Current

The current is calculated by comparing the Course Over the Surface computed after the Heading (HDG) and the Leeway, and the Course Over the Ground, calculated after the data returned by a positionning device such as a GPS.

In other words, it is the difference between the Dead Reckoning and the actual position. It is represented as a vector, having a speed and a direction.

The sphericity of the Earth introduces a lot of fun in this calculation!
We will use the loxodromic formulas, as, starting from a know position, we are looking for the point we can reach by having a constant heading with a given speed.

Dead Reckoning

Calculating the Dead Reckoning introduces the concept of Increasing Latitude, which is the very same one used to build the Mercator charts. Increasing Latitude is also known as Distance to Equator, or Mercator Latitude.
For a given latitude L, the Increasing Latitude L_c is given by the following formula:

      
        L_c = Ln( tg (Pi/4 + L/2) )

The Latitude L is expressed in Radians, as well as the result, which is to be multiplied by 180/Pi to be read in degrees. The Increasing Latitude is the one to consider for any trigonometric calculation used to determine the Dead Reckoning, as it reflects the ratio between meridians and parallels at a given latitude. The Earth is not flat!
Let's take an example:
We want to calculate L_c for an actual latitude of 38°.


 L = 38° = 38 x (Pi/180) rd = 0.663225 rd
 tg(Pi/4 + L/2) = 2.05030
 L_c = ln(2.05030) =  0.71798799 rd = 41.1376°

Notice that tg(Pi/4) = 1, and ln(1) = 0. Increasing latitude of 0 is 0, which is fine. But tg(90) is infinite. As a result, on a Mercator Chart, the poles are not represented. The major interest of the Mercator projection is that between two points, a straight line on the chart will represent the loxodromic (rhumb line) route (constant heading).
It actually makes more sense to consider the Increasing Latitude as a distance, rather than as an angle, as it can excess 90° (over 66.5° of actual latitude)...

Now we know what the Increasing Latitude is, and the formula for the Dead Reckoning can be exposed the following way:

We start with the following data:

L : Latitude we start from
G : Longitude we start from
d : distance travelled in Nautical Miles
r : route (heading) for d

        
    deltaL = (d / 60.0) * cos(r)        <= Divided by 60 to turn the miles in degrees
    L2 = L + deltaL;
    IncLat_1 = increasingLatitude(L);   <= See formula above for Increasing Latitude
    IncLat_2 = increasingLatitude(L2);
    deltaIncLat = IncLat_2 - IncLat_1;
    deltaG = deltaIncLat * tg(r);
    G2 = G + deltaG;

The value we are looking for - the Dead Reckoning value - is the point located in L2, G2.

Now, knowing where we should be (Dead Reckoning), and were we are (positionning device), the difference between those two positions is introduced by the current. Knowing what the distance is between those two positions, and knowing what the time interval used to calculate the dead reckoning was, it is then pretty easy to calculate the speed and direction of the current.

Note: In order to smooth this calculation, it could be a good idea to compute the dead reackoning through several logged points - like at least 20 - to get rid of the variations introduced in the heading by the waves or the driver... The current would then be calculated each 20 points.

Any comment ?