shown to be useful for the estimation of water quality parameters [7,37], the nonlinear components, such as exponential, logarithmic, and power transformations of each visible and infrared band of MODIS were also chosen as input data for the ANN. For each reflectance $\rho$, $ln\left(\rho \right),exp\left(\rho \right),{\rho }^2,{\rho }^3,{\rho }^{-1},{\rho }^{-2},{\rho }^{-3}$ were calculated and scaled to a 0–1 range in each dimension.

We applied the ANN for estimation of the TP concentration. The structure of the ANN is shown in Figure 2. The first layer is the input layer. The reflectance of Bands 1 to 19 and the nonlinear transformations were input to the ANN model. Layers 2 to 6 are fully connected layers. The number of kernels of the five layers were 40, 40, 20, 20, and 10, respectively. Exponential linear units (ELU) [48] were applied in these five fully connected layers as the activation function. The dropout technique was applied from Layers 3 to 5 to prevent the ANN from overfitting [49]. The last layer outputted the predicted TP concentration. The loss of ANN was determined by mean squared error (MSE). The ANN was trained with a stochastic gradient descent. The early stop strategy was applied to the training process to avoid overfitting.

The output results are represented as $T{P}_p^k=f\left(B_{I\_J}^k\right),k=1,2,\dots ,N$, where $T{P}_p^k$ means the predicted TP concentration of the k-th sample, N is the number of samples, f means the function of ANN, B represents the reflectance $\rho$ of all bands and their nonlinear transformations, I represents all bands, and J represents all transformations of each band.

The estimation performance of ANN was evaluated by the determination coefficient (R${}^2$) [50] and root mean squared error (RMSE):

$$R^2=1-\frac{{\sum }_{k=1}^N{(T{P}_m^k-T{P}_p^k)}^2}{{\sum }_{k=1}^N{(T{P}_m^k-{\overline{TP}}_m)}^2}$$

(2)

$$RMSE=\sqrt{\frac{{\sum }_{k=1}^N{(T{P}_m^k-T{P}_p^k)}^2}{N}},$$

(3)

where $T{P}_m^k$ is the measured TP concentration of the k-th sample and ${\overline{TP}}_m$ is the mean value of in situ data.

3.3. Causal Inference

Causal inference, which is used to find the causal relationship between a cause and its effect, was introduced to explore the sensitive bands of remote-sensing imagery with regard to TP concentration. Finding the sensitive bands can not only improve the interpretability of our proposed ANN model, but is also beneficial for studying the lakes’ characteristics. It assists researchers to further improve the estimation accuracy and observe the TP concentration.

Causality exists in machine learning [38]. The authors of [42] learned causality between text features and vocabulary in recurrent neural networks. The authors of [51] used causal inference in Bayesian Additive Trees. To find the importance of each MODIS band to the TP concentration estimation, one of the features in the input layer of ANN was set as the treatment of causal inference under the potential outcome framework. The prediction error of each sample was set as the outcome. Without changing any parameter of ANN, the effect of each feature can be calculated when the treatment is set to 0.

For each sample of TP concentration estimation, the prediction error is $A{E}_k=|T{P}_m^k-T{P}_p^k|$. For all samples, the mean absolute error (MAE) is:

$$MAE=\frac{1}{N}\sum _{k=1}^N\left(\right|T{P}_m^k-T{P}_p^k\left|\right).$$

(4)

If one feature in the nonlinear transformation layer of the ANN is set to zero, the predicted TP result is $T{P}_{p,\backslash i\_j}^k=f\left(B_{I\_J\backslash i\_j}^k\right)$, and the prediction error is:

$$A{E}_{k,\backslash i\_j}=|T{P}_{p,\backslash i\_j}^k-T{P}_m^k|,$$

(5)

where $\backslash i\_j$ means the j-th transformation of the band i is set to zero.

The individual treatment effect (ITE) is calculated as:

$${\tau }_{k,\backslash i\_j}=A{E}_k-A{E}_{k,\backslash i\_j}=|T{P}_m^k-T{P}_p^k|-|T{P}_{p,\backslash i\_j}^k-T{P}_m^k|.$$

(6)

For all samples of the ANN model, the mean treatment effect is calculated as:

$${\tau }_{\backslash i\_j}=\mathbb{E}[A{E}_k-A{E}_{k,\backslash i\_j}]=\mathbb{E}\left[A{E}_k\right]-\mathbb{E}\left[A{E}_{k,\backslash i\_j}\right]=MAE-MA{E}_{\backslash i\_j}.$$

(7)

Thus, the rate of change is:

$${\eta }_{\backslash i\_j}=\frac{{\tau }_{\backslash i\_j}}{MAE}\times 100\%=1-\frac{MA{E}_{\backslash i\_j}}{MAE}\times 100\%.$$

(8)

${\eta }_{\backslash i\_j}>0$ means introducing the j-th transformation of the band i to ANN will cause a bad effect on TP estimation, and vice versa. A smaller ${\eta }_{\backslash i\_j}$ means the transformation $B_{i\_j}$ is more important for the estimation of the TP concentration. The results of the TP estimation and causal inference are shown in the next section.

4. Results and Discussions

To validate the effectiveness of our methods, TP concentration estimation in two study areas were tested. The traditional methods based on band combinations were also tested for comparison. The experiments of causal inference were conducted to explore sensitive bands to the TP concentration. Spatial–temporal TP concentration maps were drawn for the analysis of the distribution and trend of TP concentration in the studied lakes.

4.1. Results of TP Estimation

The designed ANN model was tested in both Lake Okeechobee and Lake Erie. The ratio of the training set to the test set in each lake was set to 80%:20%. Stochastic gradient descent was used to train the ANN model, the learning rate was set to 0.1, and the decay was set to $1.0\times {10}^{-7}$. For each lake, the ANN model was run five times and the average result was calculated to reduce the effect caused by random initialization. The ANN model achieved good TP estimation results, which are presented in

Table 1. The estimation results of the linear regression and ANN-based methods in Lake Okeechobee.

Methods	Training Set		Test Set
	${\mathit{R}}^2$	RMSE (mg/L)	${\mathit{R}}^2$	RMSE (mg/L)
Linear regression of band combinations	0.74	0.031	0.59	0.038
ANN without nonlinear components	0.78	0.032	0.70	0.038
ANN with nonlinear components	0.86	0.026	0.73	0.037

and

Table 2. Estimation results of linear regression and ANN-based methods in Lake Erie.

Methods	Training Set		Test Set
	${\mathit{R}}^2$	RMSE (μg/L)	${\mathit{R}}^2$	RMSE (μg/L)
Linear regression of band combinations	0.65	4.6	0.47	5.7
ANN without nonlinear components	0.66	4.5	0.57	5.1
ANN with nonlinear components	0.84	3.1	0.73	4.1

.

In Lake Okeechobee, the $R^2$ of the training set was over 0.86 and the RMSE was 0.026 mg/L. In the test set, $R^2$ was over 0.73 and RMSE was 0.037 mg/L. In Lake Erie, our results were $R^2=0.84$, $RMSE=3.1\mathsf{\mu }\mathrm{g}/\mathrm{L}$ in the training set, and $R^2=0.73$, $RMSE=4.1\mathsf{\mu }\mathrm{g}/\mathrm{L}$ in the test set. The distributions of the predicted TP concentrations in the test set using the acquired ANN model are shown in Figure 3. The results demonstrate that our proposed approaches can effectively estimate TP concentration and are suitable for different lakes.

To evaluate ANN’s performance, we compared its results with that of traditional empirical methods. Combining with all visible and infrared band reflectance, linear regression was applied for the estimation of TP concentration. To demonstrate the effect of nonlinear components, ANN-based experiments without the nonlinear transformation of band reflectance were also conducted for comparison. Estimation results are shown in Table 1 and Table 2.

We found that the predicted TP concentration was not in good agreement with the in situ data when using the linear regression of band combinations to model the relationship between TP concentration and remote-sensing reflectance. The determination coefficients in the test set of Lake Okeechobee and Lake Erie were both not satisfied. When comparing the results of the statistical regression to that of the ANN, the ANN model outperforms the traditional empirical methods. With the nonlinear components of band reflectance, the ANN-based experiment performs better.

4.2. Results of Causal Inference

To explore the sensitivity of each MODIS band to the TP concentration, according to causal inference, each feature of the input layer in our proposed ANN model was set to zero, whereas other features and parameters of the ANN model were kept unchanged. The change rate of prediction error was calculated. Each setting was tested five times. The results of the two research areas are shown in

Table 3. Band sensitivity to TP concentration in Lake Okeechobee. Smaller values mean higher sensitivity.

Wavelength (nm)	Band	Change Rate after Setting the Component to 0								Band Average
		$\mathit{\rho }$	$\mathbf{1}/\mathit{\rho }$	$\mathit{l}\mathit{n}\left(\mathit{\rho }\right)$	$\mathit{e}\mathit{x}\mathit{p}\left(\mathit{\rho }\right)$	${\mathit{\rho }}^{\mathbf{2}}$	${\mathit{\rho }}^{-\mathbf{2}}$	${\mathit{\rho }}^{\mathbf{3}}$	${\mathit{\rho }}^{-\mathbf{3}}$
405–420	8	−1.3	−1	−1.8	−1.7	−4.8	−2.5	−12.5	−5	−3.8
438–448	9	−1	−8.3	−2.9	−1	−2	−15	−3.4	−21.7	−6.9
459–479	3	−12.9	−1.9	−12.4	−12.8	−12.1	−0.6	−10.4	−2	−8.1
483–493	10	−3.7	−12.1	−10.1	−3.4	−2.6	−19.5	−3.8	−25.7	−10.1
526–536	11	−4.6	−34.4	−13.3	−4.1	−1.5	−48.4	−0.4	−59.7	−20.8
546–556	12	−1.6	−15.9	−5	−1.4	0	−5.2	0.1	−9.4	−4.8
545–565	4	−117.7	−114.1	−146.5	−115	−84.4	−71.1	−54	−38.5	−92.7
620–670	1	−13.3	−5.3	−8.6	−13.2	−13	−9.5	−10.3	−11.2	−10.6
662–672	13h	−2.7	−7.7	−5.8	−1.9	1.3	−11.1	−0.3	−12.4	−5.1
662–672	13l	−1.7	−50.4	−14.2	−1.4	0.1	−18.1	0.3	−6.6	−11.5
673–683	14h	−15.6	−1.1	−22.2	−13.2	−0.3	−3.7	−15	−6.8	−9.7
673–683	14l	−0.1	−18.9	−7.2	0.2	0.9	−3.8	0.2	−2.5	−3.9
743–753	15	0.4	−7.2	−1	0.5	0.8	−4.2	0	−4	−1.8
841–876	2	−26.7	−31.7	−47.2	−26.7	−25.1	−6.3	−14.5	−3.1	−22.7
862–877	16	−6.9	−93.4	−25.1	−7.8	−12.5	−5.2	−6.4	−1.8	−19.9
890–920	17	0.9	−18	−14.8	1.2	3	−3.2	−1.6	−4.7	−4.7
931–941	18	0.7	−7.2	−6.8	1	−10.4	−1.5	−35	−0.9	−7.5
915–965	19	0.9	−9.6	−8	0.9	−0.4	−1.8	−1.9	−0.3	−2.5
1230–1250	5	−64.2	−7.2	−179.3	−53	3.8	−0.9	1.5	−0.3	−37.5
1328–1652	6	−50.2	−6.3	−60.8	−46.5	−10.9	−0.8	0.6	−2.4	−22.2
2105–2155	7	1.6	−2.7	−28.2	1.5	−1.8	−2.3	−0.8	−0.1	−4.1

and

Table 4. Band sensitivity to TP concentration in Lake Erie.

Wavelength (nm)	Band	Change Rate after Setting the Component to 0								Band Average
		$\mathit{\rho }$	$\mathbf{1}/\mathit{\rho }$	$\mathit{l}\mathit{n}\left(\mathit{\rho }\right)$	$\mathit{e}\mathit{x}\mathit{p}\left(\mathit{\rho }\right)$	${\mathit{\rho }}^{\mathbf{2}}$	${\mathit{\rho }}^{-\mathbf{2}}$	${\mathit{\rho }}^{\mathbf{3}}$	${\mathit{\rho }}^{-\mathbf{3}}$
405–420	8	−22.8	−37.2	−26.5	−7.1	−8.8	−76.3	−11.4	−99.1	−36.2
438–448	9	−4	−14.7	−12.2	−38.2	−8.8	−17.2	−8.6	−14.8	−14.8
459–479	3	−33.1	−28.4	−61.8	−45.6	−44.2	−28.4	−24.1	−9.3	−34.4
483–493	10	−14.2	−13.7	−45.2	−52.5	−23.3	−25	−8.3	−21.3	−25.4
526–536	11	−25.8	−19.8	−18.3	−21.6	−4.7	−20.2	−8.3	−8.3	−15.9
546–556	12	−9.2	−21.4	−52.1	−9.1	−7.5	−18	−4.1	−7.5	−16.1
545–565	4	−14.7	−25.4	−22.6	−8.2	0.6	−11.4	−3.7	−17.7	−12.9
620–670	1	−21	−10.6	−17.7	−6.4	−6.3	−12.2	−5.1	−13.2	−11.6
662–672	13h	−8.1	−31.8	−13.4	−12.2	−4.6	−8.7	−0.9	−5.3	−10.6
662–672	13l	−1.9	−5.8	−13.7	−1.7	−4.6	−8.9	0.6	−5.5	−5.2
673–683	14h	−7	−7.9	−21	−6.8	−3.6	−3.3	−2.1	−5.5	−7.2
673–683	14l	−12.4	−13.4	−7.5	−0.9	−0.7	−7	−2.9	−5.2	−6.3
743–753	15	−6.3	−22.3	−8.6	−7.5	−0.7	−0.9	−2.7	−6.9	−7.0
841–876	2	−5.8	−17.2	−12.9	−1.7	0.9	−15.7	0.7	−5.8	−7.2
862–877	16	−1.7	−36.4	−9.8	−1.3	−2.8	−6.5	0.6	−2.8	−7.6
890–920	17	−0.1	−13.4	−9.6	−8.2	−0.4	−6.4	2.2	−4.1	−5.0
931–941	18	1.2	−23.1	−3.5	0.8	−2.1	−15.2	1.4	−2.3	−5.4
915–965	19	−0.6	−25.2	0.6	1.9	0.2	−15.6	−2	−10.9	−6.5
1230–1250	5	−0.9	−5.5	−9.7	−5.7	−2.4	−2.9	0.9	−1.4	−3.5
1328–1652	6	−5	−3.2	−14.8	−7.1	−1.2	0	−0.4	−0.4	−4.0
2105–2155	7	−6.8	−5.5	−7	−6.5	−3.2	−0.9	−1.6	−0.5	−4.0

. In these two tables, $\rho$ is the band reflectance of surface water, $ln\left(\rho \right)$ and $exp\left(\rho \right)$ mean the logarithmic and exponential transformation of band reflectance, respectively.

Table 3 and Table 4 show that the sensitivity bands are different in the two lakes. The most important band to TP concentration estimation of Lake Okeechobee is Band 4 (wavelength: 545–565 nm), which belongs to the green bands. Some infrared bands also have a high sensitivity with TP concentration, such as Band 2/Band 16 (wavelength: 841–877 nm) and Band 5 (wavelength: 1230–1250 nm). In Lake Erie, the top three bands are Band 8 (wavelength: 405–420 nm), Band 3 (wavelength: 459–479 nm), and Band 10 (wavelength: 483–493 nm), which belong to the blue bands. Red and infrared bands do not have much effect on the estimation of TP concentration in Lake Erie.

The results of the causal inference experiment demonstrate that the blue, green, and infrared bands are important for the estimation of the TP concentration, and the sensitive bands are closely connected with optically active substance. The sensitive spectral bands of Lake Okeechobee and Lake Erie are different. The sensitive spectral bands may depend on transparency, turbidity, water depth, and latitude of the lakes. Lake Erie is a deep and transparent lake. In clean water, the energy of near-infrared light can be strongly absorbed [52], and the penetration ability of blue light is more powerful than the infrared bands. Thus, the sensitive bands in Lake Erie are the lights with short wavelengths. In Lake Okeechobee, the green and infrared bands have a high sensitivity with TP concentration. TP is closely associated with optically active substances, such as Chl-a and turbidity [7,8]. The reflection of green light is related to Chl-a, and reflection of the near-infrared light is related to suspended matters [53], whereas the energy of the blue bands was mainly absorbed by phytoplankton [54,55]; therefore, these bands can be utilized to estimate TP concentration. Many researchers, such as [30,56], have used the blue, green, near-infrared, or mid-infrared bands to monitor TP concentration. Their results are consistent with our results.

The sensitive bands we have identified not only increase the interpretability and transferability of our approaches, but also help researchers to observe the TP concentration of specific lakes through several bands and further improve the study of water characteristics. Remote-sensing estimation of the TP concentration of inland lakes will thus become more accurate and cost-effective.

4.3. Spatial Distributions of TP

Remote sensing provides a synoptic view of the whole water region. Using an ANN model, the TP concentration of the whole lake can be estimated. Figure 4 and Figure 5 show the estimation results of the TP concentration of both Lake Okeechobee and Lake Erie since 2000. The spatial distribution and change tendency of TP concentration can also be observed.

Figure 4 and Figure 5 show that the TP concentration of Lake Okeechobee is much higher than that of Lake Erie. The central and eastern part of Lake Okeechobee has a high concentration of phosphorus, whereas in Lake Erie, the highest concentration of TP is located in the west. The human population and urbanization are well-correlated with the distribution of TP concentration. Domestic and fertilizer-rich agricultural wastewater may contribute to the high concentration of phosphorus along the lakes.

Distributions of TP concentration are closely connected with shore land-use around the lakes. Spatial–temporal TP maps show that the TP concentration is higher to the west of Lake Erie. Detroit, the biggest city in State of Michigan, is located to the west of Lake Erie. The high density of human population and the development of the economy lead to higher TP concentration. The TP concentration maps are consistent with the results in [46].

Agricultural areas and human population are mainly located to the east and south of Lake Okeechobee. Agricultural and domestic wastewater may flow into the lake. Wetland in the west contributes to water purification. The TP concentration maps show that the TP concentration is higher in the east of Lake Okeechobee, and lower in the west. The TP concentration maps correspond to shore land-use.

Large algal blooms are closely related to high TP concentration. In Lake Okeechobee, sediments contain plentiful phosphorus. Hurricanes and heavy rainfall often cause an increase of TP concentration in this shallow lake. Algal blooms happened in May 2016 [45]. The TP maps in Figure 6 showed the TP concentration increased rapidly in summer and decreased in winter, which is consistent with the algal blooms. The TP concentration maps demonstrate that estimating TP concentration by remote sensing is valuable and beneficial for lake protection.

4.4. Future Work

Some results of the ANN-based TP estimation are not in good agreement with the in situ TP concentration. The reason may be that the spatial resolution of the MODIS images is too low to match the spatial heterogeneity of the water characteristics [36]. The scales and water-surface areas of the studied lakes are large. Researchers [4,7,29] have found that using multiple models for different parts of a large lake could yield more accurate results. We will improve the estimation results of TP concentration by utilizing multiple models and high-resolution remote-sensing imagery in the future.

5. Conclusions

TP concentration is the main index of water quality assessment. It is closely associated with water safety. We constructed an artificial neural network to estimate the long-term TP concentration based on MODIS data. We applied causal inference under the potential outcome framework to explore sensitive spectral bands to the TP concentration. We tested our ANN model in both Lake Okeechobee and Lake Erie. Compared to traditional empirical methods based on band combinations, our model achieved better results with determination coefficients of more than 0.73 in the research areas. The results show that our modeling approaches are more accurate than traditional methods, and can be applied to lakes with different inherent optical properties. Through causal inference, we found the green bands are sensitive to the TP concentration in Lake Okeechobee and blue bands are sensitive to the TP concentration in Lake Erie. We have thus provided an efficient method for the estimation of TP concentration in inland lakes. Our method is applicable to the observation of the TP concentration of inland lakes through remote sensing. The accurate monitoring of the TP concentration in inland lakes and identification of sensitive bands are important for both the study of lake characteristics and water resource management.