Spheres versus Spheroids

 Cédric H. David, CRWR

Table of Contents

 

Introduction

 

Atmospheric observational and model output datasets and Hydrologic datasets are now available on continental scales.  The temptation to link both on a large scale is becoming larger and larger. 

 

Communities of hydrologists and atmospheric scientists grew with different representations of the Earth: atmospheric scientists use a spherical Earth whereas hydrologists use a spheroidal one (see Figure 1).

 

Figure 1 Sphere and spheroid (from http://geography.sierra.cc.ca.us)

 

The different representations of the Earth bring interoperability issues between hydrologic and atmospheric science data, some of these issues are addressed in this document.

 

The following work describes how the sphere spheroid problem can be addressed in ArcGIS, with application to atmospheric sciences and hydrologic data; but can be generalized to other cases.

 

Geodetic and geographic latitudes

 

There are various ways to define latitudes and longitudes, amongst them are geodetic and geographic definitions of latitudes.  The geodetic latitude is the angle made with the equatorial plane of a normal to a tangent plane on the spheroidal earth's surface.  The geographic latitude is the angle made with the equatorial plane of a normal to a tangent plane on the spherical earth.   The main difference is that for the spherical earth the normal always passes through the center whereas for the spheroidal earth it does not. (see Figure 2).

 

Figure 2 Geodetic and geographic latitudes

 

The different representations of the Earth bring interoperability issues between hydrologic and atmospheric science data, some of these issues are addressed in this document.

 

The following work describes how the sphere spheroid problem can be addressed in ArcGIS, with application to atmospheric sciences and hydrologic data; but can be generalized to other cases.

 

 

Integration of atmospheric science models with hydrologic and land surface information

 

Most weather models run off a mathematically constructed spherical earth of a defined radius.  The integration of atmospheric science models (on spherical Earth) with hydrologic and land surface information (on a spheroidal Earth) is non-trivial.

 

The netCDF files compliant with the first version of the Climate and Forecast (CF-1) conventions are widely used in and made available by the atmospheric scientist community.  Therefore, the use of netCDF CF-1 as atmospheric data files for interdisciplinary projects seems to be the appropriate choice.  Unfortunately, the geospatial referencing of the atmospheric models whose outputs are distributed in netCDF CF-1 files is not communicated in the said files.  The geospatial referencing information in netCDF CF-1 files specifies a shape for the earth (mainly sphere for atmospheric datasets) and projection parameters (type, characteristic features, etc.) if the coordinates are projected.  Unfortunately, the radius of the sphere is not part of the requirements of the CF-1 conventions.  Careful investigation the model parameters is necessary before using the resulting outputs of any models.  The investigation will lead to specific geographic and projected coordinate systems with appropriate transformations that is crucial for the use of netCDF CF-1 files as atmospheric data for interdisciplinary projects.

 

Application: netCDF CF-1 in ArcGIS

 

For the purpose of continental water dynamics modeling, NHDPlus (USGS - USEPA) was used as hydrologic data and NAM (North American Model, National Center for Environmental Prediction) was used as atmospheric data.  NHDPlus data is available in a Geographic Coordinate System (GCS) called North American Datum 1983 (NAD83).  NAD83 is based on a spheroid.  NAM is available in a GCS based on a 6370 km radius Earth, and in a Projected Coordinate System (PCS, here Lambert Conformal Conic) based on this spherical Earth.  The NAM files used in this document are netCDF CF-1 from the 40 km NAM Conduit.

 

ArcGIS 9.2 focuses on the reading of netCDF that conforms to the CF convention.  To draw netCDF data with other map data in a GIS, ArcGIS needs to know what sphere/ellipsoid/datum to use, but the CF convention does not include this information.  Because World Geodetic System 1984 (WGS84) is the most commonly used datum, ArcGIS assumes the datum is always WGS84.  WSG84 is a spheroid.  Much science data is on the WGS84 spheroid, and if it is on a sphere, it is most likely that the latitudes and longitudes came from WGS84, and therefore really are WGS84 (Steve Kopp, Pers. Comm.).

 

There remains an open question for the validity of this assumption in the case of NAM: is the combination of NAM and NHDPlus appropriately be done in ArcGIS?  Figure 3, Figure 4 and Figure 5 show the combination of NAM and NHDPlus data in ArcGIS.

 

Figure 3 NHDPlus subbasins for Hydrologic Unit Code (HUC) 05

Figure 4 NAM40k forecast temperature field

Figure 5 Combination of NHDPlus subbasins and NAM40k for HUC 05

Possible work-around

 

ArcGIS displays the netCDF data in the Projected Coordinate System (PCS) that is defined in the netCDF file, assuming the PCS is based on WGS84.  What if the PCS was actually based in the 6370 km radius Earth?

 

ArcGIS uses the projected coordinates to display netCDF CF files.  By using the Multidimension tools / Create NetCDF feature layer tool, a table can be created.  The AddXY tool will display the created feature layer based on either the geographic or projected coordinates.  The appropriate GCS for NAM model results is Geographic Coordinate System / Spheroid Based / Sphere EMEP.   The PCS for NAM based on the projection parameters defined in the netCDF file and on a spherical Earth can be created as follows.

 

Creation of a Projected Coordinate System for North American Model

 

Use the tool Data Management Tools / Projections and Transformations / Define Projections.

 

Create a new Projected Coordinate System

The geospatial information provided in the netCDF file is the following:

 

   char Lambert_Conformal;

     :grid_mapping_name = "lambert_conformal_conic";

     :standard_parallel = 25.0; // double

     :longitude_of_central_meridian = -95.0; // double

     :latitude_of_projection_origin = 25.0; // double

     :GRIB_earth_shape = "spherical";

     :GRIB_earth_shape_code = 0; // int

     :_CoordinateTransformType = "Projection";

     :_CoordinateAxisTypes = "GeoX GeoY";

 

 

The projection is Lambert Conformal Conic, both standard parallels are 25.0 degrees and the latitude of origin is -95.0 degrees.  The shape of the Earth in NAM is defined in Sphere EMEP:

The new projection can be then defined as follows:

This projection can be saved:

Therefore, if the projection coordinates provided in the netCDF file are with regard to a sphere, and not the WGS84 spheroid, they can be used with the PCS that was created in this section.

 

Creation of a geographic transformation between Spheroid and Sphere

 

There is no predefined transformation between spheres and spheroids in ArcGIS.  For a sphere to spheroid conversion, the only alignment problem is with the latitude, longitude remains unchanged (if the spheroid is geocentric) (David Maidment, Pers. Comm.)

 

In order to create an appropriate transformation, use the tool Data Management Tools / Projections and Transformations / Create Custom Geographic Transformation. 

Figure 6 Geocentric transformation for spheroid to sphere

 

A “Geocentric_Translation" can be defined between a sphere and a spheroid (see Figure 6).  Leaving all the parameters set to zeroes will convert the latitude values from sphere to spheroid (and vice versa), the longitudes will remain unchanged.

 

Creation of a conformal transformation between Spheroid and Sphere

A conformal transformation can be defined between the geographic (longitude: l, latitude: j) and geodetic coordinates (longitude: l', latitude: j').  The formula is the following (Bugayevskiy and Snyder, “Map projections, a reference manual”, 1995):

 

Where a and C are constants, e is the eccentricity of the spheroid, F0 = F0‘ the central parallel.

Conclusion

 

Until better understanding of the parameters involved in the computation of the weather models is gained, work-arounds will be necessary for the integration of atmospheric science models (on spherical Earth) with hydrologic and land surface information (on a spheroidal Earth).  The work described here is issued from hydrologic and atmospheric science integration, but is of interest to geospatial data interoperability in general.

 

Primary Contact

 

Cédric H. David

Center for Research in Water Resources

University of Texas at Austin

 

cedric.david@mail.utexas.edu

 

These materials may be used for study, research, and education, but please credit the authors and the Center for Research in Water Resources, The University of Texas at Austin. All commercial rights reserved. Copyright 2006 Center for Research in Water Resources.