Generalized Method of Biparametric Sub Pixel Thermal Location

It is well-known that according the Dozier’s method, utilization of integral of Planks function in fusion of signals of two different channels of airborne radiometer makes it possible to compute such components of temperature field within one pixel as temperatures of the object and background. In the paper, the generalization of Dozier method is suggested. The suggested generalization of Dozier’s bispectral method named as biparametric method is applicable for static remote objects. In the suggested biparametric method, the measurements are carried out at the moments t1 and t2. It is assumed that the object temperature reaches quantity T(t1) and T(t2) at these moments. On the bases of operational data of scanning infrared radiometer, the square area of one pixel can be calculated in dependence of distance between object and radiometer. This makes it possible to carry out location of static objects from two basis points using serial single wavelengths measurements of radiation emitted by the sub pixel object.


Introduction
It is well-known that between such spheres of technical cybernetics as location, positioning and remote sensing, the firm interrelation does exist.The information theory based grounding of such an interrelation firstly is described in the work [1].
The properties of thermal location are that the pixel type structure of images used for location purposes by scanning airborne radiometers leads to inevitable errors of integrated assessment of signal.Upon thermal scanning of surface of researched object, if the latter contains two different surface materials, all radiation emitted from these materials located within one pixel will be averaged as a single pixel signal depending on wavelength of sensor's operational channel.
According to [2], utilization of integral of Planks function for different channels of airborne radiometer makes it possible to compute following parameters: 1) Radiation temperature of one of two temperature fields on sub pixel level of resolution; 2) Share of each component of temperature field within pixel (that is temperatures of the object and background).
If assume, that effect of atmosphere is lacking, the upward radiation at the input of airborne scanning radiometer will be determined as [1] ( ) where: λ ε -emissivity at the wavelength λ; ( ) ( ) λ Φ -sensor's spectral instrument function.The Planks functions of black body with temperature T at the wavelength λ is determined as ( ) where: С 1 -the first constant of Plank, equals to 3.741832 × 10 −16 W•m 2 ; С 2 -the second constant of Plank, equals to 1.438786 × 10 −2 m•k; Т-temperature, K; λ-wavelength, μm.The property of the Planks function is that if in the fixed temperature T 1 , the signal at the wavelength λ 1 surpasses the signal at the wavelength λ 2 in sufficiently higher temperature T 2 , the contrary case does occur.
This property of the Plank's function is illustrated in Figure 1, where the output signals of 3-rd and 4-th channels of AVHRR radiometer installed on the board NOAA-6 are shown.As it seen from shown graphics upon temperature above 460 K, the signal of 4-th channel (102 -116 μm) surpasses the one of 3-rd channel (35 -40 μm).
The above said property is the bases of the Dozier's method [2], according which the total radiation emitted from non-homogenous, two-parts structured pixel at the surface of researched site upon lacking of atmospheric effect can be computed as where: p-weight coefficient; 0 < p < 1; T t -objects temperature; λ 1 , λ 2 -wavelengths used for measurements.
According to the Dozier's method upon carrying out measurements at the wavelengths λ 1 and λ 2 , if the value of T b is known, the amount of p and T t can be calculated using the system of Equation ( 3).It should be noted that to remove any possible dynamic errors, all measurements upon realization of this method should be carried out synchronously.This method makes it possible to carry out the sub pixel remote identification of hidden and remote objects.

Further Development of Doziers Bispectral Method and Suggested Biparametric Generalization
The Dozier's method further was developed and modernized in works [3]- [6].
Let us consider in brief the modernized Dozier's method described in [6].In this modification the limitation imposed in [2] is characterizing the consideration of only ground component of radiation.In the work [6] the general case of remote assessment of parameters of heated remote sub pixel object is considered taking into account the effect of atmosphere.The Equation (3) in this case should be written as ( ) ( ) , 1

( ) ( )
where: L b,4 and L b,11 -atmospheric radiations at the wavelength 4 μm and 11 μm; τ 4 and τ 11 -atmospheric transfer at the pertinent wavelengths.Operationally, L b,4 should be determined by averaging the radiation of neighbor pixels, assuming that these pixels are identical on temperature.Solution of Equations ( 4), (5) relative p an T i is carried out as following.From Equations ( 4) and ( 5 8) upon T f → ∞ asymptotically go near to zero.The solution of the task is gradual increase of T f till the first component at the left side of (8) would approach zero with acceptable accuracy.Thus, the above said method named as Dozier method make it possible to calculate parameters p and T t of sub pixel heated object carrying out radiometric measurements at the wavelength λ 1 = 4 μm; λ 2 = 11 μm.The suggested threemeasured interpretation of the Dozier method is illustrated in Figure 2. The Figure 2 illustrate scheme of radiometric measurements at the wavelengths λ 1 and λ 2 at the moment t o carried out for identification of sub pixel object with temperature T to .As it is seen from three-measured diagram the line AB determines the graphical interpretation of carried out bispectral measurements.
The suggested generalization of Dozier's bispectral method named as biparametric method is applicable for static remote objects.We assume that in the time interval 1 2 t t t ∆ = − the temperature of static remove object changes from T t1 as far as T t2 (Figure 3).The measurements are carried out at the single wavelength.The characteristics of the searched object is that the function   In the suggested biparametric method the measurements are carried out at the moments t 1 and t 2 .It is assumed that the object temperature reaches quantity T(t 1 ) and T(t 2 ) at these moments.The Equations ( 4) and ( 5) in the suggested biparametric method should be written as ( ) ( ) ( ) , , 1 , From the Equation (9) we find ( ) ( ) ( ) ) ( )

Figure 1 .
Figure 1.Dependence of signals of 3-rd and 4-th channels from temperature in spectroradiometer AVHRR of satellite NOAA-6.
that the radiation parameters of background are known, then obviously both components of the left side of equation ( =where: t-time of day, is approximately known.

Figure 2 .
Figure 2. Three-measured interpretation of Dozier's method applicable for remote objects with unchanged temperature.

Figure 3 .
Figure 3. Three-measured interpretation of suggested biparametric method applicable for remote objects with changing temperature.