Method for determining long term stability

Abstract

A method of determining long term power variation, as an indication of long term stability of power output, by steps of measuring power output at successive intervals of time, determining variations in power output, from an average power output, at corresponding intervals of time, making a Fourier transform of each variation in power output, calculating a power spectral density from each Fourier transform, plotting a log of each power spectral density, extrapolating the plot to find a value, designated as a log of a power spectral density, from the plot, calculating a power spectral density from the found value of the log of the power spectral density, and making an inverse Fourier transform of the calculated power spectral density, to determine the long term power variation.

Description

FIELD OF THE INVENTION

The present invention relates to a method of determining an indication of long term stability of power output from a device such as a gravity meter.

BACKGROUND OF THE INVENTION

In the past, power output from a device such as a gravity meter had to be measured over a long term T_A, in order to determine long term stability of the power output from the device. Variations, Delta P_T2, Delta P_T3 . . . Delta P_TA from a null power output value, P, of the device would be measured over the term T_A. This period of time T_A would be needed to determine a long term variation Delta P_TA from the null power output value P. T_A is an empirically determined drift cycle time for power output variations of such devices.

The present method allows one to determine Delta P_TA, as an indication of long term stability of power output, without having to take measurements of power output over a long term T_A. The present method allows one to determine long term stability of power output of the device, even though measurements are not taken over a period of a drift cycle time, T_A.

In the present method, measurements of the power output values P_T2, P_T3 etc. of the gravity meter are made at intervals of time T₂, T₃, T₄ etc, until a length of time T₁ is reached. Here T₃ equals twice T₂. T₄ equals three times T₂. A power output plot versus time, is formed from the set of power output values.

A center line is drawn through the power output plot. This center line represents the constant power output,P, of the gravity meter. A set of power output variation values, Delta P_T2, Delta P_T3 . . . Delta P_T1 from value P is calculated. The vavalues in this set is correspond to the regular intervals of time from T₂ to T₁.

A Fourier transform of each power output variation, Delta P_T2, Delta P_T3, . . . Delta P_T1, is made. A power spectral density value PSD is calculated. PSD is the value of the Fourier transform at a particula frequency f_T, where f_T equals 1/T. For example, a power spectral density value, (PSD)_T2, is the value of the Fourier transform of Delta P_T2, at the frequency f₂=1/T₂. A power spectral density value (PSD)_T3 is the value of the Fourier transform of Delta P_T3 at the frequency f₃=1/T₃. A power spectral density value (PSD)_T1 is the value of the Fourier transform of Delta P_T1 at the frequency f₁=1/T₁.

The log of each calculated power spectral density value is ploted against the log of a frequency associated with the length of time required to find a power output variation associate with each calculated power spectral density value. The plotting is done on log-log paper. This log-log paper is an example of a log-log form.

The power spectral density log plot can be extrapolated, in order to find a log of a power spectral density value (PSD)_TA a a value that is the log of a frequency f_A=1/T_A. The value of (PSD)_TA and f_A are found from the log values of (PSD)_TA and f_A.

The inverse Fourier transform of the power spectral density value (PSD)_TA is taken in order to find the variation, Delta P_TA.

One can determine a line of slope of the power spectral density plot, and can therefore extrapolate the line of slop. plot. This determination of a lione of slope is performed by examining the power spectral densty plot generated from data taken over a length of time T₁, that is at intervals T₂, T₃, T₄ . . . T₁.

One does not have to measure a power variation, Delta P_TA, in the power output of a gravity meter at a time T_A, in order to determine the power output drift, or variation, for this time T_A. One can merely measure the slope of the power spectral density curve in order to predict the value of the power spectral density at a time, T_A, and take the inverse Fourier transform of that value.

Further one can find noise power, P_TATB, in a bandwith between f_A and f_B, where f_B is smaller than f₂. One first integrates a power density function, S(f), that gives the ploted power spectral density, (PSD)_T, between f₁ and f₂, to find a noise power P_T1T2. Then one uses the value P_T1T2 in an algorithm to find P_TATB.

SUMMARY OF THE INVENTION

A method of determining power variation Delta P_TA as an indication of long term stability of power output comprising determining power output at successive intervals of time T beginning at a first interval of time T₂ and continuing to an interval of time T₁, where time T₁ is shorter than an interval of time T_A, determining variations, Delta P_T2 to Delta P_T1, in power output, from an average power output P, at corresponding intervals of time from T₂ to T₁, making a Fourier transform of each variation in power output, calculating a power spectral density, from (PSD)_T2 to (PSD)_T1, from each Fourier transform, at a frequency f, where f equals 1 over an interval of time T associated with each Fourier transform, plotting a log of each power spectral density, from (PSD)_T2 to (PSD)_T1 at a log of the associated frequency f, on a log-log plot form, extrapolating value designated as a a log of a power spectral density, (PSD)_TA, at a log of a frequency f_A, where f_A equals 1/T_A, from the plot form, calculating the power spectral density (PSD)_TA from the log of the power spectral density (PSA)_TA, making an inverse Fourier transform of the power spectral density (PSD)_TA to determine power variation Delta P_TA.

DESCRIPTION OF THE DRAWING

FIG. 1 is a plan diagram of a test setup to determine power output of a gravity meter,

FIG. 2 is a power spectral density curve, using power output variation measurements over a length of time T₁.

FIG. 3 is an extrapolated power spectral density curve that predicts power spectral density at a long period of time T_A, where T_A is an empirically determined drift cycle time in power output of a gravity meter.

DESCRIPTION OF THE PREFERRED EMBODIMENT

In FIG. 1, a gravity meter 10 is located on a table 11. The table 11 is rigidly attached to a pedestal 12. The pedestal 12 is anchored to a concrete floor 14. The concrete floor 14 extends for 50 feet horizontally over the earth 15 in all directions from the gravity meter 10. The concrete floor 14 is solidly joined to the earth 15.

The gravity meter 10 is electrically connected to a capacitor 21 be means of an electrical cable 22, the connection being through a switch 23 and also through a full wave rectifier 24. The gravity meter 10 is shown as energized. Power output of the gravity meter 10 is nulled to a nominal zero voltage output while gravity meter 10 is measuring acceleration due to local gravity. The power out of the gravity meter 10 has a nominally nulled zero voltage value due to a constant force of gravity of 1 g.

Noise energy E₂, due to drift in the gavity meter 10, is collected by the capacitor 21 for one hour and produces a voltage V₂ in capacitor 21. The noise energy E₂, due to drift, is accumulated in capacitor 21 over an interval T₂ of one hour. That is, the energy, generated due to unstable circuit elements within the gravity meter 10, for a period T₂ of one hour, is gathered in an energy accumulator, such as capacitor 21.

A volt meter 25 is connected through an electrical cable 26 and a switch 28. Switch 23 is opened and switch 28 is closed. The voltage reading V₂ of the capacitor 21 is taken after one hour by volt meter 25. The associated frequency for this reading is f₂. f₂ equals 1/T₂.

Switch 28 is then opened and switch 23 is closed.

The voltage reading V₃ of the capacitor 21 is repeated after a second hour. The associated energy-gathering period T₃ for this latter voltage reading, V₃, is two hours. T₃ equals twice T₂. The associated frequency for the latter reading is f₃. f₃ equals 1/T₃. The energy collected after two hours is E₃.

Reading of voltages, V₂, V₃ . . . etc. of capacitor 21 are taken at periodic intervals T₂, T₃ . . . etc. The associated frequencies for the readings are f₂=1/T₂, f₃=1/T₃ . . . etc. The associated noise energies are E₂, E₃ . . . etc.

Twenty four voltage readings of capacitor 21 are taken, at one hour intervals. The twenty four voltage readings are thus obtained over a length of time of 24 hours, a time designated as T₁.

Again, an accumulation of drift energy is continued in capacitor 21 for twenty four intervals, each period being greater than the previous interval by a time T₂. The first interval, T₂, is one hour long. The twenty-fourth interval, T₁, is twenty four hours long. The twenty fourth accumulation of energy is the total noise energy E₁ that is generated by the gravity meter 10 over a twenty-four hour period T₁. The noise energy E₁, due to drift over a 24 hour period T₁, is thus determined.

A power value P_T2, where P_T2=E₂/T₂, is determined. A power value P_T3, where P_T3=E₃/T₃, is also determined. This determination is repeated until a power value P_T1, where P_T1=E₁/T₁, is determined.

The noise power values P_T2, P_T3 . . . P_T1 are plotted and a straight line drawn through the plot. This straight line represents the average noise power out of the gravity meter 10 over the twenty four one hour lengths of time.

Twenty four power variations, Delta P_T2, Delta P_T3 . . . Delta P_T1, from the straight line, are obtained.

The power variations, Delta P, are determined after reading voltage levels, V₂, V₃, V₄, . . . V₁ of the capacitor 21. Since E=CV₂, where C is the capacitance of capacitor 21, the energy values E can be determined. Then twenty four noise power values P_T2, P_T3, P_T4 . . . P_T1 are determined using the energy values E₂, E₃, E₄ . . . E₁ that are determined for the twenty four periods of time T₂, T₃, T₄ . . . T₁ during which energy, collected by capacitor 21, is measured. A frequency f₂, equal to 1/T₂, is calculated. f₂ is associated with a power variation Delta P_T2. A frequency f₃, equal to 1/T₃, is also calculated. f₃ is associated with power variation Delta P_T3. This is repeated until a frequency f₁, equal to 1/T₁, is calculated. f₁ is associated with power variation Delta P_T1.

Each of these power variations, Delta P, is operated on by a Fourier transform process. A Fourier transform, FT, for each power variation, Delta P, is found.

A power spectral density value (PSD) for each power variation is found. A power spectral density value (PSD) is the value of a Fourier transform FT of a power variation Delta P, at as associated frequency f. For instance, a power spectral density value (PSD)_T2 is found from Fourier transform, FT, of Delta P_T2 at f₂.

Power spectral density values (PSD)_T2, (PSD)_T3 . . . (PSD)_T1 are the values of the twenty four Fourier transforms of the power variation values Delta P_T2, Delta P_T3 . . . Delta P_T1 at the frequencies f₂, f₃ . . . f₁. A book on how to take a Fourier transform of a value, is entitled “Signal Analysis And Estimation” by Ronald L. Frante, John Wiley & Sons (1988). This book is incorporated herein by reference.

As shown in FIG. 2, a log of a power spectral density value (PSD) associated with each of the twenty four frequencies f₂, f₃ . . . f₁ is plotted on a log-log plot. The value of the log of a power spectral density value (PSD) is plotted at the value of the log of the frequency f used in finding (PSD).

An algorithm is generated. The algorithm describes a line through the points of the log-log plot, as shown in FIG. 2. The algorithm is used to evaluate the stability of the gravity meter 10. The algorithm is:
log (PSD)=log (k)+N log (f),
where (PSD) is power spectral density associated with an energy E collected over an interval of time T, and f equals 1/T. N is the slope of the straight line, shown in FIG. 2, drawn through the log-log plot. N is a negative number.

Thus,
log (PSD)_T1=log (k)+N log (f₁).

Also,
log (PSD)_T2=log (k)+N log (f₂).
By extrapolation of the line of FIG. 2, the point log (PSD)_TA=log (k)+N log (f_A), is reached, as shown in FIG. 3.

Also, log (PSD)=log (k)−N log (T), where (PSD) is power spectral density associated with an energy E collected over an interval of time T, and T equals 1/f.

Thus,
log (PSD)_T1=log (k)−N log (T₁).

Also,
log (PSD)_T2=log (k)−N log (T₂).

To find the value of N, log (PSD)_T2 is subtracted from log (PSD)_T1. Then log (PSD)_T1−log (PSD)_T2=N log (T₂)−N log (T₁). N=[(log (PSD)_T1−log (PSD)_T2)/(log(T₂)−log(T₁))].

To find the value of log (k), the found value for N is substituted into log (PSD)_T1=log (k)−N log (T₁). Thus,
log(k)=log(PSD)_T1=[(log(PSD)_T1−log(PSD)_T2)/(log(T₂)−log(T₁))](logT₁).

I. Determination of P_TA

Substituting the values for N and log (k) into log (PSD)_TA=log (k)−N log (T_A), the value for log (PSD)_TA is log(PSD)_TA=log(PSD)_T1+[(log(PSD)_T1−log(PSD)_T2)/(log(T₂)−log(T₁))](logT₁)−[(log(PSD)_T1−log(PSD)_T2)/(log(T₂)−log(T₁))](logT_A). The log of the power spectral density (PSA)_TA, log(PSD)_TA, due to an amount of drift after a thirty day time T_A, is thusly determined.

The power spectral density (PSA)_TA, for a time T_A, is found by taking the inverse log of log (PSD)_TA. Delta P_TA is found by taking the inverse Fourier transform of (PSA)_TA.

From the power spectral density (PSA)_TA, one can find the variation, Delta P_TA, that is, the output noise power variation from the average output noise power P, of the gravity meter 10, after a relatively long time T_A. Again, delta P_TA is found by taking the inverse Fourier transform of the power spectral density (PSD)_TA.

Further
(PSD)_T2=kf₂^N=k(1/T₂)^N=k(T₂⁻¹)^N=kT₂^−N.
(PSD)_T1=kf₁^N=K(1/T₁)^N=k(T₁⁻¹)^N=kT₁^−N.

N is the slope of the straight line drawn through the log-log plot of FIG. 2. Again N is a negative number.

The capacitor 21 collects energy when the drift of gravity meter 10 is positive. Capacitor 21 also collects energy when the drift of gravity meter 10 is negative. The capacitor 21 should be a very low noise capacitor. The values of noise energy for 24 intervals are measured. The power spectral densities are determined by taking the Fourier transforms of variations from an average power, for the 24 measured energies involved.

It is found that an algorithm, such that log of the power spectral density (PSD) equals the log of k, where k is a constant, minus N times the log of the frequency for the particular power spectral density, defines the line shown in the log-log plot of FIG. 2. N is the slope of the straight line fitted to the log-log plot of FIG. 2.

II. Determination of P_TATB

The above found value of N is used in another algorithm to find the noise power, P_TATB, in the bandwith between frequencies f_A and f_B. P_TATB=PT1T2[(f_B^N+1−f_A^N+1)/(f₂^N+1−f₁^N+1)]. P_T1T2 is found by first integrating a power spectral function S(f) from the log of the frequency f₁ to the log of the frequency f₂. S(f) is a mathematical expression generated to mathematically express the plot of the log of the power spectral density of FIG. 2. P_T1T2 is the noise power in the bandwith between frequencies f₁ and f₂. One can thus determine the noise power, P_TATB, in the bandwith between frequencies f_A and f_B.

f₁=/(24 hours) where T1 is 24 hours. f₂=1/(1 hour) where T₂ is 1 hour. f_A is 1/(720 hours) where T_A is 720 hours. f_B could be a lower frequency, such as f_B=1/(½ hour). T_B is a period of ½ hour.

In the above example:
f₁=1/T₁ T₁=24 hours
f₂=1/T₂ T₂=1 hour
f_A=1/T_A T_A=720 hours
 f_B=1/T_B T_B=½ hour

While the present invention has been disclosed in connection with the preferred embodiment thereof, it should be understood that there may be other embodiments which fall within the spirit and scope of the invention as defined by the following claims.

Claims

1. A method of determining power variation Delta PTA as an indication of long term stability of power output, comprising:

(a) determing power output at successive intervals of time T beginning at a first interval of time T2 and continuing to an interval of time T1, where time T1 is shorter than an interval of time TA;

(b) determining variations, Delta PT2 to Delta PT1, in power output, from an average power output P, at corresponding intervals of time T2 to T1;

(c) making a Fourier transform of each variation in power output;

(d) calculating a power spectral density, from (PSD)T2 to (PSD)T1, from each Fourier transform, at a frequency fT, where T is the interval of time associated with each Fourier transform;

(e) plotting of a log of each power spectral density (PSD)T2 to (PSD)T1 at a log of the associated frequency f, on a log-log plot form;

(f) extrapolating a value designated as a log of a power spectral density (PSD)TA, at a log of frequency fA, where fA equals 1/TA, from the plot form;

(g) calculating the power spectral density (PSD)TA from the log of the power spectral density (PSD)TA;

(h) making an inverse Fourier transform of the power spectral density (PSD)TA to determine power variation Delta PTA.