System and method for measuring characteristics of continuous media and/or localized targets using at least one multi-frequency sensor

[0001] 1. Field of the Invention

[0002] The invention relates to the field of detection and monitoring and specifically to a system and method for measuring the characteristics of continuous medium and/or localized targets in a predetermined volume using at least one remote sensor operating at multiple frequencies.

[0003] 2. Description of the Related Art

[0004] An important problem in the field of detection and monitoring is accurately and reliably measuring as many characteristics of a monitored object as possible. Adequate accuracy and reliability are especially important in the case of remote detection and monitoring under adverse measurement conditions. Characteristics of a monitored object produced by detection and monitoring equipment seldom represent the final product of a measurement system. Typically these characteristics are further interpreted for making decisions and/or recommendations and the decisions and/or recommendations are presented to the system's users. By reliably and accurately measuring more characteristics of the monitored object, one ensures more robust and definite decision making and significantly decreases the probability of making an incorrect decision. Best possible data interpretation is especially important for automated real-time systems.

[0005] The monitored object can be a predetermined volume in a continuous medium such as the atmosphere, lakes, rivers, the ocean, surface and subsurface terrain, the human body, a chemical reactor, or any other media. Measured characteristics of such medium in a specified volume are used, for example, in the fields of meteorology, weather forecasting, geology, agriculture, medicine, and astronomy. Additionally, measured characteristics of such medium in a specified volume are also used, for example, in monitoring the airspace around airports, in monitoring conditions in chemical and processing plants, and in monitoring other somewhat similar processes and physical configurations. Monitored objects can also be specified targets located in a predetermined volume, such as, missiles, airplanes, obstacles, defects in a product, intruders, or other specified targets; in these cases the measured characteristics of the targets are used for purposes of national defense and homeland security, collision avoidance, non-destructive product testing, business and personal protection, and the like. It should be noted that in the instant disclosure the general term “monitored object” can be construed to refer to a predetermined volume in a continuous medium or a specified target in a predetermined volume. When a specified target is located in a predetermined volume it may also be referred to as a localized target.

[0006] Existing monitoring equipment can be divided into two classes namely, single sensor equipment and multiple sensor equipment. In addition, equipment with both single and multiple sensors can operate at single or multiple frequencies. Single-frequency, single-sensor equipment, such as, for example, standard single-receiver Doppler radars and sonars operating at a single frequency have been and are still widely used for numerous applications. However, such equipment provides a relatively small amount of initial information about the monitored object. This factor significantly limits the number of characteristics of the object that can be determined and also limits the accuracy and reliability of measurements, especially at adverse conditions. Manufacturing and implementing multi-frequency single-sensor or multiple-sensor monitoring equipment capable of performing real-time operations is now possible because of the outstanding progress in electronics and computer technologies that has been made during the last several decades. When compared with single frequency monitoring equipment, multi-frequency monitoring equipment provides a dramatically larger amount of initial information about the monitored object and therefore, enables more reliable measurement of a greater number of characteristics of the monitored object with much higher accuracy under any conditions. Known examples of multi-frequency sensor configurations include arrays of receiving antennas operating at multiple distinct frequencies, which are used in spaced antenna atmospheric profiling radars, telescopes and radio-telescopes collecting signals at multiple distinguishable frequencies, surveillance radars utilizing the frequency modulated chirp pulse compression, and continuous wave radars. Monitoring equipment that obtains signals at multiple frequencies from a single sensor or multiple sensors produce a large amount of initial information about a monitored object in comparison to the information produced with equipment that obtains signals from the same sensors at a single frequency. The objective of data analysis is to accurately and reliably extract as many useful characteristics of the object as possible. All data processing methods using data at multiple frequencies obtained from single or multiple sensors are basically similar in that they utilize the same initial information: time series of signals at multiple frequencies from one or a plurality of sensors. The methods differ by the mathematical functions used for analyzing the signals, the mathematical models for relating these functions to the characteristics of the monitored object, and the assumptions that are adopted for constructing the models.

[0007] Traditional correlation function and spectra-based data processing methods for multi-frequency single or multiple sensor monitoring equipment have been widely used for decades in numerous areas of applications. At the same time, the drawbacks of these methods have been well recognized and thoroughly documented. They are as follows: (1) a poor temporal and/or spatial resolution; (2) the inability to operate in adverse measurement conditions such as external interference and strong clutter; (3) a low reliability of measurements due to the adoption of inappropriate, often too restrictive assumptions, and (4) a limitation in the number of characteristics of the monitored object that can be retrieved. For example, the variance of the vertical turbulent velocity is the only characteristic of atmospheric turbulence that can be retrieved with a spaced antenna profiler using traditional data processing methods. Another crucial problem with traditional data processing methods is referred to as the radial velocity ambiguity. This problem is inherent in remote sensors which use the Doppler spectra for measuring the radial velocity of continuous medium and/or localized targets.

[0008] Drawbacks of traditional data processing methods described hereinabove have been partly addressed by the structure function-based method. In the particular case of a spaced antenna atmospheric profiler operating at a single frequency, using structure functions allows an improvement of temporal resolution, the mitigation of external interference and clutter effects, the determination of the variances of the horizontal turbulent velocities and the horizontal momentum flux, and the derivation of operational equations by making a smaller number of less restrictive assumptions. Notwithstanding, the limitations of the structure function-based method have also been well recognized.

[0009] To understand the limitations, one should consider the definition and interpretation of structure functions for the received signals s({right arrow over (x)}₁,t) and s({right arrow over (x)}₂,t) from two sensors in close spatial locations {right arrow over (x)}₁ and {right arrow over (x)}₂ at two close times t₁ and t₂. The cross structure function of the order p is defined as follows:
D_p({right arrow over (x)},δ{right arrow over (x)},t,τ)=Δs^p({right arrow over (x)},δ{right arrow over (x)},t,τ), Δs({right arrow over (x)},δ{right arrow over (x)},t,τ)=s({right arrow over (x)},t)−s({right arrow over (x)}+δ{right arrow over (x)},t+τ)  (1)
Hereinafter: t is time, {right arrow over (x)}={right arrow over (x)}₁, t=t₁, δ{right arrow over (x)}={right arrow over (x)}₂−{right arrow over (x)}₁ and τ=t₂−t₁ are respectively the spatial separation between the sensors and the temporal separation between the signals, and the angular brackets <> denote ensemble averages. It is important that equation (1) defines only one equation of order p for a pair of sensors. One can see that D_p({right arrow over (x)}, δ{right arrow over (x)}, t, τ) is the p^th order statistical moment of the increment Δs({right arrow over (x)}, δ{right arrow over (x)}, t, τ); the latter is customarily interpreted as a band-pass filter extracting fluctuations with spatial and temporal scales |δ{right arrow over (x)}| and τ, respectively. However, it has long been established that this is not the case and that the increment is, in fact, a multi-band filter. For example, the normalized spectral transfer function of the auto increment Δs({right arrow over (x)}, 0, t, τ)=s({right arrow over (x)},t)−s({right arrow over (x)},t+τ) is 1−cos(2πfτ) with maxima occurring at multiple frequencies f=1/(2τ)+k/τ, k=0, 1, 2, . . . . Customarily, only the first band at k=0 is taken into account in the interpretation of structure functions, while others of the same intensity and bandwidth are merely ignored. The next issue is that a cross structure function is not a rigorous mathematical tool. It follows from equation (1) that the temporal D_p({right arrow over (x)}, 0, t, τ) and spatial D_p({right arrow over (x)}, δ{right arrow over (x)}, t, 0) auto structure functions at |τ|→0 and |δ{right arrow over (x)}|→0 are the first-order finite approximations of the respective temporal derivatives and spatial derivatives in the direction δ{right arrow over (x)}. The first order approximation of a cross derivative at |δ{right arrow over (x)}|→0, |τ|→0 is:

[0010] $\begin{array}{cc}\frac{{\partial }^{2}s\left(\stackrel{\rightharpoonup }{x},t\right)}{\partial t\partial \stackrel{\rightharpoonup }{x}}\approx \frac{1}{\tau \delta \stackrel{\_}{x}}\left[s\left(\stackrel{\rightharpoonup }{x}+\delta \stackrel{\rightharpoonup }{x},t+\tau \right)-s\left(\stackrel{\rightharpoonup }{x}+\delta \stackrel{\rightharpoonup }{x},t\right)-s\left(\stackrel{\rightharpoonup }{x},t+\tau \right)+s\left(\stackrel{\rightharpoonup }{x},t\right)\right]\equiv -\frac{1}{\tau \delta \stackrel{\rightharpoonup }{x}}\left\{\left[s\left(\stackrel{\rightharpoonup }{x},t\right)-s\left(\stackrel{\rightharpoonup }{x}+\delta \stackrel{\rightharpoonup }{x},t+\tau \right)\right]-\left[s\left(\stackrel{\rightharpoonup }{x},t\right)-s\left(\stackrel{\rightharpoonup }{x}+\delta \stackrel{\rightharpoonup }{x},t\right)\right]-\left[s\left(\stackrel{\rightharpoonup }{x},t\right)-s\left(\stackrel{\rightharpoonup }{x},t+\tau \right)\right]\right\}& \left(2\right)\end{array}$
One can see from equations (1) and (2) that the increment Δs({right arrow over (x)}, δ{right arrow over (x)}, t, τ) corresponds to the first bracketed term in the derivative while the second and third terms are merely ignored. Therefore, the cross structure function is a truncated representation of the cross derivative ∂²s({right arrow over (x)},t)/(∂t∂{right arrow over (x)}).

[0011] These theoretical issues lead to serious practical drawbacks when using structure function-based data processing methods with monitoring equipment operating at multiple frequencies and having a single sensor or multiple sensors. For example, in the case of an atmospheric spaced antenna profiler, the major drawbacks are as follows: (1) an inability to retrieve the vertical momentum fluxes, (2) a high sensitivity to white noise, (3) an inability to directly measure the correlation between noise from different sensors, and (4) an inability to provide more than one equation for each pair of sensors. Importantly, the structure function-based data processing methods do not allow measuring the radial velocity with single-frequency monitoring equipment in most practical sensor configurations. These drawbacks complicate the operational use of existing data processing methods and cause degradation in the performance of monitoring equipment with one or more sensors operating at multiple frequencies.

[0012] It is accordingly an object of the invention to provide a system and method for measuring one or more selected characteristics of continuous medium and/or localized targets in a predetermined volume using one or more sensors operating at multiple distinguishable frequencies, which overcome the above-mentioned disadvantages of the prior art systems and methods of this general type.

[0013] With the foregoing and other objects in view, there is provided, a system for obtaining data indicative of at least one characteristic of a continuous medium or at least one localized target located within a predetermined volume of space. The system includes a sensor configuration including at least one sensor for acquiring a plurality of signals from the continuous medium or the target. Each one of the plurality of signals is centered at a respective distinguishable frequency. The system also includes a processing circuit for obtaining the data indicative of the characteristic of the continuous medium or the target by calculating a plurality of powered weighted increments using the plurality of signals acquired by the sensor configuration. The processing circuit then relates the plurality of powered weighted increments to the characteristic of the continuous medium or the target using a plurality of models.

[0014] In accordance with an added feature of the invention, the sensor configuration includes a plurality of sensors having centers spatially separated from each other in at least one spatial direction, and the plurality of sensors is configured for concurrently acquiring the plurality of signals from the continuous medium or the target.

[0015] In accordance with an additional feature of the invention, the plurality of sensors is positioned outside the predetermined volume of space, and the plurality of signals acquired by the plurality of sensors is generated by the continuous medium or the target.

[0016] In accordance with another feature of the invention, the plurality of sensors is positioned outside the predetermined volume of space, and the plurality of signals acquired by the plurality of sensors is caused by predetermined radiation that is generated and propagated through the predetermined volume of space to induce backscatter from the continuous medium or the target at each the distinguishable frequency.

[0017] In accordance with a further feature of the invention, the at least one sensor of the sensor configuration is positioned outside the predetermined volume of space, and the plurality of signals acquired by the sensor is generated by the continuous medium or the target.

[0018] In accordance with a further added feature of the invention, the at least one sensor of the sensor configuration is positioned outside the predetermined volume of space, and the plurality of signals acquired by the sensor is caused by predetermined radiation that is generated and propagated through the predetermined volume of space to induce backscatter from the continuous medium or the target at each the distinguishable frequency.

[0019] In accordance with a further additional feature of the invention, the processing circuit is configured for increasing an amount of information extractable from the plurality of signals by modifying the plurality of signals, and subsequently obtaining the data indicative of the characteristic of the continuous medium or the target.

[0020] In accordance with another added feature of the invention, the processing circuit is configured for modifying the plurality of signals by performing at least one modification step selected from a group consisting of: band-pass filtering the plurality of signals, converting the plurality of signals from complex signals to a real signals, removing noise from the plurality of signals, removing mean values from the plurality of signals, normalizing each one of the plurality of signals with a standard deviation of the respective one of the plurality of signals, and generating virtual sensors using combinations of the plurality of signals.

[0021] In accordance with another additional feature of the invention, the processing circuit is configured for calculating the plurality of powered weighted increments for: at least one specified order, at least one specified pair of the plurality of signals acquired by the sensor, and at least one specified combination of weights.

[0022] In accordance with another further feature of the invention, the processing circuit is configured for relating the plurality of powered weighted increments to the characteristic of the medium or target by: fitting the plurality of powered weighted increments to a plurality of predetermined models, estimating a plurality of adjustable parameters in the plurality of predetermined models, and relating the plurality of adjustable parameters to the characteristic of the continuous medium or the target.

[0023] In accordance with yet an added feature of the invention, at least one of the plurality of predetermined models is formed as a decomposition into a Taylor series.

[0024] In accordance with yet an additional feature of the invention, each one of the plurality of predetermined models is constructed from at least one model selected from a group consisting of an analytically derived operational equation formed as a decomposition into a polynomial function over a selected parameter, a tabulated function obtained using a numerical simulation, and a tabulated function obtained using a physical experiment.

[0025] In accordance with yet another feature of the invention, the processing circuit is configured for increasing an accuracy of the data indicative of the characteristic of the medium or the target by analyzing multiple estimates of the data indicative of the characteristic of the continuous medium or the target.

[0026] In accordance with yet a further feature of the invention, the system includes an output circuit for outputting the data indicative of the characteristic of the continuous medium or the target.

[0027] With the foregoing and other objects in view, there is provided, a method for obtaining data indicative of at least one characteristic of a continuous medium or at least one localized target located within a predetermined volume of space. The method includes a step of using a sensor configuration having at least one sensor to acquire a plurality of signals from the continuous medium or the target. Each one of the plurality of signals is centered at a respective distinguishable frequency. The method also includes a step of obtaining the data indicative of the characteristic of the continuous medium or the target by calculating a plurality of powered weighted increments using the plurality of signals acquired by the sensor and by relating the plurality of powered weighted increments to the characteristic of the continuous medium or the target using a plurality of models.

[0028] In accordance with an added mode of the invention, the method includes: providing the sensor configuration with a plurality of sensors having centers spatially separated from each other in at least one spatial direction, and concurrently acquiring the plurality of signals from the continuous medium or the target with the plurality of sensors. The previously mentioned at least one sensor is one of the plurality of sensors that is now specified.

[0029] In accordance with an additional mode of the invention, the method includes: positioning one sensor or a plurality of sensors outside the predetermined volume of space, and when performing the step of concurrently acquiring the plurality of signals from the continuous medium or the target, acquiring signals generated by the continuous medium or the target.

[0030] In accordance with another mode of the invention, the method includes: positioning one sensor or a plurality of sensors outside the predetermined volume of space, and generating and propagating predetermined radiation through the predetermined volume of space to induce backscatter from the continuous medium or the target at each distinguishable frequency.

[0031] In accordance with a further mode of the invention, the method includes: increasing an amount of information extractable from the plurality of signals by modifying the plurality of signals, and subsequently performing the step of obtaining the data indicative of the characteristic of the continuous medium or the target.

[0032] In accordance with a further added mode of the invention, the method includes increasing an amount of information extractable from the plurality of signals by performing at least one modification step selected from a group consisting of: band-pass filtering the plurality of signals, converting the plurality of signals from complex signals to real signals, removing noise from the plurality of signals, removing mean values from the plurality of signals, normalizing each one of the plurality of signals with a standard deviation of the respective one of the plurality of signals, and generating virtual sensors using combinations of the plurality of signals. Subsequent to the step of increasing the amount of information extractable from the plurality of signals, the method includes performing the step of obtaining the data indicative of the characteristic of the continuous medium or the target.

[0033] In accordance with a further additional mode of the invention, the method includes calculating the plurality of powered weighted increments for: at least one specified order, at least one specified pair of the plurality of signals acquired by the sensor, and at least one specified combination of weights.

[0034] In accordance with another further mode of the invention, the method includes relating the plurality of powered weighted increments to the characteristic of the continuous medium or target by: fitting the plurality of powered weighted increments to a plurality of predetermined models, estimating a plurality of adjustable parameters in the plurality of predetermined models, and relating the plurality of adjustable parameters to the characteristic of the continuous medium or the target.

[0035] In accordance with yet an added mode of the invention, at least one of the plurality of predetermined models is formed as a decomposition into a Taylor series.

[0036] In accordance with yet an additional mode of the invention, each one of the plurality of predetermined models is constructed from at least one model selected from a group consisting of: an analytically derived operational equation formed as a decomposition into a polynomial function over a selected parameter, a tabulated function obtained using a numerical simulation, and a tabulated function obtained using a physical experiment.

[0037] In accordance with yet another mode of the invention, the accuracy of the data indicative of the characteristic of the continuous medium or the target is increased by analyzing multiple estimates of the data indicative of the characteristic of the continuous medium or the target.

[0038] In accordance with a concomitant mode of the invention, the data indicative of the characteristic of the continuous medium or the target is output.

[0039] The present invention is based on the development of powered weighted increments, which is a new mathematical tool for data processing. A plurality of powered weighted increments can be used with a plurality of received signals in order to obtain data indicative of selected characteristics of the medium or the target. By processing a plurality of powered weighted increments obtained using a plurality of received signals in the manner described herein, the present invention overcomes many of the problems found in existing monitoring equipment. A non-exhaustive list of such problems that have now been solved with the present invention includes: the ability to retrieve important characteristics of the monitored object, such as, for example, the vertical momentum fluxes in the case of an atmospheric profiler, the ability to directly measure the correlation between noise from different sensors, the ability to provide more than one equation for each pair of sensors, and the ability to obtain an unambiguous radial velocity.

[0040] A key feature of the present invention is using powered weighted increments to determine selected characteristics of the object from multiple received signals, rather than using correlation functions, spectra, or structure functions of the signals. The powered weighted increments enable the use of an unlimited number of equations with variable weights for a plurality of signals at a plurality of distinct frequencies in order to obtain the most efficient and reliable estimation of each selected characteristic of the monitored object. The present invention allows the estimation of selected characteristics of the medium or the target with a higher accuracy and an improved temporal and/or spatial resolution compared to prior art systems and methods. When compared to the prior art, the present invention functions more reliably in adverse conditions, allows the detection and identification of specified targets that could not be detected and/or identified otherwise, allows the retrieval of characteristics of the monitored medium that could not be retrieved otherwise, and provides measurement error for each estimated characteristic of the monitored object. The powered weighted increments are presented to mathematical models with adjustable parameters and these parameters are related to the selected characteristics of the monitored object that are to be determined. The models can be analytical expressions, tabulated results of numerical simulations or experiments, and the like, and the adjustable parameters can be estimated using standard, well-known fitting methods. The present invention enables constructing models for any type and configuration of detection and monitoring equipment, any operational mode of the detection and monitoring equipment, and any of a number of selected characteristics of the object. The preferred mathematical models are analytical operational equations in the form of decompositions into polynomial functions over a sufficiently small temporal and/or spatial separation, where the adjustable parameters are the coefficients in the decompositions. When preferred conditions for using the present invention are satisfied, the preferred models can be constructed by making a smaller number of less restrictive assumptions than would be possible with prior art methods. The present invention can be applied to detection and monitoring equipment receiving multi-frequency signals related to known characteristics of the target or medium independent of the physical nature of the signals and the configuration of the data acquisition device. Such detection and monitoring equipment can deploy passive or active remote sensors that can be mounted on a fixed or moving platform.

[0047] In describing the present invention we will begin with an introduction of the new concept of powered weighted increments and a basic overview of a preferred embodiment of the inventive method. Later on, a specific example is provided solely for the purpose of illustrating some of the features of the invention. The invention should not be construed as being limited to being applied in the manner shown in the example or in the preferred embodiment. The invention relates to a method as well as a system for implementing the method. The invention provides an important advance in processing multiple signals—the key feature of which is the use of a plurality of powered weighted increments to determine the characteristics of the object from a plurality of signals received from a single sensor or a plurality of sensors. The plurality of signals are centered at a plurality of distinguishable frequencies. Powered weighted increments provide substantial advantages over conventional correlation function, spectrum or structure function-based methods. Powered weighted increments Φ_p are a new mathematical tool for processing signals at two distinct frequencies f₁ and f₂. Powered weighted increments when obtained from a pair of sensors in spatially separated locations {right arrow over (x)}_s1 and {right arrow over (x)}_s2 at temporarily separated instances t₁ and t₂, are defined as:
Φ_p({right arrow over (x)}_s1,{right arrow over (x)}_s2,f₁,f₂,t₁,t₂,ω_x,ω_τ)={[s({right arrow over (x)}_s1,f₁,t₁)−s({right arrow over (x)}_s2,f₂,t₂)]
−ω_x[s({right arrow over (x)}_s1,f₁,t₁)−s({right arrow over (x)}_s2,f₂,t₁)]−ω_τ[s({right arrow over (x)}_s1,f₁,t₁)−s({right arrow over (x)}_s1,f₂,t₂)]}^p  (3)
The pair of sensors can include two physical sensors, two virtual sensors, or one physical and one virtual sensor. In the case when a single sensor is used, then {right arrow over (x)}_s1={right arrow over (x)}_s2 and in order to obtain powered weighted increments, Eq. (3) simplifies to:
Φ_p({right arrow over (x)}_s1,f₁,f₂,t₁,t₂,ω_x,ω_τ)={(1−ω_τ)[s({right arrow over (x)}_s1,f₁,t₁)−s({right arrow over (x)}_s1,f₂,t₂)]
−ω_x[s({right arrow over (x)}_s1,f₁,t₁)−s({right arrow over (x)}_s1,f₂,t₁)]}^p  (3a)
The angular brackets <> in the definitions (3) and (3a) denote the ensemble averages, the temporal averages over a specified time interval, or the spatial averages over a specified spatial domain. The weights −∞<ω_x, ω_τ<∞ are free parameters that may have arbitrary real values. It should be noted that both weights ω_x and ω_τ may take on zero values at the same time for calculating some of the powered weighted increments, but they both cannot always be zero when calculating a plurality of powered weighted increments for determining a particular characteristic or set of characteristics of a monitored object. The powered weighted increments (3) evaluated as |{right arrow over (x)}_s2−{right arrow over (x)}_s1|→0, |t₂−t₁|→0, and with f₁=f₂, ω_x=ω_τ=1 exactly approximate the cross derivative (2). It is important that the power p in definitions (3) and (3a) be applied before averaging. Therefore, the function Φ_p is not a linear combination of the individual increments in the square brackets. Using equation (3) or (3a) one may obtain an unlimited number of equations for a pair of signals. The system and method for measuring characteristics of continuous medium and/or localized targets is therefore based on obtaining a plurality of powered weighted increments from a plurality of received signals—a new mathematical tool for data processing.

[0048] The present invention enables the measurement of characteristics of continuous medium and/or localized targets using a sensor configuration that includes at least one physical sensor operating at a plurality of distinguishable frequencies. The present invention overcomes the above-noted problems including the inability to retrieve more characteristics of the monitored object such as the vertical momentum fluxes for the case of an atmospheric profiler, the inability to directly measure the correlation between noises from different sensors, the inability to provide more than one equation for each pair of sensors, and the inability to obtain an unambiguous radial velocity.

[0049] The invention enables one to obtain efficient and reliable estimates for each selected characteristic of the monitored object by choosing an appropriate number of equations in the form of equation (3) or (3a) in which each such equation is configured with the optimal combination of weights ω_x and ω_τ and power p for each characteristic that is to be determined. The present system and method also improves the temporal and/or spatial resolution, decreases the sensitivity to white noise and further mitigates or removes contamination of the results by external interferences and clutter.

[0050] The present system and method enable one to determine a selected set of characteristics of a predetermined object, where the number and values of multiple frequencies as well as the number and location of sensors in this system depend on the characteristics of the object that will be measured. Notably, however, at least one physical sensor operating with at least two distinguishable frequencies must be deployed. The present invention is capable of determining various characteristics of a monitored object, including but not limited to the: size, shape, visibility, speed and direction of the object's motion, rates of changes of the above characteristics, and the like.

[0051] Referring now to the figures of the drawing in detail and first, particularly, to FIG. 1 thereof, there is shown a block diagram illustrating the steps involved in a generalized exemplary embodiment of the method for measuring selected characteristics of continuous medium and/or localized targets. The system would implement these method steps. It should be noted, however, that not all shown steps must necessarily be performed. It is conceivable that in some cases performing only steps 101, 103, 104, and 105 would be sufficient and acceptable and it is intended that the invention not be necessarily limited beyond these steps. The characteristics of the continuous medium and/or localized targets would then be available for display, archiving, or transfer to another device as is indicated in step 107.

[0052] As shown in FIG. 1, at step 101 the system collects received signals from the sensor configuration. The system configuration and operational parameters and the data processing parameters may be specified at this step or may have been previously specified.

[0053] The actual signals that have been received or obtained from each sensor in the sensor configuration may be modified at step 102 to ensure efficient estimation of the selected characteristics of the medium or the target. The modification of the actual signals may include any of a number of known operations, such as, for example, combining signals from a plurality of physical sensors to generate virtual sensors, band-pass filtering of the signals, normalizing the signals, and other modification processes which will become apparent to one of ordinary skill in the art after considering the instant disclosure. Such modification steps may be performed by electronic circuitry constructed as part of the sensor configuration or may be performed by the processing circuit.

[0054] A plurality of powered weighted increments with specified orders and specified weights are then calculated at step 103 for each selected pair of signals obtained from the sensor configuration.

[0055] In step 104, the powered weighted increments that were calculated in step 103 are fit to predetermined models and adjustable parameters in the models are estimated using standard fitting techniques. It is preferable that such models be analytically derived operational equations in the form of decompositions into polynomial functions over a small parameter; it is also notable however, that such models can also be, for example, tabulated functions obtained using numerical simulations or physical experiments.

[0056] In step 105, the adjustable parameters are related to the selected characteristics of the continuous medium or localized target. Specific models for relating the adjustable parameters to the selected characteristics of the monitored object depend on: the type, configuration, and operational mode of the monitoring equipment, the underlying theory and techniques that are chosen for constructing the models of the object, and the characteristics of the object that are to be determined. The adjustable parameters are preferably related to the selected characteristics using analytical expressions, however, discretely tabulated curves may also be used for this purpose.

[0057] Multiple estimates for each selected characteristic of the monitored object may then be analyzed at step 106 to provide the best estimate for each characteristic, to obtain the accuracy for the best estimate, and, if this is required, to obtain a measure for the reliability of the best estimates. This processing step can include a statistical analysis of all available estimates for the selected characteristics of the object, a joint statistical analysis of the selected characteristics at the analyzed time interval with those from a previous time interval(s), and other analysis techniques which will become apparent to one of ordinary skill in the art after considering the instant disclosure.

[0058] The best and/or all multiple estimates for the selected characteristics of the monitored object can be displayed at step 107 in any user specified format. The set of characteristics can also be transferred to a remote location and/or archived for future use and/or reference.

Sensor Configuration

[0059] FIGS. 2A, 2B, and 3 are provided to illustrate an example of a transmitter and sensor configuration that can be used to implement the present system for measuring characteristics of continuous medium and/or localized targets. The example is intended to demonstrate the main features and advantages of the present invention compared with multi-frequency measurement systems implementing prior art data processing methods. This example is provided for illustrative purposes only and is not intended to limit the areas of application or the scope of the present invention.

[0060] The illustrative example is an atmospheric profiling radar based system that uses the single three-panel transmitting antenna T shown in FIG. 2A and the plurality of identical spatially separated physical sensors A₁, A₂, and A₃ shown in FIG. 2B to estimate selected characteristics of the atmosphere in a predetermined region of space V. The characteristic size of the transmitting antenna T is denoted by D in FIG. 2A and the characteristic size of the receiving antennas or sensors A₁, A₂, and A₃ is denoted by D_A in FIG. 2B. FIG. 3 shows that in this system, the transmitting antenna T transmits a focused beam of radio frequency energy to illuminate the volume of space that contains the predetermined volume V. The focused beam is formed from a series of pulses and each pulse has a pulse origination time at which the pulse is emanated from the transmitting antenna T. The radar operates in a multi-frequency mode, for example, in a way that is customarily used for applying the range imaging (RIM) technique. In this case each transmitted and received pulse can be considered as consisting of a small set of distinct but very close frequencies f_T1, f_T2, . . . , f_TK such as |f_Ti−f_Tj|<<f_Ti for any i,j=1, 2, . . . , K. The radar operating frequencies are customarily presented as the corresponding wavelengths λ_Ti=cf_Ti, i=1, 2, . . . , K where c is a speed of the wave propagation which is a speed of light for the radio waves.

[0061] The transmitted pulses of radio frequency energy are scattered by the refractive index irregularities in the atmosphere. Following standard terminology, these irregularities are referred to as scatterers independent of their nature and type of scattering. A portion of the scattered waves reaches the sensors A₁, A₂, and A₃ (each panel in the transmitting antenna T operates as a separate multi-frequency sensor in this system) and induces signals at multiple frequencies f_T1, f_T2, . . . , f_TK in each of the independent receiving channels, that is in each sensor. The signals are collected in the selected range gates and are processed to estimate the characteristics of the atmosphere in each predetermined range gate or predetermined volume V with a center-range height R above the ground as shown in FIG. 3. An adaptive Cartesian coordinate system {right arrow over (x)}={x, y, z} with a vertically directed z axis and x and y axes in a horizontal plane is used hereinafter, in which symbols in brackets { } denote the Cartesian components of a vector. For each predetermined volume V, the adaptive origin of the coordinate system is placed at a height R above the center of the transmitting antenna T. The bullets in FIGS. 2A and 2B indicate the antenna phase centers with the coordinates {right arrow over (x)}_T={0,0,−R} for the transmitting antenna T shown in FIG. 2A and {right arrow over (x)}_Am={x_Am, y_Am, −R} for the receiving sensors A_m, m=1, 2, and 3 shown in FIG. 2B.

[0062] The current induced in the internal resistance R_int of the matched-filter sensor A_m (m=1, 2, or 3 in this system) can be presented in standard complex form as:

[0063] $\begin{array}{cc}E\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am}},\lambda ,t\right)=I\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am}},\lambda ,t\right)+jQ\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am}},\lambda ,t\right)=C\sum _{i=1}^N\frac{W_T\left({\stackrel{\rightharpoonup }{x}}_i\right)g_T^{1\text{/}2}\left({\stackrel{\rightharpoonup }{x}}_i\right)g_{\mathrm{Am}}^{1\text{/}2}\left({\stackrel{\rightharpoonup }{x}}_i\right)\Delta n\left({\stackrel{\rightharpoonup }{x}}_i,t\right)}{r_T\left({\stackrel{\rightharpoonup }{x}}_i\right)r_{\mathrm{Am}}\left({\stackrel{\rightharpoonup }{x}}_i\right)}\exp \left\{-j\frac{2\pi }{\lambda }\left[r_T\left({\stackrel{\rightharpoonup }{x}}_i\right)+r_{\mathrm{Am}}\left({\stackrel{\rightharpoonup }{x}}_i\right)\right]\right\}& \left(4\right)\end{array}$
Here I_Am(t) and Q_Am(t) are the in-phase and quadrature components of the induced signal; the subscript i identifies the i^th scatterer; N is the number of scatterers; Δn({right arrow over (x)}_i,t) is the magnitude of fluctuation in the refractive index field (the refractive index irregularity) produced by a scatterer in the location {right arrow over (x)}_i(t)={x_i(t), y_i(t), z_i(t)} at instant t; W_T({right arrow over (x)}_i), g_T({right arrow over (x)}_i), and g_Am({right arrow over (x)}_i) are the range weighting function, the transmitting antenna gain function, and the sensor's field of view gain function, respectively; r_T({right arrow over (x)}_i) and r_Am({right arrow over (x)}_i) are distances to the scatterer from the phase centers of the transmitting antenna T and sensor A_m; λ=f_Ti, i,j=1, 2, . . . , K is the signal wavelength; and j=√{square root over (−1)}. The constant C is proportional to √{square root over (P_T/R_int)}/λ where P_T is the average power of the transmitted pulse. Equation (4) is valid for the far zone r_T≧(2D²/λ) and r_Ak≧(2D²_A/λ). To satisfy the far zone requirements, sufficiently large ranges are considered below, such as:
R>>D,D_A,σ_r  (5)
Hereinafter σ_r denotes the width of the range weighting function in Eq. (12) below.

[0064] At step 101 in FIG. 1, the present system for measuring characteristics of continuous medium and/or localized targets acquires the received signals E({right arrow over (x)}_Am, λ, t) at all deployed wavelengths λ=λ_Ti, i=1, 2, . . . , K from all deployed physical sensors A_m, m=1, 2, and 3 during a predetermined period of measurement T_meas starting at the predetermined instant t₀. Each sensor provides signals for a predetermined set of the center ranges R_l, l=1, 2, . . . , L corresponding to different volumes V_l. For a pulse wave radar, E({right arrow over (x)}_Am, λ, t) are discrete time series with the sampling time interval δt=NCI/PRF where PRF and NCI denote the transmitter pulse repetition frequency and the number of coherent integrations. All relevant operational parameters of the system such as t₀, T_meas, R_l, δt, and others are specified at step 101 in FIG. 1.

[0065] The technical parameters of the system are also specified at step 101. These parameters include the transmitter wavelengths λ_Ti, i=1, 2, . . . , K and its characteristic size D, the number of sensors, their characteristic size DA and coordinates of their centers {right arrow over (x)}_Am (m=1, 2, and 3 in this system), and others.

[0066] The data processing parameters are specified at step 101 in FIG. 1 as well. In this system, the ensemble averages in the definition (3) and other theoretical equations are substituted by temporal averages over the predetermined time interval T_av; the latter is specified at this step. The pairs of wavelengths λ_Ti, λ_Tj (i,j=1, 2, . . . , K) to be analyzed, the pairs of sensors, the orders p of the powered weighted increments, the weights ω_x, ω_τ in Eq. (3), and other parameters are specified.

Modification of Signals from Each Sensor

[0067] The actual received signals from all deployed physical sensors are modified at step 102 in FIG. 1 to ensure the most efficient extraction of useful information from multiple signals. Modification may include (although is not limited by) the following operations.

[0068] (a) The output signals from the deployed sensors in the sensor configuration 101 in FIG. 1 can already be multiple signals E({right arrow over (x)}_Am, λ_Ti, t) at distinct wavelengths λ_Ti, i=1, 2, . . . , K given by Eq. (4). The signals can also be linear weighted superpositions of multiple signals at distinct wavelengths, for example, E({right arrow over (x)}_Am, λ_T1, λ_T2, . . . , λ_TK, t)=E({right arrow over (x)}_Am, λ_T1, t)+E({right arrow over (x)}_Am, λ_T2, t)+ . . . +E({right arrow over (x)}_Am, λ_TK, t) or broadband signals E({right arrow over (x)}_Am, λ_T1, t) over the range of wavelengths Δλ from λ_T1 to λ_TK. At step 102 one can apply standard band-pass digital filters with a narrow band δλ_f≦δλ_min to obtain the signals (4) at distinct wavelengths λ_Ti, i=1, 2, . . . , K from the superposition E({right arrow over (x)}_Am, λ_T1, λ_T2, . . . , λ_TK, t) or the broadband signals E({right arrow over (x)}_Am, Δλ, t). In this case δλ_min=|λ_T1−λ_Tj|_min, i,j=1, 2, . . . , K, where δλ_min is the minimum difference between the specified wavelengths.

[0069] (b) Powered weighted increments are the most efficient analysis tool when applied to real signals. The complex signal from the in-phase and quadrature synchronous detector given by Eq. (4) can be converted into the real instantaneous signal power for each of the distinct wavelengths λ=λ_Ti, i=1, 2, . . . , K according to the following equation:
S({right arrow over (x)}_Am,λ,t)=I²({right arrow over (x)}_Am,λ,t)+Q²({right arrow over (x)}_Am,λ,t), m=1, 2, 3  (6)
The pure received power S({right arrow over (x)}_Am, λ, t) from scatterers in the predetermined volume V in FIG. 3, with no contribution from noise, clutter, or other contaminants, is referred to below as a pure received signal from range R from sensor A_m with a center {right arrow over (x)}_Am.

[0070] (c) In practical measurements, the received signal contains both pure return from a monitored object S({right arrow over (x)}_Am, λ, t) and noise n({right arrow over (x)}_Am, λ, t), therefore the practical received signal from the sensor Am at each distinct wavelength λ=δ_Ti, i=1, 2, . . . , K can be presented as
{tilde over (s)}({right arrow over (x)}_Am,λ,t)=S(x_Am,λ,t)+n({right arrow over (x)},λ,t), m=1, 2, 3  (7)
At step 102 one can apply a high-pass, low-pass, and/or band-pass filter to partly remove noise n({right arrow over (x)}_Am, λ, t) and increase the signal-to-noise ratio.

[0071] (d) To eliminate possible offsets, one can remove the mean values {tilde over (s)}({right arrow over (x)}_Am, λ, t), m=1, 2, 3, from the signals at each of the distinct wavelengths λ=λ_Ti, i=1, 2, . . . , K.

[0072] (e) To simplify computations, one can normalize the signals with their standard deviations √{square root over ([{tilde over (s)}({right arrow over (x)}_Am, λ, t)−<{tilde over (s)}({right arrow over (x)}_Am, λ, t)>]²)}, m=1, 2, 3 at each of the distinct wavelengths λ=λ_Ti, i=1, 2, . . . , K.

[0073] (f) One can increase the number of sensors in the system without having to change the hardware configuration by generating virtual sensors using combinations of the actual received signals. For example, one can generate received signals from three virtual sensors A₄, A₅, and A₆ at each of the distinct wavelengths λ=λ_Ti, i=1, 2, . . . , K in this system using the following equations:
{tilde over (s)}({right arrow over (x)}_A4,λ,t)=[I({right arrow over (x)}_A1,λ,t)+I({right arrow over (x)}_A2,λ,t)]²+[Q({right arrow over (x)}_A1,λ,t)+Q({right arrow over (x)}_A2,λ,t)]²
{tilde over (s)}({right arrow over (x)}_A5,λ,t)=[I({right arrow over (x)}_A2,λ,t)+I({right arrow over (x)}_A3,λ,t)]²+[Q({right arrow over (x)}_A2,λ,t)+Q({right arrow over (x)}_A3,λ,t)]²
{tilde over (s)}({right arrow over (x)}_A6,λ,t)=[I({right arrow over (x)}_A1,λ,t)+I({right arrow over (x)}_A3,λ,t)]²+[Q({right arrow over (x)}_A1,λ,t)+Q({right arrow over (x)}_A3,λ,t)]²  (8)
These virtual sensors with the phase centers {right arrow over (x)}_A4, {right arrow over (x)}_A5, and {right arrow over (x)}_A6 are illustrated in FIGS. 4A, 4B, and 4C. The signal power is a non-linear function of I({right arrow over (x)}_Am, λ, t) and Q({right arrow over (x)}_Am, λ, t), hence the signals {tilde over (s)}({right arrow over (x)}_A4, λ, t), {tilde over (s)}({right arrow over (x)}_A5, λ, t), and {tilde over (s)}({right arrow over (x)}_A6, λ, t) given by equations (8) are non-linear combinations of the actual signals from sensors A₁, A₂, and A₃. The virtual sensors A₄, A₅, A₆ have a larger aperture and, therefore provide a higher signal-to-noise ratio at any wavelength λ than is provided by the actual sensors A₁, A₂, and A₃. In addition, the combined signals are more strongly correlated among themselves both at the same wavelength λ_Ti and different wavelengths λ_Ti and λ_Tj (i,j=1, 2, . . . , K) than are the actual signals. Signals from the virtual sensors A₄, A₅, A₆ are treated in the same way as the signals from the actual sensors A₁, A₂, and A₃. For example, the above operations (c), (d) and (e) can also be applied to the combined signals (8).
Calculating and Analyzing the Powered Weighted Increments

[0074] The objective of step 103 is to calculate powered weighted increments for all individual sensors, pairs of sensors, gate heights, weights, and averaging time intervals that are specified in step 101. For each gate range R and averaging time interval specified at step 101, one calculates a powered weighted increment for each specified power p, specified pair of distinct wavelengths λ_Ti, λ_Tj (i,j=1, 2, . . . , or K), specified pair of sensors A_m1 and A_m2, and specified combination of weights ω_x and ω_τ. The powered weighted increment Φ_p is calculated as follows:
{tilde over (Φ)}_p({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2,t,τ,ω_x,ω_τ)={[{tilde over (s)}({right arrow over (x)}_Am1,λ_T1,t)−{tilde over (s)}({right arrow over (x)}_Am2,λ_T2,t+τ)]
−ω_x[{tilde over (s)}({right arrow over (x)}_Am1,λ_T1,t)−{tilde over (s)}({right arrow over (x)}_Am2,λ_T2,t)]−ω_τ[{tilde over (s)}({right arrow over (x)}_Am1,λ_T1,t)−{tilde over (s)}({right arrow over (x)}_Am1,λ_T2,τ)]}^p  (9)
For each selected combination of {right arrow over (x)}_Am1, {right arrow over (x)}_Am2, λ_T1, λ_T2, t, ω_x, and ω_τ, the calculated increment {tilde over (Φ)}({right arrow over (x)}_Am1, {right arrow over (x)}_Am2, λ_T1, λ_T2, t, τ, ω_x, ω_τ) provides a confined set of observations at the temporal separations τ such as ±δt, ±2δt . . . specified at step 101. The powered weighted increment for an individual sensor A_m1 is an obvious particular case of Eq. (9) at {right arrow over (x)}_Am2={right arrow over (x)}_Am1, that is:
{tilde over (Φ)}_p({right arrow over (x)}_Am1,λ_T1,λ_T2,t,τ,ω_x,ω_τ)={(1−ω_τ)[{tilde over (s)}({right arrow over (x)}_Am1,λ_T1,t)−{tilde over (s)}({right arrow over (x)}_Am1,λ_T2,t+τ)]
−ω_x[{tilde over (s)}({right arrow over (x)}_Am1,λ_T1,t)−{tilde over (s)}({right arrow over (x)}_Am1,λ_T2t)]}^p  (9a)
As previously discussed, in step 104, the powered weighted increments that were calculated in step 103 are fit to predetermined models and adjustable parameters in the models are estimated using standard fitting techniques. In a preferred embodiment of the invention, the powered weighted increments are fit to mathematical models in the analytical form of polynomial decompositions, and adjustable parameters (coefficients in the decompositions) are estimated. Implementing the preferred embodiment of the invention requires the deployed equipment to be operable in a manner so as to provide a sufficiently small temporal separation τ between signals and/or spatial separation |δ{right arrow over (x)}_m2,m1|=|{right arrow over (x)}_Am2−{right arrow over (x)}_Am1| between the sensors. One can always use a sufficiently high pulse repetition frequency and a sufficiently small number of coherent integrations to get sufficiently small temporal separations τ=±δt, ±2δt . . . , as will be seen later in the description of an example in the form of an atmospheric profiling radar based system. When the temporal requirement is satisfied, each powered weighted increment at very small τ→0 can be decomposed in the Taylor series as follows:
{tilde over (Φ)}_p({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2,t,τ,ω_x,ω_τ)={tilde over (d)}_p,0({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2,t,ω_x,ω_τ)
+τ{tilde over (d)}_p,1({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2,t,ω_x,ω_τ)
+τ²{tilde over (d)}_p,2({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2,t,ω_x,ω_τ)+O(τ³)  (10)
Decomposition (10) is an example of a preferred model that can be used with the powered weighted increments, where the coefficients {tilde over (d)}_p,0, {right arrow over (d)}_p,1, and {right arrow over (d)}_p,2 are the adjustable parameters. In performing step 104 according to the preferred embodiment, the coefficients can be estimated for each calculated increment {tilde over (Φ)}_p({right arrow over (x)}_Am1, {right arrow over (x)}_Am2, λ_T1, λ_T2, t, τ, ω_x, ω_τ) as the best-fit parameters using any of the standard fitting procedures, for example, the least squares as a maximum likelihood estimator. In most practical cases the coefficients {tilde over (d)}_p,0, {tilde over (d)}_p,1, and {tilde over (d)}_p,2 in Eq. (10) can be analytically related to the selected characteristics of the monitored object as will be explained in describing the preferred manner of implementing step 105. An example of such analytical relations, which are referred to below as operational equations, is given in the next section for the case of an atmospheric profiling radar based system.

[0075] When the spatial requirement is satisfied, each powered weighted increment at very small |δ_{{right arrow over (x)}m2,m1}|→0 can be decomposed in the Taylor series as follows:
{tilde over (Φ)}_p({right arrow over (x)}_Am1,{right arrow over (x)}_Am2,λ_T1,λ_T2t,τ,ω_x,ω_τ)={tilde over (d)}_p,x0({right arrow over (x)}_Am1,λ_T1,λ_T2,t,τ,ω_x,ω_τ)
+δ{right arrow over (x)}_m2,m1{tilde over (d)}_p,x1({right arrow over (x)}_Am1,λ_T1,λ_T2,t,τ,ω_x,ω_τ)
+δ{right arrow over (x)}_m2,m1²{tilde over (d)}_p,x2({right arrow over (x)}_Am1,λ_T1,λ_T2,t,τ,ω_x,ω_τ)+O(|δ{right arrow over (x)}_m2,m1|³)  (10a)
Decomposition (11a) is another example of the preferred model for the powered weighted increments where the coefficients {tilde over (d)}_p,x0, {tilde over (d)}_p,x1, and {tilde over (d)}_p,x2 are the adjustable parameters. At this step 104, the coefficients can be estimated for each calculated increment {tilde over (Φ)}_p({right arrow over (x)}_Am1, {right arrow over (x)}_Am2, λ_T1, λ_T2, t, τ, ω_x, ω_τ) as the best-fit parameters using any of the standard fitting procedures. In most practical cases the coefficients {tilde over (d)}_p,x0, {tilde over (d)}_p,x1, and {tilde over (d)}_p,x2 in Eq. (10a) can be analytically related to the selected characteristics of the monitored object.
Estimating the Selected Characteristics of the Monitored Object

[0076] The objective of step 105 is to estimate all of the selected characteristics of the monitored object by relating the adjustable parameters that were estimated in step 104 to the selected characteristics of the monitored object. In the present system and method, specific equations for relating the adjustable parameters to the characteristics of the object depend on: the type, configuration, and operational mode of the monitoring equipment, the underlying theory and techniques that are chosen for constructing the models of the monitored object, and the characteristics of the object that are to be determined. For the particular case of an atmospheric profiling radar based system shown in FIGS. 2A, 2B, 3, 4A, 4B and 4C, the models are decompositions (10) and the adjustable parameters are coefficients {tilde over (d)}_p,0, {tilde over (d)}_p,1, and {tilde over (d)}_p,2. Operational equations for obtaining the selected characteristics of the atmosphere in the predetermined volume of space from the coefficients {tilde over (d)}_p,0, {tilde over (d)}_p,1, and {tilde over (d)}_p,2 that were estimated with the decomposition (10) are derived and discussed in the following portions of this section.

[0077] The Powered Weighted Increments for a Pair of Multi-Frequency Sensors

[0078] To get the expression for the instantaneous power of the pure received signal from sensor A_m at wavelength λ=λ_Ti, i=1, 2, . . . , K at instant t, one can combine equations (4) and (6) as follows:

[0079] $\begin{array}{cc}S\left({\overrightarrow{x}}_{\mathrm{Am}},\lambda ,t\right)=C^2\sum _{i=1}^N\sum _{k=1}^{N}A\left({\overrightarrow{x}}_i\right)A\left({\overrightarrow{x}}_k\right)\Delta n\left({\overrightarrow{x}}_i,t\right)\Delta n\left({\overrightarrow{x}}_k,t\right)\cos \left[\varphi \left({\overrightarrow{x}}_i\right)-\varphi \left({\overrightarrow{x}}_k\right)\right]A\left({\overrightarrow{x}}_i\right)=\frac{W_T\left({\overrightarrow{x}}_i\right)g_T^{1/2}\left({\overrightarrow{x}}_i\right)g_{\mathrm{Am}}^{1/2}\left({\overrightarrow{x}}_i\right)}{r_T\left({\overrightarrow{x}}_i\right)r_{\mathrm{Am}}\left({\overrightarrow{x}}_i\right)},\varphi \left({\overrightarrow{x}}_i\right)=\frac{2\pi }{\lambda }\left[r_T\left({\overrightarrow{x}}_i\right)+r_{\mathrm{Am}}\left({\overrightarrow{x}}_i\right)\right]& \left(11\right)\end{array}$
Hereinafter m=1, 2, . . . , 6, which includes both the physical sensors A₁, A₂, A₃ shown in FIG. 2B and the virtual sensors A₄, A₅, A₆ shown in FIGS. 4A, 4B, and 4C. An argument t is omitted whenever it is not confusing, for example, {right arrow over (x)}_i(t) is denoted as {right arrow over (x)}_i in equations (4) and (11).

[0080] Let us consider any pair of sensors A_m1 and A_m2 at (m1≠m2)=1, 2, . . . , 6 and specify Eq. (11) for signals from these sensors at two distinct wavelengths λ_Ti=λ and λ_Tj=λ+δλ where δλ=λ_Tj−λ_Ti (i,j=1, 2, . . . , or K) and |δλ|<<λ. Without losing generality, one can consider the x axis of the adaptive Cartesian coordinate system in FIG. 3 to be directed along the baseline δ{right arrow over (x)}_m2,m1={right arrow over (x)}_Am2−{right arrow over (x)}_Am1; in this case the separation between the phase centers of the sensors is δ{right arrow over (x)}_m2,m1={δx_m2,m1, 0, 0}. The range weighting function, the transmitting antenna gain function, and the sensor's field of view gain function can be approximated in the chosen coordinate system by the standard expressions:

[0081] $\begin{array}{cc}W\left({\stackrel{\rightharpoonup }{x}}_i\right)=\exp \left(-\frac{z_i^2}{4{\sigma }_r^2}\right),g_T^{1\text{/}2}\left({\stackrel{\rightharpoonup }{x}}_i\right)=\exp \left(-\frac{x_i^2+y_i^2}{4{\sigma }^2}\right),g_{\mathrm{Am}}^{1\text{/}2}\left({\stackrel{\rightharpoonup }{x}}_i\right)=\exp \left(-\frac{{\left(x_i-x_{\mathrm{Am}}\right)}^2+{\left(y_i-y_{\mathrm{Am}}\right)}^2}{4{\sigma }_A^2}\right),i=1,2,\dots ,N& \left(12\right)\end{array}$
The range width σ_r depends on the width of the transmitted pulse. The transmitter beam width σ and the sensor's field of view width σ_A can be approximated by the standard expressions:
σ≈γλR/D, σ_A≈γλR/D_A  (13)
Here γ, γ_A are constant for a given radar; in general γ≈γ_A. It follows from inequalities (5) and Eq. (13) that for all i=1, 2, . . . , N:
r_T({right arrow over (x)}_i){right arrow over (r)}_Am({right arrow over (x)}_i)≈R², |{right arrow over (x)}_Am−{right arrow over (x)}_T|/σ, |δ{right arrow over (x)}_m2,m1|/σ<<1  (14)
One can combine equations (5) and (11)–(14) to present a signal S({right arrow over (x)}_Am1, λ, t) as:

[0082] $\begin{array}{cc}S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)=C_s\sum _{i=1}^N\sum _{k=1}^N\Delta n_i\Delta n_k{B}_i{B}_k\cos \left({\varphi }_{\mathrm{ik}}\right)& \left(15\right)\end{array}$
Here Δn_i=Δn({right arrow over (x)}_Am1, {right arrow over (x)}_i, λ, t) is a reflectivity of a scatterer in location {right arrow over (x)}_i(t) for a wavelength λ at the instant t for the sensor A_m1 with center {right arrow over (x)}_Am1; the insignificant small terms are neglected hereinafter, and

[0083] $\begin{array}{cc}\begin{array}{c}{B}_i=\exp \left(-\frac{{\alpha }^2}{4}\frac{x_i^2}{{\sigma }^2}-\frac{{\alpha }^2}{4}\frac{y_i^2}{{\sigma }^2}-\frac{1}{4}\frac{z_i^2}{{\sigma }_r^2}\right),{\alpha }^2=1+\frac{{\sigma }^2}{{\sigma }_A^2}\\ {\varphi }_{\mathrm{ik}}={\varphi }_i-{\varphi }_k,{\varphi }_i=\frac{2\pi \gamma \sigma }{D}\frac{x_i^2+y_i^2}{{\sigma }^2}+\frac{2\pi {\sigma }_r}{\lambda }\frac{z_i}{{\sigma }_r},C_s=\frac{C^2}{R^4}\end{array}& \left(16\right)\end{array}$
The magnitudes of α are different for the physical sensors A₁, A₂, A₃ and virtual sensors A₄, A₅, A₆; practical implications of the difference are discussed below. Let us denote the instantaneous velocity of i^th scatterer at instant t by {right arrow over (W)}({right arrow over (x)}_i,t)={U({right arrow over (x)}_i,t), V({right arrow over (x)}_i,t), W({right arrow over (x)}_i,t)}, and make the key assumptions for deriving operational equations for the atmospheric profiling radar based system.

[0084] Assumption 1: The characteristics of scatterers Δn_i, x_i(t), y_i(t), z_i(t), U({right arrow over (x)}_i,t), V({right arrow over (x)}_i,t), W({right arrow over (x)}_i,t) at i=1, 2, . . . , N are the locally statistically stationary random processes.

[0085] It follows from this assumption that the ensemble averages of these characteristics are independent of time t and the random vector {right arrow over (W)}({right arrow over (x)}_i,t) can be presented as:
{right arrow over (W)}({right arrow over (x)}_i,t)={right arrow over (W)}_i+{right arrow over (w)}_i(t), {right arrow over (w)}_i=0  (17)
Hereinafter {right arrow over (W)}_i={right arrow over (W)}({right arrow over (x)}_i,t) and {right arrow over (w)}_i(t)={u_i(t), v_i(t), w_i(t)} describe the mean and turbulent motion of the i^th scatterer. Neglecting the small terms O(τ²) at τ→0, one can define the location of the i^th scatterer at instant t+τ as:
{right arrow over (x)}_i(t+τ)={right arrow over (x)}_i(t)+τ{right arrow over (W)}_i(t)  (18)
Neglecting the terms O(|δ{right arrow over (x)}_m2,m1|²,τ²) at a small separation |δ{right arrow over (x)}_m2,m1| and τ→0, the reflectivity of i^th scatterer in location {right arrow over (x)}_i(t+τ) for a wavelength λ+δλ at instant t+τ for the sensor A_m2 with center {right arrow over (x)}_Am2 for (m2≈m1)=1, 2, . . . 6 can be estimated as follows:

[0086] $\begin{array}{cc}\Delta n\left[{\stackrel{\rightharpoonup }{x}}_{\mathrm{Am2}},{\stackrel{\rightharpoonup }{x}}_i\left(t+\tau \right),\lambda +\delta \lambda ,t+\tau \right]=\Delta n_i+\frac{\partial \Delta n_i}{\partial t}\tau +\frac{\partial \Delta n_i}{\partial \lambda }\delta \lambda +\frac{\partial \Delta n_i}{\partial x_{\mathrm{Am1}}}\delta x_{\mathrm{m2},\mathrm{m1}}& \left(19\right)\end{array}$
The derivative ∂Δn_i/∂t describes the instantaneous rate of change in the reflectivity of the i^th scatterer at instant t for the sensor A_m1. The rate depends on the rate of change in the scatterer's shape, size, orientation, and/or material. The derivative δΔn_i/δλ describes the rate of change in the reflectivity of the i^th scatterer at instant t with a varying wavelength. The rate depends on the scatterer's material and shape, the wavelength λ, and the scatterer's location in the volume V at the instant t. The spatial derivative ∂Δn_i/∂x_Am1 in Eq. (19) characterizes the rate of change in the reflectivity of the i^th scatterer at instant t with the sensor A_m1 being moved in x direction. The rate depends on the scatterer's shape, size, orientation, and/or material as well as the scatterer's location in the volume V at the instant t. The terms with derivatives in Eq. (19) are very small at |δ{right arrow over (x)}_m2,m1|/R<<1, |δλ|<<λ, and τ→0 in most atmospheric conditions and they are neglected below.

[0087] One can use inequality (5) and equations (11)–(14), (18), (19), and neglect the terms O(τ²) and other insignificant terms to present the pure signal from sensor A_m2 for the wavelength λ+δλ at the instant t+τ as:

[0088] $\begin{array}{cc}S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am2}},\lambda +\delta \lambda ,t+\tau \right)=C_s\sum _{i=1}^N\sum _{k=1}^N\Delta n_i\Delta n_k{B}_i{B}_k\cos \left({\varphi }_{\mathrm{ik}}-{\phi }_{\mathrm{ik}}-{\eta }_{\mathrm{ik}}+{\psi }_{\mathrm{ik}}-{\zeta }_{\mathrm{ik}}\right)& \left(20\right)\end{array}$
Here:

[0089] $\begin{array}{cc}\begin{array}{cc}{\phi }_{\mathrm{ik}}={\phi }_i-{\phi }_k,& {\phi }_i=\alpha \frac{\delta x_{\mathrm{m2},\mathrm{m1}}}{g_D}\frac{x_i}{\sigma },g_D=\frac{\alpha D}{2\pi \gamma }\\ {\eta }_{\mathrm{ik}}={\eta }_i-{\eta }_k,& {\eta }_i=\frac{\delta \lambda }{g_{\lambda }}\frac{z_i}{{\sigma }_r},g_{\lambda }=\frac{{\lambda }^2}{4\pi {\sigma }_r}\\ {\psi }_{\mathrm{ik}}={\psi }_i-{\psi }_k,& {\psi }_i-\alpha \frac{U_i\tau }{g_D}\frac{x_i}{\sigma }+\alpha \frac{V_i\tau }{g_D}\frac{y_i}{\sigma }+4\pi \frac{W_i\tau }{\lambda }\\ {\zeta }_{\mathrm{ik}}={\zeta }_i-{\zeta }_k,& {\zeta }_i=\alpha \frac{\delta x_{\mathrm{m2},\mathrm{m1}}}{g_D}\frac{U_i\tau }{\sigma }\end{array}& \left(21\right)\end{array}$
One can see from equations (15), (16) and (20), (21) that S({right arrow over (x)}_Am1, λ, t) is independent of {right arrow over (x)}_Am1. The signal S({right arrow over (x)}_Am2, λ+δλ, t+τ) depends only on δ{right arrow over (x)}_m2,m1 while it is independent of {right arrow over (x)}_Am1 and {right arrow over (x)}_Am2, and so are the powered weighted increments.

[0090] The powered weighted increments (9) for a pair of pure signals S({right arrow over (x)}_Am1, λ, t) and S({right arrow over (x)}_Am2, λ+δλ, t+τ) can be presented in a non-dimensional form as follows:

[0091] $\begin{array}{cc}{\Phi }_p\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,\tau ,{\omega }_x,{\omega }_{\tau }\right)=\frac{1}{{\left({\sigma }_{\mathrm{Am1}}^2\right)}^{p/2}}\langle {\left\{[S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)-S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am2}},\lambda +\delta \lambda ,t+\tau \right)]-{\omega }_x\left[S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)-S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am2}},\lambda +\delta \lambda ,t\right)\right]-{\omega }_{\tau }\left[S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)-S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda +\delta \lambda ,t+\tau \right)\right]\right\}}^p\rangle {\sigma }_{\mathrm{Am1}}^2=\langle {\left[S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)-\langle S\left({\stackrel{\rightharpoonup }{x}}_{\mathrm{Am1}},\lambda ,t\right)\rangle \right]}^2\rangle & \left(22\right)\end{array}$
To compare the present method with prior art methods, only the second order powered weighted increments at p=2 will be considered, and the subscript p=2 is omitted below. One can substitute the expressions (15) and (20) for pure received signals into Eq. (22), and get the following equation for the second order powered weighted increments:

[0092] $\begin{array}{cc}\Phi \left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,\tau ,{\omega }_x,{\omega }_{\tau }\right)=\frac{1}{{\theta }_0}\langle [\sum _{i=1}^N\sum _{k=1}^N\Delta n_i\Delta n_k{B}_i{B}_k{\left\{\left[\cos \left({\varphi }_{\mathrm{ik}}\right)-\cos \left({\varphi }_{\mathrm{ik}}-{\phi }_{\mathrm{ik}}-{\eta }_{\mathrm{ik}}+{\psi }_{\mathrm{ik}}-{\zeta }_{\mathrm{ik}}\right)\right]-{\omega }_x\left[\cos \left({\varphi }_{\mathrm{ik}}\right)-\cos \left({\varphi }_{\mathrm{ik}}-{\phi }_{\mathrm{ik}}-{\eta }_{\mathrm{ik}}\right)\right]-{\omega }_{\tau }\left[\cos \left({\varphi }_{\mathrm{ik}}\right)-\cos \left({\varphi }_{\mathrm{ik}}-{\eta }_{\mathrm{ik}}+{\psi }_{\mathrm{ik}}\right)\right\}\right]}^2\rangle {\theta }_0=\langle {\left[\sum _{i=1}^N\sum _{k=1}^N\Delta n_i\Delta n_k{B}_i{B}_k\cos \left({\varphi }_{\mathrm{ik}}\right)\right]}^2\rangle -{\langle \sum _{i=1}^N\sum _{k=1}^N\Delta n_i\Delta n_k{B}_i{B}_k\cos \left({\varphi }_{\mathrm{ik}}\right)\rangle }^2& \left(23\right)\end{array}$
Equations (23), (16), and (21) contain seven random variables characterizing each scatterer i=1, 2, . . . , N at instant t: namely, the coordinates x_i, y_i, z_i; the velocity components U_i, V_i, W_i; and the reflectivity Δn_i. Each one of these variables describes a physically different characteristic of the scatterer, therefore, they can be considered as statistically independent random values for the same and different scatterers. However, the velocity components can be correlated both for the same scatterer and different scatterers. One can see from definitions (16) and (21) that |φ_ik|=O(σ_r/λ, σ/D)>>1 are very large, |φ_ik|, |η_ik|=O(1) are of the unity order. One can also see from the definitions that |ψ_ik|, |ζ_ik|∝τ and |ψ_ik|, |ζ_ik|=O(τ)<<1, |ζ_ik|/|ψ_ik|=O(U_iτ/σ)<<1 are very small at τ→0. Using these considerations and applying lengthy but straightforward mathematical manipulations, the powered weighted increments (23) at very small τ→0 can be presented as follows:
Φ(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,τ,ω_x,ω_τ)=d₀(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,ω_x,ω_τ)
+η_τd₁(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,ω_x,ω_τ)
+Θ_τ²d₂(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,ω_x,ω_τ)+O(η_τ³), η_τ=τ/δt  (24)
The values Φ, d₀, d₁, d₂, and η_τ in this equation are non-dimensional, and the coefficients d₀, d₁, and d₂ for the pure signals are as follows:

[0093] $\begin{array}{cc}{d}_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,{\omega }_x,{\omega }_{\tau }\right)=2\left[{\left(1-{\omega }_x\right)}^2-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)\right]-\frac{2}{{\theta }_0}\sum _{i=1}^N\sum _{k=1}^N\left\{\langle \Delta n_i^2\rangle \langle \Delta n_k^2\rangle \langle B_i^2{B}_k^2[\left(1-{\omega }_x\right)\left(1-{\omega }_x-{\omega }_{\tau }\right)\cos ({\phi }_{\mathrm{ik}}+{\eta }_{\mathrm{ik}})+{\omega }_{\tau }\left(1-{\omega }_x\right)\cos \left({\phi }_{\mathrm{ik}}\right)-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)\cos \left({\eta }_{\mathrm{ik}}\right)]\rangle \right\}{\theta }_0=\sum _{i=1}^N\sum _{k=1}^N\langle \Delta n_i^2\rangle \langle \Delta n_k^2\rangle \langle B_i^2\rangle \langle B_k^2\rangle & \left(25\right)\\ d_1\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,{\omega }_x,{\omega }_{\tau }\right)=-\frac{2}{{\theta }_0}\sum _{i=1}^N\sum _{k=1}^N\{\langle \Delta n_i^2\rangle \langle \Delta n_k^2\rangle {B_i^2}_k^2\times {\psi }_{\mathrm{ik}}\left[\left(1-{\omega }_x-{\omega }_{\tau }\right)\sin \left({\phi }_{\mathrm{ik}}+{\eta }_{\mathrm{ik}}\right)+{\omega }_x{\omega }_{\tau }\sin \left({\phi }_{\mathrm{ik}}\right)-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)\sin \left({\eta }_{\mathrm{ik}}\right)\right]\rangle \}& \left(26\right)\\ d_2\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,{\omega }_x,{\omega }_{\tau }\right)=\frac{1}{{\theta }_0}\sum _{i=1}^N\sum _{k=1}^N\left\{\langle \Delta n_i^2\rangle \langle \Delta n_k^2\rangle \langle B_i^2{B}_k^2\times [\left(1-{\omega }_x-{\omega }_{\tau }\right){\left({\psi }_{\mathrm{ik}}-{\zeta }_{\mathrm{ik}}\right)}^2\cos \left({\phi }_{\mathrm{ik}}+{\eta }_{\mathrm{ik}}\right)+{\omega }_x{\omega }_{\tau }{\psi }_{\mathrm{ik}}^2\cos \left({\phi }_{\mathrm{ik}}\right)-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right){\psi }_{\mathrm{ik}}^2\cos \left({\eta }_{\mathrm{ik}}\right)-{{\omega }_x\left({\psi }_{\mathrm{ik}}-{\zeta }_{\mathrm{ik}}\right)}^2]\rangle \right\}& \left(27\right)\end{array}$
It is important to note that equations (24)–(27) are very generic. They are derived for a volume-scattering model using only assumption 1. It is also important that each coefficient in the decomposition (24)–(27) describes physically different characteristics of the monitored object. One can see from (16), (19), and (21) that for a particular case of atmospheric scatterers, d₀ depends on their size, shape, content and spatial distribution inside the predetermined volume V although it is fully independent of the scatterer's motion. It is shown below that d₁ depends on the mean speed of the atmosphere in the volume V although it is independent of turbulence. Atmospheric turbulence affects only coefficient d₂.

• [0094] Coefficient d₀ in Eq. (25) depends on the scatterer's reflectivity Δn_i(t) and parameters B_ik, φ_ik and η_ik. One can see from definitions (16) and (21) that parameters φ_ik and η_ik depend directly on coordinates x_i(t), y_i(t) and z_i(t) while the coordinates characterize the spatial distribution of scatterers inside the illuminated volume at instant t. In a general case, Eq. (25) also contains the instantaneous derivatives from Eq. (19) that are neglected in the given example of the profiling radar based system. The reflectivity Δn_i(t) and its derivatives ∂n_i(t)/∂t, ∂n_i(t)/∂λ, ∂n_i(t)/∂x_Am1 characterize the scatterer's size, shape, content, orientation, as well as the rate of changes in the above parameters. Therefore, coefficient do provides indicators for identifying the monitored object in the present system.

[0095] Eqs. (25), (16) and (21) cannot be used directly in practical measurements because they contain an infinite number of unknown variables: the instantaneous reflectivity Δn_i(t) and coordinates x_i(t), y_i(t), z_i(t) of each scatterer i=1, 2, . . . , N. To get an operational equation, one should make more specific assumptions, choose a more specific model for a monitored object, or both. As an example, one can consider a single thermal plume in the center of the illuminated volume with the Gaussian spatial distribution of scatterers inside the plume:

[0096] $\begin{array}{cc}{P}_{\mathrm{m2},\mathrm{m1},z}\left(x_i,z_i\right)=\frac{1}{2\pi {\sigma }_{\mathrm{m2},\mathrm{m1}}{\sigma }_z}\exp \left(-\frac{x_i^2}{2{\sigma }_{\mathrm{m2},\mathrm{m1}}^2}-\frac{z_i^2}{2{\sigma }_z^2}\right),i=1,2,\dots ,N& \left(28\right)\end{array}$
Here P_m1,m2,z(x_i,z_i) is the joint probability density distribution of the i^th scatterer location in the vertical direction and along an arbitrary horizontal baseline δ{right arrow over (x)}_m2,m1 considered in the present system. The values σ_m2,m1<<σ and σ_z<<σ_r are the plume widths in the directions δ{right arrow over (x)}_m2,m1 and z, respectively; σ_m2,m1 and σ_z are unknown parameters of the object to be determined in this example. One can combine equations (16), (21), (25) and (28) and derive the following equation:

[0097] $\begin{array}{cc}{d}_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,{\omega }_x,{\omega }_z\right)=2\left[{\left(1-{\omega }_x\right)}^2-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)-\left(1-{\omega }_x\right)\left(1-{\omega }_x-{\omega }_{\tau }\right)\exp \left(-\frac{\delta x_{\mathrm{m2},\mathrm{m1}}^2}{g_D^2}\frac{{\alpha }^2{\sigma }_{\mathrm{m2},\mathrm{m1}}^2}{{\sigma }^2}\right)\exp \left(-\frac{\delta {\lambda }^2}{g_{\lambda }^2}\frac{{\sigma }_z^2}{{\sigma }_r^2}\right)-{\omega }_{\tau }\left(1-{\omega }_x\right)\exp \left(-\frac{\delta x_{\mathrm{m2},\mathrm{m1}}^2}{g_D^2}\frac{{\alpha }^2{\sigma }_{\mathrm{m2},\mathrm{m1}}^2}{{\sigma }^2}\right)+{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)\exp \left(-\frac{\delta {\lambda }^2}{g_{\lambda }^2}\frac{{\sigma }_z^2}{{\sigma }_r^2}\right)\right]& \left(29\right)\end{array}$
Eq. (29) is a non-linear equation with respect to two unknown variables σ_m2,m1 and σ_z. The powered weighted increments allow splitting the equation into two independent equations for each unknown variable by choosing the appropriate weights ω_x and ω_τ. For example, one can assign the weight ω_x=1 and simplify Eq. (29) as follows:

[0098] $\begin{array}{cc}{d}_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,1,{\omega }_{\tau }\right)=2{\omega }_{\tau }^2\left[1-\exp \left(-\frac{\delta {\lambda }^2}{g_{\lambda }^2}\frac{{\sigma }_z^2}{{\sigma }_r^2}\right)\right]& \left(30\right)\end{array}$
It follows from Eq. (30) that the plume's width in the vertical direction σ_z can be estimated with the following expression:

[0099] $\begin{array}{cc}{\sigma }_z=\sqrt{-\ln \left[1-d_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,1,{\omega }_{\tau }\right)/\left(2{\omega }_{\tau }^2\right)\right]}\frac{g_{\lambda }}{\delta \lambda }{\sigma }_r& \left(31\right)\end{array}$
Eq. (31) is an example of an operational equation for the given example of the profiling radar based system. It analytically relates the selected characteristic of the monitored object, the plume width σ_z, to the coefficient d₀ in the decomposition (24) of measured powered weighted increments for a pair of multi-frequency sensors A_m1 and A_m2, (m2≠m1)=1, 2, . . . 6. Moreover, the right-hand-side of this equation is independent of δ{right arrow over (x)}_m2,m1 hence Eq. (31) can be directly applied to a single sensor at δ{right arrow over (x)}_m2,m1=0.

[0100] To estimate the plume width σ_m2,m1 in the direction δ{right arrow over (x)}_m2,m1 using Eq. (29), one can choose any combination of weights ω_x≠1 and ω_τ≠0 which satisfy the condition 1−ω_x−ω_τ=0. For example, the pair of weights ω_x=ω_τ=½ simplifies Eq. (29) as:

[0101] $\begin{array}{cc}{d}_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,\frac{1}{2},\frac{1}{2}\right)=\frac{1}{2}\left[1-\exp \left(-\frac{\delta x_{\mathrm{m2},\mathrm{m1}}^2}{g_D^2}\frac{{\alpha }^2{\sigma }_{\mathrm{m2},\mathrm{m1}}^2}{{\sigma }^2}\right)\right]& \left(32\right)\end{array}$
with the solution:

[0102] $\begin{array}{cc}{\sigma }_{\mathrm{m2},\mathrm{m1}}=\sqrt{-\ln \left[1-d_0\left(\delta {\stackrel{\rightharpoonup }{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,\frac{1}{2},\frac{1}{2}\right)\right]}\frac{\lambda }{\delta x_{\mathrm{m2},\mathrm{m1}}}\frac{R}{2\pi }& \left(33\right)\end{array}$
Eq. (33) is another example of an operational equation for the given example. It analytically relates the selected characteristic of the monitored object, the plume width σ_m2,m1, to the coefficient d₀ in the decomposition (24) of measured powered weighted increments for a pair of sensors A_m1 and A_m2, (m2≠m1)=1, 2, . . . 6. The width σ_m2,m1 of the thermal plume can be estimated along all available baselines (A₁,A₂), (A₂,A₃), (A₁,A₃) as well as (A₁,A₅), (A₂,A₆), (A₃,A₄) because d₀ does not depend on α in this case.

[0103] The values σ_m2,m1 and σ_z comprehensively characterize the size and shape of the plume although it is much smaller than the sizes σ and σ_r of the illuminated volume. This simple example illustrates the major advantages of the present system and method.

[0104] First, Eqs. (31) and (33) had been derived with the method of the present invention for a specified object given by Eq. (28) using the only one natural and generic assumption 1. Prior art methods would require a larger number of more restrictive assumptions for the derivation. Second, Eq. (31) can be used only at δλ≠0, that is with single or multiple multi-frequency sensors. One cannot measure the plume width in the vertical direction σ_z unless multi-frequency sensors are deployed. Third, one can get multiple estimates for each measured value σ_z and σ_m2,m1 by varying the weights ω_x and ω_τ. For example, numerous weights ω_τ≠0 can be chosen for getting multiple estimates of σ_z with Eq. (31). Similarly, one can choose numerous combinations of weights ω_x≠1 and ω_τ≠0 such as 1−ω_xω_τ=0 and get multiple operational equations for σ_m2,m1 for each baseline δ{right arrow over (x)}_m2,m1 similar to Eq. (33). The utility of multiple estimates and their practical applications are discussed below in the detailed description of step 106. The most important advantage of the present system and method is their flexibility and practical efficiency. Using the optimal combinations of weights ω_x and ω_τ in the powered weighted increments (9), one can get the most accurate, simple and practically efficient operational equation(s) for each predetermined characteristic of the monitored object such as Eqs. (31) and (33).

[0105] Similarly to Eq. (25), Eq. (26) for the coefficient d₁ contains an infinite number of unknown variables. In addition to the instantaneous reflectivity Δn_i(t) and coordinates x_i(t), y_i(t) and z_i(t), a generic Eq. (26) contains the instantaneous velocity components U_i(t), V_i(t) and W_i(t) for each scatterer i=1, 2, . . . , N. To turn Eq. (26) into an operational equation, one should make more specific assumptions, choose a more specific model for a monitored object, or both. For the given example of the atmospheric profiling radar based system, two rather generic and natural assumptions are sufficient for deriving the operational equation.

[0106] Assumption 2: The spatial distribution of scatterers inside the illuminated volume V is statistically uniform in any direction; that is:
P_m2,m1(x_i)=const, P_z(z_i)=const, i=1, 2, . . . , N  (34)

[0107] Assumption 3: All scatterers in the illuminated volume move with the same mean velocity components; that is:
U_i=U, V_i=V, W_i=W, i=1, 2, . . . , N  (35)
Note that assumption 2 replaces the model (28) that was used in the previous subsection. One can combine equations (16), (17), (21), (25), (26), (34), and (35) to derive the following expressions:

[0108] $\begin{array}{cc}{d}_0\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\mathrm{\delta \lambda },{\omega }_x,{\omega }_{\tau }\right)=2\left[{\left(1-{\omega }_x\right)}^2-{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)-\left(1-{\omega }_x\right)\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }-{\omega }_{\tau }\left(1-{\omega }_x\right)e_{\mathrm{m2},\mathrm{m1}}+{\omega }_{\tau }\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\lambda }\right]& \left(36\right)\\ e_{\mathrm{m2},\mathrm{m1}}=\exp \left(-\frac{\delta x_{\mathrm{m2},\mathrm{m1}}^2}{g_D^2}\right),e_{\lambda }=\exp \left(-\frac{{\mathrm{\delta \lambda }}^2}{g_{\lambda }^2}\right)& \\ d_1\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\mathrm{\delta \lambda },{\omega }_x,{\omega }_{\tau }\right)=-16\mathrm{\pi \gamma }\frac{\langle W\rangle \delta t}{D}\frac{{\sigma }_r}{\sigma }\frac{\mathrm{\delta \lambda }}{\lambda }\left(1-{\omega }_x-{\omega }_{\tau }\right)\left(e_{\mathrm{m2},\mathrm{m1}}-{\omega }_{\tau }\right)e_{\lambda }-8\frac{\langle U_{\mathrm{m2},\mathrm{m1}}\rangle \delta t}{g_D}\frac{\delta x_{\mathrm{m2},\mathrm{m1}}}{g_D}\left[\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\lambda }+{\omega }_x{\omega }_{\tau }\right]e_{\mathrm{m2},\mathrm{m1}}& \left(37\right)\end{array}$
One can see that coefficient d₁ depends only on the mean velocities <W> and <U_m2,m1> although it is independent of turbulence. One should recall that equations (19), (21), (23), and (25)–(27) are derived for the adaptive coordinate system with the x axis along the baseline δ{right arrow over (x)}_m2,m1. Notations σ_m2,m1 in Eqs. (29), (32), (33) and e_m2,m1, <U_m2,m1> in Eqs. (36), (37) as well as other similar notations below emphasize that the measured values are those along δ{right arrow over (x)}_m2,m1, for example, <U_m2,m1> is the projection of the mean velocity of atmospheric scatterers on the direction δ{right arrow over (x)}_m2,m1. Eq. (37) is a linear equation with two unknown variables <W> and <U_m2,m1>, therefore an unlimited number of the pairs of weights ω_x and ω_τ can be applied for getting multiple estimates for each of the unknowns. The most efficient are, however such pairs that provide separate equations for each of the unknown variables.

[0109] One can get reliable multiple estimates for the mean horizontal velocity <U_m2,m1> using multiple pairs of weights which satisfy the following conditions: (1−ω_x−ω_τ)(e_m2,m1−ω_τ)=0 and (1−ω_x−ω_τ)e_λ+ω_xω_τ≠0. For example, one can derive the following expression for a pair ω_x=ω_τ=½ using Eqs. (36) and (37):

[0110] $\begin{array}{cc}\langle U_{\mathrm{m2},\mathrm{m1}}\rangle =-\frac{d_1\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,1/2,1/2\right)}{2\left[1-2d_0\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\delta \lambda ,1/2,1/2\right)\right]}\frac{g_D}{\delta x_{\mathrm{m2},\mathrm{m1}}}\frac{g_D}{\delta t}& \left(38\right)\end{array}$
Eq. (38) is another example of an operational equation for the given example of implementing the present invention. It analytically relates the selected characteristic of the monitored object, a projection of the mean velocity of atmospheric scatterers on the direction δ{right arrow over (x)}_m2,m1 to the coefficients d₀ and d₁ in the decomposition (24) of measured powered weighted increments for a pair of sensors A_m1 and A_m2, (m2≠m1)=1, 2, . . . 6. One can measure projections of the mean horizontal velocity on different baselines δ{right arrow over (x)}_m2,m1, (m2≠m1)=1, 2, . . . 6 with the present system. Any two non-parallel baselines are sufficient for estimating the mean horizontal velocities <U> and <V> in any specified coordinate system, for example, a geophysical system with the x axis extending towards the east and the y axis extending towards the north. With any two non-parallel baselines, one can therefore estimate the mean horizontal speed <V_h> which is invariant of the coordinate system:
V_h=√{square root over (U²+V²)}  (39)
One can consider the baselines (A₁,A₂), (A₂,A₃), (A₁,A₃) and (A₄,A₅), (A₅,A₆), (A₄,A₆). However, one cannot apply Eqs. (38) to mixed pairs like (A₁,A₅), (A₂,A₆), or (A₃,A4). To simplify the equations, the same value of α within a pair was considered above; equations for different α can be easily derived in a similar way.

[0111] One can also get reliable multiple estimates for the mean vertical velocity <W> using multiple pairs of weights which satisfy the opposite conditions, namely (1−ω_x−ω_τ)(e_m2,m1−ω_τ)≠0 and (1−ω_x−ω_τ)e_λ+ω_xω_τ=0. For example, at ω_x=e_λ/(1+e_λ) and ω_τ=e_λ one can obtain from Eq. (37) the following expression:

[0112] $\begin{array}{cc}\langle W\rangle =-\frac{d_1\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\mathrm{\delta \lambda },e_{\lambda }/\left(1+e_{\lambda }\right),1/e_{\lambda }\right)}{\begin{array}{c}8\mathrm{\pi \gamma }{e}_{\lambda }\left(1+e_{\lambda }\right)d_0(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\\ \mathrm{\delta \lambda }+\mathrm{\delta \lambda },e_{\lambda }/\left(1+e_{\lambda }\right),1/e_{\lambda })\end{array}}\frac{{\sigma }_r}{\sigma }\frac{\lambda }{\mathrm{\delta \lambda }}\frac{D}{\delta t}& \left(40\right)\end{array}$
Another important example is the estimation of the mean vertical velocity <W> using a single multi-frequency sensor. In the case of δ{right arrow over (x)}_m2,m1=0 and ω_x=0, using Eq. (37) one can derive the following expression:

[0113] $\begin{array}{cc}\langle W\rangle =-\frac{d_1\left(0,\lambda ,\lambda +\mathrm{\delta \lambda },0,{\omega }_{\tau }\right)}{8\mathrm{\pi \gamma }\left[2{\left(1-{\omega }_{\tau }\right)}^2-d_0\left(0,\lambda ,\lambda +\mathrm{\delta \lambda },0,{\omega }_{\tau }\right)\right]}\frac{{\sigma }_r}{\sigma }\frac{\lambda }{\mathrm{\delta \lambda }}\frac{D}{\delta t}& \left(41\right)\end{array}$
Eqs. (40) and (41) are two more examples of operational equations for the given example of implementing the present invention. The equations analytically relate the selected characteristic of the monitored object, the mean vertical velocity <W>, to the coefficients d₀ and d₁ in the decomposition (24) of measured powered weighted increments for a pair of sensors A_m1 and A_m2, (m2≠m1)=1, 2, . . . 6, and for a single sensor, respectively. It is important that the mean vertical velocity can be measured only at δλ≠0, that is only possible with single or multiple multi-frequency sensors. It is also important that Eqs. (40) and (41) for multiple and single multi-frequency sensors do not have the radial velocity ambiguity (that is, ambiguity of the vertical velocity for the case of profiling radars). The ambiguity is inherent in measuring the radial velocity in frequency domain using the Doppler spectra. The present system and method operate in the temporal domain and, therefore are unaffected by the periodicity.

[0114] The example further illustrates the major advantages of the present system and method. First, Eqs. (38), (40), and (41) had been derived with the method of the present invention using natural and generic assumptions 1, 2 and 3. Prior art methods would require a larger number of more restrictive assumptions for the derivation. For example, the most advanced correlation function-based technique for spaced antenna radars, the Holloway-Doviak method requires eight significantly more restrictive assumptions for deriving the only one operational equation for a single pair of sensors. Second, Eqs. (40) and (41) can be used only at δλ≠0, that is with multi-frequency single or multiple sensors. One cannot use these or similar equations for measuring <W> unless multi-frequency sensors are deployed. Third, one can get multiple estimates for each measured value <U_m2,m1> and <W> by varying the weights ω_x and ω_τ. The most important advantage of the present system and method is their flexibility and practical efficiency. Using the optimal combinations of weights ω_x and ω_τ in the powered weighted increments (9), one can get the most accurate, simple and practically efficient operational equation(s) for each predetermined characteristic of the monitored object such as Eqs. (38) and (40), (41).

[0115] The rates of changes depend on the monitored object and specific equations for measuring the rates also depend on the type and configuration of monitoring equipment. For the given example of the atmospheric profiling radar based system, the monitored object is the atmosphere in the predetermined volume, and the rates mainly depend on atmospheric turbulence that changes the relative location of scatterers with respect to each other. One can see from equations (29), (36), and (37) that turbulence does not affect coefficients d₀ and d₁ in decomposition (24). On the contrary, coefficient d₂ is heavily dependent on turbulence. To derive an operational equation for d₂, it is sufficient to apply assumptions 1, 2, 3, and one more rather generic assumption.

[0116] Assumption 4: The statistical characteristics of turbulent motion are the same for all scatterers in the illuminated volume; that is:
u_i²=u², v_i²=v², w_i²=w²
u_iv_i=uv, u_iw_i=uw, v_iw_i=vw, i=1, 2, . . . , N  (42)
One can combine equations (16), (17), (21), (26), (34), (35), (39), and (42) to derive the following expression:

[0117] $\begin{array}{cc}\begin{array}{cc}{d}_2\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\mathrm{\delta \lambda },{\omega }_x,{\omega }_{\tau }\right)=8\left[\left(4{\pi }^2\frac{\langle w^2\rangle \delta t^2}{{\lambda }^2}+\frac{{\langle V_h\rangle }^2\delta t^2}{g_D^2}\right){\mu }_w-\frac{\left(2{\langle U_{\mathrm{m2},\mathrm{m1}}\rangle }^2+\langle u_{\mathrm{m2},\mathrm{m1}}^2\rangle \right)\delta t^2}{g_D^2}\frac{\delta x_{\mathrm{m2},\mathrm{m1}}^2}{g_D^2}{\mu }_u-2\mathrm{\pi \alpha }\frac{\langle u_{\mathrm{m2},\mathrm{m1}}w\rangle \delta t^2}{\mathrm{\sigma \lambda }}\frac{\delta x_{\mathrm{m2},\mathrm{m1}}}{g_D}{\mu }_{\mathrm{uw}}\right]{\mu }_w={\mu }_u-{\omega }_{\tau }\left(1-{\omega }_x-w_{\tau }\right)e_{\lambda }-{\omega }_x,{\mu }_u=\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }+{\omega }_x{\omega }_{\tau }{e}_{\mathrm{m2},\mathrm{m1}}{\mu }_{\mathrm{uw}}=\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }-{\omega }_x& \end{array}& \left(43\right)\end{array}$
The mean horizontal velocities <V_h> and <U_m2,m1> are estimated independently using coefficients d₀ and d₁ hence Eq. (43) contains only three unknown values: <w²>, u^m2,m12, and u_m2,m1w. One can get multiple estimates for all characteristics of atmospheric turbulence by applying Eq. (43) to different pairs of sensors A_m1, A_m2 and varying the weights ω_x and ω_τ for each selected pair. Numerous combinations of ω_x and ω_τ provide numerous equations for estimating the unknowns. However, a blind approach is inefficient and its accuracy could be extremely poor. One can see that the right-hand-side terms in Eq. (43) dramatically differ in magnitude. The first two terms with a multiplier μ_w are much larger than the third one with a multiplier μ_u, and the third term is much larger than the fourth with a multiplier μ_uw. As demonstrated below, the powered weighted increments enable one to overcome this problem by selecting specific weights ω_x and ω_τ for each selected single sensor A_m1 or pair of sensors A_m1, A_m2 and to ensure the most efficient and accurate measurement of each characteristic of turbulence.

[0118] For example, the variance of the vertical turbulent velocity <w²> can be accurately estimated using each single sensor separately; that is at δ{right arrow over (x)}_m2,m1=0, e_m2,m1=1. One can then estimate <w²> with the following equation:

[0119] $\begin{array}{cc}\langle w^2\rangle =\frac{d_2\left(0,\lambda ,\lambda +\mathrm{\delta \lambda },{\omega }_x,{\omega }_{\tau }\right)}{32{\pi }^2\left(1-{\omega }_{\tau }\right)\left[\left(1-{\omega }_x-{\omega }_{\tau }\right)e_{\lambda }-{\omega }_x\right]}\frac{{\lambda }^2}{\delta t^2}-\frac{{\lambda }^2}{4{\pi }^2{g}_D^2}{\langle V_h\rangle }^2& \left(44\right)\end{array}$
The efficient pairs of weights for estimating <w²> with Eq. (44) are, for example ω_x=0, ω_τ=−1 and ω_x=1, ω_τ=0.
To get reliable and accurate estimates for u^m2,m12, one should get rid of the first two terms in the right-hand-side of Eq. (43). It can be easily achieved by selecting the weights that satisfy two conditions: μ_w=0 and μ_u≠0 which lead to the following equations:

[0120] $\begin{array}{cc}{\omega }_x=\frac{\left(1-{\omega }_{\tau }\right)\left(e_{\mathrm{m2},\mathrm{m1}}-{\omega }_{\tau }\right)e_{\lambda }}{1-{\omega }_{\tau }{e}_{\lambda }-{\omega }_{\tau }{e}_{\mathrm{m2},\mathrm{m1}}+e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }},\omega \ne \frac{\left(1-{\omega }_{\tau }\right)e_{\lambda }}{e_{\lambda }-{\omega }_{\tau }}& \left(45\right)\end{array}$
One can get multiple reliable and accurate estimates of u^m2,m12 with any pair of weights ω_x and ω_τ satisfying equations (45) because the third term in Eq. (43) is much larger than the fourth one. The most efficient is, however the choice of pairs of weights that give μ_u≠0, μ_w=0, and at the same time μ_uw=0, for example, ω_τ=0 and ω_x=e_De_λ/(1+e_De_λ). Using these weights in Eq. (43), one can obtain the following expression:

[0121] $\begin{array}{cc}\langle u_{\mathrm{m2},\mathrm{m1}}^2\rangle =-2{\langle U_{\mathrm{m2},\mathrm{m1}}\rangle }^2-\frac{\begin{array}{c}\left(1+e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }\right)d(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\\ \mathrm{\delta \lambda },0,e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }/\left(1+e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }\right))\end{array}}{8e_{\mathrm{m2},\mathrm{m1}}{e}_{\lambda }}\frac{g^2}{\delta x_{\mathrm{m2},\mathrm{m1}}^2}\frac{g^2}{\delta t^2}& \left(46\right)\end{array}$
One can apply Eq. (46) to any three non-parallel baselines, for example (A₁,A₂), (A₂,A₃), (A₁,A₃) and (A₄,A₅), (A₅,A₆), (A₄,A₆), and get estimates for the variances of the horizontal turbulent velocities <u²>, <v²> and the horizontal momentum flux <uv> with the standard trigonometric relations in any specified Cartesian coordinate system.

[0122] The most difficult task is to estimate a projection of the vertical flux because the term with u_m2,m1w in the right-hand-side of Eq. (43) is much smaller than the others. The only way of getting reliable and accurate estimates is to get rid of all terms in the right-hand-side of Eq. (43) except for the last one. To do so, one should select such weights ω_x and ω_τ that would satisfy three conditions: μ_w=0, μ_u=0, and μ_uw≠0.

The conditions can be satisfied by choosing the weights ω_τ and ω_x which satisfy the equations:

[0123] $\begin{array}{cc}{\omega }_x=\frac{\left(1-{\omega }_{\tau }\right)e_{\lambda }}{e_{\lambda }-{\omega }_{\tau }},{\omega }_{\tau }^2=1,{\omega }_x\ne 0\mathrm{at}{\omega }_{\tau }=1& \left(47\right)\end{array}$
The equations are satisfied at ω_τ=−1 and ω_x=2e_λ/(1+e_λ) which lead to the following expression:

[0124] $\begin{array}{cc}\langle u_{\mathrm{m2},\mathrm{m1}}w\rangle =\frac{\left(1+e_{\lambda }\right)d_2\left(\delta {\overrightarrow{x}}_{\mathrm{m2},\mathrm{m1}},\lambda ,\lambda +\mathrm{\delta \lambda },-1,2e_{\lambda }/\left(1+e_{\lambda }\right)\right)}{32\mathrm{\pi \alpha }\left(1-e_{\mathrm{m2},\mathrm{m1}}\right)e_{\lambda }}\frac{g_D}{\delta x_{\mathrm{m2},\mathrm{m1}}}\frac{\mathrm{\sigma \lambda }}{\delta t^2}& \left(48\right)\end{array}$
One can apply Eq. (48) to any two non-parallel baselines and get multiple reliable and accurate estimates for the variances of the vertical momentum fluxes <uw> and <vw> with the standard trigonometric relations in any specified Cartesian coordinate system.

[0125] Eqs. (44), (46), and (48) are other examples of operational equations for the given example of implementing the present invention. The equations analytically relate the selected characteristics of the monitored object, namely, the characteristics of atmospheric turbulence <w²>, u^m2,m12, and u_m2,m1w to the coefficients d₀, d₁, and d₂ in the decomposition (24) of measured powered weighted increments for a single multi-frequency sensor A_m1 or a pair of multi-frequency sensors A_m1 and A_m2, (m2≠m1)=1, 2, . . . 6. These equations clearly demonstrate two remarkable features that distinguish the present, powered weighted increments-based method from all prior art methods: one can get numerous operational equations for each multi-frequency sensor A_m1 or pair of sensors A_m1, A_m2 by varying the weights ω_x and ω_τ, and one can select specific weights for each selected sensor or pair of sensors for the most efficient and accurate measurement of each predetermined characteristic of the monitored object.

[0126] Operational equations (31), (33), (38), (40), (41), (44), (46), and (48) were derived for pure signals from atmospheric scatterers in the illuminated volume. The right-hand-side of these equations contain characteristics of the atmosphere that are intended to be estimated with the present system and method for measuring characteristics of continuous medium and/or localized targets. The equations contain coefficients d₀, d₁, and d₂ of the powered weighted increments of the pure signal power with no noise or other contaminants; the latter are always present in the actual and combined signals. For the equations to be applicable to practical measurements, one should relate the powered weighted increments for pure signals (22) to those for signals with noise (9) for all deployed wavelengths λ_Ti (i=1, 2, . . . , K).

[0127] It is well known that any differential data processing method is not affected by clutter, low-frequency radio interference, or other contaminants with sufficiently large temporal scale T_cor. One can see from equations (2) and (3) that the powered weighted increments belong to the class of differential processing tools and, therefore are unaffected by contaminants with large T_cor>>δt. However, differential tools such as structure functions and powered weighted increments are highly sensitive to white noise with very small or zero temporal scale. For this reason, only white noise n({right arrow over (x)}_Am,t) in Eq. (7) is considered below, and the standard assumption about such noise is the following.

[0128] Assumption 5: The pure received signal S({right arrow over (x)}_Am, λ, t) and the noise n({right arrow over (x)}_Am, λ, t) are uncorrelated.

This assumption and a definition of white noise can be formalized for any pair of deployed wavelengths λ=λ_Ti and λ+δλ=λ_Tj, δλ=λ_Tj−λ_Ti (i,j=1, 2, . . . , K) as follows:

[0129] $\begin{array}{cc}\langle S\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t+\tau \right)\rangle =\langle S\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda +\mathrm{\delta \lambda },t+\tau \right)\rangle =\langle S\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am2}},\lambda ,t+\tau \right)\rangle =\langle S\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am2}},\lambda +\mathrm{\delta \lambda },t+\tau \right)\rangle =0\mathrm{at}\mathrm{any}\tau \langle n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t+\tau \right)\rangle =\langle n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda +\mathrm{\delta \lambda },t+\tau \right)\rangle =\langle n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)\rangle =\langle n\left({\overrightarrow{x}}_{\mathrm{Am1}},\lambda ,t\right)n\left({\overrightarrow{x}}_{\mathrm{Am2}},\lambda +\mathrm{\delta \lambda },t+\tau \right)\rangle =0\mathrm{at}\tau \ne 0& \left(49\right)\end{array}$
For multi-frequency sensors operating at close frequencies |δλ|<<λ, one can also assume that:
n²({right arrow over (x)}_Am1,λ)=n²({right arrow over (x)}_Am1,λ+δλ)=n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am1,λ+δλ)
n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)=n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ+δλ)  (50)
One can combine equations (7), (22), (49), and (50) with Eq. (9) at p=2 to derive the following expression where δ(τ) is the Kroneker function:
σ_Am1²Φ(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,τ,ω_x,ω_τ)={tilde over (Φ)}(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,τ,ω_x,ω_τ)
−(1+ω_x²+2ω_τ²−2ω_x−2ω_τ+2ω_xω_τ)n²({right arrow over (x)}_Am1,λ)−(1+ω_x²)n²({right arrow over (x)}_Am2,λ)
+2δ(τ)(ω_x²+ω_x+ω_τ+ω_xω_τ)n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)  (51)
Eq. (51) relates the powered weighted increments of pure signals to those of signals with noise; the latter can be calculated directly for received signals with Eq. (9). However, the equation contains three new unknown variables: the second order moments of noise n²({right arrow over (x)}_Am1,λ), n²({right arrow over (x)}_Am2,λ) and the correlation n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ). One can see that all terms with noise in Eq. (51) are independent of τ hence they contribute only to the coefficients {tilde over (d)}₀(δ{right arrow over (x)}_m2,m1, λ, λ+δλ, ω_x, ω_τ) in Eq. (10) while they affect neither coefficients {tilde over (d)}₁(δ{right arrow over (x)}_m2,m1, λ, λ+δλ, ω_x, ω_τ). It follows from Eqs. (9) and (49)–(51) that:
{tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,λ,λ+δλ,ω_x,ω_τ)=σ_Am1²d₀(δ{right arrow over (x)}_m2,m1,λ, λ+δλ,τ,ω_x,ω_τ)
+(1+ω_x²+2ω_τ²−2ω_x−2ω_τ+2ω_xω_τ)n²({right arrow over (x)}_Am1,λ)+(1ω_x²)n²({right arrow over (x)}_Am2,λ)
−2δ(τ)(ω_x²+ω_x+ω_τ+ω_xω_τ)n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)  (52)
It follows from a generic Eq. (25) that d₀≡0 at ω_x=1 and δλ=0 which gives:
{tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,═,λ,1,ω_τ)==2ω_τ²n²({right arrow over (x)}_Am1,λ)+2n²({right arrow over (x)}_Am2,λ)
−4δ(τ)(1ω_τ)n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)  (53)
One can get numerous independent linear equations for n²({right arrow over (x)}_Am1,λ), n²({right arrow over (x)}_Am2,λ) and n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ) by varying the weight ω_τ in Eq. (53). For example, the equations at ω_τ=1, 0, and −1 are as follows:
n²({right arrow over (x)}_Am1,λ)+n²({right arrow over (x)}_Am2,λ)−4n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)
={tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,λ,λ,1,1)/2, τ=0
n²({right arrow over (x)}_Am1,λ)+n²({right arrow over (x)}_Am2,λ)={tilde over (d)}₀(δ{right arrow over (x)}m2,m1,λ,λ,1,1)/2, τ≠0
n²({right arrow over (x)}_Am2,λ)−2n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ)={tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,λ,λ,1,0)/2, τ≠0
n²({right arrow over (x)}_Am2,λ)={tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,λ,λ,1,0)/2, τ≠0
n²({right arrow over (x)}_Am1,λ)+n²({right arrow over (x)}_Am2,λ)={tilde over (d)}₀(δ{right arrow over (x)}_m2,m1,λ,λ,1,−1)/2, τ=0, τ≠0  (54)
Therefore, one can get five independent linear equations (54) from Eq. (53) using just three weights ω_τ=1, 0, and −1. These equations are more than sufficient for reliably estimating three unknown values n²({right arrow over (x)}_Am1,λ), n²({right arrow over (x)}_Am2,λ) and n({right arrow over (x)}_Am1,λ)n({right arrow over (x)}_Am2,λ) for a pair of sensors A_m1 and A_m2. Eqs. (54) can be directly applied to a single sensor A_m1 at {right arrow over (x)}_Am1={right arrow over (x)}_Am2.