## Estimating Path Loss on Marine Non-Line-of-Sight Paths

### by James Hebert

A common concern among recreational boaters is the range of their VHF Marine Band radio. In a previous article we explored the range of communication between stations which were in a line-of-sight path, and we used a model for path loss based on assumptions of free space propagation. We revisit this topic to analyze communication paths in which there is a non-line-of-sight path between stations. This requires a new model for estimating the path loss. Using both observed signal levels and predicted signal levels, we derive a formula for estimating the path loss. Using this prediction for path loss, we then deduce the range of radio communication between two typical small boats on a non-line-of-sight radio path.

Radio communication range is determined by the overall gain and loss in a communication circuit. The signal level that is present at a distant receiver can be computed if certain elements of the circuit are known. A general formula for determining the signal power Pr available at the receiver input is

```   Pr = Pt - Lp + Gt + Gr - Lt - Lr

where

Pt = transmitter power output
(dBm or dBW, same units as Pr)

Lp = path loss between antennas (dB)

Gt = transmit antenna gain (dBi)

Gr = receive antenna gain (dBi)

Lt = transmission line loss between transmitter
and transmit antenna (dB)

Lr = transmission line loss between receive antenna

Most of these factors are easily discovered, but path loss (Lp) has to be estimated. For the case in which the path between transmitter and receiver is in free space, that is, where there is no intervening atmosphere or terrain, we can estimate the path loss in dB from the theoretical path loss in free space:

```   Lp = 36.6 + 20log(f) + 20log(d)  (f in MHz, d in miles)
```

### Non-Line of Sight Paths

The free space path loss is sometimes used to calculate the path loss for paths where the antennas of the receiver and transmitter are in line of sight, that is, the radio horizons of the receive and transmit stations overlap. It would be interesting to see if a calculation can be derived for a non-line-of-sight (non-LOS) path, that is, one where the radio horizons of the transmit and receive stations did not overlap and the propagation was over the radio horizon. Calculating over-the-radio-horizon or non-LOS path loss requires consideration of the features of the intervening earth. There are many estimates and models available, but they are often based on paths over land and uneven terrain. I am not aware of any predictions specifically for paths over long distances of water.

Propagation at VHF frequencies is generally considered to be line-of-sight, but some allowance is given for atmospheric refraction. A normal allowance for refraction is to increase the radius of the earth by a factor of 4/3, in effect stretching the radio horizon beyond the optical horizon. In the case of air above water, there can often be a temperature difference or boundary layer which enhances the refraction more than normal. Propagation over water may produce even more refraction, producing paths with are longer than expected.

Propagation of radio signals in the real world also includes the influence of signals arriving by more than just the direct path between stations. Signals may arrive from paths in which the signal has been reflected from boundaries or obstacles in the path. In the case of a path over water, the surface of the water provides a reflector that may create secondary paths. Signals which arrive by different paths have different phase relationships, and the received signal becomes the sum of the multiple signals. Often the phase of a reflected signal can be inverted with respect to the direct signal, leading to a tendency for the two signals to cancel or to have the resulting summed signal reduced in amplitude. Because of all of these factors, the path loss on a real world communication circuit will tend to be higher than the predicted path loss in free space.

### Observed Signal Levels

In a previous article we considered a particular VHF radio path between Sister Bay, Wisconsin and Leland, Michigan, a distance of about 66-miles and with some intervening terrain. We have verified that NOAA Weather Radio broadcasts from WXN69 can be received in Leland. We can use this data to derive an estimate of the path loss, which we can then use to perhaps deduce a formula for a general estimate of path loss at this frequency and distance.

Using our general equation from above, we fill in the known factors as follows:

```    Pt = 1,000 watts (per WXN69 engineer)
= 10log(1000) + 30
= 60 dBm

Gt = 8 dBi (estimated but typical for such stations)

Gr = 2 dBi	(a unity gain marine half-wavelength antenna)

Lt = 1.5 dB (estimated for 400-feet of very low loss line)

Lr = 1 dB (20-feet of RG-58/U)```

Next we must estimate the receive signal level. We know the receiver sensitivity threshold was at least one micro-volt or a Pr of -107 dBm. Since our actual reception provided a fairly decent signal, we estimate the signal was about ten decibels above that level, and we will use a Pr of -97 dBm. Now we solve for path loss (Lp):

```   Lp = Pt + Gt + Gr - Lt - Lr - Pr
```

and substitute our known values

```   Lp = 60 + 8 + 2 - 1.5 -1 - (-97)
Lp = 164.5 dB
```

From our known values or observed values, we have deduced the actual path loss observed, 164.5 dB. We compare this value with the predicted path loss for a free space path of 66 miles:

```   Lp = 36.6 + 20log(f)   + 20log(d)  (f in MHz, d in miles)
= 36.6 + 20log(156) + 20log(66)
= 36.6 + 20(2.19)   + 20(1.82)
= 36.6 + 43.86      + 36.4
= 116.8
```

The observed path loss is greater than the estimated free space path loss by 47.7 dB. We assume that the influence of distance was greater, and accounts for all the difference between predicted and observed. Therefore the observed path loss factor due to distance must have been

```   Distance factor = free space distance factor + 47.7 dB
Distance factor = 36.4 + 47.7
Distance factor = 84.1 dB
```

We can deduce the proper relationship between path loss and distance if we take our original estimate of the distance factor and compare it to the difference from observed estimate. In our free space estimate, we use a factor of 20log(d) for the distance. In our real world observed path loss the influence of distance must have been

```   xlog(66) = 84.1
```

Solving this for x

```   xlog(66) = 84.1
x        = 84.1 / log(66)
= 84.1 / 1.82
= 46.2
```

This implies the path loss equation should be something like

```   Lp = 36.6 + 20log(f) + 46.2log(d)  (f in MHz, d in miles)
```

Checking our result using the new distance factor:

```   Lp = 36.6 + 20log(156)   + 46.2log(66)  (f in MHz, d in miles)
= 36.3 + 20(2.19)     + 46.2(1.82)
= 36.3 + 43.8         + 84.1
= 164.2
```

We now have derived a formula for predicting path loss based on some observed signal strength of non-LOS paths in the real world.

### Predicted Signal Levels

We can also use the NOAA coverage predictions as a source of data to deduce the path loss estimate. We note from the coverage map for NOAA Weather Broadcast station WXN69 that the predicted signal strength should be +18 dBuV at approximately 45 miles distance (about the point on the map where the color shading changes to green from white.) This corresponds to a Pr of -89 dBm. Using this figure in our path loss equation, we see

```   Lp = Lp = 60 + 8 + 2 - 1.5 -1 - (-89)
Lp = 156.5 dB
```

This is a variance from the free space model of 156.5 - 116.8 = 39.7 dB. Applying this again to the distance factor, we get

```   xlog(66) = 36.4 + 39.7
xlog(66) = 76.1
x        = 76.1 / log(66)
x        = 76.1 / 1.82
x        = 41.8
```

Our predicted path loss from this observation would thus be

```   Lp = 36.6 + 20log(f) + 41.8log(d)  (f in MHz, d in miles)
```

We see that for real world non-LOS paths, the path loss due to distance is going to be much higher than estimated in the free space model:

```   Free space model = 20log(d)
Observed value   = 46.2log(d)
NOAA prediction  = 41.8log(d)
```

These observations are in good agreement with an often used estimate for real world paths over flat terrain that uses a distance factor of 40log(d).

### Typical Boat-to-Boat Communication

We now investigate a communication circuit between two typical recreational boats. We will use our new path loss estimation to see what the maximum range might be. For two typical small boats, we use the following parameters:

```   Pt = 25 watts
= 10log(25) + 30
= 44 dBm

Gt = 3 dBi (typical marine 4-foot antenna)

Gr = 3 dBi (typical marine 4-foot antenna)

Lt = 1 dB (20-feet of RG-58/U)

Lr = 1 dB (20-feet of RG-58/U)

Pr = -89 dBm (18 dB above receiver threshold, a good signal level)
```

We now have all factors for the communication circuit except the path loss. Solving for path loss:

```   Lp = Pt + Gt + Gr - Lt - Lr - Pr
Lp = 44 + 3  + 3  - 1  - 1  - (-89)
Lp = 137 dB
```

Using an estimate of path loss based on a distance factor of 40log(d) (often used for paths over flat terrain), we can discover the distance that corresponds to a path loss of 137 dB:

```   Lp  = 36.6 + 20log(f)   + 40log(d)  (f in MHz, d in miles)
137 = 36.6 + 20log(156) + 40log(d)
137 = 36.6 + 43.8       + 40log(d)
56 =                     40log(d)
```

Solving for d

```   40log(d) = 56
log(d) = 56/40
d  = 10^1.4
d  = 25.1 miles
```

This result implies that two recreational vessels with typical radio installations should be able to communicate at a distance of 25 miles even if not within each other's radio horizon. If we use the greater distance factor, 46.2log(d), based on our observations above, we find the range as

```   46.2log(d) = 56
log(d) = 56/46.2
d  = 10^1.2
d  = 15.8 miles
```

### Calculating Influence of Antenna Gain on Range

We now have a reasonable basis to predict path loss as a function of distance. We can use this to assess the effect on communication range which occurs with a change in signal level. For example, if we increase the the effective radiated power in the system by 3-dB, we should be able to increase the path loss by 3-dB and maintain the same signal levels.

The equation for path loss (Lp) contains an element for distance (d) with this relationship:

```   Lp = Xlog(d)
```

We have used various values for X as follows:

```   Free space model = 20log(d)
Observed value   = 46.2log(d)
NOAA prediction  = 41.8log(d)
Terrain model    = 40log(d)
```

Using the Terrain model and solving for Lp we have

```  40log(d) = Lp
log(d) = Lp/40
d  = 10^(Lp/40)
```

If we have a path with a distance of, say 10 miles, we now look for how much longer we can make the path before the path loss increases 3-dB:

```   Lp = 40log(d), find loss at d = 10 miles
Lp = 40log(10)
Lp = 40
Lp = 40 + 3, add 3-dB to loss
Lp = 43

d = 10^(Lp/40), find distance for Lp = 43 dB
d = 10^(43/40)
d = 10^(1.075)
d = 11.875 miles
```

The increase of 3-dB in the transmitted signal allows the path to increase to 11.875 miles from 10 miles with the same received signal strength. In general we can say that with an addition of 3-dB of signal, the path will increase in distance by 18.8 percent. We check this by solving for d when Lp=3:

```   d = 10^(3/40)
d = 10^0.075
d = 1.188
```

### Other Estimates of Antenna Gain Influence

A marine antenna manufacturer once published the following formulas for determining the range of communication as function of the antenna gain:

```6db Antennas:
Square Root of Height (in feet) above water x 1.15 = Range in miles

9db Antennas:
Square Root of Height (in feet) above water x 1.52 = Range in miles
```

This says the range increase for 3-dB more signal in the circuit will be in the ratio of:

```   ratio = 1.52 / 1.15
ratio = 1.32
```

We can use this to work backwords to find what assumption that the antenna manufacturer was using with regard to path loss in this case:

```Given:
d = 1.32
Lp = 3

and relationship is

Lp = X log(d)

solve for X

3 = X log(1.32)
X = 3 / log(1.32)
X = 3 / 0.1206
X = 24.88
```

This implies that the antenna manufacturer is using an equation for path loss in which the distance factor is 24.88 log(d). As we have seen above this is almost the free space theoretical value for path loss on a line-of-sight path.

### Conclusions

Based on a combination of theoretical and observed path loss calculation, it seems reasonable that two recreational vessels with typical radio installations should be expected to be able to communicate at a range of 16 to 25 miles, even if they are in a non-LOS path. This assumes they are using 25-watt transmitters, typical marine antennas, and have receivers with good sensitivity which are not impared by any local interference. It also assumes the path is over water and without any intervening obstacles.

For further discussion on this topic please use the SMALL BOAT ELECTRICAL discussion which has already been started.

DISCLAIMER: This information is believed to be accurate but there is no guarantee. We do our best!

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