appj.adoc

Appendix A: Coordinate Interpolation Methods

The general description of the compression by coordinate subsampling is given in section [compression-by-coordinate-subsampling]. This appendix provides details on the available methods for compression by coordinate subsampling.

The definitions and guidance given here allow an application to compress an existing data set using coordinate subsampling, while letting the creator of the compressed dataset control the accuracy of the reconstituted coordinates through the degree of subsampling, the choice of interpolation method and by setting the computational precision.

Furthermore, the definitions given here allow an application to uncompress coordinate and auxiliary coordinate variables that have been compressed using coordinate subsampling. The key element of this process is the reconstitution of the full resolution coordinates in the domain of the data by interpolation between the subsampled coordinates, the tie points, stored in the compressed dataset.

The appendix is organised in a sections on Section J.1, "Common Definitions and Notation", Section J.2, "Common Conversions and Formulas", Section J.3, "Interpolation Methods" and finally two sections with step procedures Section J.4, "Coordinate Compression Steps" and Section J.5, "Coordinate Uncompression Steps".

Common Definitions and Notation

The coordinate interpolation methods are named to indicate the number of dimensions they interpolate as well as the type of interpolation provided. For example, the interpolation method named linear provides linear interpolation in one dimension and the method named bi_linear provides linear interpolation in two dimensions. Equivalently, the interpolation method named quadratic provides quadratic interpolation in one dimension and the interpolation method named bi_quadratic provides quadratic interpolation in two dimensions.

When an interpolation method is referred to as linear or quadratic, it means that the method is linear or quadratic in the indices of the interpolated dimensions.

For convenience, an interpolation argument s is introduced, calculated as a function of the index in the target domain of the coordinate value to be reconstituted. In the case of one-dimensional interpolation the variable is computed as:

s = s(ia, ib, i)
  = (i - ia)/(ib - ia)

where ia and ib are the indices in the target domain of the tie points A and B and i is the index in the target domain of the coordinate value to be reconstituted.

Note that the value of s varies from 0.0 at the tie point A to 1.0 at tie point B. For example, if ia = 100 and ib = 110 and the index in the target domain of the coordinate value to be reconstituted is i = 105, then s = (105 - 100)/(110 - 100) = 0.5.

In the case of two-dimensional interpolation, the interpolation arguments are similarly computed as:

s1 = s(ia1, ib1, i1)
   = (i1 - ia1)/(ib1 - ia1)
s2 = s(ia2, ic2, i2)
   = (i2 - ia2)/(ic2 - ia2)

where ia1 and ib1 are the first dimension indices in the target domain of the tie points A and B respectively, ia2 and ic2 are the second dimension indices in the target domain of the tie points A and C respectively and the indices i1 and i2 are the first and second dimension indices respectively in the target domain of the coordinate value to be reconstituted.

The target domain is segmented into smaller interpolation subareas as described in [compression-by-coordinate-subsampling-tie-points-and-interpolation-subareas].

For one-dimensional interpolation, an interpolation subarea is defined by two tie points, one at each end of the interpolation subarea. However, the tie points may be inside or outside the interpolation subareas as shown in Figure J.1. When interpolation methods are applied for a given interpolation subarea, it must be ensured that reconstituted coordinate points are only generated inside the interpolation subarea being processed, even if some of the tie point coordinates lie outside of that interpolation subarea. See also description in [compression-by-coordinate-subsampling-tie-points-and-interpolation-subareas].

Figure J.1. One-dimensional interpolation subareas, one including and one excluding tie point A.

For two-dimensional interpolation, an interpolation subarea is defined by four tie points, one at each corner of a rectangular area aligned with the domain axes, see Figure J.2.

Figure J.2. Two-dimensional interpolation subarea.

For the reconstitution of the uncompressed coordinate and auxiliary coordinate variables the interpolation method can be applied independently for each interpolation subarea, making it possible to parallelize the computational process.

The following notation is used:

A variable staring with v denotes a vector and v.x, v.y and v.z refer to the three coordinates of that vector.

A variable staring with ll denotes a latitude-longitude coordinate pair and ll.lat and ll.lon refer to the latitude and longitude coordinates.

For one-dimensional interpolation, i is an index in the interpolated dimension, tpi is an index in the subsampled dimension and is is an index in the interpolation subarea dimensions.

For two-dimensional interpolation, i1 and i2 are indices in the interpolated dimensions, tpi1 and tpi2 are indices in the subsampled dimensions and is1 and is2 are indices in the interpolation subarea dimensions. Dimension 1 is the dimension with index values increasing from tie point A to tie point B, dimension 2 is the dimension with index values increasing from tie point A to tie point C.

Note that, for simplicity of notation, the descriptions of the interpolation methods in most places leave out the indices of tie point related variables and refer to these with a and b in the one-dimensional case and with a, b, c and d in the two-dimensional case. In the two-dimensional case, a = tp(tpi2, tpi1), b = tp(tpi2, tpi1+1), c = tp(tpi2+1, tpi1) and d = tp(tpi2+1, tpi1+1) would reflect the way the tie point data would be stored in the data set, see also Figure J.2.

Common Conversions and Formulas

	Description	Formula
sqrt	Square Root	s = sqrt(t)
atan2	Inverse Tangent of y/x	a = atan2(y, x)
fll2v	Conversion from geocentric (latitude, longitude) to three-dimensional cartesian vector (x, y, z)	(x, y, z) = fll2v(ll) = (cos(ll.lat)*cos(ll.lon), cos(ll.lat)*sin(ll.lon), sin(ll.lat))
fv2ll	Conversion from three-dimensional cartesian vector (x, y, z) to geocentric (latitude, longitude)	(lat, lon) = fv2ll(v) = (atan2(v.z, sqrt(v.x * v.x + v.y * v.y)), atan2(v.y, v.x))
faz2v	Conversion from (azimuth, zenith) angles to three-dimensional cartesian vector (x, y, z)	(x, y, z) = faz2v(az) = (sin(az.zenith) * sin(az.azimuth), sin(az.zenith) * cos(az.azimuth), cos(az.zenith))
fv2az	Conversion from three-dimensional cartesian vector (x, y, z) to (azimuth, zenith) angles	(azimuth, zenith) = fv2az(v) = (atan2(y, x), atan2(sqrt(x * x + y * y), z))
fplus	Vector Sum	(x, y, z) = fplus(va, vb) = (va.x + vb.x, va.y + vb.y, va.z + vb.z) (x, y, z) = fplus(va, vb, vc) = (va.x + vb.x + vc.x, va.y + vb.y + vc.y, va.z + vb.z + vc.z)
fminus	Vector Difference	(x, y, z) = fminus(va, vb) = (va.x - vb.x, va.y - vb.y, va.z - vb.z)
fmultiply	Vector multiplied by Scalar	(x, y, z) = fmultiply(r, v) = (r * v.x, r * v.y, r * v.z)
fcross	Vector Cross Product	(x, y, z) = fcross(va, vb) = (va.y*vb.z - va.z*vb.y, va.z*vb.x - va.x*vb.z, va.x*vb.y - va.y*vb.x)
fdot	Vector Dot Product	d = fdot(va, vb) = va.x*vb.x + va.y*vb.y + va.z*vb.z

Interpolation Methods

Linear Interpolation

Name
interpolation_name = "linear"

Description
General purpose one-dimensional linear interpolation method for one or more coordinates

Interpolation parameter terms
None.

Coordinate compression calculations
None.

Coordinate uncompression calculations
The coordinate value u(i) at index i between tie points A and B is calculated from:

u(i) = fl(ua, ub, s(i))
     = ua + s*(ub-ua)

where ua and ub are the coordinate values at tie points A and B respectively.

Bilinear Interpolation

Name
interpolation_name = "bi_linear"

Description
General purpose two-dimensional linear interpolation method for one or more coordinates.

Interpolation parameter terms
None.

Coordinate compression calculations
None.

Coordinate uncompression calculations
The interpolation function fl() defined for linear interpolation above is first applied twice in the interpolated dimension 2, once between tie points A and C and once between tie points B and D.

It is then applied once in the interpolated dimension 1, between the two resulting coordinate points, yielding the interpolated coordinate value u(i2, i1):

uac       = fl(ua, uc, s(ia2, ic2, i2))
ubd       = fl(ub, ud, s(ia2, ic2, i2))
u(i2, i1) = fl(uac, ubd, s(ia1, ib1, i1))

Quadratic Interpolation

Name
interpolation_name = "quadratic"

Description
General purpose one-dimensional quadratic interpolation method for one coordinate.

Interpolation parameter terms
Optionally the term w, specifying a numerical variable spanning the interpolation subarea dimension.

Coordinate compression calculations
The expression

w = fw(ua, ub, u(i), s(i))
  = (u - (1 - s) * ua - s * ub)/(4 * (1 - s) * s)

enables the creator of the dataset to calculate the coefficient w from the coordinate values ua and ub at tie points A and B respectively, and the coordinate value u(i) at index i between the tie points A and B. If the number of points in the interpolation subarea (ib - ia + 1) is odd, then the data point at index i = (ib + ia)/2 shall be selected for this calculation, otherwise the data point at index i = (ib + ia - 1)/2 shall be selected.

Coordinate uncompression calculations
The coordinate value u(i) at index i between tie points A and B is calculated from:

u(i) = fq(ua, ub, w, s(i))
     = ua + s * (ub - ua + 4 * w * (1 - s))

where ua and ub are the coordinate values at tie points A and B respectively and the coefficient w is available as a term in the interpolation_parameters, or otherwise defaults to 0.0.

Quadratic Interpolation of Geographic Coordinates Latitude and Longitude

Name
interpolation_name = "quadratic_latitude_longitude"

Description
A one-dimensional quadratic method for interpolation of the geographic coordinates latitude and longitude, typically used for remote sensing products with geographic coordinates on the reference ellipsoid.

Requires a pair of latitude and longitude tie point variables, as defined unambiguously in [latitude-coordinate] and [longitude-coordinate]. For each interpolation subarea, none of the tie points defining the interpolation subarea are permitted to coincide.

By default, interpolation is performed directly in the latitude and longitude coordinates, but may be performed in three-dimensional cartesian coordinates where required for achieving the desired accuracy. This must be indicated by setting the location_use_3d_cartesian flag within the interpolation parameter interpolation_subarea_flags for each interpolation subarea where interpolation in three-dimensional cartesian coordinates is required.

The quadratic interpolation coefficients cea = (ce, ca), stored as interpolation parameters in the product, describe a point P between the tie points A and B, which is equivalent of an additional tie point in the sense that the method will accurately reconstitute the point P in the same way as it accurately reconstitutes the tie points A and B. See Figure J.3 and Figure J.4.

Although equivalent to a tie point, the coefficients ce and ca have two advantages over tie points. Firstly, they can often be stored as a lower precision floating point number compared to the tie points, as ce and ca only describes the position of P relative to the midpoint M between the tie points A and B. Secondly, if any of ce and ca do not contribute significantly to the accuracy of the reconstituted points, it can be left out of the data set and its value will default to zero during uncompression.

The coefficients may be represented in three different ways:

• For storage in the dataset as the non-dimensional coefficients cea = (ce, ca), referred to as the parametric representation. The component ce is the offset projected on the line from tie point B to tie point A and expressed as a fraction of the distance between A and B. The component ca is the offset projected on the line perpendicular to the line from tie point B to tie point A and perpendicular to the plane spanned by va and vb, the vector representations of the two tie points, and expressed as a fraction of the length of A x B.

• For interpolation in three-dimensional cartesian coordinates as the coefficients cv = (cv.x, cv.y, cv.z), expressing the offset components along the three-dimensional cartesian axes X, Y and Z respectively.

• For interpolation in geographic coordinates latitude and longitude as the coefficients cll = (cll.lat, cll.lon), expressing the offset components along the longitude and latitude directions respectively.

The functions fq() and fw() referenced in the following are defined in Quadratic Interpolation.

Interpolation parameter terms
Optionally, any subset of terms ce, ca, each specifying a numerical variable spanning the interpolation subarea dimension.

The mandatory term interpolation_subarea_flags, specifying a flag variable spanning the interpolation subarea dimension and including location_use_3d_cartesian in the flag_meanings attribute.

Coordinate compression calculations
First calculate the tie point vector representations from the tie point latitude-longitude representations:

va = fll2v(lla)
vb = fll2v(llb)

Then calculate the three-dimensional cartesian representation of the interpolation coefficients from the tie points va and vb as well as the point vp(i) at index i between the tie points A and B. If the number of points in an interpolation subarea (ib - ia + 1) is odd, then the data point at index i = (ib + ia)/2 shall be selected for this calculation, otherwise the data point at index i = (ib + ia - 1)/2 shall be selected.

The three-dimensional cartesian interpolation coefficients are found from:

cv = fcv(va, vb, vp(i), s(i))
   = (fw(va.x, vb.x, vp(i).x, s(i)),
      fw(va.y, vb.y, vp(i).y, s(i)),
      fw(va.z, vb.z, vp(i).z, s(i)))

Finally, for storage in the dataset, convert the coefficients to the parametric representation:

cea(is) = (ce(is), ca(is))
        = fcv2cea(va, vb, cv)
        = (fdot(cv, fminus(va, vb))/gsqr,
           fdot(cv, fcross(va, vb))/(rsqr*gsqr))

where

vr   = fmultiply(0.5, fplus(va, vb))
rsqr = fdot(vr, vr)
vg   = fminus(va, vb)
gsqr = fdot(vg, vg)

The interpolation parameter term interpolation_subarea_flags(is) shall have the flag location_use_3d_cartesian set if the interpolation subarea intersects the longitude = 180.0 or if the interpolation subarea extends into latitude > latitude_limit or latitude < -latitude_limit. The value of latitude_limit is set by the data set creator and defines the high latitude areas where interpolation in three-dimensional cartesian coordinates is required for reasons of coordinate reconstitution accuracy. The latitude_limit is used solely for setting the flag location_use_3d_cartesian, and is not required in a compressed dataset.

Coordinate uncompression calculations
First calculate the tie point vector representations from the tie point latitude-longitude representations:

va = fll2v(lla)
vb = fll2v(llb)

Then calculate the three-dimensional cartesian representation of the interpolation coefficients from the parametric representation stored in the dataset using:

cv = fcea2cv(va, vb, cea(is))
   = fplus(fmultiply(ce, fminus(va, vb)),
           fmultiply(ca, fcross(va, vb)),
           fmultiply(cr, vr))

where

vr   = fmultiply(0.5, fplus(va, vb))
rsqr = fdot(vr, vr)
cr   = sqrt(1 - ce(is)*ce(is) - ca(is)*ca(is)) - sqrt(rsqr)

If the flag location_use_3d_cartesian of the interpolation parameter term interpolation_subarea_flags(is2, is1) is set, use the following expression to reconstitute any point llp(i) between the tie points A and B using interpolation in three-dimensional cartesian coordinates:

vp(i)  = fqv(va, vb, cv, s(i))
       = (fq(va.x, vb.x, cv.x, s(i)),
          fq(va.y, vb.y, cv.y, s(i)),
          fq(va.z, vb.z, cv.z, s(i)))
llp(i) = fv2ll(vp(i))

Otherwise, first calculate latitude-longitude representation of the interpolation coefficients:

cll = fcll(lla, llb, llab)
    = (fw(lla.lat, llb.lat, llab.lat, 0.5),
       fw(lla.lon, llb.lon, llab.lon, 0.5))

where

llab = fv2ll(fqv(va, vb, cv, 0.5))

Then use the following expression to reconstitute any point llp(i) between the tie points A and B using interpolation in latitude-longitude coordinates:

llp(i) = (llp(i).lat, llp(i).lon)
       = fqll(lla, llb, cll, s(i))
       = (fq(lla.lat, llb.lat, cll.lat, s(i)),
          fq(lla.lon, llb.lon, cll.lon, s(i)))

Figure J.3. With the expansion coefficient ce = 0 and the alignment coefficient ca = 0, the method reconstitutes the points at regular intervals along a great circle between tie points A and B.

Figure J.4. With the expansion coefficient ce > 0 and the alignment coefficient ca > 0, the method reconstitutes the points at intervals of expanding size (ce) along an arc with an alignment offset (ca) from the great circle between tie points A and B.

Biquadratic Interpolation of Geographic Coordinates Latitude and Longitude

Name
interpolation_name = "bi_quadratic_latitude_longitude"

Description
A two-dimensional quadratic method for interpolation of the geographic coordinates latitude and longitude, typically used for remote sensing products with geographic coordinates on the reference ellipsoid.

Requires a pair of latitude and longitude tie point variables, as defined unambiguously in [latitude-coordinate] and [longitude-coordinate]. For each interpolation subarea, none of the tie points defining the interpolation subarea are permitted to coincide.

The functions fcv(), fcv2cea(), fcea2cv(), fcll(), fqv() and fqll() referenced in the following are defined in Quadratic Interpolation of Geographic Coordinates Latitude and Longitude. As for that method, interpolation is performed directly in the latitude and longitude coordinates or in three-dimensional cartesian coordinates, where required for achieving the desired accuracy. Similarly, it shares the three different representations of the quadratic interpolation coefficients, the parametric representation cea = (ce, ca) for storage in the dataset, cll = (cll.lat, cll.lon) for interpolation in geographic coordinates latitude and longitude and cv = (cv.x, cv.y, cv.z) for interpolation in three-dimensional cartesian coordinates.

The parametric representation of the interpolation coefficients, stored in the interpolation parameters ce1, ca1, ce2, ca2, ce3 and ca3, is equivalent to five additional tie points for the interpolation subarea as shown in Figure J.5, which also shows the orientation and indices of the parameters.

Interpolation parameter terms
Optionally, any subset of terms ce1 and ca1, each specifying a numerical variable spanning the subsampled dimension 2 and the interpolation subarea dimension 1.

Optionally, any subset of terms ce2 and ca2, each specifying a numerical variable spanning the interpolation subarea dimension 2 and the subsampled dimension 1.

Optionally, any subset of terms ce3 and ca3, each specifying a numerical variable spanning the interpolation subarea dimension 2 and the interpolation subarea dimension 1.

The mandatory term interpolation_subarea_flags, specifying a flag variable spanning the interpolation subarea dimension 2 and the interpolation subarea dimension 1 and including location_use_3d_cartesian in the flag_meanings attribute.

Coordinate compression calculations
First calculate the tie point vector representations from the tie point latitude-longitude representations:

va = fll2v(lla)
vb = fll2v(llb)
vc = fll2v(llc)
vd = fll2v(lld)

Then calculate the three-dimensional cartesian representation of the interpolation coefficients sets from the tie points as well as a point vp(i2, i1) between the tie points. If the number of points in the first dimension of the interpolation subarea (ib1 - ia1 + 1) is odd, then the data point at index i1 = (ib1 + ia1)/2 shall be selected for this calculation, otherwise the data point at index i1 = (ib1 + ia1 - 1)/2 shall be selected. If the number of points in the second dimension of the interpolation subarea (ic2 - ia2 + 1) is odd, then the data point at index i2 = (ic2 + ica)/2 shall be selected for this calculation, otherwise the data point at index i2 = (ic2 + ia2 - 1)/2 shall be selected.

Using the selected (i2, i1), the three-dimensional cartesian interpolation coefficients are found from:

s1    = s(ia1, ib1, i1)
s2    = s(ia2, ic2, i2)
vac   = fll2v(ll(i2, ia1))
vbd   = fll2v(ll(i2, ib1))
cv_ac = fcv(va, vc, vac, s2)
cv_bd = fcv(vb, vd, vbd, s2)
cv_ab = fcv(va, vb, fll2v(ll(ia2, i1)), s1)
cv_cd = fcv(vc, vd, fll2v(ll(ic2, i1)), s1)
cv_zz = fcv(vac, vbd, fll2v(ll(i2, i1)), s1)
vz    = fqv(vac, vbd, cv_zz, 0.5)
vab   = fqv(va, vb, cv_ab, 0.5)
vcd   = fqv(vc, vd, cv_cd, 0.5)
cv_z  = fcv(vab, vcd, vz, s2)

Finally, before storing them in the dataset’s interpolation parameters, convert the coefficients to the parametric representation:

cea1(tpi2, is1)   = fcv2cea(va, vb, cv_ab)
cea1(tpi2+1, is1) = fcv2cea(vc, vd, cv_cd)
cea2(is2, tpi1)   = fcv2cea(va, vc, cv_ac)
cea2(is2, tpi1+1) = fcv2cea(vb, vd, cv_bd)
cea3(is2, is1)    = fcv2cea(vab, vcd, cv_z)

The interpolation parameter term interpolation_subarea_flags(is2, is1) shall have the flag location_use_3d_cartesian set if the interpolation subarea intersects the longitude = 180.0 or if the interpolation subarea extends into latitude > latitude_limit or latitude < -latitude_limit. The value of latitude_limit is set by the data set creator and defines the high latitude areas where interpolation in three-dimensional cartesian coordinates is required for reasons of coordinate reconstitution accuracy. The latitude_limit is used solely for setting the flag location_use_3d_cartesian, and is not required in a compressed dataset.

Coordinate uncompression calculations
First calculate the tie point vector representations from the tie point latitude-longitude representations:

va = fll2v(lla)
vb = fll2v(llb)
vc = fll2v(llc)
vd = fll2v(lld)

Then calculate the three-dimensional cartesian representation of the interpolation coefficient sets from the parametric representation stored in the dataset:

cv_ac = fcea2cv(va, vc, cea2(is2, tpi1))
cv_bd = fcea2cv(vb, vd, cea2(is2, tpi1 + 1))
vab   = fqv(va, vb, fcea2cv(va, vb, cea1(tpi2, is1)), 0.5)
vcd   = fqv(vc, vd, fcea2cv(vc, vd,  cea1(tpi2 + 1, is1)), 0.5)
cv_z  = fcea2cv(vab, vcd, cea3(is2, is1))

If the flag location_use_3d_cartesian of the interpolation parameter term interpolation_subarea_flags is set, use the following expression to reconstitute any point llp(i2, i1) between the tie points A and B using interpolation in three-dimensional cartesian coordinates:

llp(i2, i1) = fv2ll(fqv(vac, vbd, cv_zz, s(ia1, ib1, i1)))

where

s2    = s(ia2, ic2, i2)
vac   = fqv(va, vc, cv_ac, s2)
vbd   = fqv(vb, vd, cv_bd, s2)
vz    = fqv(vab, vcd, cv_z, s2)
cv_zz = fcv(vac, vbd, vz, 0.5)

Otherwise, first calculate latitude-longitude representation of the interpolation coefficients:

llc_ac = fcll(lla, llc, fv2ll(fqv(va, vc, cv_ac, 0.5)))
llc_bd = fcll(llb, lld, fv2ll(fqv(vb, vd, cv_bd, 0.5)))
llab   = fv2ll(vab)
llcd   = fv2ll(vcd)
llc_z  = fcll(llab, llcd, fv2ll(fqv(vab, vcd, cv_z, 0.5)))

Then use the following expression to reconstitute any point llp(i2, i1) in the interpolation subarea using interpolation in latitude-longitude coordinates:

llp(i2, i1) = fqll(llac, llbd, cl_zz, s(ia1, ib1, i1))

where

s2    = s(ia2, ic2, i2)
llac  = fqll(lla, llc, llc_ac, s2)
llbd  = fqll(llb, lld, llc_bd, s2)
llz   = fqll(llab, llcd, llc_z, s2)
cl_zz = fcll(llac, llbd, llz)

Figure J.5. The parametric representation of the interpolation coefficients cea = (ce, ca), stored in the interpolation parameters ce1, ca1, ce2, ca2, ce3 and ca3, is equivalent to five additional tie points for the interpolation subarea. Shown with parameter orientation and indices.

Coordinate Compression Steps

Step	Description	Link
1	Identify the coordinate and auxillary coordinate variables for which tie point and interpolation variables are required.
2	Identify non-overlapping subsets of the coordinate variables to be interpolated by the same interpolation method. For each coordinate variable subset, create an interpolation variable and specify the selected interpolation method using the interpolation_name attribute of the interpolation variable.	[compression-by-coordinate-subsampling-interpolation-variable]
3	For each coordinate variable subset, add the coordinates variable subset and the corresponding interpolation variable name to the the coordinate_interpolation attribute of the data variable.	[compression-by-coordinate-subsampling-coordinate-interpolation-attribute]
4	For each coordinate variable subset, identify the set of interpolated dimensions and the set of non-interpolated dimensions.	[compression-by-coordinate-subsampling-dimensions]
5	For each set of the interpolated dimensions, identify the continuous areas and select the interpolation subareas and the tie points, taking into account the required coordinate reconstitution accuracy when selecting the density of tie points.	[compression-by-coordinate-subsampling-tie-points-and-interpolation-subareas]
6	For each of the interpolated dimensions, add the interpolated dimension, the corresponding subsampled dimension and, if required by the selected interpolation method, its corresponding interpolation subarea dimension to the tie_point_mapping attribute of the interpolation variable.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-dimension-mapping]
7	For each of the interpolated dimensions, record the location of each identified tie point in a tie point index variable. For each interpolated dimension, add the tie point index variable name to the tie_point_mapping attribute of the interpolation variable.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-index-mapping]
8	For each of the target coordinate and auxillary coordinate variables, create the corresponding tie point coordinate variable and copy the coordinate values from the target domain coordinate variables to the tie point variables for the target domain indices identified by the tie point index variable. Repeat this step for each combination of indices of the non-interpolated dimensions.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-index-mapping]
9	For each of the target coordinate and auxillary coordinate variable having a bounds attribute, add the bounds_tie_points attribute to the tie point coordinate variable and create the bounds tie point variable. For each continuous area, copy the selected set of bounds tie points values from the target domain bounds variable to the bounds tie point variable for the target domain indices identified by the tie point index variable. Repeat this step for each combination of indices of the non-interpolated dimensions.	[compression-by-coordinate-subsampling-interpolation-of-cell-boundaries]
10	If required by the selected interpolation method, follow the steps defined for the method in Section J.3, "Interpolation Methods" to create any required interpolation parameter variables. As relevant, create the interpolation_parameters attribute and populate it with the interpolation parameter variables.	[compression-by-coordinate-subsampling-interpolation-variable] Section J.3, "Interpolation Methods"
11	Optionally, check the consistency of the original coordinates and the reconstructed coordinates and add a comments attribute to one or more of the tie point coordinate variables reporting key figures like maximum error, mean error, etc.

Coordinate Uncompression Steps

Step	Description	Link
1	From the coordinate_interpolation attribute of the data variable, identify the coordinate and auxillary coordinate variable subsets, for which tie point interpolation is required, and the interpolation variable corresponding to each subset.	[compression-by-coordinate-subsampling-coordinate-interpolation-attribute]
2	For each coordinate variable subset, identify the interpolation method from the interpolation_name attribute of the interpolation variable.	[compression-by-coordinate-subsampling-interpolation-variable]
3	For each coordinate variable subset, identify the set of interpolated dimensions and the set of non-interpolated dimensions from the tie_point_mapping attribute of the interpolation variable.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-dimension-mapping]
4	From the tie_point_mapping attribute of the interpolation variable, identify for each of the interpolated dimensions the corresponding subsampled dimension and, if defined, the corresponding interpolation subarea dimension.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-dimension-mapping]
5	From the tie point index variables referenced in the tie_point_mapping attribute of the interpolation variable, identify the location of the tie points in the corresponding interpolated dimension.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] [compression-by-coordinate-subsampling-tie-point-index-mapping]
6	For each of the interpolated dimensions, identify pairs of adjacent indices in the tie point index variable with index values differing by more than one, each index pair defining the extend of an interpolation subarea in that dimension. A full interpolation subarea is defined by one such index pair per interpolated dimension, with combinations of one index from each pair defining the interpolation subarea tie points.	[compression-by-coordinate-subsampling-tie-points-and-interpolation-subareas]
7	As required by the selected interpolation method, identify the interpolation parameter variables from the interpolation variable attribute interpolation_parameters.	[compression-by-coordinate-subsampling-interpolation-parameters]
8	For each of the tie point coordinate and auxillary coordinate variables, create the corresponding target coordinate variable. For each interpolation subarea, apply the interpolation method, as described in Section J.3, "Interpolation Methods", to reconstitute the target domain coordinate values and store these in the target domain coordinate variables. Repeat this step for each combination of indices of the non-interpolated dimensions.	[compression-by-coordinate-subsampling-tie-point-mapping-attribute] Section J.3, "Interpolation Methods"
9	For each of the tie point coordinate and auxillary coordinate variables having a bounds_tie_points attribute, add the bounds attribute to the target coordinate variable and create the target domain bounds variable. For each interpolation subarea, apply the interpolation method to reconstitute the target domain bound values and store these in the target domain bound variables. Repeat this step for each combination of indices of the non-interpolated dimensions.	[compression-by-coordinate-subsampling-interpolation-of-cell-boundaries]
10	If auxiliary coordinate variables have been reconstituted, then, if not already present, add a coordinates attribute to the data variable and add to the attribute the list of the names of the reconstituted auxiliary coordinate variables.	[coordinate-system]