Mordred

Some interesting work on examining the data presented. I got curious on your comment "Gaussian beam divergence". Atmospheric optics isn't my strong suit. However I do recall collimation correlations with atmospheric turbulance via Gaussian Schell model beam. Though never really studied it in detail.

Yes I'm familiar with some of Wolframs work. Though certainly not in great detail. Ok so you are simulating Newton gravity along x axis distance. Judging from above. This shows x axis, but what about the y and z axis? I look through your links I'm unclear which is your work or others. you may find these articles helpful. Cell Automata and Physics. http://www.google.ca/url?sa=t&source=web&cd=1&ved=0ahUKEwi0js7IrpzPAhUGy2MKHefsA-8QFggbMAA&url=https%3A%2F%2Farxiv.org%2Fpdf%2Fphysics%2F9907013&usg=AFQjCNHja_3J6NFDSGI3PzKbHvDALS-3mQ&sig2=ymSvYw3QO-umfbRFt5yLKA And "What are the hidden Quantum laws behind Newtons laws" http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi0js7IrpzPAhUGy2MKHefsA-8QFgggMAE&url=http%3A%2F%2Farxiv.org%2Fpdf%2Fphysics%2F9904036&usg=AFQjCNGiDpIcBEmWePTYr-qmDmD_O8qx-g&sig2=n2oIRg0Qj90JeDaINLgbKw both articles gives some excellent examples in Newton gravity. particularly in the 3d regime with CA having 26 neighbors. 2D only 8 ( see second article)

Ok look forward to seeing to it. At least you understand the importance of the math apects in your modelling. Even wrong directions teach (generality, not aimed at you). You may have manipulated the variant quantities. (observer aspects) but will also need to include your invariant quantites. Ideally by using a ds^2 line element, to show the departures.

Probably because as posted its too difficult to follow accurately. You need to find a more detailed way to explain the above. That is easily readable. People lose patience with difficult to read posts. "this post takes too much effort to translate. Let someone else worry about it" The quoted section is probably the typical response...

Understood GR is tricky to learn. I caught the cross post aspect. However just learning the tensor usages without knowing how to use tensors can give clues. ie anytime you see [latex]\eta [/latex] just remember Minkowskii geometry.

I wouldn't know,its too tricky to try and read your last post. You don't need every decimal place. That why we invented scientific notation. Secondly the above is pointless, it doesn't tell us the formula you used. Coincidence on a few numbers isn't substantial proof your formulas is correct. Particularly since you still ignore the thermodynamics in the FLRW. (you completely missed the details of what changes and why.. in the different line elements I posted earlier. (the volume changes, not the time aspects, in terms of dilation. In the FLRW metric) You claim to understand these line elements but keep chasing a garden path... Rather amusing actually, considering I showed the departures of redshift formulas beyond the Hubble limit... The other aspect you ignored is in those three line elements. Geodesic light paths follow completely different relations. These deviations alone should tell you the difference between gravitational redshift and cosmological redshift. The two formulas are not direct matches as they are derived from two seperate line elements. The first is time dilation relations. The second volume change. Again you chose to ignore this to chase your garden path... most importantly the standard time dilation formula does not work, beyond Hubble limit due to apparent velocity greater than c. I gave you those corrections. You assume the numbers above are correct as you do not accept current distance values. However you can't simply compare numbers. We must compare the formulas and see if those formulas match known physics. Derivable with known physics.

Ok mathematically I have no problem with using math to describe reality. This is essentially done with physics. Without having to search your links is there specific formulas you want to look at? The Newton limit we can already derive using math. So without having to search your links. Can you post the math in your solution? The extension of the Bohr model should also be looked at.

Excellent. Now the formulas you tend to see for GW waves is for an observer away from a BH in an Newton approximated geometry. (An observer on Earth). A GW wave polarizes in the x and y axis. The direction of propogation is the z axis. It doesn't alter the mass term per se, but it does induce pressure changes. So in this manner a GW wave through the energy/mass relation. Can induce additional gravity via the stress tensor. In the formula above the stress tensor changes will be included under the "h" tensor. (I think of the h tensor as the permutation tensor)

Newtonian gravity can be derived from GR. Its a special treatment where the effects of gravity is significantly low. Mainly no time dilation. Once you start having time dilation you depart from Newtonian gravity approximation. Does that help or did it confuse ? [math]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/math] The [latex]\eta [/latex] specifies an approximately Euclidean geometry. More specifically Minkowskii geometry. This doesn't hold true for geometry near a blackhole

Most likely the evaperation duct. The average height for the temperature range was roughly 2 metres. Assuming I ran the calcs correct. The effect is that the light path curvature would be less than the curvature of the Earth. This causes signal entrapment and skipping. Either way it indicated conditions not favorable to the test. On the interest of the next test. It is difficult to guage how much those involved learned. Both from previous test and discussion. I recommend we look first at methodology. Please describe how you plan to undertake the next test. Break it down to each stage. From there we can look to improve this stage first. Then look at addressing human error corrections etc. As Studiot mentioned include your control points.

1) methodology 2) human error 3) instrument error 4) refraction in order. Every experiment has the same sequence. The corrective measures not with standing

excellent post MrMaker +1

I got about as far as the first ten minutes. Then I recalled a lesson. Systematic errors get increasingly worse at later measurement points and increased distance in this case. After seeing the uncertainties in the first two points that was enough for me. Hopefully the next attempt takes and applies the advice given. Its a common mistake to overly trust a laser. One may be aware of atmospheric effects but not aware of how much a tiny variation increases over distance in angles. As well as other optical effects. (simply calculate how much a 1% change in angle will change the height measured over 100 metres let alone 1 km.) Most surveyer handbooks I've looked over including their equipment manuals recommend breaking down your total distance to increments of less than 100 metres. (coincidentally refraction is the primary reason). At least on the ones including level correction. That a lot of measurement points and time over a 7 km lake let alone a 10 km survey.

equipment datasheets are definitely handy. I can also use them to look for some of these affects mentioned. The effect of the elevator duct varies depending on equipment frequencies. Glad to see someone that can help double check the equations involved for this application. I'm going down memory lane on everything involved, in terms of refraction, scatter etc. The more I think of this test, the more I keep thinking that two emitters may be a good idea. I keep looking at the pros and cons involved. Assuming parallel transport placement. Safety issues addressed (lol). KEY NOTE use two distinct frequencies. PROS. 1) detection in deviation in parallel transport can be used to find changes in refractive index. 2) if the two lasers are mounted in tandem vertically we can use geometry relations between the two beams to assist in leveling. Combined with a grid backboard we can break the leveling aspects down to straightforward trig. This will help correct leveling errors in 3D if done correctly 3) the changes in frequency and loss of parallel transport can be used to calculate the refraction index. (this in principle is used in equipment today. Usually through a known sample) however a known sample isn't needed just more complex. If not. CONS 1) added equipment and complexity, adds to potential systematic errors 2) greater number of control point data to collect. (measurements between the two laser beams for example, as well as receiving frequencies.) Even without worrying about refraction, etc the advantages in PRO 2 is in my opinion worth the added cons The above may work better if the two lasers were aimed to a common point. Forming two sides of our triangle, this can help in leveling and distance from emitter calcs. (KEY NOTE do not mount the two lasers onto the same mount backboard. We don't want to add material expansion/contraction problems)

nice procedure on that survey paper.

No need, there are far easier and more reliable means to measure a lake level than optics. There is also ways to reduce the corrections by good control points as well as reliable reference points. LOL back to that whole multi-laser idea it can be used as a control point but the math may or may not be simpler. Ideally we always prefer to eliminate the need for the corrections.

Probably the easiest way is to find the effective Earth radius.."k" I'll try to run you through the calculations.. First what is the Effective Earth radius ? well its a hypothetical radius where the curvature path of our signal matches the curvature of the Earth. For our situation we can ignore variations in chemical composition and use the refractive index of air as unity value 1. Its extremely close... So what we now need is the refractive index. For the refractive gradient we define a new unit capital N. [latex]N=(n-1)*10^6[/latex] [latex]k=\frac{1}{(1+a\frac{dn}{dh})}[/latex] when k=-157 N units/km the ray path and the effective Earth path effectively match. The ray follows the curvature of the Earth. Your three main zones can be distinquished by values of k. This site covers the basic fundamentals but a key formula on this link is [latex]N=77.6\frac{P}{T}+3.37*10^5\frac{e}{T+2}[/latex] the left hand side of the + sign is dry term, the right hand side the wet term. the water vapor vapor pressure "e" can be calculated via [latex]e=H*\frac{6.11121exp(\frac{17.502t}{t+240.97}}{100}[/latex] where H is the relative humidity %, t is temp in degrees celcius (careful some of the other formulas use kelvin), e_s is the saturation vapor pressure in hPa http://www.mike-willis.com/Tutorial/PF6.htm its a basic tutorial. It doesn't detail the regions well. The different refractive conditions can be characterize by the following using k and the effective earth radius value -157 N/km normal air conditions [latex]k=\frac{1}{(1+a\frac{dn}{dh})/157}[/latex] here dn/dh is roughly -39 N/km which equates to roughly k=4/3 subfraction is [latex]4/3> k> 0[/latex] superfraction [latex]\infty>k>4/3[/latex] Ducting is [latex]\infty<k<0[/latex] this is where your evaperation duct conditions lie within. For the duct height we need to correlate the frequency aspects the link sort of hints that lol. under roughing ( you will need the laser frequency)

Excellent answer. That is extremely well defined...

Lets run a simple example of data to gain mathematical proof. Test laser hitting a backboard. What is the angle of the backboard to the laser? Well we can use the reflection angle and the incoming angle of the laser to confirm the relation of the backboard to the laser. Good ole Pythagoras... This is what you want to do at every measurement point. We can take this further to assist in refraction. Establish a coordinate system For the test your looking at, you have a couple to choose from. If you want a Flat coordinate system use 3d Cartesian coordinates where Pythagoras holds true between two reference points. [latex]x^2+y^2+z^2 [/latex] If you want the metric for a curved metric use polar coordinates. So we have defined our metric. Now lets start modelling our atmosphere. Ideal gas law applications. (density, pressure,temperature,humidity,chemical composition) Lets start with density Well again we have another choice. Static or dynamic distribution. Static works well if there isn't any inherent direction of motion. (wind). so we can use the above metric for a Flat and static distribution. A uniform distribution (homogeneous and isotropic) will be [latex]\rho_x^2+\rho_y^2+\rho_z^2[/latex] However this isn't the case. The lower elevations has a higher density to the higher elevations. We have an inverse relation. So we show this relations along the y axis. As were only dealing with elevation change. If you plot the change at each y coordinate you can establish the rate of change at each elevation. (As were static we don't need the vector direction we only need the magnitude.) this rate of change will depends on the other factors mentioned above as well as good ole gravity. These are just starting steps to model atmospheric refraction. I hope you can start to see the kind of mathematical detail a good paper will include... This detail such as the first example can help eliminate measurement errors and uncertainties. Via differential geometry... When you apply the refraction formulas, you apply them under a metric with corrections due to the degrees of freedom inherent of an ideal gas. Added degrees of freedom include temperature, pressure, humidity, chemical composition. (though the latter two can argued one and the same) depending on modelling choices.

Yes the evaperation ducting can cause a huge number of optical problems. Even having multiple potential laser paths. This can easily cause motion in every direction with a preferred direction due to wind movement. (think of the effect moving water can have). In the evaperation duct the humidity approaches 100% as you approach the water surface. During the test. If possible at each point record the temperature, at laser elevation and surface water temp. humidity, and wind speed and direction. Ideally the atmosphere chemical composition would also be useful. Oxygen, nitrogen, carbon dioxide %, etc is useful data for location refraction values. The atmospheric pressure is also needed to fine tune calcs. This data can be used to further analyze your dataset. Also include the technical datasheet for the equipment your using. (ie the laser.. ). The emitted wavelength etc is important... In terms of beam dispersion, the evaperation duct can easily increase the beam radius. (extremely easy) I would recommend on your backboard you have reference grid lines both horizontal and vertical. This way we remove the need to pixel count etc the beam radius from photos etc. Remember the more data you provide, the greater your accuracy becomes. It also reflects diligents to important potential influences and steps to mitigate or acccount for their influence. For example if you sample the above list at each point. You can fine tune refraction data at each measurement point. ( physicists studying your paper, just love these details). Of course they will also want to see your calculations. (To that end, I'm still putting together the list of formulas with correction formulas for locality). Its quite intense... Remember a good paper, will contain sufficient data, that doesn't require studying a photograph etc. 1) Appropriate formulas 2) accurate data, the more measurement data the better 3) A detailed list of possible measurement errors and the corrections. (including the related calculations that the corrections are effective) 4) good reference papers, including previous studies. There is a rule of thumb, physicists usually only accept mathematical proof, that corrections are in place... They won't accept a verbal only description. Nor will they take anyones word that corrective measures are in place. Unless they can mathematically confirm the corrective measures... For example an image isn't sufficient. What distortions are inherent to the image? Just a side note, personally if I'm studying a paper proposing a new model and datasets. I personally won't accept any paper that doesn't include a detailed list or error margins on critical data points. When you get down to it, most of your paper should detail possible errors, corrections and means to mathematically confirm those corrections. They will not accept anyones word on any aspect.

Actually there may be an aspect to consider here. Over large bodies of water there is a specific type of Ducting. Evaperation duct. I'm still looking into this but if I'm right your chosen height of the laser may have been too low... Did you perchance recall the wind speed in knots? the evaperation duct height calculation requires that detail. The back of envelope naval chart I'm looking at gives roughly 3 metres in height for the water temp/air temp in the video at zero knots. Best time of day for sampling when the temperature of the atmosphere most closely matches the surface temperature of the water. Reduces the potential evaperation duct height due to humidity gradient. Bigger boat and higher height above water surface highly recommended to get to a more uniform elevation distribution. (with as little wind as possible)

Physics is all about what you can mathematically define. In this case the thermodynamic state of this "self named region" requires a metric modelling. As well as the corresponding thermodynamic degrees of freedom of the system state being described. Hence it is an undefined region, until you can properly define it I wouldn't call myself an expert in atmospheric refraction. It's been awhile since I last practiced those particular equations. To be 100% honest with you, I honestly feel learning how to calculate the best times yourself and show the refraction corrections on your paper is an important step in eliminating error margins. I can certainly help explain how those equations work. (once you understand them, they can easily be programmed) There are calcs available but if you don't understand how pressure density and temperature affects refraction and how to mathematically model such in an atmosphere they won't do much good. Most calculators are typically designed with the basic formulas. They don't often have elevation corrections etc. I'll post some metric examples when I get a chance. Takes a bit of time to post.

Well glad to see the approach to eliminate systematic errors. I've decided to pick up a couple of cheap lasers for myself lol. I want to test my idea... (curiosity killed the cat...)

Lol a fixed mount of the additional lasers would help. Quite frankly the safety aspects should have been addressed in the first place.... Including anyone on the shoreline and other boaters

by the way Dark thanks for this link. http://aty.sdsu.edu/~aty/explain/atmos_refr/bending.html