1.1.1 Orientation Coordinate Systems

Transcription

1 1.1.1 Orientation Coordinate Systems The velocity measurement is a vector in the direction of the transducer beam, which we refer to as beam coordinates. Beam coordinates can be converted to a Cartesian coordinate system (XYZ) by knowing the beam orientation, or the flow can be presented in Earth normal coordinates (ENU- East, North and Up) if the instrument is equipped with a compass and tilt sensor. A description on how to transform the coordinate system is described in the next section. The coordinate systems are defined as follows: Beam coordinates are defined as the raw velocity measurements measured independently along the three (four) beams of the instrument. A positive velocity is directed in the same direction as the beam points. Beam 1 is marked with an X on most instrument heads; the exception is the AQD head, where beam 3 is marked with X. On the Vector and the Vectrino, Beam 1 is defined as the arm with the black / red marking (placed opposite of the engraved head ID on the Vector). XYZ coordinates are the locally fixed orthogonal coordinate system that is oriented relative to the instrument head. A positive velocity in the X-direction goes in the direction of the X-axis arrow, and the X-axis points in the same direction as beam 1. Use the right-hand-rule to remember the notation conventions for vectors. Use the first (index) finger to point in the direction of positive X-axis and the second (middle) finger to point in the direction of positive Y. The positive Z-axis will then be in the direction that the thumb points. ENU coordinates are defined in an earth coordinate system, where E represents the East-West component, N represents the North-South component and U represents the Up-Down component. This is also a right-handed orthogonal system. The internal compass is mounted so that when the X marking points towards magnetic North, the heading is 0. ENU and XYZ coordinates are the most practical when handling data. Beam coordinates are primarily useful for higher-level turbulence calculations and for dealing with phase wrapping issues. You may specify the preferred coordinate system in the Deployment Planning menu.

2 Feil! Det er ingen tekst med den angitte stilen i dokumentet.-1: Coordinate systems. a) Beam coordinates, b) XYZ coordinates and c) ENU coordinates. The figure below shows an assortment of velocimeters probes and instrument heads. For the Vector probe (see rightmost illustration, upper part, of the figure below), one may notice that the X component is predominantly measured by beam 1 with nearly equal contributions from beam 2 and beam 3. This makes sense since the XYZ coordinate system is aligned with X pointing along one beam / receiver arm. Beams 2 and 3 are at some angle α to the X-axis and measure a component of X proportional to cos α. For the Y-component, beam 1 contributes to zero (or very near zero) because the Y-component is perpendicular to beam 1. Finally, the Z component is an equal combination of all three beams since the Z-axis is aligned with the central transducer and each beam is at the same angle to the Z-axis. Note that this applies to other head configurations as well.

3 Feil! Det er ingen tekst med den angitte stilen i dokumentet.-24: The Vectrino, Vector, AWAC and Aquadopp (symmetric head) XYZ coordinate systems and beam numbering as defined relative to the probes/beams. The Vectrino in the upper middle is the side-looking version. Beam 1 of the Vectrino has a red marking, a black marking defines beam 1 for the Vector (not shown here). For the Vector, beam 1 is also the arm opposite of the engraved head ID. Beam 1 point in the direction of positive X-axis; the Z- direction is towards the electronics of the instruments. The Y direction can be found in accordance with the right-hand rule. Beam 1 of the AWAC and Aquadopp is marked with an X engraved on the housing, barely visible in the AWAC in the lower middle. The Vector, Vectrino and Vectrino Profiler measure velocity components parallel to their three bi-static beams, or in beam components. They can report data in beam or XYZ coordinate systems, the Vector can in addition report data in ENU. The XYZ coordinates are relative to the probe and independent of whether the instruments point up or down.

4 Coordinate Transforms The transform of coordinate system from beam to XYZ or ENU coordinates is an important step when examining Doppler velocity data. Understanding coordinate transforms is valuable when interpreting velocity data, fixing problems in a data set, and ultimately, obtaining the highest quality data. Below there are three cases where a coordinate transform can be of use. 1) To transform raw along beam velocities to XYZ coordinate system 2) To transform beam or XYZ coordinates to geographical (ENU) coordinates 3) To transform East and North velocities to Speed and Direction. Before describing these three cases more thoroughly, a couple of points is important to be aware of concerning 2) and 3): The instrument will use the tilt sensor to interpret the sign on the transformation matrix correctly. If you are doing the transforms in post-processing, you will need to consider the instrument orientation to obtain the correct transformation matrix. Note: For the Vector, the instrument is defined to be in the UP orientation when the probe is pointing down, i.e. when the communication cable end of the canister is on the top of the canister. For the other instruments that are used vertically, the UP orientation is defined to be when the head is on the upper side of the canister. When the status bit is zero, the instrument is pointing up. The transformation matrix must be recalculated every time the orientation (heading, pitch and roll) are updated. 1. Transform raw along beam velocities to instrument fixed coordinate system: The XYZ coordinate system of your instrument is defined by the transformation matrix. The information needed to convert from Beam to XYZ coordinates is thus a set of beam velocity measurements and the instrument s 3 x 3 (or 4 x 4 for the Vectrinos) transformation matrix. Each instrument has its own unique version of the transformation matrix, based on azimuth and horizontal angles of each beam, and has been compensated for possible small misalignments that may have occurred during casting. Azimuth (Az) is the vertical angle of each beam in degrees, and angle (An) is the vector of horizontal angle of each beam in degrees (referenced to X). The transformation matrix is constructed like this: sin(az(1)) cos (An(1)) sin(az(1)) sin(an(1)) cos(az(1)) T = [ sin(az(2)) cos(an(2)) sin(az(2)) sin(an(2)) cos(az(2)) ] sin(az(3)) cos(an(3)) sin(az(3)) sin(an(3)) cos(az(3)) In the past the transformation matrix was output as an integer and had to be scaled by 4096 to be the appropriate value for actually performing transforms. Now, most instruments will report the transformation matrix pre-scaled so you do not need to divide by 4096 anymore. It is easy to tell if you need to or not, if the number is a floating point value, you do not need to scale the transformation matrix. For simplicity, let us refer to it like this:

5 T 11 T 12 T 13 T xyz = [ T 21 T 31 T 22 T 32 T 23 ] T 33 The T xyz matrix is reported in the *.hdr file, accessible when performing a binary to ASCII conversion in the software. Each row of the matrix represents a component in the instrument s XYZ coordinate system, starting with X at the top row. Each column represents a beam. The third and fourth rows of the Vectrino transformation matrix represent the two estimates of vertical velocity (Z1 and Z2) produced by the instrument. To transform velocities measured in beam coordinates to XYZ coordinates, you simply have to multiply the beam velocities with the transformation matrix. Recall the basics of matrix multiplication: b 1 T 11 T 12 T 13 Vx [ T 21 T 22 T 23 ] [ b 2 ] = [ Vy] T 31 T 32 T 33 b 3 Vz Where T ij represents the elements in the Transformation matrix, b i are the beam velocities and Vx, Vy, Vz are the transformed velocities. When writing out the matrix multiplication above, the following system of equations are obtained: T 11 b 1 + T 12 b 2 + T 13 b 3 = Vx T 21 b 1 + T 22 b 2 + T 23 b 3 = Vy T 31 b 1 + T 32 b 2 + T 33 b 3 = Vz In these equations one can see how each orthogonal velocity component (XYZ) is a combination of the various beam velocities, as introduced in the section above. 2. Transformation from instrument fixed coordinates to earth referenced coordinates In order to get the measured velocities referenced to earth coordinates (ENU) it is necessary to have information about the instrument s orientation in space. The tilt sensor measures the orientation of the transducer head with respect to the direction of the gravitational pull. Heading, pitch and roll are output in degrees. cos(hh) sin(hh) 0 Heading = [ sin (hh) cos (hh) 0] Where hh (heading) is defined as π (heading 90). The -90 subtraction has to do with how the compass is 180 oriented relative to the instrument axis. For our instruments, X is aligned with N on the compass. The subtraction of 90 rotates the heading into mathematical convention for sine and cosine as used above. cos (pp) sin(pp) sin (rr) cos(rr) sin (pp) Tilt = [ 0 cos (rr) sin (rr) ] sin (pp) sin(rr) cos (pp) cos(pp) cos (rr)

6 Where pp (pitch) is defined as π (pitch) and rr (roll) as π (roll) The resulting transformation matrix then becomes the multiplication of the transformation matrix presented in the header, the heading matrix and the tilt matrix: T 11 T 12 T 13 cos(hh) sin(hh) 0 cos (pp) sin(pp) sin (rr) cos(rr) sin (pp) T enu = [ T 21 T 22 T 23 ] [ sin (hh) cos (hh) 0] [ 0 cos (rr) sin (rr) ] T 31 T 32 T sin (pp) sin(rr) cos (pp) cos(pp) cos (rr) This resulting T enu matrix can be used to transform Beam coordinates to ENU coordinates. Simply perform a matrix multiplication as described above to transform the velocities. The table below gives an overview of the calculation that needs to be done in order to transform coordinate system from the system data were collected in, to the system that is now desirable. Has/Wants Beam XYZ ENU Beam T xyz * Beam velocities T enu* Beam velocities XYZ (T xyz) -1 * XYZ velocities T enu * (T xyz) -1 * XYZ velocities ENU (T enu) -1 * ENU velocities T xyz * (T enu) -1 * ENU velocities 1. Relations between radial speeds and true velocity components in ENU coordinates: With the foundation of transforming coordinates from point 1) and 2), a description on how to use this to obtain velocity as the more comprehensible Speed and Direction follows. Since the compass heading is of interest, the Z velocity can be ignored. Speed and direction are the magnitude and resultant angle referenced to an earth coordinate system from vector addition using the horizontal velocity components. This is also known as Euclidean Vectors, that is, it is a physical quantity having both speed and direction. There are two different methods of describing a Euclidean Vector, as East and North or as Speed and Direction. The East and North presentation of this vector can be calculated using the following formulas: East = cos(hh) Vx sin(hh) Vy North = sin(hh) Vx + cos(hh) Vy These equations can be found from the T enu matrix. hh is the heading, as defined above. It is assumed that the tilt of the instrument is close to zero. To convert East and North to Speed and Direction, use the following formulas: Current speed (Pythagoras): East 2 + North 2 Excel and Matlab: SQRT(East*East+North*North) Current direction: The calculation is based on the familiar Theta = arctan(x/y). In order to know the correct quadrant that the computed angle belong in, one has to use the four-quadrant invers tangent (arctan2). The 180/π converts from radians to degrees. The modulo (mod) is taken in order to avoid negative degrees.

7 Excel: MOD(ARCTAN2(East;North)*180/PI,360), Matlab: mod(atan2(north, East)*180/pi, 360) Note that the current direction is defined as towards.

8 Compass update rate The transformation between the beam and XYZ coordinates can be accomplished with no loss of information, because the transformation matrix is used. When converting to ENU coordinates, the compass update rate sets the limit to which coordinate transforms can be performed. The magnetometer measures the magnitude of three components of the earth's magnetic field at a user-specified rate. All Nortek products (with compass) combine this information to compute the instrument's heading, and then use this to correct the measured velocities to earth coordinates. Compass is typically sampled at 1 Hz, meaning the transformation matrix is updated with pitch, roll and heading information every second. If the current measurement data rate is 4 Hz, it means the four samples in each second share the same heading and attitude and there the same transformation. If onboard averaging is performed and the instrument is moving during the average interval, it is not possible to recover valid XYZ data from an ENU average velocity. With the Vector, it is therefore not recommended to collect internally averaged beam or XYZ velocity data because of the possibility of bias due to a moving instrument Pitch, Roll and Heading In their raw format, currents are measured along each of the three beams. In order to get the information referenced to earth coordinates (ENU) it is therefore necessary to detect the instrument s orientation in space. Attitude sensors, such as pitch, roll and compass heading (if equipped) are therefore used to aid in the transformation needed to correct for the instrument s attitude and motion. Note that the transforms from beam to XYZ or ENU corrects for tilt (and compass) in the sense that the final coordinate system is aligned with the gravitational axis (and the magnetic north pole). However, the data are not automatically corrected for the vertical mapping of the cells. When the instrument is tilted during deployment, the measurement cell from e.g. beam 1 may not correspond to the same measurement cell of e.g. beam 2, as sketched on the next page. In this example, depth cell 6 from the first beam corresponds to depth cell 7 of the second beam when the instrument is tilted 15. In all current profilers, the velocity is measured in fixed cells along each acoustic beam. The length of each cell ( cell size ) is defined as a time interval multiplied by the speed of sound, which is then projected onto the vertical axis. However, since the beam axis is not vertical (but 25 ), the size of the cell will not be the same in beam 1 as beam 2 when tilted. For this reason, there will be residual errors in the current profile. The residual error can be characterized by: Smearing of shear. The shear layer will look thicker than it really is, since the measurements are retrieved at different depths. Apparent vertical velocities. In areas of shear, there will appear to be a vertical velocity that is in fact an artifact of the processing. It is possible to remap the velocity cells for each beam and thereby minimize the residual error. This is not done in the instruments for the simple reason that the mapping is non-linear. Remapping is included as processing options in the Storm and Surge software. Depth cell mapping matches the cells at equal depth by using the information from the tilt sensor when computing the velocity, to maintain the assumption of horizontal homogeneity of the current velocity. Reprocessing with the software will also ensure that the

9 shear data are as accurate as possible. The best solution, however, is to make sure the instrument is level during deployment. Tilt degrades data in ways that are not always recoverable, such as increasing the thickness of the surface sidelobe layer and in some case reducing the effective range of your instrument. Feil! Det er ingen tekst med den angitte stilen i dokumentet.-3: Tilting of the instrument (here 15 ) result in measurements from different cells at the same depth. Sometimes deployment frames tilt excessively or even fall over. If the instrument s tilt reading is 20 or less, your data should be within the specifications. Tilt readings between 20 and 30 affect the data accuracy in a way that is likely to make the data fail to meet the specifications. Data acquired during tilts exceeding 30 are in general not reliable and should be discarded. Using an AWAC with the AST option, a maximum tilt of 5 within the vertical should be targeted.