OrthoRadar Theory

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Parameters Required for Orthorectification

SAR image orthorectification requires certain information about the sensor and the SAR image. Different sensors (TerraSAR-X, RADARSAT-2, and so forth) express these parameters in different ways and in different units. To simplify the design of our SAR tools and easily support future sensors, all SAR images and sensors are described using our SAR node model. The sensor-specific parameters are converted to a SAR node model on import or direct-read.

The following table lists the parameters of the SAR model and their units. These parameters can be viewed in SAR Model tab in SAR Model Properties (OrthoRadar) dialog.

SAR Parameters Required for Georeferencing

Parameter

Description

Units

sensor

Sensor that produced the image

orbit_direction

Ascending (S->N) or descending (N->S)

data_type

Data type

mode

Original sensor acquisition mode

coord_sys

Coordinate system for ephemeris ('I' = Inertial, 'F' = Fixed Body or Earth rotating)

year

Year of image data collection

month

Month of image data collection

day

Day of image data collection

doy

Greenwich Mean Time (GMT) day of year of image data collection

num_samples

Number of samples in each line in the image (in range)

num_lines

Number of lines in the image (in azimuth)

first_pt_secs_of_day

Time of first ephemeris point (This parameter is updated during orbit adjustment)

seconds of day

first_pt_org

Original time of first ephemeris point

seconds of day

time_interval

Time interval between ephemeris points (This parameter is updated during orbit adjustment)

seconds

time_interval_org

Original time interval between ephemeris points in seconds

seconds

image_start_time

Image start time (azimuth direction)

seconds of day

image_end_time

Image end time (azimuth direction)

seconds of day

image_duration

Time duration of images (azimuth direction)

seconds

semimajor

Semimajor spheroid axis of Earth model used during SAR processing

meters

semiminor

Semiminor spheroid axis of Earth model used during SAR processing

meters

target_height

Assumed height of scene above Earth model used during SAR processing

meters

look_side

+90 = right-looking, -90 = left-looking

degrees

local_incidence_angle

Incidence angle relative to a flat horizontal terrain at scene center (0 degrees = vertical reference, 90 degrees = horizontal reference)

degrees

wavelength

Wavelength of sensor

meters

range_sampling_frequency

Range sampling rate

Hz

azimuth_sampling_frequency

Azimuth sampling rate

Hz

range_processing_bandwidth

Range processing bandwidth

Hz

azimuth_processing_bandwidth

Azimuth processing bandwidth

Hz

range_window_func

Range window function

range_window_coefficient

Range window coefficient

azimuth_window_func

Azimuth window function

azimuth_window_coefficient

Azimuth window coefficient

fdc_early_azimuth

Doppler centroid in early azimuth

fdc_late_azimuth

Doppler centroid in late azimuth

range_pix_spacing

Slant or ground range pixel resolution

meters

azimuth_line_spacing

Azimuth line resolution

meters

near_slant_range

Slant range to near range pixel

meters

num_pos_pts

Number of ephemeris points provided

projection

Ground or slant range projection (either PAR_GROUND or PAR_SLANT)

gnd2slt_coeffs[6]

Coefficients used in polynomial transform from ground to slant range plane

time_dir_pixels

Time direction for increasing pixel (range) direction

time_dir_lines

Time direction for increasing azimuth (line) direction

rsx

Spacecraft X position (Earth Fixed Body coord system) in meters

meters

rsy

Spacecraft Y position (Earth Fixed Body coord system) in meters

meters

rsz

Spacecraft Z position (Earth Fixed Body coord system)

meters

vsx

Spacecraft X velocity (Earth Fixed Body coord system)

meters per second

vsy

Spacecraft Y velocity (Earth Fixed Body coord system)

meters per second

vsz

Spacecraft Z velocity (Earth Fixed Body coord system)

meters per second

Calculated Ephemeris Coefficients

orbit_state

Status flag which indicates if the orbit has been adjusted

rs_coeffs[9]

Coefficients used to model the sensor orbit positions as a function of time

vs_coeffs[9]

Coefficients used to model the sensor orbit velocities as a function of time

Subset Information Relative to the Original SAR Image

sub_unity_subset

Status flag indicating that the entire image is present

sub_range_start

Starting range sample of the current raster relative to the original image

sub_range_end

Ending range sample of the current raster relative to the original image

sub_range_degrade

Range sample compression factor of the current raster relative to the original image

sub_range_num_samples

Range number of samples of the current raster image

sub_azimuth_start

Starting azimuth line of the current raster relative to the original image

sub_azimuth_end

Ending azimuth line of the current raster relative to the original image

sub_azimuth_degrade

Azimuth line compression factor of the current raster relative to the original image

sub_azimuth_num_lines

Azimuth number of lines of the current raster image

Algorithm Description

Overview

The rectification process consists of several steps:

  • ephemeris modeling and refinement (if GCPs are provided)
  • sparse mapping grid generation
  • output formation (including terrain corrections)

Each of these steps is described in detail in the following sections.

Ephemeris Coordinate System

The positions and velocities of the spacecraft are internally assumed to be in an Earth Fixed Body coordinate system. If the ephemeris are provided in inertial coordinate system, OrthoRadar converts them from inertial to Earth Fixed Body coordinates.

The Earth Fixed Body coordinate system is an Earth-centered Cartesian coordinate system that rotates with the Earth. The x-axis radiates from the center of the Earth through the 0 longitude point on the equator. The z-axis radiates from the center of the Earth through the geographic North Pole. The y-axis completes the right-handed Cartesian coordinate system.

Ephemeris Modeling

The platform ephemeris is described by three or more platform locations and velocities. To predict the platform position and velocity at some time (t):

radar_ephemeris_modeling_equation1

Where:

Rs = sensor position

Vs = sensor velocity

radar_ephemeris_modeling_equation2

To determine the model coefficients radar_ephemeris_model_coefficients, first do some preprocessing. Select the best three consecutive data points prior to fitting (if more than three points are available). The best three data points must span the entire image in time. If more than one set of three data points spans the image, then select the set of three that has a center time closest to the center time of the image.

Once a set of three consecutive data points is found, model the ephemeris with an exact solution.

Form matrix A:

radar_ephemeris_matrix_a

Where t1, t2, and t3 are the times associated with each platform position.

Select t such that t = 0.0 corresponds to the time of the second position point.

Form vector b:

radar_ephemeris_vector_b_equation

Where Rs,x(i) is the x-coordinate of the i-th platform position (i =1: 3).

We wish to solve Ax = b where x is:

radar_ephemeris_x_a1_a2_a3_t

To do so, use LU decomposition. The process is repeated for: Rs,y, Rs,z, Vs,x, Vs,y, and Vs,z

SAR Imaging Model

Before discussing the ephemeris adjustment, it is important to understand how to get from a pixel in the SAR image (as specified by a range line and range pixel) to a target position on the Earth [specified in Earth Centered System (ECS) coordinates or x, y, z]. This process is used throughout the ephemeris adjustment and the rectification itself.

For each range line and range pixel in the SAR image, the corresponding target location (Rt) is determined. The target location can be described as (lat, lon, elev) or (x, y, z) in ECS. The target can either lie on a smooth Earth ellipsoid or on a smooth Earth ellipsoid plus an elevation model.

In either case, the location of Rt is determined by finding the intersection of the Doppler cone, range sphere, and Earth model. In order to do this, first find the Doppler centroid and slant range for a given SAR image pixel.

Let i = range pixel and j = range line.

Time

Time T(j) is thus:

radar_orthoradar_time_equation

Where:

T(0) = image start time

Na = number of range lines

tdur = image duration time

Doppler Centroid

The computation of the Doppler centroid fd to use with the SAR imaging model depends on how the data was processed. If the data was deskewed, this value is always 0. If the data is skewed, then this value may be a nonzero constant or may vary with i.

Slant Range

The computation of the slant range to the pixel i depends on the projection of the image. If the data is in a slant range projection, then the computation of slant range is straightforward:

radar_slant_range_equation

Where:

Rsl(i) = slant range to pixel i

rsl = near slant range

radar_slant_range_delta_r_sr_symbol= slant range pixel spacing

If the projection is a ground range projection, then this computation is potentially more complicated and depends on how the data was originally projected into a ground range projection by SAR processor.

Intersection of Doppler Cone, Range Sphere, and Earth Model

To find the location of the target Rt corresponding to a given range pixel and range line, the intersection of the Doppler cone, range sphere, and Earth model must be found. For an ellipsoid, these may be described as follows:

radar_intersection_doppler_cone_range_sphere_equation

Where:

Rs and Vs = platform position and velocity respectively

Vt = target velocity ( = 0, in this coordinate system)

Re = Earth semimajor axis

Rm = Earth semiminor axis

The platform position and velocity vectors Rs and Vs can be found as a function of time T(j) using the ephemeris equations developed previously.

The following figure graphically illustrates the solution for the target location given the sensor ephemeris, doppler cone, range sphere, and flat Earth model:

Doppler Cone

radar_doppler_cone_diagram

Ephemeris Adjustment

There are three possible adjustments that can be made: along track, cross track, and radial. In OrthoRadar, the along track adjustment is performed separately. The cross track and radial adjustments are made simultaneously. These adjustments are made using residuals associated with GCPs. Each GCP has a map coordinate (such as lat, lon) and an elevation. Also, an SAR image range line and range pixel must be given. The SAR image range line and range pixel are converted to Rt using the method described previously (substituting htarg = elevation of GCP above ellipsoid used in SAR processing).

The along track adjustment is computed first, followed by the cross track and radial adjustments. The two adjustment steps are then repeated.

For more information, see SAR Geocoding: Data and Systems Gunter, Schreier, Ed.

Orthorectification

The ultimate goal in orthorectification is to determine, for a given target location on the ground, the associated range line and range pixel from the input SAR image, including the effects of terrain.

To do this, there are several steps. First, take the target location and locate the associated range line and range pixel from the input SAR image assuming smooth terrain. This places you in approximately the correct range line. Next, look up the elevation at the target from the input DEM. The elevation, in combination with the known slant range to the target, is used to determine the correct range pixel. The data can now be interpolated from the input SAR image.

Sparse Mapping Grid

Select a block size M. For every Mth range line and Mth range pixel, compute Rt on a smooth ellipsoid (using the SAR Earth model), and save these values in an array. Smaller M implies less distortion between grid points.

Regardless of M and the total number of samples and lines in the input SAR image, always compute Rt at the end of every line and for the very last line. The spacing between points in the sparse mapping grid is regular except at the far edges of the grid.

Output Formation

For each point in the output grid, there is an associated Rt. This target should fall on the surface of the Earth model used for SAR processing, thus a conversion is made between the Earth model used for the output grid and the Earth model used during SAR processing.

The process of orthorectification starts with a location on the ground. The line and pixel location of the pixel to this map location can be determined from the map location and the sparse mapping grid. The value at this pixel location is then assigned to the map location. The following figure illustrates this process.

Sparse Mapping and Output Grids

radar_sparse_mapping_output_grid_diagram

Conversion to Real / Imaginary Data

The formula used for converting a block of complex 1-layer data into a 2-layer IQ (real/imaginary) output datastack, or converting a block of 2-layer input data into a 2-layer IQ (real/imaginary) output datastack is shown here. The conversion is available in Radar Conversions dialog. The magnitude and phase data is represented in the figure below.

real = magnitude * cos(phase)

imaginary = magnitude * sin(phase)

The above mathematical notation is 100% equivalent, simply expressed in a different form.

For radar systems, a complex number implies that the representation of a signal, or data file, needs measures of both magnitude and phase. In the context of digital SAR, a complex number can be also be represented by the real in-phase component (I) and the imaginary quadrature component (Q). The role of complex numbers is an essential part of the signal as signal phase is used to obtain high resolution.

Source: European Space Agency, 2010a and MathResources, 2010

  • The set {magnitude, phase}, or {r,greek_theta_symbol} represents coordinates in polar form.
  • The set {real, imaginary}, or { I, Q } represents coordinates in Cartesian form.

Magnitude and Phase Data as shown in the complex plane

radar_conversion_imaginary_complex_numbers_graph

Where:

Real = Real axis

Imaginary = Imaginary axis

r = Magnitude is defined as a dimensionless number describing the length of the backscatter vector in the complex domain.

phase = Relative interferometric phase angle in radians in the range 0 to 2*greek_pi_symbol, measured from the positive Real axis towards the backscatter vector.