An Empirical Algorithm using Derivative Difference for Estimating Chlorophyll-A in Case-II Water

Water quality management includes measurement of quantity and quality. The abundance of phytoplankton in water body represents the physical condition and chemical constituents. As Chlorophyll-a exists in all types of phytoplankton, naturally the choice for water quality measurement is estimation of Chl-a concentration. The Chl-a concentration is estimated using various spectral reflectance algorithms such as single band regression, band ratio, three-band ratio, four-band ratio etc have been developed and is being used. Subsequently, the first order derivative ratio and second order derivative ratio methods are also used in some studies. Though such algorithms provide water quality measures, new algorithms are being introduced to improve estimation accuracy. In this paper, a new algorithm ‘Derivative Difference’ is proposed. It is based on the Chlorophyll concentration variation with shape of reflectance spectrum. The derivative data represents the slope of the reflectance data at different wavelength. The difference in derivative values at selected two different wavelengths were correlated with measured values to estimate Chl-a concentration. Obtained results were compared with the values obtained from band ratio method and derivative ratio methods. The algorithm is found to be better in some conditions.


Introduction
The deterioration of water quality of inland water bodies is a serious ecological and social problem as they are the water resources for drinking, domestic, agricultural and industrial purposes. Management of quality of water bodies is essential to provide water to the ever increasing population and changing lifestyle. Water management activities usually include protecting water bodies from polluting elements, saving water and distributing it without wastage, i.e. managing the quality and quantity of water. While quantity management is maintaining capacity of water body by preventing encroachment and sedimentation, quality management is controlling the physical, chemical and bio-objects within the specified limit to enable usability of the water. al., 2009). In case of ocean water (Case-I), the reflectance is dominated by the chlorophyll and the variation of reflectance can be attributed to the Phytoplankton abundance variation only. In contrast, the reflectance of in-land water (Case-II) is considerably affected by the optical properties of inorganic suspended matter and Colored Dissolved Organic Matter (CDOM). Hence the algorithms with bluegreen bands used in case-I water is not suitable to case-II water. As CDOM absorption and particulate scattering decrease with increase in wavelength and become negligible in Red-infrared region, Red and Near Infrared (NIR) spectral region is extensively used to estimate Chl-a in Case-II water (Yu et al., 2014). As the physical and chemical components vary from lake to lake, the spectral bands suitable to measure the Chl-a for a particular lake may not suit to another lake. For every lake, suitable spectral bands are to be selected by comparing the estimated values from spectral bands combinations with laboratory-measured values.
The conventional method of quantifying Chl-a measurement is collecting samples from different locations of water body and measuring the chlorophyll-a concentration using spectrophotometer. This method needs field sampling, usage of chemical etc, which are time consuming, so difficult to perform for large area water bodies. Measuring certain optical properties like reflectance in different wavelength provides the concentration of the chlorophyll instantly. The spectral reflectance methods are used to estimate Chl-a in Aerial, Satellite and in situ measurements. The aerial surveys collect the spectral reflectance of the water bodies of small area and study the relation between reflectance spectrum with the chlorophyll-a. The satellite remote sensing provides a near real time, large scale and spatially continuous data in fixed time frequency, at less cost.
Many algorithms are developed to estimate the abundance of these constituents in water bodies like ocean, coastal, lakes and rivers based on spectral reflectance. The Chl-a estimation algorithms can be divided into two different categories: (i) Multiband models in the forms of ratios of different band combinations which utilizes the absorption bands of Chl-a in the red and NIR spectral region (Ruddick et al., 2001); and (ii) algorithms that use the Chl-a fluorescence emission around 685 nm (Gower et al., 1999).
The first group of algorithms are based on multispectral imaging. In this group, many algorithms have been developed using Red and NIR region spectral bands to estimate Chl-a concentration in eutrophic and turbid case II waters (Huang et al., 2014;Han et al., 2014). It covers single band, band ratio model (Hoge et al., 1987), three band models (Sathyendranath et al., 1989;Chen et al., 2013;Duan et al., 2010a, Tian et al., 2014, four band model (Le et al., 2009), five band model (Gohin et al., 2002), normalised differential chlorophyll index (NDCI) (Mishra and Mishra, 2012) etc. Some band ratio algorithms developed for the Chlorophyll estimation in terrestrial vegetation have also demonstrated in estimating Chl-a in water (Dall'Demo et al., 2003). As multispectral imageries are collected with less number of wide spectral bands, they cannot reflect minute variations like intensity variation in particular wavelength, intensity maximum shifting from wavelength to wavelength. As their output is integrated reflectance, change in reflectance with respect to narrow spectral bands is submerged in wider band.
The second group of algorithms are mainly based on Hyperspectral data, which provides almost continuous spectral measurements with hundreds of narrow spectral bands. They are: peak position near 700 nm (Gitelson, 1992); peak magnitude above baseline; area above baseline; first order derivative (Rundquist et al., 1996), second order derivative (Shi et al., 2007) and derivative ratio model (Tsai and Philpot, 1998) etc. Artificial neural network based algorithm (Chauhan et al., 2005), spectral decomposition method (Zhang et al., 2014), weighted algorithm (Yi, 2013), Spectral Vector Machine (SVM) based model ) and three-band reflectance difference model (Duan et al., 2010b) are also used to estimate the chlorophyll-a concentration.
In this paper, a new algorithm called Derivative difference is introduced to estimate the Chl-a in Case-II water. The effectiveness of this algorithm in case-II water bodies is demonstrated using spectroradiometer data.

Methods
Among many factors affect the reflectance values of water, atmospheric absorption, CDOM, Organic Substance Sediments (OSS) are important. When we calculate the derivative values from reflectance, the effect of slowly changing factors with respect to wavelength such as atmosphere and CDOM are eliminated (Philpot, 1991). The derivative values provide the slope of reflective spectrum. These derivative values and derivative ratio values are used to estimate chlorophyll in many studies.
In the present study, as a new measure, the difference of derivative values at two different wavelengths has been used to estimate the chlorophyll-a. Obtained results are compared with the estimated values from both band ratio and derivative ratio methods. The estimated values obtained using this algorithm using spectroradiometer data were verified with measured values as well. Method, study area and results are described in following sections.

Study Area
The study area selected for this purpose is the Halasuru Lake, which is located at the heart of Bangalore city, India spread over the latitudes of 12 o 58' 41" N and 12 o 59' 15" N and longitudes of 77 o 36' 57" E and 77 o 37' 22" E, respectively. The area of the lake is approximately 125 Acres and is 930 m above the sea level. The depth of the lake varies from 3 m to 15 m and average depth of the lake is 5m. The main source of water to this lake is rainwater. It also receives direct industrial and domestic wastewaters from the surrounding area after aerated. The increased settlements near the lake is the cause for the pollution of the Lake. The satellite view of Halasuru Lake is shown in Figure 1.

In-situ Reflectance Measurement using Spectroradiometer
In-situ measurements were carried out with spectro-radiometer at different locations of the lake. Sacchi disk measurement was also carried out to measure the Sacchi depth to avoid bathymetry radiation. The depth of lake at sample collection locations was much more than the Sacchi depth. The specular reflection from the water was avoided by carrying out the fieldwork between 9.30 AM and 11.00 AM local time. The locations of sample collection in Halasuru Lake are marked and shown in Figure 1.
Hyperspectral reflectance was measured using Analytical Spectral Devices, Field spec Pro spectroradiometer with a spectral resolution of one nm and the spectral range from 350 nm to 1800 nm. During the measurement, the instrument was held 0.6 m above the water surface manually, 0.5 m away from the boat and the probe was directed vertically down towards water. During the measurement at each location, sampling of the radiance was carried out for 25 times and the final output is produced after averaging these values. Five such outputs were collected at each location. The radiance plots from those outputs from same locations were compared and the datasets with large errors were rejected. Good data sets were averaged to generate final radiance data. This procedure was followed for measurement of water radiance and reference plate (Lambertian reflector) radiance as well. When radiance of reference plate was measured, the reference plate was kept parallel to the water surface in sun light and the measurement probe was held normal to the surface. The Latitude and Longitude of sample collection location were recorded from a GPS system output.
… (1) Where L w () water leaving radiance, L cal () scattered by reference plate and R cal () reflectance of reference panel. At each sampling location, both water leaving radiance and reference radiance were measured without time gap.

Laboratory Measurements
Water samples were collected at a depth of 10 to 15 cm below the surface immediately after the reflectance measurements. Chl-a in the water was measured using the laboratory method (Jeffrey and Humbprey, 1975;Arar, 1997).

Reflectance Spectra Analysis
As the reflectance variation due to Chl-a concentration is found to be between 350 to 800 nm and the absorption of radiation by water is more beyond 800 nm, hence spectro-radiometer data with wavelength between 350 and 800 nm was selected for study. The spectral reflectance of the samples collected from different locations is shown in Figure 2.
The reflectance spectra magnitudes in the region of 350 to 450 nm are low due to the high absorption of water in this region. The maximum absorption is seen at 440 nm. The reflectance spectra show a clear reflectance peak in the green region between 550 nm and 570 nm. After 570 nm the reflectance comes down steadily upto 630 nm and at 640 nm, an inflection is observed. The red absorption is peak around 670 nm. This is due to the combined effect of absorption of chlorophyll and water. After this, the reflection steadily increases up to the region around 705 nm (NIR) and this high reflectance is due to the fluorescence of chlorophyll-a (Neville and Gower, 1997). After this, the reflection decreases to the lowest level due to the absorption of water. Also, around 760 nm a small peak is seen in reflection spectra.

Figure 2:
Reflectance spectra of all samples

Reflectance Band Ratio Methods
As the reflectivity in the NIR region R NIR is due to the fluorescence of chlorophyll and is directly proportional to chlorophyll contents and the dip in reflectance in Red region R Red is due to the absorption of Chl-a, and inversely proportional to reflectance in Red region, the Ratio R NIR /R Red will be directly proportional to the chlorophyll concentration. Hence, the ratio of NIR to Red reflectance will represent the chlorophyll-a concentration.

First Order Derivative Values
The derivative values are calculated using following equation: Where, … Where R n+1 , R n-1 are the reflectance at n+1 and n-1. These values provide the change in reflection with respect to small change in wavelength (1 or 2 nm).

Regression Analysis between Derivative Values and Chl-a
The derivative values of reflectance spectrums are correlated with measured chlorophyll values to estimate Chlorophyll-a (Huang et al., 2010). In this study, the derivative values are smoothened to reduce the noise and correlated with Chl-a values. The maximum correlation value 0.69 was found at wavelength of 678 nm. Smoothening was done with linear running mean filter with different window size. Following figure shows the derivative values smoothened by different window sizes.

The First Order Derivative Ratio Method
The derivative ratio method combines the benefits of the ratio and derivative methods. In this method the ratio of first derivative at two different wavelengths is correlated with measured values of chlorophyll-a. The ratio of first derivative can be computed as P mn = … (3) Where,  The high correlation was found with the wavelengths of 460 nm and 473 nm.

The Second Order Derivative Method
The higher order derivative methods for spectral analysis is proposed by Philpot (1991). The second order derivative values were used for chlorophyll estimation in many studies (Shi, 2007). The ratio of Second derivative is computed as Where, P mn is the ratio of second derivative, is the second derivative at n and is the second derivative at m. As the second order derivative values have noise and spikes, smoothening is recommended to get better results.

The Second Order Derivative Ratio Method
The second order ratio method is used to estimate the chl-a Concentration.
The ratio of Second derivative is computed as P mn = … (5) Where, P mn is the ratio of second derivative, is the second derivative at n and is the second derivative at m.

Derivative Difference Method
In this study, a new algorithm called Derivative Difference method is introduced to estimate Chlorophyll-a. In this algorithm the difference between derivative values at different wavelengths were correlated with the measured Chl-a values.

Difference of First Order Derivative
The difference of first order derivative can be computed as P' mn = … (6) Where, P mn is the Difference of first order derivative, is the derivative at n and is the derivative at m.

Difference of Second Order Derivative
The difference of Second order derivative can be computed as P" mn = … (7) Where, P" mn is the difference of second order derivative, is the second order derivative at n and is the second order derivative at m.
This method was used on raw data, first order derivative data and second order derivative data, individually. The results are tabulated and shown below in Table 6-8.    Table 9 compiles the effectiveness of the derivative difference methods w.r.t to derivative ratio and band ratio methods.

Conclusion
Chlorophyll concentration was estimated with band ratio, derivative ratio and derivative difference methods. The suitable wavelength sets vary slightly with smoothening window size. From Table 9, it is found the 1 x 5 window provides better correlation values. It is noticed that while proposed band difference method provided a poor correlation with raw data, it provides improved high correlation results with derivative data. The results of proposed difference method on derivative data has provided highest determination coefficient uniformly in 1st order and 2nd order derivative difference methods. It is to be noted from Tables 1-8, the correlation coefficients due to band ratio and derivative methods are poorer and mostly effective for a narrow range of wavelengths; on the other hand the correlation coefficients due to derivative difference methods provide very good measure across all wavelength bands.