Ground Balance Theory

[Ferric Toes]

Hi Eric,

ok, I have checked the point, that you fit the exponent between two readings (Amplitude(t) = A1, Amplidute(2*t) = A2). Is this correct? Or how do you calculate the exponents in the last picture?

If I have understood it correct,
(1) B*t^b = A1
(2) B*(2*t)^b = A2

solved 1+2 for b:
b = log2(A2/A1) = log(A2/A1) / log(2)

But if I put the parameters into the formula however, I get different values.
Aziz

The exponents in my data are arrived at by taking the reading at 10uS delay and for convenience, setting it at a round figure with the sensitivity control. Usually 1000, 500, or 200 for weaker specimens. There is a both a rotary switch for delay and a toggle switch that exactly doubles whatever delay is set on the rotary switch. The sample is removed, which causes the unit to auto-zero. The toggle switch is then set to x2 (20uS) and the new reading taken. If the slope is exactly -1 then the second reading should be exactly half the first i.e. 500, 250, or 100, in the case of the above initial settings. In the case of the Tuff with a first reading of 1000 at 10uS, then a second reading of 495 at 20uS, we get 1000/495, then divided by 2 again gives 1.01 which is the slope. Repeat at 30uS delay and first reading is 325 and second (60uS) 155. 325/155/2 = 1.048, so we can see that the exponent has increased. This process is repeated for all delays on the rotary switch and then doubled on the toggle switch. Dead easy really with no complex maths. Only realised a couple of days ago that I could do this easily for a quick test; A 10 - 20uS calculation, 30 - 60, and a 50 - 100. This will give 3 data points to show if the exponent reduces, increases, or stays flat across the measurement range.

One point to note is that the sample widths are always the same as the delay. i.e. 10uS delay, 10uS sample, 15uS delay, 15uS sample and so on such that the 100uS delay has a 100uS sample. Pulse frequency has also dropped proportionally so that integration time remains the same. The delay figure should strictly be the mid point of the sample pulse rather than the time to the start edge. 10uS delay to the start edge is really 15uS to the mid point a.s.o.

Eric.

[Ferric Toes]

Electrically conductive ferrite material.

--Dave J.

Hi Dave,

Nice try, but the rods that exhibit viscosity are >20M end to end, while the ones that I use as PI probe cores measure about 3k.

Tiva canyon Tuff is interesting and there is quite a bit of research being done by various organisations. The Institute for Rock Magnetism has a couple of good articles for download starting vol.14 No.3, 2004. I was lucky enough to get some a few years ago when starting some magnetic soil measurements.

Eric.

[Midas]

Brilliant! That is just what I was hoping someone would do. My graphing does a best fit, so the exponent is averaged over the whole curve, but the fit is pretty good. I always noticed that the 10uS sample was a bit higher in amplitude than it should be and wondered if it was an instrumental problem, but I could not see what was going on further down i.e. change of slope. Now with your results and what I have measured over the past two days, I see that it is not an instrument problem, and I will explain why.

Firstly there are no timing errors throughout the range of delays. At the shortest delay (10uS) the clock is running at 100kHz and measured today on a precision frequency meter with an oven controlled reference, is accurate to 1Hz. Likewise, the TX and sample widths are all as accurate. This being so if measurements are taken at 10uS, 20uS, 40uS and 80uS there should be a factor of 2 between each succeeding pair for a slope of -1. e.g. say 500/10uS, 250/20uS, 125/40uS, and 62.5 at 80uS. I also did the intermediate values 15 - 30, 25 - 50uS etc. Having done this I also plotted a downward trend in exponent as you have for the Oz Rock and Red Hill soil. Basically, I took the shorter delay reading, halved it, and divided it by the longer delay reading. e.g. (1000/2)/494 = 1.0121 for the first reading of the Tuff discussed below.

Now, it could be argued that there is some, as yet, undiscovered error in the instrument - but for the following. I plotted the results for Tiva Canyon Tuff, which although derived from volcanic ash has some interesting properties. The main one being a very high frequency dependence of susceptibility, >20% resulting in a high viscosity reading relative to its dc susceptibility. With this material the trend of the exponent goes the other way i.e. it increases at late times. To my thinking, this rules out an instrumental error.

To improve the S/N figure for the Red Hill and Tiva Canyon material I used 20gm samples instead of 10gm. This almost doubles the signal throughout the range. Theoretically it should double exactly but the bigger pot is poking slightly out of the sensor coil where the flux density will be a bit less. I repeated the measurements a few times and it is consistent.

As you say, we are dealing with very small changes here. The Tuff goes from a slope of -1.0121 between 10&20uS to -1.055 between 50&100uS. Plotting out the full results for the Tuff and doing a best fit curve, it almost looks as though the exponent has its own power law of +0.026.

Eric.

Hi Eric,

That method for determining exponents actually only holds true for an exponent of 1. Anything else and it delivers incorrect results, try it on a generated curve and see.
Aziz I'm not sure what your method is but I'm pretty sure its also not correct. Nice work on the least error fit though, better than mine.

My method pretty simple exploiting the fact that the slope of the log\log plot is equal to the exponent. So, using the first set of Redhill points as an example: a=(log(204)-log(134))/(log(30)-log(20))

Applying it to the Tuff there's not much of a trend. It does go up broadly to start with but very erratically and the final exponent is actually less than the first. I'm starting to lean towards measurement error.

[Ferric Toes]

Hi Eric,

That method for determining exponents actually only holds true for an exponent of 1. Anything else and it delivers incorrect results, try it on a generated curve and see.
Aziz I'm not sure what your method is but I'm pretty sure its also not correct. Nice work on the least error fit though, better than mine.

My method pretty simple exploiting the fact that the slope of the log\log plot is equal to the exponent. So, using the first set of Redhill points as an example: a=(log(204)-log(134))/(log(30)-log(20))

Applying it to the Tuff there's not much of a trend. It does go up broadly to start with but very erratically and the final exponent is actually less than the first. I'm starting to lean towards measurement error.

Well it's all good clean fun, but all I need is the slope to 2 decimal places, so I might as well stick to my graphing software exponent calculation. 2DP is an order greater than has been done before, and I have eliminated many of the inaccuracies in previous instrumental results. Anything further may be of interest to environmetal magnetism scientists, but the primary interest here is knowledge for effective ground balancing in a PI metal detector.

-1.05 it is.

Eric.

[Ferric Toes]

Just out of interest I tried log log graphing in sections of the Tiva plot. i.e 1,2,3, - 2,3,4, - 3,4,5 etc and getting the graph calculation for each. I got 1.014, 1.023, 1.037, 1.039, 1.053 ..... which are not far removed from my original results. I still think the effect is genuine, particularly as the RH Soil and Oz rock go the other way.

Eric.

[Aziz]

Thanks for the clarification guys.
I'll update the data for the Tiva Tuff.

BTW, the Oz Rock can't be modelled (fitted) with less error.
Both approximation functions
G1(t) = B*t^b and
G2(t) = -B * (Exp(-t/T1) - Exp(-t/T2)) / (t*Log(T2/T1))
doesn't fitt well in the measured data. The error at the beginning and at the end of the samples (early and late samples) is quite high.

Cheers,
Aziz

[Aziz]

Look at this.
G2(t) approximates the decay behaviour of Tiva Tuff much better (less error).



Excel table (zipped):
Decay_Eric_TivaTuff01.zip

Aziz

[Aziz]

The Freakin Oz Rock

Now look at the freakin Oz Rock this time. And you will understand, that the Oz Rock is difficult to tame and it will screw your GB. All the nice presented formulas in the scientific papers can be thrown into the garbage bin now.



Excel table (zipped):
Decay_Eric_OzRock01.zip

I can't get better approximations with less errors. Even my own solver software does not deliver them.
Maybe Eric makes a more accurate measurement once again? (like the Tiva Tuff)

Cheers,
Aziz

[Aziz]

Aziz's Magic Formula G(t) = B*(t+p)^b


My magic formula is the best fitting and approximating formula so far!
G(t) = B*(t+p)^b


I'll show you all the three formulas in comparison soon.

Aziz

[Aziz]

Now lets see the best fitting results:

Red Hill Soil:


The freakin Oz Rock:


Tiva Tuff:


And the freakin Excel tables (zipped, three tables):
ExcelTables01.zip

Bugger me!, keep digging every signal!!!
(Could be a hotrock! *LOL*)

Cheers,
Aziz

[Ferric Toes]

Hi Aziz,

Thanks for doing all this work. You had better summarise in words though, so we can all understand what you have found. I will re-measure some Oz rock samples as accurately as I can and post them shortly. I don't think a detailed magnetic study has been done before on Oz goldfield ironstone, so its properties will not be in any paper or textbook. Pity the sample I gave to Yoga Das as few years ago was not investigated. I will get in touch with someone else I know who might be able to get a mineralogical analysis done.

Eric.

[Ferric Toes]

Here are two superimposed plots of Oz Rock (green) and Tiva Tuff (red). Both plots commenced at 1000 units and placed on top of each other so only seen as one at the start point. The curve fit for the Oz rock was too far out in relation to the rest of the points, so ceases just below the 1000 dot. This indicates a faster initial decay for the Oz rock. The linear decay plot shows the true difference between both curves which gradually come together at late times. I get a slope calculation of -1.094 for Oz compared to -1.016 for Tiva for the portion between 10 and 20uS. You can see how the red dots drift below the line between 20 and 50uS and how some red dots drift above in the Tiva plot. This is to do with the opposing changes of slope that have been noted with these two different materials.

Does it matter? Not for most ground, but Oz ground is still a challenge that needs to be addressed.

Eric.

[Aziz]

Hi Aziz,

Thanks for doing all this work. You had better summarise in words though, so we can all understand what you have found. I will re-measure some Oz rock samples as accurately as I can and post them shortly. I don't think a detailed magnetic study has been done before on Oz goldfield ironstone, so its properties will not be in any paper or textbook. Pity the sample I gave to Yoga Das as few years ago was not investigated. I will get in touch with someone else I know who might be able to get a mineralogical analysis done.

Eric.

You're welcome Eric and no problem at all. I would do even more if you would provide me with more data. I would like the numerical list of the measured data (sample time, measurement reading) as it makes the import to the Excel table easier for me.

--- sorry for ranting ---
BTW, I have successfully prevented Mr. Bill Gates to get more rich. My new Excel version 2010 did require a registration and I didn't buy it. I have cracked the damn thing. Yes! I didn't pay more money to the B$%"$%§"$%. This B§$%"§$%§ is one of the big Monsanto share holders. This B$$"§$%"§%& is earning money by letting people hunger or die!
--- Ok, sorry for ranting, sorry, but don't censor it ok Mr. chief? ---
Aziz

[Midas]

How about trying a blend of Tiva and Redhill and\or Oz? We can see if the trend of the mix fits neatly between the two

Aziz: Open office calc, just as good I reckon. Should fit into your anti establishment ethos quite nicely too.

[Aziz]

How about trying a blend of Tiva and Redhill and\or Oz? We can see if the trend of the mix fits neatly between the two

Aziz: Open office calc, just as good I reckon. Should fit into your anti establishment ethos quite nicely too.

Thanks for the tip. I don't want to pay M$ more money to make the B$"§$%"$% Mr. Bill Gates getting more rich.
I have payed them really enough money.
But I would like to use the cracked M$ Excel 2010.
BTW, don't buy the Windows 8. Wait! Wait for the improved version. *LOL*

Aziz