The P6/98 seismic bin grid exchange format described a special case of the Affine Geometric Transformation. The method is also described in the SEG-Y revision 1 seismic data exchange format.
In practice bin grid definitions are based on an identified map grid. As such the example bin grids in the EPSG Dataset, coordinate transformation codes 6918 and 6919, are described with the map grid (projected CRS) as source CRS. However to retain consistency with the syntax used in the description of the affine general transformations in the previous sections of this document, in the formulas below the bin grid is the affine source CRS and the map grid is the affine target CRS.
For the P6 seismic bin grid case of the affine geometric transformation
• the base coordinate reference system is a map grid (projected CRS) which by definition has orthogonal axes with the same units;
• the bin grid axes are orthogonal;
and one or both of the following may apply:
• the origin of the bin grid may be assigned non-zero bin grid coordinates;
• the bin grid units may increase in increments other than 1, i.e. IncSX and IncSY may not be unity.
a) P6 Right-handed Seismic Bin Grid Transformation (EPSG Dataset coordinate operation method code 9666)
The P6/98 exchange format defined the bin grid axes to be named I and J, where the I-axis is rotated 90° clockwise from the J-axis (when viewed from above the plane containing the two axes). This is sometimes described as 'right-handed'.
Figure 16 – Relationship of right-handed bin grid to map grid The defining parameters are:
P6/98 term Equivalent EPSG Dataset term
Bin grid origin (Io) Ordinate 1 of evaluation point in source CRS (XSO)
Bin grid origin (Jo) Ordinate 2 of evaluation point in source CRS (YSO)
Map grid easting of bin grid origin (Eo) Ordinate 1 of evaluation point in target CRS (XTO)
Map grid northing of bin grid origin (No) Ordinate 2 of evaluation point in target CRS (YTO)
Scale factor of bin grid (SF) Point scale factor (k)
Nominal bin width along I axis (I_bin_width) Scale factor for source coordinate reference system first axis (MX)
Nominal bin width along J axis (J_bin_width) Scale factor for source coordinate reference system second axis (My)
Grid bearing of bin grid J axis (θ) Rotation angle of source coordinate reference system axes (θ)
Bin node increment on I axis (I_bin_inc) Bin node increment on I-axis Bin node increment on J axis (J_bin_inc) Bin node increment on J-axis
In this method the terms XS, YS, MX and MY in the orthogonal case of the Affine Geometric Transformation
formulas are replaced by (XS – XSO), (YS – YSO), (MX / IncSX) and (MY / IncSY) respectively. Thus the
transformation from bin grid to map grid (source to target coordinate reference system in Affine Geometric Transformation formula nomenclature) is:
VT = VTO + R2 * k * S2 * V2
where, as in the orthogonal case of the Affine Geometric Transformation method:
XT XTO cos θ sin θ
VT = VTO = and R2 =
YT YTO –sin θ cos θ
MX / IncSX 0 XS – XSO
S2 = and V2 =
0 MY / IncSY YS – YSO
That is,
XT XTO cos θ sin θ MX / IncSX 0 XS – XSO
= + * k * *
YT YTO –sin θ cos θ 0 MY / IncSY YS – YSO
Algebraically:
XT = XTO + [(XS – XSO) * cos θ * k * MX / IncSX] + [(YS – YSO) * sin θ * k * MY / IncSY]
YT = YTO – [(XS – XSO) * sin θ * k * MX / IncSX] + [(YS – YSO) * cos θ * k * MY / IncSY]
Using the symbol notation in the P6/98 document these expressions are:
I_bin_width
0
E EO cos θ sin θ / I_bin_inc I – IO
= + * SF * *
N NO –sin θ cos θ
0 J_bin_width J – JO
/ J_bin_inc
or
E = EO + [(I – IO) * cos θ * SF * I_bin_width / I_bin_inc]
+ [(J – JO) * sin θ * SF * J_bin_width / J_bin_inc]
N = NO – [(I – IO) * sin θ * SF * I_bin_width / I_bin_inc]
+ [(J – JO) * cos θ * SF * J_bin_width / J_bin_inc]
For the reverse transformation (map grid to bin grid): VS = (1/k) * S2 –1 * R2 -1 * (VT – VTO) + VSO
or
XS IncSX / MX 0 cos θ –sin θ XT – XTO XSO
= 1/k * * * +
YS 0 IncSY / MY sin cos θ YT – YTO YSO
or algebraically:
XS = {[( XT – XTO) * cos θ – (YT – YTO) * sin θ ] * [IncSX / (k * MX)]} + XSO
YS = {[(XT – XTO) * sin θ + (YT – YTO) * cos θ] * [IncSY / (k * MY)]} + YSO
Using the symbol notation in the P6/98 document these reverse expressions are: I I_bin_inc /
0 cos θ –sin θ E – EO IO = 1/SF * I_bin_width * * +
J
0 J_bin_inc / sin cos θ N– NO JO
I_bin_width
I = IO + {[(E – EO) * cos θ – (N – NO) * sin θ ] * [I_bin_inc / (SF * I_bin_width)]}
J = JO + {[(E – EO) * sin θ + (N – NO) * cos θ] * [J_bin_inc / (SF * J_bin_width)]}
Example:
This example is given in the P6/98 document. The bin grid is based on projected CRS WGS 84 / UTM Zone 31N. The origin of the bin grid is defined to be at E = 456781.0, N = 5836723.0. The point scale factor at this point is 0.99984.
The bin width on the I-axis (XS axis) is 25 metres, whilst the bin width on the J-axis (YS axis) is 12.5 metres.
The origin of the grid has bin values of 1,1.
In the map grid, the bearing of the bin grid I and J axes are 110° and 20° respectively. Thus the angle through which the bin grid axes need to be rotated to coincide with the map grid axes is +20 degrees.
The transformation parameter values are:
Parameter EPSG Affine
Transformation symbol
P6 symbol Parameter value
Bin grid origin I XSO Io 1
Bin grid origin J YSO Jo 1
Bin grid origin Easting XTO Eo 456781.00 m
Bin grid origin Northing YTO No 5836723.00 m
Scale factor of bin grid k SF 0.99984
Bin Width on I-axis MX I_bin_width 25 m
Bin Width on J-axis MY J_bin_width 12.5 m
Map grid bearing of bin grid J-axis θ θ 20 deg
Bin node increment on I-axis IncSX I_bin_inc 1
Bin node increment on J-axis IncSY J_bin_inc 1
Calculation of map grid coordinates for centre of bin with coordinates: I = 300, J = 247:
Easting = XT = XTO + [(XS – XSO) * cos θ * k * MX / IncSX] + [(YS – YSO) * sin θ * k * MY / IncSY]
= 456781.000 + 7023.078 + 1051.544 = 464855.62 m.
Northing = YT = YTO – [(XS – XSO) * sin θ * k * MX / IncSX] + [(YS – YSO) * cos θ * k * MY / IncSY]
= 5836723.000 - 2556.192 + 2889.092 = 5837055.90 m.
Calculation of bin grid coordinates for this map grid location 464 855.62mE, 5 837 055.90mN: Bin grid I = XS = XSO + {[( XT – XTO) * cos θ – (YT – YTO) * sin θ ] * [IncSX / (k * MX)]}
= 300 bins,
Bin grid J = YS = YSO + {[(XT – XTO) * sin θ + (YT – YTO) * cos θ] * [IncSY / (k * MY)]}
= 247 bins
Further examples of input and output may be found in test procedure 5209 of the test dataset which accompanies OGP Publication 430, Geospatial Integrity in Geoscience Software (GIGS).
b) P6 Left-handed Seismic Bin Grid Transformation (EPSG Dataset coordinate operation method code 1049)
The P6/98 exchange format constrained the bin grid axes to be 'right-handed', i.e. by definition the bin grid I- axis orientation is 90° clockwise from the J-axis when viewed from above the plane of the coordinate system. The P6/11 format extends the P6/98 format to allow for a 'left-handed' bin grid in which the I-axis orientation is 90° counter-clockwise from the J-axis when viewed from above the plane of the coordinate system. (Using the P6/98 format this can be accomplished through giving the Bin node increment on I-axis parameter a negative value).
Figure 17 – Relationship of left-handed bin grid to map grid
The transformation from bin grid to map grid is: VT = VTO + R2 * k * S3 * V2
where
MX / IncSX * –1 0
S3 =
0 MY / IncSY
and other matrices are as defined earlier, or
I_bin_width
0
E EO cos θ sin θ / –I_bin_inc I – IO
= + * SF * *
N NO –sin θ cos θ
0 J_bin_width J – JO
/ J_bin_inc
E = EO – [(I – IO) * cos θ * SF * I_bin_width / I_bin_inc]
+ [(J – JO) * sin θ * SF * J_bin_width / J_bin_inc]
N = NO + [(I – IO) * sin θ * SF * I_bin_width / I_bin_inc]
+ [(J – JO) * cos θ * SF * J_bin_width / J_bin_inc]
For the transformation from map grid to bin grid: VS = (1/k) * S3 –1 * R2 -1 * (VT – VTO) + VSO or I –I_bin_inc / 0 cos θ –sin θ E – EO IO = 1/SF * I_bin_width * * + J
0 J_bin_inc / sin cos θ N– NO JO
I_bin_width
or
I = IO – {[(E – EO) * cos θ – (N – NO) * sin θ ] * [I_bin_inc / (SF * I_bin_width)]}
J = JO + {[(E – EO) * sin θ + (N – NO) * cos θ] * [J_bin_inc / (SF * J_bin_width)]}
Example:
This example is given in the OGP P6/11 user guide. The base map grid is a projected CRS NAD27 / BLM 16N (ftUS) upon which the origin of the bin grid is defined at E = 871200 ftUS, N = 10280160 ftUS. As the survey was acquired on the map grid, the bin grid scale factor at the bin grid origin is chosen to be 1.
The bin width on the I-axis (XS axis) is 82.5 US survey feet, the bin width on the J-axis (YS axis) is 41.25 US
survey feet. The origin of the bin grid has bin values of I=5000, J=0.
In the map grid, the bearing of the bin grid I and J axes are 250° and 340° respectively. Thus the angle through which the bin grid axes need to be rotated to coincide with the map grid axes is +340 degrees. The transformation parameter values are:
Parameter EPSG Affine
Transformation symbol
P6 symbol Parameter value
Bin grid origin I XSO Io 5000
Bin grid origin J YSO Jo 0
Bin grid origin Easting XTO Eo 871200 ftUS
Bin grid origin Northing YTO No 10280160 ftUS
Scale factor of bin grid k SF 1
Bin Width on I-axis MX I_bin_width 82.5 ftUS
Bin Width on J-axis MY J_bin_width 41.25 ftUS
Map grid bearing of bin grid J-axis θ θ 340 deg
Bin node increment on I-axis IncSX I_bin_inc 1
Calculation for centre of bin with coordinates: I = 4700, J = 247:
Easting = XT = XTO – [(XS – XSO) * cos θ * k * MX / IncSX] + [(YS – YSO) * sin θ * k * MY / IncSY]
= 890 972.63 ftUS.
Northing = YT = YTO + [(XS – XSO) * sin θ * k * MX / IncSX] + [(YS – YSO) * cos θ * k * MY / IncSY]
= 10 298 199.29 ftUS.
Reverse calculation for this point 890 972.63 ftUS E, 10 298 199.29 ftUS N:
Bin grid I = XS = XSO – {[( XT – XTO) * cos θ – (YT – YTO) * sin θ ] * [IncSX / (k * MX)]}
= 4700 bins,
Bin grid J = YS = YSO + {[(XT – XTO) * sin θ + (YT – YTO) * cos θ] * [IncSY / (k * MY)]}
= 247 bins