Method of calibration of remote spectrometric meter of heavy metal concentration on land plots
In the papers of many researchers it is noted  that the traditional methods of evaluation of heavy metal distribution in soil by the selection of sampling points and their further laboratory analysis and chemical identification have high cost and require long-term material preparation. As opposed to them, the methods of spectrometric analysis, which are integral, do not have these disadvantages. However, the direct identification of heavy metals in the capacity of inorganic contaminants of soil is complicated. The reason lies in the low concentration of these substances in soil and absence of clear specific spectral features in visible and near infrared ranges of observations. In addition, there are different combinations of heavy metals with organic substances and associations with hydroxides and carbonates. The spectral measurements of their concentration make it possible to determine indirectly the presence of heavy metals and identify their composition. Authors of the paper  inform that the hyperspectral measuring equipment allows obtaining sufficiently full and continuous spectral information, which is suitable for the detection of more detailed spectral properties of surface and mineralogical content of soil. This ability can be used for the monitoring of soil contamination.
The method of reflection spectroscopy has high performance in comparison with the chemical methods of analysis and identification. Sufficiently intense spectral features of different soil components lie within the spectral range of 400—2500 nm. Results of carried out experimental studies  showed that concentrations of heavy metals, in particular Pb, Zn, Mn, correlate well with the value of ratio between the intensities of reflection spectrums at the wavelengths of 610 and 500 nm – U (610) /U (500), and concentrations of Ni and Cr have better correlation with the estimate of inclination of the spectral intensity function line at the point corresponding to the wavelength of 980 nm.
There is information  that the intensity of reflecting spectral radiation of soil varies in inverse proportion to the concentration of heavy metals. The reflection spectrums of soils, which contain such minerals as pyrite (having impurities of Со, Ni, As, Cu and Au), jarosite (usually containing impurities of Na, Al and Pb), copiapite (typical impurities of Cu and Al), ferrihydrite (impurities of Сd, Zn and Pb) and goethite (typical impurity of Mn), are given in Fig. 1.
It should seem that the installation of spectrometric device onboard the aircraft will make it possible to solve the task of monitoring of territory surface easily in order to determine the concentration of heavy metals in soil. However, use of airborne spectrometric devices for remote detection and estimation of the degree of variations of heavy metal concentration on the ground does not allow obtaining the information with high level of reliability for a number of reasons.
The problems of equipment calibration and absence of identification criteria refer to the factors, which reduce the level of reliability of airborne measurement information.
The first listed problem occurs during online calibration of airborne meters. Firstly, the objects with different reflective properties (vegetation and soil) get in the field of view of airborne spectral systems. Secondly, use of only one parameter (concentration of heavy metals in soil) for the estimation of soil contamination is not sufficient; there is one more indirect parameter, at least, which is not taken into account during calibration – content of heavy metals in plants.
The second problem consists in the absence of criterion, which could take into account the ratio between the territories of overgrown and non-overgrown sites within the test field.
The purpose of this paper consists in the development of the method of online calibration of airborne spectral systems during the remote estimation of heavy metal content on the ground using the methods of remote probing, which would not have aforementioned disadvantages of existing calibration methods.
In order to solve the calibration task, let us assume that in the first approximation the field of view of spectral radiometric meter (not taking into account geometric distortions) covers the rectangular site ABCD, in which the areas with vegetation and areas without vegetation are observed (Fig. 2). In other words, the site is composite, its total area is SABCD and total area of vegetation sites is S (1). Share of vegetation in the field of tested site ABCD can be determined using the weight coefficient α1:
The total radiation flux U1 (λ0) gets on the input of spectral radiometer from plants at the wavelength λ0. The total flux U2 (λ0) gets on the input of spectral radiometer from soil, which is not overgrown with vegetation, at the wavelength λ0. Then, the total spectrometric signal, which was obtained at online calibration by airborne meter at the fixed wavelength λ0, can be estimated on the basis of the following expression:
It is well known that reflective properties of vegetation also strongly depend on the content of heavy metals in them. For example, the reflection spectrum of rice leaves depending on the content of cadmium in them is given in Fig. 3. Interesting results were obtained by authors of the paper ; they established that with the growth of plant contamination degree within the range of red wavelength the reflected signal increases and in near IR region it decreases. In order to take into account the influence of heavy metal concentration on calibration, we assume that CV refers to the concentration of heavy metals in vegetation and CS refers to the concentration of heavy metals in soil. Then, the total spectrometric signal can be expressed as follows:
Authors of the paper  discovered that strong correlation dependence is observed between the degree of soil contamination with heavy metals CS and degree of contamination of the vegetation growing on contaminated ground CV. These observations confirm the results, which were obtained by authors of the papers concerning various types of plants (Fig. 4). On the basis of the analysis of obtained dependences we can believe that it has linear character:
where β is correlation coefficient; a is constant. Then, the total spectrometric signal can be expressed as follows:
With known values α1, λ0, β, а, the equation (1) allows determining the value CS on the basis of measured parameter . We will show that obtained equation (1) allows performing online calibration of the remote meter of heavy metal concentration in soil with consistency of α1, β and а and with proper selection of λ0. The proper selection of the wavelength λ0 means the measurements in red region, in which with the increase of soil contamination degree the signal from vegetation will grow and signal from "bare" ground will decrease. Therefore, in simplified case considered task comes to the determination of polynomial extremum:
where U1 (CS) means growing function CS; U2 (CS) is decaying function. Extremum of the expression (2) can be considered as the factor for the calibration of remote meter of heavy metal concentration in soil.
It should be noted that theoretically the presence of extremum in scalar convolutions of the type (2) with paraphrase nonlinear functions U1 (CS) and U2 (CS) of its arguments is conclusively established fact in theory of multiobjective optimization. In order to obtain the specific expression for CS, at which the parameter UƩ (CS) could reach the extremum value, we will expand the functions U1 (CS) and U2 (CS) into Taylor’s series and confine ourselves to their representation in the form of the first three terms of series:
Taking into account the expressions (2—4) we can obtain the following quadratic equation:
Solution of the quadratic equation (5) allows determining the value CS, at which the spectral radiometer reading onboard the carrier will reach extremum during the flight of the carrier of measurement equipment above the sites with fixed values of constant equation coefficients. Taking into account obtained results, the following algorithm of online calibration of remote meter of heavy metal concentration on ground is suggested:
• Continuous determination of the values of constant coefficients of equation (5) during the flyover of the carrier of spectrometric equipment.
• Fixation of the moment, at which output signal of spectral radiometer reaches the extremum value.
• Determination of the value of heavy metal concentration at the moment of maximum of radiometer output signal in case of solution of equation (5).
DISCUSSIONS AND CONCLUSIONS
Thus, suggested calibration method of remote measurement of heavy metal concentration in soil during the carrier movement provides the generation of extremum signal at the output of airborne spectral radiometer. Analytical determination of extremum character using the analysis method of the second derivative of expression (2) on the basis of CS taking into account the expressions (3—4) results in the cumbersome expressions and it is not specified here. It should be noted that suggested calibration method of meter of heavy metal concentration at studied composite site will make it possible to obtain more reliable estimations in relation to the total contamination of studied site with heavy metals because it takes into account the properties and correlation between the components of this site.
The main conclusions of carried out studies are outlined below:
• Calibration method of the remote meter of total contamination with heavy metals in relation to the composite site, which consists of the area of non-overgrown ground and area with vegetation, is suggested.
• Algorithm of implementation of the calibration method of remote meter of heavy metal concentration at considered composite sites is developed.