Performance Measures

characterized by redundancy of information among the wavelet coefficients. This redundancy, on account of correlation between coefficients, is intrinsic to the wavelet-kernel and not a characteristic of the analysed signal. As an alternative, for practical applications (as in the study of noise reduction models for communication systems and image and signal compression), DiscreteWavelet Transform (DWT) is usually preferred [90]. The DWT scale and position are based on the power of two (dyadic scales and positions) and can be defined as [88]:

W(a, b)D = 2−j/2

Z j=J

j=1

ψ∗(2−j/2− k)f(t) dx (3.2) where the real numbers j and k are the integers that control the wavelet dilation and translation, respectively. In the DWT, a signal S passes through the low pass filter Lo and the high pass filter Ho followed by down sampling (keeping only one data point out of two) with both filters. At decomposition level n, the high pass filter followed by down sampling produces the detail information D(n) of the signal while the low pass filter associated with the scaling function produces the approximation A(n). This decomposition (filtering and down sampling) process continues until the desired level is achieved [89].

The DWT is generally a non-redundant transform and, accordingly, only a minimally required number of wavelet coefficients are preserved at each level of decomposition which, as a further consequence, enables reconstruction of the original signal from a reduced number of wavelet coefficients. While this property is useful in applications such as data and image compression, this type of non-redundant DWT, however, is prone to shift sensitivity and is therefore an undesirable feature when applied to problems related to singularity detection, forecasting and nonparametric regression [90].

3.2

Performance Measures

The forecast performance of all developed models was evaluated using the following measures of goodness of fit:

HAAR

In this section, we present the discrete wavelets like Haar Wavelet and Daubechies Wavelet for implementation in a still image compression system. This is implemented in software using MATLAB Wavelet Toolbox. The experiments and results is carried out on .jpg format images. These results provide a good reference for application developers to choose a good wavelet compression system for their application.

Haar functions have been used from 1910 when they were introduced by the Hungarian mathematician Alfred Haar [91]. Haar wavelet is discontinuous, and resembles a step function. It represents the same wavelet as Daubechies db1. Haar used these functions to give an example of an orthonormal system for the space of square integrable function on the unit interval [0,1] 3.3.

Haar Transform (b, a)(t) = √1 af _{x − b} a  . (3.3)

In Haar’s system, a particular family of piecewise linear functions can be used as a basis for representing any given function of finite energy. He used the pattern χ(k)

n (s) with n and k

representing what are now referred to as scale and time. χ0(s) is simply 1 on the interval [0, 1], and χ1(s) is 1 from [0,1₂) and −1 from (1₂,1]. In a more modern phrasing, χ0(s) is φ(t), the scaling function, and χ(k)

22 3. Wavelet Artificial Neural Networks (WANN)

Figure 3.3. HAAR Wavelet

As scale increases by integers, the wavelet functions at that scale become shorter in time by a factor of 2, and amplitude higher by a factor of√2 (preserving norm). The k parameter delays these shortened functions in time. Taken together, an expression relates the wavelet functions at higher scale to the lowest-scaled version, or mother wavelet:

ψj,k(t) = 2j/2ψ(2jt − k) (3.4)

Haar’s insight was that, as this set of functions are orthogonal to one another and have unit norm, we can project a function onto these scaled and shifted χ functions and obtain a complete description of the function in this new domain. Here instead of t (or s), we have j, k (or n, k). A synthesis expression for a time-domain function could be written as

f(t) =X

j,k

aj,kψj,k(t)

And thanks to the orthonormality of the basis functions, we can determine the scaling coefficients aj,k with inner products, or projections, onto those basis functions:

aj,k = hψj,k(t), f(t)i

And finally, those Hilbert space projections have a simple integral form: hψj,k, f(t)i =

Z ∞

−∞ψ ∗

j,k(t)f(t) dx

(In this classic orthonormal wavelet basis, the same basic idea applies as in the Fourier example earlier, but with a different change of variables instead of switching between pure time and pure frequency.)

The ANN and WANN models were trained, validated and tested. As an example, the performance of ANN and WANN models for physico-chemical parameters are presented in the results section.

23

Chapter 4

State-of-the-Art

Contents

4.1 Emerging applications areas satellite image requirements . . . . 23 4.1.1 Remote Sensing Data Processing . . . 23 4.2 Related Works . . . . 25

The importance of lakes and reservoirs leads to the high need for monitoring lake water quality both at local and global scales. The aim of the study was to test suitability of Sentinel- 2 Multispectral Imager’s (MSI) and Landsat data for mapping different lake water quality parameters. Data of chlorophyll-a (Chl a), Total Suspended Solids and Transparency from two small and large lakes were compared with band ratio algorithms derived from Sentinel-2 Level-1C and atmospherically corrected.

4.1

Emerging applications areas satellite image requirements

we conduct an in-depth study of the existing work in principle, O conjunto de dados in situ foi limitado em número, mas cobriu um gama de propriedades ópticas da água. The results allow us to assume that Sentinel-2 and Landsat will be a valuable tool for lake monitoring and research, especially taking into account that the data will be available routinely for many years, the imagery will be frequent, and free of charge.

4.1.1 Remote Sensing Data Processing

The free availability of data from the United States Geological Survey (USGS), as well as the temporal resolution of 16 days of the OLI sensor, provide certain advantages over other sensors. Orthorectified and terrain corrected Level1 OLI imagery was obtained. Imagery of the USGS website is processed by the Level 1 Product generation System (LPGS) and is provided in GeoTIFF format with UTM projection and WGS84 datum. The fieldwork data were collected approximately three days before and/or after the overpass of the OLI sensor for all scenes. The main characteristics of the OLI sensor are summarized in Table 4.1 and the main characteristics of the Sentinel-2, MSI (Multispectral Imager) sensor are summarized in Table 4.2

The atmospheric correction of satellite measurements in aquatic ecosystems is very important for the reason that a large part of radiation detected by the sensor is backscatter from the atmosphere. Thus, to properly identify the pixel content in an image in terms of water quality, the atmospheric correction presents a critical step in data processing of satellite images [80].

The atmospheric correction was performed using Dark Object Subtraction (DOS) method using the Semi-Automatic Classification Plugin (SACP). SACP is a free plugin for QGIS (open

24 4. State-of-the-Art

Table 4.1. L8/Operational Land Imager (OLI) bands with wavelength and ground sampling distance (GSD).

Band Wavelength (nm) Range GSD (m) 1 (Coastal/aerosol) 433–453 30 2 (Blue) 450–515 30 3 (Green) 525–600 30 4 (Red) 630–680 30 5 (NIR) 845–885 30 6 (SWIR 1) 1560–1660 30 7 (SWIR 2) 2100–2300 30 8 (PAN) 500–680 15 9 (CIRRUS) 1360–1390 30

Table 4.2. Sentinel-2 Satellite Sensor Specifications

Band Wavelength (nm) Range GSD (m)

Band 1 – Coastal aerosol 0.433 60

Band 2 – Blue 0.490 10

Band 3 – Green 0.560 10

Band 4 – Red 0.665 10

Band 5 – Vegetation Red Edge 0.705 20 Band 6 – Vegetation Red Edge 0.740 20 Band 7 – Vegetation Red Edge 0.783 20

Band 8 – NIR 0.842 10

Band 8A – Narrow NIR 0.865 20

Band 9 – Water vapour 0.945 60

Band 10 – SWIR – Cirrus 1.375 60

Band 11 – SWIR 1.610 20