Analysis of Radar Derived Wind Velocity with Ocean Interference using 2D Imaging Techniques

EE262 Two-Dimensional Imaging Final Project Report

Tao Chu Huimin Huang

Hyatt Errol Moore IV Dept of Electrical Engineering

Stanford University Stanford, CA 94305

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Table of Content:

I. Abstract………………………………………………………………………….3

II. Introduction and background……………………..……………………………..4 III. Mathematical basis for implementation…………………………………………6

IV. Simulation and results…...……………………………………………..………16

V. Discussion of error……………………………………………………………..30 VI. Conclusion……………………………………………………………………...32 VII. Reference……………………………………………………………………….33

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I. Abstract

Airborne radar is frequently used over the ocean for measurement of wind velocity from the Doppler shift of rain droplets. In the case of wind velocity measurements over the ocean, a problem of ocean surface interference arises in the Doppler measurements near the surface of the ocean. In this region, wind field estimates from the rain droplets may be contaminated by contributions of scattering from the ocean surface. This contamination occurs as a natural consequence of the limited directivity of the radar antenna and the large difference in power between ocean surface and the rain echo. In this project we will simulate the radar measurements seen by an aircraft in this situation and develop a novel solution to the interference problem. We hope to analyze the affects of the interference, how accurate our model is, and how well our solution works to recreate the true wind speed we have simulated. Topics that we intend to use covered in this course include: FFT’s for Doppler shift modeling, ring impulse sampling and interpolation of sample data, Fraunhoffer approximation, and coherent imaging (due to Doppler estimation). We attempt to solve the problem using a comprehensive adaptive signal processing approach, by first simulating the true wind data and ocean interference using three dimensional data. This data will be converted to raw radar data that will be summed and then sampled, and then Fourier transformed to Doppler information to be processed to wind speed. The processed wind speed will be compared to the true wind speed from our simulation to as a gauge to the accuracy of our proposed technique.

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II. Introduction and Background Airborne radar is frequently used over the ocean for measurement of wind velocity from the Doppler shift of rain droplets. This type of instruments can provide high resolution wind measurements, and are useful for studying the inner structures of atmospheric phenomena. There is, however, a substantial problem in this method that arises near the ocean’s surface. The problem of ocean surface interference is brought about due to Doppler measurements near the surface of the ocean. In this region, wind field estimates from the rain droplets may be contaminated by contributions of scattering from the ocean surface. This contamination occurs as a natural consequence of the limited directivity of the radar antenna and the large difference in power between ocean surface and the rain echo.

In this project we will simulate measurements observed by such a radar system, and determine the corresponding wind speed and direction gathered from our modeled atmosphere. We will then analyze the effects of the interference due to noise and ocean interference and then attempt to remove this interference using a least squares optimized solution. We will conclude with a summary of results. Pulsed Doppler radar systems are common instruments for measurement of meteorological targets such as clouds and rain drops and are frequently used as moving target detection systems. In this problem, the collection of rain drops are the moving targets, they act as the tracers of the wind. Measuring the Doppler frequency of the rain drops allows estimation of the two horizontal wind components. Vertical wind components are not addressed in this project. The received spectrum is Doppler shifted proportionally to the line of sight, or radial velocity between the radar and the target. Radial velocity is defined as the rate of change of range with respect to time along the radar beam. The Doppler radar can measure the Doppler frequency shift (defined below) and therefore determine the radial velocity of the target. The quantity we measure is known as the mean Doppler velocity, which is the first moment or the mean of the Doppler spectrum. This informs us how fast an object is coming toward(+fD) or going away(-fD) with respect to the radar.

Using radial velocity measurements, Vr, in order to retrieve two components of horizontal wind field, we need multiple measurements of Vr, this technique is known as the Velocity Azimuth Display or simply VAD and will be discussed in further detail in the next section. This technique is straight forward and used quite often in the meteorological community. In order to make our simulations feasible and time well spent, we have simplified our problem from the real world situation to a more manageable version with the following assumptions:

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First, we assume that no roll, pitch, or yaw is introduced in the aircraft motion, and that the aircraft travels in a straight line. Further, we assume perfectly circular scan for a pencil beamed antenna pattern pointed an angle off nadir. In practice, the aircraft does exhibit changes in roll, pitch, and yaw and the radar does not make perfect circular scans. Secondly rain droplets are assumed to all uniform in size and carried by wind that only traverses the horizontal plane. In reality, the rain droplets vary in size and thus have different radar returns. Furthermore, there is vertical component to the wind that does exist in practice, at times quite significant (though not seen as significant of a factor under normal circumstances esp. when compared to the horizontal wind components). Thirdly, we assume the return from the ocean surface is due to the Bragg scattering mechanism only and that the ocean surface is relatively flat from the point of view of the radar. Typically, one cannot assume that ocean surface to be flat, as it exhibits dynamic motion due to gravity and longitudinal waves, where radar returns are further distorted by the aircraft’s changing elevation and orientation. Lastly, wind speeds near the aircraft are assumed to be uncorrelated with the interference caused by the ocean surface. In practice, the ocean’s motion will be somewhat correlated with that of the wind at the ocean surface, and that the wind at each altitude is correlated with the wind interference above and below it. However, we found it reasonable to assume that any perceivable correlation to the ocean’s surface to that of the wind speed will have disappeared at the plane’s altitude (~3000m).

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III. Mathematical basis for implementation The hypothetical Radar system to be used to address our problem would be a C-band (5GHz) high resolution airborne system designed to measure wind profiles in the intervening atmosphere as shown in Figure 1. The radar has a single slant beam off nadir and it conically scans the volume below as the plane moves forward. Its data acquisition capabilities would allow it to obtain I & Q data or digitized version of the complex amplitude the antenna observes of the returned radar echo. From the raw data it is then possible to obtain the Doppler spectra then calculation of wind speed and direction. Table 1 highlights the important radar system parameters.

Figure 1. Geometry for the proposed problem

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For simulation purposes we approach the problem as three cases with increasing complexity:

Case 1: This is the simplest case where the antenna has perfect directivity (no contamination from the side lobes) there would be no interference from the sea surface with our measurements. Case 2: The antenna has a circular aperture and hence the measurements from the main lobe and side lobes beamwidth have to be taken into account. Case 3: There is interference from the ocean surface in the measurement.

Fixed parameters Incidence Angle (θ) 30o Radar wavelength (λ = c/f) 6 cm Pulse Repitition Frequency (PRF) 18.432 KHz Range Resolution (c*tau/2) 34.64 m Pulse width (tau) 230 ns Radar beamwidth (HPBW) 3o Frequency of scanning (rotation/minute) 60 RPM Variable parameters Azimuth scan angle (wrt nose of aircraft) φ Altitude (ocean surface h = 0) h

Table 1. Summary of key radar parameters

The sea surface is taken to be at altitude = 0 m. We take 100 measurements at intervals of ~35 m each. Hence, the range of altitude (different from slant range) is from 0 – 3000 m. The aircraft is assumed to be flying at a high enough altitude such that the Fraunhoffer approximation is applicable to these measurements.

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Case 1: In this scenario, we assume that the effects of the interference from the ocean surface are negligible. We also assume that the main lobe is a perfect delta function and that the side lobes also have negligible contributions.

Figure 2. Geometry for Case 1

The antenna aperture is:

Intensity amplitude of the antenna could be found using the Fraunhoffer approximation with the Fourier Transform relationship :

The intensity pattern of the antenna in terms of the azimuth angle is:

Let a = 1

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Figure 3. Antenna pattern of rect(r) aperture

Antenna with perfect directivity with no sidelobes:

Figure 4. Simplified antenna pattern for analysis in Case 1

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Within the sampled volume of the pencil beam, we designated a total of 100 range gates. For each range gate, the complex amplitude from the sum of return of all rain drops within the range gate volume is captured by the antenna. Altitude, h extends from 0m – 3000m. Since the antenna is mounted at a fixed angle, θ = 30o, the range distance, r varies from 0m –3464m. This is evaluated using the relation: Range distance, r = h/cos θ = h/cos(30) = 1.1547h. We model the distribution of wind velocity using two Gaussian distributions: one for wind speed and the other for wind direction. At each altitude, these distributions share the same mean, and Gaussian variations occur over angle φ. The mean at different altitudes is calculated using a linear relation: mean(h) = m*h + c, where m and c are constants. We sample this wind velocity distribution using a ring impulse to get a sifted distribution b(h,φ). Taking the mean of this distribution b(h,φ) gives us the complex amplitude r at a given h (a value between 0 - 3000m) and φ (a value between 0 – 360o). Taking measurements along the pencil beam at a fixed φ gives a vector of complex amplitude r, with each element corresponding to the complex amplitude at each of the 100 range gates: Complex amplitude at fixed φ, r = [ r1 r2 … r100] The antenna is rotated to collect 18,432 range measurements for φ = 0 o to 360 o, i.e. we get one complex amplitude vector r for every 0.0195 o increment. The resultant complex amplitude matrix is 100 x 18,432: Matrix of complex amplitudes, C = [ r1 r2 … r18,432 ], where r1 corresponds to the vector r at φ = 0. See Figure 5 for this matrix.

Figure 5 Matrix of complex amplitudes and its FFT

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Doppler frequency, fD, is a quantity that depends on the phase difference of at least two complex amplitude signals. One way to obtain fD is by taking FFT of a time series data within a relatively small spatial variation in φ and observe the resulting spectrum, the first moment or the mean of this Doppler spectrum is the radial Doppler velocity. A reasonable size for taking FFT is for each 5o increments in φ which translates to a 256 data points for 72 of 5o chunks for each altitude. Choosing 256 point FFT is for a good balance between achieving fine spectral resolution and low quantized noise. First, take the 256-point discrete Fourier transform (DFT)/FFT across each segment of the measurement matrix. The frequency that corresponds to the peak of the spectrum gives us the Doppler frequency, see Figure 5. We thus obtain a matrix of Doppler frequencies, D, which is 100 x 72 shown in Figure 6.

Figure 6 Matrix of Doppler frequencies and VAD

This Matrix D allows us to plot the variation of fD with azimuth angle φ for each altitude See Figure 6. The shape of this Doppler velocity curve within a 360o scan follows a sinusoidal behavior considering a homogeneous wind field per altitude is across all φ Based on this graph, we are able to find the wind speed and wind direction for each altitude. This method for retrieving 2-D wind profile is called the Velocity Azimuth Display (VAD) where:

Wind direction = 360 – Angle corresponding to maximum fD = 360 – φmax Wind speed = (λ/2)fDmax

We are thus able to create a vertical profiles of wind speed and wind direction versus altitude using these relationships.

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Case 2: In Case 1, we considered an ideal case for the antenna pattern and neglected contributions from the non-ideal antenna main lobe and side lobe contributions. However, in a more realistic case, the complex amplitudes that are measured are indeed, weighted sums of the back-scattered amplitudes from the main lobe and all its side lobes, see Figure 7. For this case we continue to assume that interference from the sea surface is negligible that all radar measurements are radar return that of the atmosphere rain droplet backscatters.

Figure 7. Geometry for Case 2 & Case 3

The antenna pattern now is more complex than in Case 1:

Figure 8. Antenna pattern for Case 2 and 3

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Case 3: In this final case, we account for interference from the ocean surface. This contribution is modeled using an exponent that decreases with increasing altitude, h:

Ocean surface interference = , where a is a constant, R= radial distance. Because of this ocean surface effect, the matrix of complex amplitudes, C, that we measure includes the back-scatter from the ocean surface. The effects get greater as we approach lower altitudes as illustrated below:

Figure 10. Effects of sea surface interference on complex amplitude measurements Because an incorrect fD is obtained, this affects the result of wind speed and direction using the VAD method, especially for the lower altitudes namely below h = 300 m. To rectify this, we will do a cross-correlation of the complex quantity data taken from low altitudes, where interference from the sea surface is significant, with modified data taken from a high altitude, where interference from the sea surface is negligible. This data is modified according to hypothesized relations between wind speed and altitudes. Since the back scatter from the rain drops is uncorrelated with the back scatter from the sea surface, taking the cross-correlation will attenuate the effects of the sea surface interference. Using the assumption given that the interference from the ocean was uncorrelated with the wind closes to the aircraft an optimization filter based on the autocorrelation of the data can be built so as to optimize the error in the least mean square sense. The least mean square (LMS) algorithm will be used as the method of choice for solving this problem. The LMS algorithm makes an assumption about the mean square error. Instead

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of collecting all of the data and forming an autocorrelation matrix to solve for a quadratic (least squares) solution directly, it makes an estimate of the performance/quadratic surface at each step. This estimate, an error defined as the difference between the input and the desired output, is used to adaptively adjust filter weights which are linearly combined with the input data. This is a practical method to use in a real time systems. A diagram of the of the LMS adaptive linear combiner as a traversal filter is shown in Figure 11. The error ε, the difference between the desired response d and the filter output is defined as

Where the subscript k represents the kth sample. X is a column vector representing the input an W is the corresponding weight vector that is multiplied by the input components of X. The weights are adaptively updated at each iteration by approximating the gradient of the assumed quadratic surface (which is a necessary assumption in order to optimize it in the least squares sense). The approximation to the gradient at iteration k is defined as

The weights are then updated at each iteration according to

where µ regulates the step size and directly impacts the speed and stability of convergence.

w0k   w1k   wLk  

Input  xk  

Desired  Response  dk  

Error  εk  

+  +  

Output  yk  

…  

+  

+  

Z-­‐1   Z-­‐1   Z-­‐1  

     

∑  ∑  

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Figure 11. LMS adaptive linear combiner Assuming that the wind speed near the aircraft is uncorrelated with the interference seen at the ocean’s surface, it is possible to use this wind data as the desired data and the data nearest the ocean’s surface as the input data. The adaptive filter adjusts its filter weights in order to minimize the mean square error between data with the highest concentration of ocean interference and that of the lowest concentration of ocean interference. Upon convergence, the error produced in this case will just be ocean interference which could not adapt to the uncorrelated wind data at the higher elevation. This error data is a close approximation to the ocean interference at the surface level which and is thus subtracted from the sampled data at the surface level to obtain a better estimate of the true wind speed.

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IV. Simulation and Results Due to the large number of ranges simulated, only the results from the surface level (and sometimes aircraft level) were selected to be shown visually in this section. Section V provides a tabulated format of the results of sampling every tenth range gate for each case in order to provide another perspective of analysis. Wind Modeling and Sampling The wind was generated as described in Part III. A snapshot of the modeled wind field for the ocean surface without interference from the ocean or noise is shown in the following figure.

Figure 4-1

It is clear that the wind is going in the same direction at the same speed. The x, y axis represent the cartesian coordinate system on the horizontal plane at the surface level. In this case, the z-axis represents altitude. The data shown in the figures has been subsampled for plotting purposes in order to convey understanding to the reader. Using simulation resolution of an entire plane would result in a blurred plot with the vector arrows merging into a single blue image.

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Figure 4-2

The effect of adding gaussian noise to the wind field is shown in Figure 4-2. The speed of the wind was normalized in this case to have a maximum value of unity.

Figure 4-3

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The effect of adding ocean interference is shown in Figure 4-3, and the effect of adding Gaussian noise to both the wind and the ocean interference is shown in Figure 4-4.

Figure 4-4

It is clear that the ocean interference will change the direction and magnitude seen by the radar, in our simulation, at the surface level. The direction and magnitude do not effect the wind returns as drastically at higher levels, and are even negligible near the aircraft, due to the attenuating effects of altitude change as described in part III.

Figure 4-5

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Figur 4-5 shows the response from our ring sampling. Figure 4-6 gives a better idea of our ring sampling in terms of being closer to the resolution actually used.

Figure 4-6

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Case 1: The purpose of this case was to ensure that atmophseric model used and proposed method of aquiring wind speed from Doppler shifted radar returns worked in a perfect scenario as outlined in part III. Specifically, we assumed that there was no interference or return from the ocean and that we had perfect sampling of the rain droplets at each range gate. Figure 4-7 gives the work flow diagram that was followed in this first case, and which was built upon for the following cases of the simulation

Figure 4-7. work flow for case 1 The wind was generated and sampled as explained above. The true wind speed and direction was specified for each range gate according to the model described in previous section. Sampling was done per arc, and showng in Figure 4-7, and the following steps were repeated in a straight forward manner until the circular scan was completed. A plot of the measured wind speed at each arc section is shown in Figure 4-8. Recall that each arc section covered 5 degrees of the entire 360 degree circle that was scanned. The result of applying VAD to these measured data is shown with the plot. It can be seen that the method used resulted in a perfect solution for this case.

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Figure 4-8

Each range gate resulted in a perfect find of the true wind speed at that range as seen in section V. Satisfied that the simulation was behaving as desired, noise was added to the measurements as explained in Section III. Figures 4-9 and 4-10 show the result of adding noise. The experimental data is no longer as smooth as the original data, resulting in some error from the true wind.

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Figure 4-9

Figure 4-10

The degree of error for each case is provided in Section V.

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Case 2: In this case, the contribution of the antenna’s sidelobes were taken into account. This was done by taking the FFT of the antenna pattern and multiplying it by the sampled data in the frequency domain as discussed in Section III. The flow diagram for this case is shown in Figure 4-11.

Figure 4-11 work flow for Case 2

Figure 4-12

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Figure 4-12 shows the effect of the antenna’s sidelobes. In general, because of the method of taking the mean of each sampled section of per arc, the effects of the sidelobes from the antenna pattern were averaged out through making their effect relatively harmless to the overall sampling of the data. Figure 4-13 shows an example of measured wind speed for the sidelobe interference in the presence of noise. The sidelobes had a filtering effect on the noisy data, which helped smooth the sampled sinusoidal curve.

Figure 4-13

Tabulated results for the ranges can be found in Section V.

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Case 3: In the third case, the contribution due to ocean interference was taken into account. The interference was produced to give a wind speed reading of 40 m/s at 60 degrees in the absence of any wind.

Figure 4-14

Figure 4-14 shows the effects of ocean interference to our true wind measurements in the absence of noise. The interference dominates the radar’s wind speed measurement in this case giving a false wind reading. Next, the noise interference and the antenna pattern are taken into account. Figure 4-15, shows the effects of ocean interference with noise, and Figure 4-16 shows the effects of ocean interference in noise with the antenna pattern taken into account.

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Figure 4-15

Figure 4-16

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The above figures show how the interference from the ocean dominates measurements of the true wind at the surface level. Due to the fourth power decay of the interference, as described in Section III however, the effects of the ocean quickly diminish after the first few range gates. The effect of the ocean’s interference on the measured wind’s speed and direction profile are shown in Figure 4-17 and Figure 4-18.

Figure 4-18

Figure 4-19

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Figure 4-20

Figure 4-21

Figure 4-20 and Figure 4-21 provide a closer look at the measured wind’s profile, due to ocean interference, and lower altitudes (i.e. higher range gate readings).

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Figure 4-22

Figure 4-22 shows the results of adapting over this range. Using the assumption given that the interference from the ocean was uncorrelated with the wind closes to the aircraft an optimization filter based on the autocorrelation of the data can be built so as to optimize the error in the least mean square sense. The least mean square (LMS) algorithm was used as the method of choice to adaptively solve this problem. It was found only to be necessary to use near the ocean’s surface where the interference was greatest, and not at higher altitudes where the interference became negligible. There is still some error at the surface level, as seen in Figure 4-22 and discussed in Section V, however it is substantially better than the original case.

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V. Discussion of Error The most interesting errors were found at the last range gates, or at the lowest altitudes. Figures 4-18 to 4-21 demonstrated that the error due to ocean interference was negligible at higher altitudes. Tables 5-1 & 5-2 thus contains simulated results of wind speed and direction for every 10 range gates from one to 90 for completeness and contains range gates 96-100 for focused error analysis near the ocean’s surface.

Range Gate Altitude

True Speed

Measured Speed

Filtered Speed

Filtered Speed Error

Experimental Speed Error

1 2970 178.3 177.64 - - 0.4% 10 2700 163 163.02 - - 0.0% 20 2400 146 145.56 - - 0.3% 30 2100 129 129.48 - - 0.4% 40 1800 112 111.67 - - 0.3% 50 1500 95 95.04 - - 0.0% 60 1200 78 78.24 - - 0.3% 70 900 61 60.98 - - 0.0% 80 600 44 44.28 - - 0.6% 90 300 27 27.09 - - 0.3% 91 270 25.3 25.51 - - 0.8% 92 240 23.6 23.72 - - 0.5% 93 210 21.9 22.25 - - 1.6% 94 180 20.2 20.59 - - 1.9% 95 150 18.5 18.77 - - 1.5% 96 120 16.8 17.27 18.83 12.1% 2.8% 97 90 15.1 15.30 17.20 13.9% 1.3% 98 60 13.4 13.13 14.74 10.0% 2.0% 99 30 11.7 9.41 13.10 12.0% 19.6%

100 0 10 30.29 9.64 3.6% 200.0%

Table 5-1

The LMS method worked considerably well for the surface altitude, and provided decent results for the next altitude measured. However, higher measurements did not exhibit improvement over the measured error, and in fact became worse than the unaltered measured error itself. Due to the rapid decaying nature of the error, its interference falls off quickly causing the measurement to rapidly decline. The large interference near the surface was found to be necessary for our implementation of the LMS algorithm, which attempts to minimize the error in the least squares sense by finding some correlation with the desired signal. However, as the interference from the ocean dies out, there is less information to adapt to. Instead of subtracting out the ocean interference, the algorithm begins to adapt to parts of the signal itself, which are then removed, thereby increasing the error. Thus it was necessary to shut off the adaptive algorithm above 60m. This will not likely be seen as a perfect solution, and would likely need some fine tuning in actual implementation.

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Range Gate Altitude

True Direction

Measured Direction

Filtered Direction

Filtered Direction Error

Experimental Direction Error

1 2970 235 245.00 - - 4.3% 10 2700 235 235.00 - - 0.0% 20 2400 235 240.00 - - 2.1% 30 2100 235 240.00 - - 2.1% 40 1800 235 240.00 - - 2.1% 50 1500 235 235.00 - - 0.0% 60 1200 235 245.00 - - 4.3% 70 900 235 235.00 - - 0.0% 80 600 235 240.00 - - 2.1% 90 300 235 235.00 - - 0.0% 91 270 235 240.00 - - 2.1% 92 240 235 230.00 - - 2.1% 93 210 235 235.00 - - 0.0% 94 180 235 230.00 - - 2.1% 95 150 235 225.00 - - 4.3% 96 120 235 235.00 230.00 2.1% 0.0% 97 90 235 235.00 230.00 2.1% 0.0% 98 60 235 230.00 250.00 6.4% 2.1% 99 30 235 235.00 235.00 0.0% 0.0%

100 0 235 60.00 230.00 2.1% 74.5%

Table 5-2 Another difficulty that arose in our model was the linear nature of our wind profile. A quadratic model would have likely been better suited to true atmospheric conditions, in which case our results and implementation would likely differ. We believed that the addition of noise would help account for atmospheric fluctuations, but this was an assumption we made. A more rigorous atmospheric model would likely be needed for future analysis.

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VI. Conclusion The model presented here represents a relatively simple model of a complex problem that is found today in attempting to acquire wind speed measurements from an aircraft’s radar reflection of rain droplets while flying over an ocean’s surface. Several assumptions were made and stated in the formulation of this problem in order to simplify the problem into manageable portions, while leaving room for useful analysis. Further analysis of this problem would include curve fitting techniques as a method for handling noise. Larger errors from measured speed were often due to noise spikes near the top and bottom of the VAD curves. The effect of these spikes could likely be reduced by first fitting a sinusoid to the pattern first and then taking the wind measurements from there. Several imaging techniques were used in this project. A short list of these techniques includes: sampling and interpolation, antenna apertures and side lobe, noise handling, Fourier transforms, Doppler shifting, and optimization in the least squares sense.

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VII. References Journal Papers: Heymsfield, G.M., J.R. Carswell, L. Li, D. Schaubert, J. Creticos, 2006. “Development of the High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP)”, Eos Trans. AGU, 87(52), Fall Meet. Suppl., Abstract IN21A-1202. Esteban-Fernandez, D., E. Kerr, A. Castells, S. Frasier, J. Carswell, P. Chang, P. Black, F. Marks, 2005. "IWRAP: the Imaging Wind and Rain Airborne Profiler for remote sensing of the ocean and atmospheric boundary layer within tropical cyclones," IEEE Trans. Geosci.& Rem. Sensing, 43(8), 1775-1787. Browning, K.A., and R. Wexler, 1968: “The determination of kinematic properties of a wind field using Doppler radar”. J. Appl. Meteor., 8, 105-113. Lhermitte, R.M., and D.A. Atlas, 1961: “Precipitation motion by pulse Doppler radar.” Proc. 9th Weather Radar Conf., Boston, Amer. Meteor. Soc., Boston, 498-503. Lehmann, V. and G. Teschke, 2001. “Wavelet based methods for improved wind profiler signal processing”, Annales Geophysicae 19, 825-836. Text Books: Bracewell, R.N., Two Dimensional Imaging, Prentice Hall, 1995 Bracewell, R.N., The Fourier Transform and Its Applications (McGraw-Hill, 1965, 2nd ed. 1978, revised 1986) Widrow, B and Stearns, S. Adaptive Signal Processing, Prentice Hall, Inc., Englewood Cliffs, New Jersey 07632, 1985 Doviak, R, Zrnic, D. Doppler Radar and Weather Observations, Academic Press, San Diego, CA, 1993 Ulaby, F. T., Moore, R. K., Fung, A. K. Microwave Remote Sensing Active and Passive Volume II, Artech House, Inc Norwood, MA 02062, 1982