P. M. A. De Jong, J. J. Van der Laan, A. C. In ‘t Veld, M. M. Van Paassen, and M. Mulder, “Wind-Profile Estimation Using Airborne Sensors,” Journal of Aircraft, vol. 51, no. 6, pp. 1852–1863, 2014.
Wind Profile Estimation using Airborne Sensors P.M.A. de Jong∗, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands
J.J. van der Laan† Sim-Industries, Hub van Doorneweg 8, 2171 KZ Sassenheim, The Netherlands
A.C. in ’t Veld‡, M.M. van Paassen§, M. Mulder¶ Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands
Wind is one of the major contributors to uncertainties while flying Continuous Descent Operations. When aircraft are issued a required time of arrival over the runway threshold, as is foreseen in some of the future air traffic system concepts, the onboard availability of both dependable and accurate wind estimates becomes a necessity for spacing. This paper presents a method for real-time estimation of a wind profile in the terminal maneuvering area, based on data transmitted by nearby aircraft. The algorithm produces high resolution and real-time wind profile estimates, usable for accurate trajectory prediction to improve Continuous Descent Operations and spacing performance. The wind estimation algorithm is tested with Mode-S derived meteorological data from Amsterdam Airport Schiphol. It combines multiple measurements from different aircraft to estimate the current wind profile using a Kalman filter. Using these wind observations ∗
PhD-Student, Control and Simulation Section, Faculty of Aerospace Engineering, Delft University of Technology. Kluyverweg 1, 2629 HS Delft, The Netherlands. AIAA Student Member. E-mail:
[email protected] † Chief Simulator Engineer, Sim-Industries, Hub van Doorneweg 8. 2171 KZ Sassenheim, The Netherlands ‡ Assistant Professor, Control and Simulation Section, Faculty of Aerospace Engineering, Delft University of Technology. Kluyverweg 1, 2629 HS Delft, The Netherlands. AIAA Member § Associate Professor, Control & Simulation Section, Faculty of Aerospace Engineering, Delft University of Technology. Kluyverweg 1, 2629 HS Delft, The Netherlands.AIAA Member ¶ Professor, Control & Simulation Section, Faculty of Aerospace Engineering, Delft University of Technology. Kluyverweg 1, 2629 HS Delft, The Netherlands.AIAA Senior Member 1 of 28
data, the algorithm showed a root mean square in the estimation error of 1.35 KTS, which is lower than the observed root mean square measurement error root mean square of 1.94 KTS, reducing the uncertainty when used in trajectory prediction. Another simulation study investigating aircraft spacing showed a clear benefit of using accurate wind estimates along the own trajectory by reducing the spacing time error and improving control space.
I.
Introduction
Studies on new approach procedures, such as Tailored Arrivals,1 Optimum Profile Descent2, 3 into San Francisco International Airport and Los Angeles International Airport, and Time and Energy Managed Operations4, 5 have shown that a sufficiently accurate wind estimate is required for accurate descent profile prediction. Other studies investigated the effect of wind on, for example, time-of-arrival control,6–8 predicting Top of Descent9 for idle descents and self-spacing10 and concluded that for accurate trajectory predictions, an accurate knowledge of the prevailing winds is required.11–16 Today, pilots mostly rely on wind reports and meteorological wind-charts, such as NOAA Rapid Refresh, which are defined on a course 13 kilometer grid and are updated at most once every hour. Based on these charts, pilots insert winds aloft information data into the onboard Flight Management System (FMS) for use by the Trajectory Predictor (TP) to perform fuel and time predictions. Different studies have turned to different solutions to improve wind information in the FMS, ranging from a simple profile17 based on the wind measured onboard the aircraft and the wind reported at the runway,6 to up-linking high resolution wind grids.1, 18–20 Mondoloni used statistical data and Kalman filtering techniques21 to predict wind for use in trajectory prediction. Others have used aircraft radar tracks22, 23 to estimate the windfield at an aircraft’s position and use Kalman filtering to reduce the effects of measurement noise. Some solutions offer a high-resolution profile but do not provide a high update rate1, 18–20 or only provide an estimate for a certain area.23 To construct high resolution and real-time updated wind profile estimates for use during trajectory prediction, a novel wind estimation algorithm, referred to as Airborne Wind Estimation Algorithm (AWEA), has been developed.10, 24–26 The rationale of AWEA is to take advantage of the fact that in the envisioned future air traffic environments (SESAR and NextGen), aircraft will be equipped with a data-link system, such as Automatic Dependent Surveillance - Broadcast (ADS-B), that could broadcast and receive measured atmospheric data. This allows aircraft to gather reliable atmospheric information from nearby aircraft at short time intervals.
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AWEA constructs tailor-made wind estimation profiles using Kalman filtering. This allows to reduce noise from these measurements and to relate the various received measurements to the own trajectory. The benefit of this approach is that all aircraft produce information of the wind close to the own aircraft and, as such, act as airborne sensors, increasing the resolution and update rate of wind estimates significantly. This allows trajectory predictors to use the latest available and accurate wind estimates when predicting trajectories. The paper is structured as follows: Section II covers background information regarding wind estimation, trajectory prediction and atmospheric measurements. Section III gives an overview of the concept of operation and principles used in this research. Section IV contains the mathematical description of the new wind estimator algorithm. Section V presents results of two case-studies based on either derived real-life wind data or simulated wind data to review the performance of the new wind estimator. Finally, conclusions are drawn in Section VI and the paper ends with a set of recommendations for further research.
II.
Background
This section discusses how wind is estimated onboard aircraft and how wind estimates affect trajectory predictions. The section continues with today’s approach to wind estimation used for trajectory prediction and possible ways for improvement. II.A.
Wind Estimation
Fig. 1 shows an overview of the kinematics of an aircraft in horizontal flight and with a horizontal wind acting upon the aircraft. In this figure, the a-subscript denotes the air reference frame, the g-subscript denotes the Earth reference frame and the k-subscript denotes the aircraft kinematic reference frame.27 The aircraft groundspeed Vg equals the horizontal component of the Vk relative to the Earth. The wind speed vector Vw is calculated onboard the aircraft by resolving the speed vectors Va , the True Airspeed (TAS), and groundspeed Vg , see Fig. 1. The TAS is determined using the Air Data Computer (ADC) and pitot-static system for input of impact pressure, static pressure and total air temperature. The groundspeed is calculated by the Inertial Reference Unit and GPS. The vector calculation to obtain the wind speed is given by: Vw = Vg − Va
(1)
Fig. 1 shows that a horizontal headwind, Vw , decreases the horizontal component of Vk
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Xg χk χa
Vw
Vk
Xg
αT
Va
Yg γk
γk
γa Vk
χw Va
Vw Zg Figure 1. The effect of wind on an aircraft’s trajectory during descent.
or aircraft groundspeed. When the aircraft is descending, the same windfield increases the flight-path angle relative to the Earth, γk , provided the aircraft descends at a given speed. Thus an aircraft will require a shorter distance to descend from the same altitude relative to a situation where the aircraft experiences no wind or a tailwind. When the windfield varies with altitude, the motion of the aircraft relative to the air varies as well28, 29 affecting both the acceleration and descent angle change due to this wind gradient. All velocity vectors must be measured with sufficient accuracy to accurately determine the wind since horizontal winds are relatively small compared to the true airspeed and groundspeed. It also requires measurements of the aircraft body angles and angle of attack. However, during descent the aircraft flight-path angle is relatively small, causing the aircraft’s velocity component to be largest in horizontal direction. For this reason, the calculations can be simplified to include only TAS, groundspeed and heading and track angles by neglecting any vertical wind components.30 From this point onward, wind is only considered in horizontal direction in this paper. II.B.
Applications of Wind Estimation
The FMS predicts trajectory parameters, such as time and fuel estimates, by performing calculations that require estimates of the windfield along the trajectory towards the arrival airport to improve the accuracy of these calculations. An error in estimating the windfield will result in an error of the predicted groundspeed, deceleration and flight-path angle. In turn, the flight-path angle error affects the predicted groundspeed and predicted vertical trajectory. Hence, the accuracy of the trajectory, both temporal and spatial, is greatly influenced by the wind estimation error.11, 12 Today’s FMSs are able to store wind and temperature information at waypoints contained
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in the flightplan during cruise. For climb and descent, the FMS has typically five altitude points available that can contain wind information and linearly interpolates between these altitudes31 and extrapolates beyond the altitude limits. Due to the low resolution of available data points (both horizontal and vertical), the FMS introduces an interpolation uncertainty additional to the wind prediction error. The FMS uses the sensed wind to correct the predicted winds by calculating a correction from the difference between the sensed and predicted wind.32 To improve the onboard Controlled-Time of Arrival function used in SESAR’s Initial 4D project, the amount of available levels for which wind and temperature information could be supplied was increased to ten levels.8, 33 During actual flight trails, aircraft aimed at reaching a metering fix within a 10 seconds accuracy. The results showed decreased on-time performance when the inserted wind information deviated from the experienced wind but remained within the prescribed accuracy margin. II.C.
Current Methods for Wind Estimation
The winds aloft data can be entered manually from paper or automatically through Aircraft Communication and Reporting System (ACARS) to simplify operations. The data is generally provided by airline dispatch who uses winds aloft data from various meteorological sources. These forecast winds (and temperature) aloft data are generated four times a day for various worldwide locations at several altitudes. The winds aloft data are generated using complex numerical weather models that use observational data from radiosondes, satellites and aircraft as input. The predicted data are valid until 6, 12 or 24 hours beyond their issued time and thus the spatial and temporal resolution is low. However, for longer flights, it could be beneficial to update the winds aloft data when new data is available through a data-link. To address the low update rate, Boeing developed Wind Updates 18 in an aim to reduce fuel. This service sends tailored updated wind information to each individual aircraft subscribed to its service. Pilots accept the received data set through the ACARS control panel to automatically update the wind data in the FMS. However, the system does not include additional altitude intervals at which wind data can be stored but intelligently selects the best altitude intervals representing the forecast wind. A similar system is AVTECH’s Aventus NowCast.19 II.D.
Atmospheric Measurements using Airborne Aircraft
Onboard aircraft, atmospheric parameters are calculated by the ADC. This data can be retrieved by other aircraft or ground-stations using various techniques, detailed in this section.
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II.D.1. AMDAR To increase the accuracy of numerical weather prediction, the WMO in the 1970s proposed the Aircraft Meteorological Data Relay (AMDAR)34 project to use aircraft sensory data to improve the number of observations used in (global) weather forecast models. The sensory data is transmitted automatically through ACARS or a satellite-based equivalent and has been active since the late 1990s.34 The AMDAR panel foresees the use of Automatic Dependent Surveillance (ADS) for broadcasting of meteorological data in the future. The observed atmospheric data is combined into a single AMDAR report34 that contains position (latitude and longitude), altitude, time, temperature, (horizontal) wind direction and speed, turbulence (vertical acceleration), humidity and icing, phase of flight, a number of aircraft state variables such as roll and pitch angles, and an aircraft identifier. II.D.2. Mode-S A novel method uses aircraft transponder data to increase the number of atmospheric observations for estimating windfields.35, 36 The Secondary Surveillance Radar radars at major airports receive Mode-S Elementary Surveillance (Mode-S ELS) data from all aircraft within range containing velocity and position data. These Mode-S ELS data messages are different from AMDAR data since these do not include direct wind and temperature measurements. However, aircraft transponders featuring Mode-S Enhanced Surveillance (Mode-S EHS) are able to relay additional information containing various Downlink Aircraft Parameters integrated in BDS registers upon request that can be used to derived temperature and wind estimates. These additional BDS registers contain flight level, Mach number, roll, heading and track angles, and TAS. The tracking radar complements the received messages with groundspeed, track and position information. The combined information is then used to derive the wind vector from TAS and groundspeed whereas temperature information is derived from TAS and Mach number as temperature measurements are not included in the Mode-S message.35 II.D.3. Automatic Dependent Surveillance (ADS) Using ADS, aircraft broadcast state data at specified time intervals using a data-link system such as Mode-S Extended Squitter (Mode-S ES), Universal Access Transceiver or VHF Data Link. The specification of Automatic Dependent Surveillance - Broadcast (ADS-B) prescribes the Air Referenced Velocity (ARV) report which contains TAS and heading information, while groundspeed and ground track are available from the State Vector (SV) report.30 Using the data from these reports, the wind vector30 can be resolved. However, according to de Leege,23 only few aircraft currently broadcast the ARV report resulting in 6 of 28
a low availability of wind-derived data using ADS-B. Furthermore, as SV and ARV reports are broadcast at different rates (respectively 1 s and 30 s),30 which will result in additional wind error. Additionally, Automatic Dependent Surveillance - Contract (ADS-C) messages contain meteorological data upon request as agreed upon in a ‘contract’ between a ground-station and aircraft. ADS-C reports consist of several aircraft data blocks,37, 38 including a meteorological data block and air vector and ground vector blocks. The contract specifies sampling rate and content of the ADS-C reports. The meteorological data block contains wind speed and direction, temperature, humidity and turbulence data. Combined with the basic data block that contains spatial information, a full meteorological report is received. ADS-C messages without meteorological data can still be useful when only the ground and air vector blocks are received by deriving the wind vector from remaining data.36 II.E.
Future Needs
Aircraft primarily use rather coarse wind charts as input in the FMS which does not offer sufficient accuracy for today and tomorrow’s demand for accurate flying of 4D RNP trajectories.39 System Wide Information Management (SWIM)39 is also expected to share meteorological data between users. By using the received meteorological observations from aircraft within range, an improved estimate of the prevailing winds can be constructed and frequently updated when new observations are received. Combining this estimate with knowledge of the own trajectory, an aircraft specific wind estimate can be constructed along this profile. When the trajectory is unknown, for example on the ground, a wind profile or 3D grid for an entire area could be estimated.
III.
AWEA: Airborne Wind Estimation Algorithm
This section discusses the basic principles of the new wind estimation algorithm, AWEA. AWEA is specifically developed for use onboard aircraft to improve onboard trajectory prediction but can also be used on the ground for estimation of an entire windfield. III.A.
AWEA Working Principles
AWEA uses airborne measurements from the own aircraft and other nearby aircraft to construct a wind profile along the own trajectory. The wind profile is modeled using a stochastic model. The advantage of such a model is that it does not rely on the modeling of physical processes that govern wind behavior but instead uses observations and stochastic principles to estimate any profile shape.
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AWEA uses all incoming wind observation data from other aircraft within its range of reception. These observations are grouped together into equal altitude intervals. Measurements could be received through ADS-B, using a capable ADS-B-IN receiver, or ADS-C as long as sufficient data is transmitted to derive a wind measurement. Next, the noise in the received data is filtered using a Kalman filter.40 The purpose of the filter is to reduce the noise components from the measured wind data and to assign smaller weights to measurements that were taken at a larger distance from the own trajectory. The Kalman filter state vector consists of estimates at these altitude intervals. The smoothed data is used to construct a wind estimate for every altitude interval over a short time interval. These estimated wind speeds and directions are then combined to form the estimated wind profile using linear interpolation. The complete process is repeated at every time step including newly collected data. Every step of this iterative process will be discussed in more detail in the following sections. AWEA runs separately for all meteorological variables for simplicity and to avoid nonlinear state and observation dynamics of an Extended Kalman Filter and multiplication effects of noise in speed and direction. For all variables, the state vector would be equal in size but the noise matrices contain different values to account for the differences in accuracy of the respective sensors. Running the algorithm for wind speed and direction separately, yields wind speed and wind direction profile estimates along the own 2-dimensional (altitude, alongtrack-distance) trajectory. For the remainder of this paper, only wind speed observations are discussed but the same algorithm principles hold for other meteorological sensory data, such as temperature or pressure. The benefit of using aircraft data is that the observed meteorological data is spatially and temporally concentrated around the busiest routes at cruise altitudes and in the Terminal Maneuvering Areas (TMAs) around major airports, where aircraft typically follow the same arrival and departure routes based on RNAV or Air Traffic Control (ATC) vectoring. Inevitably, the accuracy in the wind estimation at less crowded airports or during low traffic hours will be lower. This is not expected to cause major issues since in these situations the demanded runway capacity is usually substantially less than the maximum available capacity and lower trajectory prediction accuracy is acceptable. To account for the distance between the location of the measurement and the own trajectory (or grid-point in case of a ground-based station), the Kalman filter optionally uses this distance to adjust the influence of the observation on the new estimate by adding distancebased uncertainty to the measurement. This distance d is calculated by the shortest distance between the measurement and the planned position of the aircraft where it reaches the measurement altitude. An arbitrary 3-dimensional approach path of the own aircraft and tracks of other aircraft within the Amsterdam Airport Schiphol TMA is illustrated in Fig. 2. Also 8 of 28
Observation 3 Observation 2 Observation 1 d3
d2
d1
d4 Observation 4
Figure 2. Own track of an aircraft (with aircraft symbol). The tracks of two departing aircraft (Observations 1 & 2) and two arriving aircraft (Observations 3 & 4) are also displayed. The asterisks indicate observations from other aircraft and their projections upon the own trajectory.
shown in this figure are trajectories and observations from other arriving and departing aircraft. The relative distance d between the observation and the projection upon the own trajectory are indicated with dotted-lines. This relative distance is included in the Kalman filter as explained in Section IV. III.B.
Implementation Methods
AWEA can be run onboard the aircraft, where data are processed by the FMS, or run on ground, as seen in Fig. 3. AWEA as an aircraft-based implementation, see Fig. 3(a), uses observations from the own and surrounding aircraft, and optionally complemented with a profile received from a ground station. The airborne observations contain distance information and are used to alter the current estimate based upon the aircraft’s trajectory to the runway. This allows for tailored wind profile estimation and increases the update rate of the estimation as the algorithm receives frequent updates from other aircraft. A ground-based implementation, as seen in Fig. 3(b), could work as follows: a ground station uses AWEA and receives meteorological observations to continuously estimate a wind profile representative for the entire TMA. When aircraft enter the TMA, they receive the latest estimated wind profile to update planning of the final segment of the descent. Updated wind profiles could be sent more frequently but this will require significant bandwidth. The ground station could estimate the wind along 3D grid-points using distance information between a grid-point and an aircraft. An intelligent algorithm, using aircraft intent, 9 of 28
could create a tailored wind profile for each aircraft entering the TMA. Alternatively, to simplify the implementation, distance information can be ignored to obtain a wind profile representative of the entire TMA. Hence, all aircraft entering the TMA at the same time will receive the same profile. A ground-based implementation of AWEA provides ATC with an estimated wind profile to support accurate vectoring of aircraft. Moreover, aircraft without ADS-C capabilities or with insufficient FMS computational power to use AWEA could receive an estimated profile constructed by a ground station.10 The update rate and accuracy will be lower compared to an onboard estimator but at least wind data will be available with an improved resolution. This way, more aircraft benefit from the airborne ‘sensors’ nearby. Aircraft entering TMA
Aircraft entering TMA tn−1 7→ tnow
tn−1 7→ tnow Aircraft on Approach tn−2 7→ tnow
ADS-B: SV,ARV Distance
Aircraft on Approach
IAF
IAF
tn−2 7→ tnow
ADS-B: SV,ARV Distance tnow ADS-B: SV,ARV
GroundADS-B receiver
tnow ADS-B: SV,ARV
ACARS Profile uplink
GroundADS-B receiver
GroundProcessing
(a) AWEA Aircraft-based.
ACARS Profile uplink
GroundProcessing
(b) AWEA Ground-based.
Figure 3. Various methods of implementing AWEA.
IV.
AWEA Algorithm Principles
This section discusses the principles of the estimation algorithm running onboard an aircraft to estimate the wind speed and resides inside the FMS. However, the derivations are similar for use in a ground station that would construct the wind profile for an entire area or 3D grid. IV.A.
Main Steps of the Algorithm
The algorithm consists of the following steps to construct a wind profile estimation: Step 1: Determine the nominal flight profile of the own aircraft. This profile is the intended route as currently planned. Along this profile, the wind profile will be estimated. The altitude resolution of the state vector is chosen to provide the desired altitude resolution; here set to 500 ft. It is noted that the accuracy of the estimation depends on the number of measurements and the spatial and temporal spread of the incoming 10 of 28
data. Using smaller intervals increases the accuracy of the estimates slightly but increases algorithm calculation times. Step 2: Generate an initial estimate. This initial estimate can be based upon a nominal profile provided by a ground station, or constructed using a wind measurement from the own aircraft. Based on this own measurement, a standard logarithmic wind profile is constructed according to the power-law:41
Vw (h) = Vw0
h h0
!p
(2)
In this equation, Vw varies with altitude and depends on the wind velocity Vw0 at a reference altitude h. The exponent p is an empirically derived constant and has a value of 1/7.41 Step 3: Update the Kalman filter. Although all aircraft broadcast data at the same rate, they do so at different times. Moreover, the Kalman filter considers measurements at each update step to be broadcast at the same moment in time. Therefore, the Kalman filter is updated at 1 Hz intervals to allow for a fast update of the estimates. This final step is discussed in more detail in the next section. IV.B.
AWEA Kalman Filter
AWEA uses a Kalman filter to reduce measurement noise and relate the incoming measurements to the own trajectory. The result of the filter is an estimated wind profile at arbitrary altitude intervals along the own trajectory. The general state and measurement equations are given by: x(k) = A(k|k − 1)x(k − 1) + w(k − 1) y(k) = C(k)x(k) + v(k)
(3)
In the state equation, x(k) is the wind speed vector at time k including all observations up to time k − 1. A(k|k − 1) is the transition matrix and w(k − 1) is the process noise. In the measurement equation, y(k) is the measured wind speed vector at time k, C(k) is the measurement matrix and v(k) the measurement noise. AWEA does not include system (wind) dynamics in the estimation model, such that only filtering is applied. Therefore, the transition matrix A reduces to the identity matrix, A(k|k − 1) = I(n × n), where n is the number of altitude intervals (states) to be filtered. The random variables w and v are process and measurement noise, respectively. Both
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noise sources are assumed to be constant, white, zero mean noise signals with, n
Q = E wwT n
R = E vv T
o
o
(4)
The elements of the Kalman filter comprise of five consecutive steps, at each time instant k in the following fashion, 1. The filter starts to predict the state estimate xˆ and estimate covariance P . The estimated state vector xˆ is an n × 1 vector, where n is the number of altitudes, depending on the chosen resolution (in this case, the resolution is set at 500 ft, depicted as Flight Levels). The state estimate, x ˆ (k|k − 1), initially equals the previous estimate and is: xˆ(k|k − 1) =
vw0
vw05
vw10
vw15
...
vw100
T
(5)
The prediction error covariance matrix, P (n × n), is predicted using the process noise covariance matrix Q(n × n) and no model dynamics: P (k|k − 1) = P (k − 1|k − 1) + Q(k − 1)
(6)
In this equation P (k|k − 1) denotes the covariance of the state estimate at time k using all observations up to time k−1. The initial values of the P matrix are set high to assign more weight to incoming measurements as the accuracy of the initial wind estimate is uncertain. 2. When a new observation comes in, the wind speed or direction are stored, along with the altitude and position at which the observation was recorded (for example at 1,700 ft, N52◦ 14’36.96” E4◦ 37’8.76”, indicated as vw1,700 ), and processed at the new update cycle of the filter: 1,700 (7) y(k) = vw The observations are used to construct the observation model matrix C(m × n), where m indicates the number of received observations. The C matrix determines which elements of the state vector can be updated when a measurement comes in, corresponding to the two points that are closest to the altitude of the measurement. The weights in the matrix are determined based on the altitude difference between the states of interest and the measurement and are linearly interpolated. In this example, with a sample observation recorded at 1,700 ft (hence, between 1,500 ft and 2,000 ft), the C matrix 12 of 28
is filled as: C(k) =
0 0 0 0.6 0.4 . . . 0
(8)
The observations also determine the measurement noise covariance matrix R(m × m) as R depends on the shortest, horizontal, distance between the location of the measurement and the own track d in nautical miles (see Fig. 2),
alt 0 R0 + Kw · d(k) 0 R0alt + Kw · d(k) R(k) = .. .. . .
0
0
0 ... 0 ... .. . . . .
0 0 .. .
0 . . . R0alt + Kw · d(k)
(9)
where Kw is a scaling parameter, varying between 0 and 1, that determines the influence of an observation based upon the horizontal distance between the measurement and the own trajectory at the same altitude. For our example of a single measurement at 1,700 ft, the R matrix is: R(k) =
R01,700
+ Kw · d(k)
(10)
3. With the observation matrix, C, the current estimate for the altitude of the observation can be calculated, and its value can be compared to the measured value to determine the innovation, e (m × 1): yˆ(k|k − 1) = C(k)ˆ x(k|k − 1) e(k) = y(k) − yˆ(k|k − 1)
(11) (12)
This approach assures that an incoming measurement influences only the states in its (vertical) vicinity as shown in Fig. 4. Using the C and R matrices, the innovation covariance matrix, S(m×m), is calculated: S(k) = C(k)P (k|k − 1)C(k)T + R(k)
(13)
where P (k|k − 1) is the prediction error covariance matrix calculated in Eq. (6). 4. The next step determines the Kalman gain, K(n × m), by using the observation model, state covariance and innovation covariance matrices. The Kalman gain is based on the relative magnitudes of the uncertainties in the current estimate and the new measure-
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Observation y (1,700 ft)
Update Matrices C and R
State Estimate
New State Estimate
10,000 ft
10,000 ft
9,500 ft
9,500 ft
2,000 ft
2,000 ft
Kalman Gain
1,500 ft 1,000 ft
e
(P, Q, S, K)
1,500 ft 1,000 ft
500 ft
500 ft
0 ft
0 ft
Figure 4. Schematic of a state update in which the darker elements of the new state vector were updated.
ment:
P (k|k − 1)C(k)T P (k|k − 1)C(k)T = K(k) = S(k) C(k)P (k|k − 1)C(k)T + R(k)
(14)
5. In the final step, the innovation is multiplied by the Kalman gain yielding the updated state estimate x ˆ (k|k): xˆ(k|k) = xˆ(k|k − 1) + K(k) · e(k)
(15)
The prediction error covariance matrix P is updated using the Kalman gain and observation model: P (k|k) = (I − K(k)C(k)) P (k|k − 1) (16) A high uncertainty in the current estimate (high P ) and much confidence in the accuracy of the measurement (low S) yields a high value of the Kalman gain, which in turn assigns a large weight to the incoming observation in updating the state estimate through the innovation: The opposite holds for a low uncertainty in the current estimate (low P ), and little confidence in the accuracy of the measurement (high S) which yields a low value of the Kalman gain. This results in a small weight for the incoming observation when updating the state estimate. When no new observations are received, there will be no innovation, and the loop is reduced to updating the prediction error covariance P . In this way, the uncertainty about an estimate increases when time passes without new incoming measurements. This ensures that more weight is assigned to newly received observations to alter the estimate. Hence, AWEA relies on stable weather when no new measurements are received as no update to the 14 of 28
state estimate is performed. A schematic of the algorithm loop is shown in Fig. 5, illustrating that a prediction step is driven by a new time step and an update step is driven by a newly received measurement. Observations
T +∆T
Predict State
①
Wind Speed → y Altitude → C Position → d
x ˆ(k|k−1)=ˆ x(k−1|k−1)
and Predict Estimate Covariance P (k|k−1)=P (k−1|k−1)+Q(k−1)
Predict New Data Available?
Y es
No
②
③
Update Matrices C and R (Eqs. (8,10)) Calculate Innovation e(k)=y(k)−C(k)ˆ x(k|k−1)
and Calculate Innovation Covariance S(k)=C(k)P (k|k−1)C(k)T +R(k)
④
Calculate Kalman Gain K(k)=P (k|k−1)C(k)T S(k)−1
⑤ State and Covariance Unchanged x ˆ(k|k)=ˆ x(k|k−1),P (k|k)=P (k|k−1)
Update State Estimate x ˆ(k|k)=ˆ x(k|k−1)+K(k)e(k)
and Update Estimate Covariance P (k|k)=(I−K(k)C(k))P (k|k−1)
Update
Figure 5. AWEA Kalman Filter schematic containing all five steps.
The output of the algorithm consists of estimates at the given altitude intervals along the planned trajectory which represent the evolution of the wind speed or direction, starting from the current position until the time T minutes ahead at the end of trajectory. After each estimation step, the complete wind profile is constructed through linear interpolation of the estimates per altitude interval.
V.
AWEA Performance Evaluation
This section discusses the performance of the AWEA algorithm using two case studies. First, the wind estimator performance in an off-line simulation is analyzed. For these simulations, measurement data from aircraft based upon actual Mode-S EHS radar soundings — courtesy of the Royal Netherlands Meteorological Institute — in the vicinity of Amsterdam Airport Schiphol were used. The data set was calibrated and corrected using methods described by de Haan.35 Second, results of a fast-time simulation study into aircraft spacing10 in a varying windfield are analyzed to investigate the effect of improved wind profile estimation on trajectory predictions.
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V.A.
Case Study 1: Simulations using Mode-S Enhanced Data
For this analysis, the available observations consisting of over 305,000 radar soundings were considered to be the true wind speed. One aircraft was selected to represent the own aircraft which estimates a wind speed profile using AWEA. The algorithm estimates the wind from TMA entry at 10,000 ft down to the runway and estimated an initial profile using the powerlaw of Eq. (2), see Fig. 6(a). A normally distributed noise (mean: 0, std: 1.94 KTS, such that measurement errors are within a 3σ value of ± 5.83 KTS, which corresponds to a typical wind measurement error34 of 3 m/s) was added to the data set, to represent noisy incoming measurements. These simulated measurements form the input for the AWEA algorithm onboard the aircraft. Time: 07:38:47
Time: 07:45:55 12, 000
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6, 000 d: 2.27 NM
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15
(a) Wind speed update process, h = 10,025 ft.
25
30
(b) Wind speed update process, h = 5,775 ft.
Time: 07:49:42
Time: 07:55:10
12, 000
12, 000
10, 000
10, 000 d: 63.57 NM
d: 126.06 NM
8, 000
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20
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d: 17.81 NM
6, 000
6, 000
d: 96.94 NM
4, 000
4, 000 Own Measurement
Current Actual Wind 2, 000 d: 0.83 NM
Current Actual Wind
d: 21.75 NM
2, 000
Wind Estimate
Wind Estimate
Own Measurement
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Previous Estimated Wind
Observations
Observations
0
0 0
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5
10
15
20
25
30
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(c) Wind speed update process, h = 3,050 ft.
(d) Wind speed update process, h = 1,025 ft.
Figure 6. Estimated wind profile evolving throughout the descent (R0 =10, Kw =0.4).
Fig. 6 shows the dynamic evolution of the wind profiles over time. When the aircraft descends through 10,000 ft, the initial estimate (solid line), shown in Fig. 6(a) is based on the aircraft’s own measurement of the prevailing wind and equals the logarithmic profile 16 of 28
discussed in Section IV. For this initial estimate, the initial prediction error matrix P is set high as no observations had been received yet. This yields a high Kalman gain, giving a high weight to the first set of observed data. Fig. 6(b), Fig. 6(c) and Fig. 6(d) show profiles constructed at later time steps during the descent. In these figures, the stars represent the actual incoming measurements, and the solid line is the new wind profile estimate, while the dashed line represents the previous estimate. From these figures, it can be seen that the distance between the measurement affects the magnitude of change made in the wind estimate and that a measurement only affects the estimate of altitudes within the altitude interval through the C matrix. V.A.1.
Performance accuracy
Fig. 7(a) shows estimated and true wind versus altitude throughout the entire descent. At 2,000 ft and 3,000 ft the aircraft flew level and the wind estimate and true wind therefore show a range of values for wind speeds. 2.2
10, 000 9, 000
2 8, 000
RMS Wind Speed [KTS]
Altitude [ft]
7, 000 6, 000 5, 000 4, 000 3, 000
1.8
1.6
1.4
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2, 000
Mean: 1.3484
Mean: 1.9421
Estimated
Measured
1 True Wind
1, 000
Estimated Wind 0 0
5
10
15
20
25
0.8
Wind Speed [KTS]
(a) True wind and estimated wind throughout the (b) RMS of the wind speed estimation error and entire descent. measurement error. Figure 7. Wind estimation performance including mean and 95% confidence interval(R0 =10, Kw =0.4).
To verify the robustness of the estimation process, a bootstrap sample extracted from the simulation is used to calculate the mean and 95% confidence interval of the Root Mean Square (RMS) of the estimation error using 10 different noise realizations. For this combination of R0 and Kw , the mean RMS of the estimation error is 1.35 KTS see Fig. 7(b). For comparison, the mean RMS of the (simulated) measurement deviation is depicted on the right in the same figure and equals 1.94 KTS. As such, AWEA is effective in reducing the effects of measurement noise while employing measurements from surrounding aircraft to construct an estimated wind profile.
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V.A.2.
Kalman Filter Observation Model
According to Kalman filter theory, the accuracy of an estimate depends on the quality of the system model, the accuracy of the available measurements and the number of measurements. AWEA does not contain system dynamics, hence, the accuracy of the wind estimation depends on the measurements and actual prevailing wind. Therefore, it is hypothesized that the Kalman filter estimation error will decrease as more data become available. Important parameters calculated in the Kalman filter (as described in Section IV) are the error covariance matrix P and Kalman gain K. The evolution of these parameters is shown in Fig. 8. Shown in these figures are incoming wind observations from other aircraft flying at altitudes between 500–1500 ft. Clearly seen in Fig. 8(a) is the increasing value of the error covariance at ≈ 7 : 41 : 00hr when measurements are received farther away from the altitude of interest and hence have a small coefficient in the C-matrix. Once a new and significant measurement is received (≈ 7 : 46 : 00), it is assigned a high weight and the uncertainty in the current estimated value drops as seen by the reduction of P . For a lower update rate of the Kalman filter, the P matrix would have increased less, assigning less weight to the new measurement. Hence, when using a lower update rate, the Q matrix should be adjusted accordingly, while the R matrix could include a time dependent factor similar to addition of distance information. At certain moments during descent, the value of the Kalman gain is slightly negative. These negative values result from the applied observation model and altitude resolution. The used observation model relates altitude intervals through linear interpolation resulting in correlations in the prediction error covariance matrix. These, in turn, can result in negative values for the Kalman gain. Alternative observations models or altitude intervals could produce estimates without negative Kalman gain values, for instance by limiting the effect of distant measurements through truncating of coefficients in the C-matrix. V.A.3.
Effect of distance information
Multiple simulations were run for varying values of R0 and Kw to investigate the effect of both parameters on the performance of the AWEA algorithm. All conditions were simulated for 10 different Gaussian noise realizations, added to the wind measurements. The simulated values of R0 were 0.1; 1; 5; 10; 50; 100; 200; and Kw was varied between 0 (all measurements have equal uncertainty) and 1 (full distance include) in steps of 0.1. The performance of the algorithm is assessed by calculating the RMS wind estimation error at the current position. The RMS is calculated by determining the estimated value at every time step for which a true wind value is available and including this in the summation for the RMS value at that point.
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60
1.2 C-Matrix
Covariance Estimate
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Value [-]
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−10 40 36: 07:
00 40: 07:
20 43: 07:
40 46: 07:
00 50: 07:
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−0.2 40 36: 07:
40 56: 07:
20 53: 07:
(a) Error covariance P , measurements and estimated wind speed.
00 40: 07:
20 43: 07:
40 46: 07:
00 50: 07:
40 56: 07:
20 53: 07:
Time [hh:mm:ss]
(b) C-matrix value and Kalman gain.
Figure 8. Kalman Filter parameters for observations at 1,000 ft.
Besides the current estimated RMS, the RMS value of an estimate (prediction) of the winds at 1,000 ft along the trajectory is calculated by determining the current estimate for 1,000 ft and subtracting the actual value at 1,000 ft once the aircraft reaches this point. 8
8 Prediction RMS
6
5
4
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7
Wind Estimation RMS [KTS]
Wind Estimation RMS [KTS]
Prediction RMS
Current RMS
7
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1 −0.2
0
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0.6
0.8
1
1.2
Kw [-]
(a) R0 = 1.
(b) R0 = 10.
Figure 9. Mean and 95% confidence interval of RMS values for the current position (cross) and position along route (circle) including a quadratic fitted model.
Fig. 9 shows that by increasing R0 , the RMS of predicted estimates decrease while the RMS of the estimates at the current position increase slightly. This latter result was to be expected as the R matrix represents the uncertainty (covariance) in a measurement and hence the current estimate will primarily depend on the own measurement which will have increased uncertainty for larger values of R0 . Alternatively, the reduced RMS values of the predicted estimates for larger values of R0 show that the prediction benefits from increasing the uncertainty. This can be explained as the prediction is primarily based on measurements
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from other aircraft at that altitude and at different altitudes. As the wind is likely to change over time, increasing the modeled uncertainty thus makes sense. Evident from these figures is that both RMS values benefit from introducing distance information into the R matrix. Moreover, increasing R0 further leads to R0 to be dominant and hence Kw has less effect on the found RMS values. Furthermore, the quadratic fitted model for R0 = 1 shows that the RMS for Kw = 0 is an outlier and reduces the goodness of fit. Removing this value from the fitted model results in a better fit. This shows the importance of distance information for low values of R0 and hence that the estimates benefit from assigning ‘uncertainty’ to an observation located farther away. V.B.
Case Study 2: Aircraft Spacing and Wind Estimation
A simulation study into aircraft spacing for approach, investigated the use of the AWEA algorithm to improve spacing between in-trail aircraft.10 In this study, aircraft use Interval Management42 and receive an Estimated-Time of Arrival at the runway of the assigned lead aircraft while ATC instructs the aircraft to achieve a time spacing goal with the lead aircraft at the same runway. The own aircraft continuously adjusts its speed profile to achieve the spacing goal with respect to that lead aircraft. This study uses the Airborne Spacing for Terminal Arrivals (ASTAR) algorithm43 to calculate speed adjustments to achieve the spacing goal using a 4D TP and speed-control law. AWEA is expected to improve the accuracy of the TP and consequently improve aircraft spacing accuracy. Five different implementations of wind estimation were compared of which four implementations used AWEA as seen in Table 1. Table 1. Wind estimation implementations used in simulation study.
Name
Description
Charts Wind estimation using wind charts Ground 10 Ground-based AWEA algorithm with ADS-B broadcast rate of 10 seconds Ground 30 Ground-based AWEA algorithm with ADS-B broadcast rate of 30 seconds Air 10 Aircraft-based AWEA algorithm with ADS-B broadcast rate of 10 seconds Air 30 Aircraft-based AWEA algorithm with ADS-B broadcast rate of 30 seconds
The wind charts implementation ‘flattened’ the simulated wind for each scenario into a single static wind profile at 4 altitude levels: ground level, 2,500, 5,000 and 10,000 ft and represents the average of a dynamic wind scenario over less than three hours. Wind estimates between these levels were obtained through linear interpolation. The ground-based AWEA implementation (see Fig. 3(b)) used broadcast wind measurements from aircraft within the own TMA to construct a single wind profile that represents the windfield in the entire TMA. As such, any dependency of the position of the measurements is removed and hence the R matrix is constant. In this implementation, the ground 20 of 28
station broadcasts the latest estimated wind profile to an aircraft upon TMA entry. This profile is stored in the FMS and not updated with new measurements. The aircraft-based implementation (see Fig. 3(a)) works similarly except this implementation combines the received ground-based profile with measurements from other aircraft to continuously update the onboard profile. In these measurements, the positional information from the measurements is included in the construction of the R matrix, see Eq. (10). Both AWEA implementations were evaluated twice using different measurement broadcast rates to investigate its effect on wind estimation. This means that aircraft broadcast measurements once every ten seconds or once every thirty seconds. The simulated winds were constructed using actual AMDAR data.10 Based upon this simulated windfield, all arriving aircraft sensed the current wind speed and direction includ¯ V = 0 KTS, σV = 0.7 KTS; X ¯ χ = 0◦ , σχ = 2.4◦ ). ing a measurement error (X W W W W V.B.1.
Spacing performance
Besides the wind estimation implementations, various modifications to the spacing algorithm were evaluated in a within-subjects design which are not within the scope of this paper. Each scenario was repeated 50 times, using a different set of disturbances: a) timing at TMA entry; b) aircraft type arrival sequence; c) wind errors. The mean spacing accuracy is the mean of the spacing error of all aircraft for a single scenario. Results of a one-way repeated measures Analysis of Variance showed a significant main effect (Greenhouse-Geisser corrected, χ2 (9) = 116.173, p < 0.001) on the mean spacing accuracy (F1.835,89.920 = 23.618, p < 0.001). A Bonferroni corrected post-hoc analysis revealed that the AWEA implementations performed significantly better compared to using wind charts, see Fig. 10(a). Analysis showed no significant differences between the AWEA implementations (see Fig. 10(a)). Fig. 10(b) shows the RMS error in estimating the wind along the trajectory. AWEA significantly reduces the RMS in estimating the prevailing wind. As normality was violated, a Friedman test was performed and confirmed this significant effect (χ2F (4) = 183.728, p < 0.001). A post-hoc analysis, using a Wilcoxon Signed Rank test and a Bonferroni correction of 0.05/10, revealed that all implementations performed significantly different from another. These results show that using additional observations from airborne aircraft to update the onboard profile yields the smallest wind estimation error. Although the effect of using AWEA on spacing accuracy and wind estimation errors proved statistically significant, the actual value of the spacing improvement is in the order of 0.15 seconds. Another aspect worth considering is time in loss of separation, which represents the cumulative time that an aircraft violates the minimum separation requirement of 3 NM during an approach.10 During this experiment, aircraft were instructed to achieve spacing 21 of 28
8 0.6 7 Mean RMS Wind Error [KTS]
Mean Spacing Error [s]
0.4
0.2
0
−0.2
−0.4
6 5 4 3 2 1
−0.6 Charts
Ground 10
Ground 30
Air 10
0
Air 30
(a) Mean spacing accuracy per scenario.
Charts
Ground 10
Ground 30
Air 10
Air 30
(b) Mean RMS of wind estimation.
Figure 10. Spacing accuracy and separation violations for different wind estimation implementations.
at the runway threshold and as such no active spacing was performed throughout the entire descent which could lead to loss of separation during the descent. aircraft were not instructed to perform a missed approach procedure when separation was violated. The analyzed data for time in loss of separation was not uniform distributed over all runs and did not allow the use of parametric or non-parametric tests. Fig. 11(a) shows the mean time in loss of separation for all scenarios wind separation violations per wind estimation implementation and their respective 95% confidence intervals. For 52% of all runs per implementation, no separation violations were found and were therefore not included in Fig. 11(a). This figure shows that aircraft that relied on wind charts violated separation more often than aircraft that used AWEA wind estimates. This implies that less accurate wind estimated reduced the available control space for the ASTAR algorithm to achieve the spacing goal at the runway threshold. In a real operational scenario, loss of separation would result in a missed approach which negatively affects runway throughput. Updating the wind profile also increased the mean number of speed changes additional to the nominal speed changes to fly the nominal profile as seen in Fig. 11(b). As normality was violated, a Friedman test on these additional speed changes was used and showed a significant main effect (χ2F (4) = 94.910, p < 0.001). The subsequent post-hoc Wilcoxon Signed Rank tests showed that the wind charts performed significantly different (p < 0.001) from the aircraft-based implementation while the wind charts showed no significant difference with the ground-based implementation (p > 0.622). Similarly, a significant difference between the ground-based and aircraft-based implementations was found. The broadcast rate, however, showed no significant influence (p > 0.404).
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400
22 20 Average Commanded Speed Changes [-]
350
Time in Loss of Separation [s]
300 250 200 150 100
18 16 14 12 10 8 6 4 2
50
0 0
Charts
Ground 10 Ground 30
Air 10
Charts
Ground 30
Ground 10
Air 10
Air 30
Air 30
(a) Cumulative time in loss of separation.
(b) Mean number of commanded speed changes.
Figure 11. Spacing accuracy for different wind estimation implementations.
V.B.2.
Relevance of Distance Information
To investigate the influence of distance information in the R matrix, Fig. 12 shows wind profiles for a single aircraft during approach. These wind profiles shows the wind at different altitudes during the descent from 10,000 ft to the runway. These profiles include the ‘true’ wind and the estimated winds using the ground-based AWEA, aircraft-based AWEA and a wind profile based on an aircraft-based AWEA using measurements from other aircraft but without distance information. 10, 000 True Wind
9, 000
AWEA Ground AWEA A/C
8, 000
AWEA A/C no dist.
Altitude [ft]
7, 000 6, 000 5, 000 4, 000 3, 000 2, 000 1, 000 0
0
5
10
15 20 Wind Speed [KTS]
25
30
Figure 12. Different profile estimates and true wind for a single descending aircraft.
Fig. 12 shows that the ground-based implementation provides a good estimate of the prevailing winds. However, the airborne-based AWEA often provides better results. Also shown is the influence of distance information as using this information provides additional smoothing since the ‘no distance’ implementation shows increased variations. This was to be 23 of 28
4 3 2 1 0 −1 −2 −3 69
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Wind Estimation Error [KTS]
expected as using the distance information assigns distant measurements with an increased uncertainty. Finally, Fig. 13 shows the wind estimation error throughout the arrival sequence for the ground-based and aircraft-based implementations. The results show only small differences but the aircraft-based implementation reduces the variation and outliers in the actual wind estimation best.
Aircraft in the Sequence [-]
4 3 2 1 0 −1 −2 −3 67
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Wind Estimation Error [KTS]
(a) Ground-based Algorithm.
Aircraft in the Sequence [-] (b) Airborne-based Algorithm. Figure 13. Wind estimation error throughout the arrival sequence.
The results of this experiment showed statistically improved, but limited in size, spacing performance. The results also showed that AWEA improved control space compared to using wind charts in a simulated windfield. However, no significant difference between the various types of AWEA implementations was observed.
VI.
Conclusions
An algorithm is proposed that estimates wind profiles using measurements of meteorological data from nearby aircraft. The algorithm has the benefit of producing high-fidelity, 24 of 28
high-resolution and user-tailored wind profiles. Using actual wind observations from aircraft transponder data, the influence of distance on the quality of the estimation was evaluated and showed to reduce the estimation errors. Simulations showed that the algorithm was able to reduce measurement noise from 1.94 KTS to 1.35 KTS. Moreover, the estimated wind profiles can be used for predicting wind at locations farther along the own trajectory with minor errors. Airborne Wind Estimation Algorithm was also evaluated in a simulation study that investigated trajectory prediction and aircraft spacing in a varying windfield and shows that AWEA improves spacing performance. Today, the availability of actual meteorological data, through ADS-B data included is still limited. Once these reports become more widely available or through other systems such as System Wide Information Management, AWEA could be adapted and evaluated in a real-time environment onboard an aircraft. Future work should also investigate alternative observation models and investigate the use of an extended Kalman filter to filter wind speed and direction simultaneously and use a non-linear observation model. Moreover, the state vector could be defined as a wind speed in the Earth reference frame and be applied to a 3D grid to obtain a windfield estimate.
Acknowledgments The authors would like to thank dr. Siebren de Haan and Jan Sondij (both KNMI) for providing the wind observations used in this research. dr. Qiping Chu and dr. Christiaan Tiberius (both TU Delft) are acknowledged for their support on tuning the algorithm.
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