**Background**

Read up on compass variation or declination. For example, this NOAA site provides some useful information.

Mariners use the magnetic variation to compute the difference between True north (i.e., aligned with the grid on the chart) and Magnetic north (i.e., aligned with the compass.)

The essential use case here is "What's the compass variation at a given point?" The information is printed on paper charts, but it's more useful to simply calculate it.

There are two magnetic models: the US Department of Defense World Magnetic Model (WMM) and the International Association of Geomagnetism and Aeronomy (IAGA) International Geomagnetic Reference Field (IGRF).

A packaged solution is geomag7.0. This includes both the WMM and the IGRF models. This is quite complex. However, it does have "sample output", which amount to unit test cases.

The essential spherical harmonic model is available separately as a small Fortran program, igrf11.f.

Which leads us to reverse engineering this program into Python.

**TDRE Approach**

The TDRE approach requires having some test cases to drive the reverse engineering process toward some kind of useful results.

The geomag7.0 package includes two "Sample Output" files that have the relevant unit test cases. The file has column headings and 16 test cases. This leads us to the following outline for the unit test application.

```
class Test_Geomag( unittest.TestCase ):
def __init__( self, row ):
super( Test_Geomag, self ).__init__()
self.row= row
def runTest( self ):
row= self.row
if details:
print( "Source: {0:10s} {1} {2:7s} {3:10s} {4:10s} {5:5s} {6:5s}".format( row['Date'], row['Coord-System'], row['Altitude'], row['Latitude'], row['Longitude'], row['D_deg'], row['D_min'] ),
file=details )
date= self.parse_date( row['Date'] )
lat= self.parse_lat_lon( row['Latitude'] )
lon= self.parse_lat_lon( row['Longitude'] )
alt= self.parse_altitude(row['Altitude'] )
x, y, z, f = igrf11syn( date, lat*math.pi/180, lon*math.pi/180, alt, coord=row['Coord-System'] )
D = 180.0/math.pi*math.atan2(y, x) # Declination
deg, min = deg2dm( D )
if details:
print( "Result: {0:10.5f} {1} K{2:<6.1f} {3:<10.3f} {4:<10.3f} {5:5s} {6:5s}".format( date, row['Coord-System'], alt, lat, lon, str(deg)+"d", str(min)+"m" ),
file=details )
print( file=details )
self.assertEqual( row['D_deg'], "{0}d".format(deg) )
self.assertEqual( row['D_min'], "{0}m".format(min) )
def suite():
s= unittest.TestSuite()
with open(sample_output,"r") as expected:
rdr= csv.DictReader( expected, delimiter=' ', skipinitialspace=True )
for row in rdr:
case= Test_Geomag( row )
s.addTest( case )
return s
r = unittest.TextTestRunner(sys.stdout)
result= r.run( suite() )
sys.exit(not result.wasSuccessful())
```

The Test_Geomag class does two things. First, it parses the source values to create a usable test case. We've omitted the parsers to reduce clutter. Second, it produces details to help with debugging. This is reverse engineering, and there's

**lots**of debugging. It depends on a global variable, details, which is either set to sys.stderr or None.

This suite() function builds a suite of test cases from the input file.

The unit under test isn't obvious, but there's a call to the igrf11syn() function where the important work gets done. We can start with this.

```
def igrf11syn( date, nlat, elong, alt=0.0, coord='D' ):
return None, None, None, None
```

This lets us run the tests and find that we have work to do.

**Reverse Engineering**

The IGRF11.F fortran code contains this IGRF11SYN "subroutine" that does the work we want. The geomag 7.0 package has a function called shval3 which is essentially the same thing.

Both are implementations of the same underlying "13th order spherical harmonic series" or a "truncated series expansion".

The Fortran code contains numerous Fortran "optimizations". These are irritating hackarounds because of actual (and perceived) limitations of Fortran. They fall into two broad classes.

**Hand Optimizations**. All repeated expressions were manually hoisted out of their context. This is clever but makes the code particularly obscure. It doesn't help when local variables are named ONE, TWO and THREE. Bad is it is, not much needs to be done about this. Python code looks a bit like Fortran code, so very little needs to be done except add `math.` to the various function calls like sort, cos and sin.**Sparse Array Chicanery**. There are actually two spherical harmonic series. The older 10-order and the new 13-order. Each model has two sets of coefficients:*g*and*h*. These form two half-matrices plus a vector. The old models have 55*g*values in one matrix, 55*h*values in second matrix, and a set of 10 more*g*values that form some kind of vector; 160 values. The new models have 91 g, 91*h*and 13*g*in the extra vector; 195 values. There are 23 sets of these coefficients (for 1900, 1905, ... 2015). The*worst*case is 23×195=4,485 values. This appears to be too much memory, so the two matrices and vectors are optimized into a single opaque collection of 3,256 numbers and delightfully complex set of index calculations.

**Phase 1.**Do the smallest "literal" transformation of Fortran to Python.

This means things like this:

- Transforming the subroutine into a Python function with multiple return values.
- Reasoning out the overall "steps". There's a bunch of setup followed by the essential series calculation followed by some final calculations.
- Locating and populating the global variables.
- Reformatting the
**if**statements. - Removing the GOTO's. Either make them separate functions or properly nest the code.
- Reformatting the
**do**loop. - Handling the 1-based indexing. In almost all cases, Fortran "arrays" are best handled as Python dictionaries (
**not**lists).

Once this is done, there are some remaining special-case discrepancies. Most of these are tacit assumptions about the problem domain that turn out to be untrue. For example, the Geodetic, Geocentric features seemed needless. However, they're not handled trivially, and need to be left in place. Also, conversion of signed values in radians to degrees and minutes isn't trivial.

This leads to passing all 16 unit tests with the single opaque collection of 3,256 numbers and delightfully complex set of index calculations.

**Phase 2.**Optimize so that it makes some sense in Python.

This involves unwinding the index calculations to simplify the array. The raw coefficients are available (igrf11coeffs.txt) and they have a sensible structure that separates the two matrices very cleanly. The code uses the combined matrix (called gh) in a very few places. The index calculations aren't obvious at all, but a few calls to print reveal how the matrix is accessed.

Given (1) unit tests that already work and (2) the pattern of access, it's relatively easy to hypothesize a dictionary by year that contains a pair of simple dictionaries, g[n,m] and h[n,m], for the coefficients.

**Cleanup and Packaging**

Once the tests pass, the package -- as a whole -- needs to be made reasonably Pythonic. In this case, it means a number of additional changes. For example, converting the API from degrees to radians, supplying appropriate default values for parameters, providing convenience functions.

Additionally, there are Python ways to populate the coefficients neatly and eliminate global variables. In this case, it seemed sensible to create a Callable class which could load the coefficients during construction.

Additionally, there are Python ways to populate the coefficients neatly and eliminate global variables. In this case, it seemed sensible to create a Callable class which could load the coefficients during construction.

Note that there's little point in profiling to apply further optimizations. The legacy Fortran code was already meticulously hand optimized.