Is Visibility Statute or Nautical? Decide for Yourself

In a kayak, it's often good to know the distance to some destination or object. The human visual system is pretty bad at this task. I can easily recall many instances of paddling for hours without any apparent gain on large distant targets. It always feels that progress should be faster…

Kayak navigation books such as Burch or Ferrero provide some help. They give a simple formula for calculating distances at which objects with given elevations above the sea level will be seen. The range of visibility in miles is equal to the square root of elevation as measured in feet. For example, a light house 100 feet above the sea level will be visible from 10 miles away. The formula is for an observer at the sea level.

In a kayak, you are not at sea level. For an observer in a kayak cockpit, you can add 1.5 miles to the visibility range of an object as derived from the formula. One-and-a-half miles is the square root of the eye-level elevation above the surface of the water. On average, paddler's eyes are just over two feet above the surface. Here are some immediate applications—if you can see beach goers feet as they walk close to the water, you're roughly within 2 miles off-shore. That's 1.5 plus a little bit added for the elevation of the beach. Two kayaks will lose sight of each other when they are separated by three miles—one-and-a-half miles of visibility from each side.

This is where I got tripped up. The formula for the visibility range is supposed to give you statute, not nautical miles. Burch, then, suggests that the visibility range as determined this way is underestimated by about 15%. What this means is that 10-mile estimate for an object with 100' of elevation should actually be 11.5 miles instead. Funny, that just happens to be the distance in nautical miles. So what is going on here? Exactly how much off is the square root approximation?

Time to dust off high school trigonometry books. First, let's track down the mathematical solution to the problem. Then, we will have grounds to decide if the limits of visibility are statute or nautical and how precise they are.

Generalized shape of the Earth is basically a sphere. For our purposes, the sphere can be further reduced to a circle. An average radius from the center to the surface is approximately 3,440 nautical or 3,959 statute miles.

Any line that just touches the outside of the circle can be used to mark the range of visibility—objects whose height is above the line will be visible, the ones below will be obscured by the horizon. Another line can be drawn from the point where the visibility line touches the outside of the circle to the center of the Earth. This line will have two known properties: (1) its length will be Earth's radius and (2) its angle to the visibility line will be 90 degrees. The Earth's radius and the visibility line can be viewed as two sides of a right triangle. The hypotenuse of this triangle is equal to the Earth's radius plus the elevation of the object above the sea level. What we have here is a simple case of Pythagorean Theorem with two known sides of the right triangle. We can solve for the unknown third side. In our context, a² = c² - b²
can be written as Visibility² = (Radius + Elevation)² – Radius².


Plugging in the numbers and plotting the results of this equation we get the following picture. If the result of the square root solution is read in nautical miles, then it underestimates the actual distance by 6%. In statute miles, the distance is underestimated by just over 18%.

There it is clear as day: mathematically, the square root of elevation (measured in feet) is much closer to geographical visibility expressed in nautical rather than statute miles.

So why, then, is the formula is presented as offering the solution in statute miles? I don't know the answer to this one…

Here are some thoughts. Mathematical solution gives the distance at which the top of the object will reach the line of visibility. So the question is if you be able to see the top as soon as it reaches the line? To me, it depends. If the object is a mountaintop, the answer is most likely "No!" The tip of the mountain will have to get above the line at least somewhat to be seen. A mountain that is 400 feet above the sea level will reach the line of visibility when the distance is 21.3 nautical miles or 24.5 statute miles. Square root formula tells you that, when you first see the tip of the mountain, you are 20 miles away. By the numbers, when you are actually 20 nautical miles away, the mountain is already 47 feet above the line of visibility. When you are 20 statute miles away, the mountain will extend by 133'. When will you actually see the mountaintop? I don't know but, personally, I would rather be closer than I think I am. To me, nautical mile interpretation is a more conservative one and seems to be more appropriate.

If you are paddling at night and it is a light that you're looking for, you will see the glow well before it reaches the theoretical line. The moment the actual light emerges from beneath the horizon will be determined more precisely than in the previous scenario. Again, if the height of the light is known, it's square root is more appropriately interpreted as the distance in nautical, not statute, miles.

What do you think?

Rules for Tides: Thirds, 50/90, Twelfths

Confused yet? I was! So I decided to work it out for myself once and for all. In the process, I found a way to inject some method into this madness. I have not seen the rules differentiated this way anywhere in the literature so I hope this will help clear things up for folks.

If you know the definitions for these rules, you can readily skip the next section and find the disambiguation in the Q and A part. The Math section will go into the mathematical details of how I figured this for myself. If you are a math type, you may find several hours of amusement therein. If you're not, it is likely to confuse you right back to where you started or worse.

DEFINITIONS

Here's a typical example of two rules mixed up into one: "The Rule of Thirds states that relative to the total maximum current speed, the current jumps 50% the first hour, 90% the second hour, 100% the third hour.  The current then decelerates to slack in the same order."  You can find this on page 90 of Sea Kayaking: Rough Waters (2007, Heliconia) by Alex Matthews.

Let's ask Gordon Brown—a recent kayak reference volume from a highly reputed BCU coach. On page 168 if his Sea Kayak (2006, Pesda) book he gives this for the Rule of Thirds: "… over the period of the first hour the current will flow at one third of its maximum rate …, for the second hour it will flow at two-thirds and the third hour at … three-thirds."  Hence the ubiquitous 1:2:3:3:2:1 abbreviation of the rule which stands for 1/3 of the max current rate in the first hour, 2/3 in the second hour, 3/3 during hours three and four, and then down to 2/3 in the fifth hour and 1/3 during the last hour of the cycle.

The 50/90 Rule gives you "… the speed of the current at the end of each hour." Starting from slack, the current will flow at 50% of its maximum speed at the end of the first hour, 90% at the end of the second hour and full 100% or maximum speed at the end of the third hour.  It will then slow down in the same steps: 90% at the end of the fourth hour, 50% fifth and back to slack at the end of the 6-hour period.  The full Rule of 50/90 should be stated as 0/50/90/100/90/50/0.

I could not find a reference to the Rule of Twelfths in Brown's volume so let's go to page 178 of David Burch's Fundamentals of Kayak Navigation (1999, Globe Pequot): "The rule of twelfths [is a] method of determining the tide height between high and low waters." In the six-hour period that separates the low and the high waters, the overall level will rise/fall 1/12^th of the full tidal range in the first hour, 2/12^ths in the second hour, 3/12^ths in the third hour. We know from the 50/90 Rule that after three hours the current reaches is maximum speed and starts to slow down. For the rest of the flood/ebb cycle the rise/fall of the water will be 3/12^ths in the fourth hour, 2/12^ths in the fifth and 1/12^th in the last sixth hour.  Here's a crown jewel or you: the Rule of Twelfths is also frequently presented as 1:2:3:3:2:1—same as Rule of Thirds! Fonfusing? Well, yes, you should be!!!

Burch also has definition of 50/90 Rule on page 226 as a "rule to estimate the effect of changing tidal current on net progress." Upon a quick scan, he does not seem to have the Rule of Thirds in his authoritative reference guide but there is a table in his text which gives "the constant current speed that is equivalent to the changing current of the cycle."

All of these definitions help me separate the Rule of Twelfths from the other two. Rule of Twelfths is about the height of the water while 50/90 and Thirds are both about the current. Horizontal and vertical dimensions.  Burch helps some by introducing the term 'net progress' but I think he has it associated with the wrong rule!

Q and A

Q: Are rules of Twelfths, Thirds, and 50/90 all dealing with the same thing? And, if so, can they be used interchangeably?
A: No! and No!

Q: I've heard that the Rule of Twelfths is different from the rules of Thirds and 50/90?
A: Yes! The Rule of Twelfths is about the rise and fall of the water levels at various stages of the tide while the latter two deal with current. Think VERTICAL versus HORIZONTAL dimension.

Q: That helps. So the rules of Thirds and 50/90 are basically interchangeable then, right?
A: No they are very different, although I've frequently seen write-ups using the two without distinction.

Q: But you said they were both about the current. What is the difference between them, then?
A: Think of it this way: Rule of 50/90 is used to estimate current speed at the end of each hour of the six-hour tidal period. The Rule of Thirds, on the other hand, is used to calculate distances that the current travels in full one-hour increments or drift. The first one describes the speed of the current at a single point in time, while the latter helps estimate what happens to an object affected by the current over a period of one hour.

Q: Wait, wait ... I don't get it. Is there a graph for this or something?
A: Great idea! Let's try a graph.

• Rule of 50/90: instantaneous speed of the current observed at each of the six hours in the tidal period ON THE HOUR.
  
• Rule of Thirds: cumulative distance the current travels DURING THE ENTIRE LENGTH OF EACH OF THE HOURS in the six-hour tidal period.
  
• Rule of Twelfths: cumulative change in the height of the water DURING THE ENTIRE LENGTH OF EACH OF THE HOURS in the tidal period.

Q: I think I get it but can you give an example?
A: We need to have some input information before the illustration can work. Let's say that High Water (HW) is at noon and Low Water (LW) is at 6pm. Reference materials specify that between noon and 6pm the water level will drop from 12' to 0' above the chart datum and the maximum ebb current speed will be 6 knots.

     The Rule of Twelfths will tell you that from noon to 1pm—the first hour of ebb tide—the water level will drop 1/12^th of the total change in water levels between HW and LW. Since 1/12^th of 12' is 1 foot, the water will drop from 12' to 11' between noon and 1pm. Between 1pm and 2pm the water will drop an additional 2/12^ths—that's 2 feet. At 2pm the water height will be two feet less than 11' or 9'. By 3pm the water will drop another 3/12^ths and will stand at 6'. Then another 3/12^ths by 4pm = 3'. Another 2/12^ths by 5pm = 1'. The final 1/12^th will drop the water level to 0' at 6pm.

     The 50/90 Rule helps estimate current speed at the top of the hour. At noon the speed is 0% of maximum—that's slack. At 1pm the current will pick up to 50% of its maximum flow or 3 knots (6*50%=3). At 2pm the current will flow at 90% of its maximum level or 5.4knots (6*90%=5.4). At 3pm 100% = 6 knots and then back to 5.4 knots at 4 pm, 3 knots at 5pm and another slack at 6pm.

     Finally, we can use the Rule of Thirds to estimate how far the current will travel during each of the hours of the tidal period. Between noon and 1pm the current will travel 1/3^rd of the maximum current speed or 2 miles (nautical). During the second hour of the ebb—between 1pm and 2pm—the current will travel 2/3^rds or 4 miles. Between 2pm and 3pm the current will drift the most or 3/3^rds = 6 miles. Then back in descending order—by 4pm another 6 miles of drift, by 5pm another 4 miles and, finally, by 6pm the drift will add another two miles.  Don't read the following mathematical sentence in italics if you understand the differences between the rules—it may confuse you. For those who are not afraid of math, drift or distance traveled by the current is the same as AVERAGE current speed during the hour.

Q: That's great but my current speeds are in kilometers per hour.  Can I still use the Rule of Thirds or 50/90?
A: Absolutely!  Only the units of measurement change in the Rule of Thirds.  Instead of nautical miles you will get kilometers.  If the speed of the current is given in MPH, then you get distances in statute rather than nautical miles.

Q: Wait, I just lost it again, I use the Rule of Thirds to estimate what and 50/90 what?
A: 50/90 Rule is for estimating the current speed on top of each hour in the 6-hours tide cycle. Rule of Thirds is for estimating the distance that the current will carry an object during each of the 6 hours in the cycle. If you want to know when the current will get too fast for you paddle against, use 50/90 Rule. If you want to know how far you will drift with the current use the Rule of Thirds.

Q: How precise are these estimates?
A: Glad you asked! These are all rules of thumb. As all rules of thumb, these ones describe idealized situations. Time between HW and LW will not be exactly 6 hours. For diurnal tides it will be around 12!  Divide the time between HW and LW into six equal intervals and you can still use the rules.  Tidal flow is influenced greatly by shoreline dynamics and other factors on Earth as well as in the skies. Importantly, peak current speed normally lags somewhat behind the the midways between HW and LW due to friction.  It is not uncommon to find that slack is mismatched with HW by hours.  Use these rules as guidelines.

Q: Why doesn't Burch reference the Rule of Thirds and why does he have that complicated table to calculate drift? He is the ultimate authority in the field of kayak navigation after all. That makes me uncomfortable with the Thirds Rule.
A: Well, it probably should. Burch probably thought that the rule of thirds is too far off the actual estimates for the drift that are obtained mathematically (see the last section for details). It estimates almost 10% too much for the first hour and is off by between 3% and 5% during the remaining two thirds. Quite a gross approximation if you ask me but would you rather use 26/70/96 rule?

DA MATH

For the present calculations of floods I used the following sine function to get the flow throughout the tidal cycle:

Flow = MaxFlow * sin(2*π*Period^-1*Time)

where Flow is the proportion of maximum current speed or actual current speed, Period is the length of the entire tidal period or 12 hours for tides and Time is the clock variable. MaxFlow constant can be used to set the output to the actual speed of the current.  When MaxFlow=1 we get proportion of maximum flow or the speed of the current when maximum flow is one knot.  A constant can also be added to Time variable.  This constant will allow one to manipulate the Y intercept.

  

Here are the results that this function yields:



Note how the numbers of 50% and 86.6% fit the 50/90 Rule pretty close.

The numbers don't work nearly as well for the Rule of Thirds. Mathematically the drift/average current speeds are 25.6%, 69.9% and 95.5% rather than 33.3%, 66.7% and 100% predicted by the rule.  For 6 knot max current, the Rule of Thirds will predict 1*6/3 + 2*6/3 + 3*6/3 = 2 + 4 + 6 = 12 nautical miles of drift.  The actual average current speed over the three-hour period is 0.713 so over the the entire 3-hour period the drift would come out to be 3hr*6knots*71.3% = 12.8 nautical miles.  Numbers are reasonably close when added up but deviate from the ideal scenario when taken hour by hour.

The average speeds for each of the one-hour intervals can be obtained by integrating the area under the harmonic curve for each hour. In this exercise they were calculated using simulated data instead. A theoretical dataset was generated using the above harmonic function with 1,000 data points for each hour. That's roughly one data point for every 3 seconds.  A simple average of all speeds was used.

Let me know if this helps clear things up or if it is useless.  Suggestions for improvement?  E-mail karovaldas@gmail.com.  I am especially curious about the graphical representation of three rules.