Patent Publication Number: US-8543350-B2

Title: Synthetic vibration isolation system for freefall gravimeter

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority under 35 U.S.C. §119 from U.S. Provisional Patent Application Ser. No. 61/236,023 entitled “SYNTHETIC VIBRATION ISOLATION SYSTEM FOR FREEFALL INTERFEROMETRIC GRAVIMETER,” filed Aug. 21, 2009, which is incorporated by reference in its entirety for all purposes. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
    
    
     BACKGROUND 
     1. Field 
     The present disclosure generally relates to systems and methods of measuring of acceleration due to gravity and, more particularly, gravimeters using signal processing to provide synthetic vibration isolation. 
     2. Description of the Related Art 
     Gravimetry is a branch of science concerned with measurement of gravitational fields. Precision measurement of the earth&#39;s gravitational field is important both for fundamental scientific research and for the exploration of oil and mineral resources. Gravity is usually measured in units of acceleration. One commonly used unit of acceleration is the “g”, where 1 g is the standard value of earth&#39;s gravitational acceleration at sea level and, more precisely, has a value of 9.80665 meters per second squared (9.80665 m/s 2 ) as declared at the 3rd Conférence Générale des Poids et Mesures (CGPM) in 1901. The derived unit of a “micro-g”, i.e. a millionth of a g or 9.80665 μm/s 2 , is sometimes used when dealing with small variations in gravity. Another unit of acceleration used extensively in the science of gravimetry is the “gal”, named in honor of the Italian physicist Galileo Galilei. The gal is defined as 1 centimeter per second squared (1 cm/s 2 ). 
     Precision measurements of gravity are also used to map the shape of the earth, which is the science of geodesy, as the value of the acceleration due to gravity changes with distance from the center of the earth. Post-glacial rebound, the visco-elastic response of the Earth to the melting of the large ice sheets of the ice ages, currently is expressed in vertical crustal motions on the order of one millimeter per year over the entire Earth. The crustal rebound exceeds 1 centimeter per year in regions of North America and Europe, where the major ice sheets existed 20,000 years ago. The change in gravity caused by a change in the distance of a point on the surface of the Earth from the center of the mass is approximately 3 microgal per centimeter. The ability to measure a change in gravity of 1 microgal therefore corresponds to a resolution of 3 millimeters in vertical crustal motion, assuming that other causes of changes in gravity can be properly accounted for. As the gravitational acceleration on the Earth&#39;s surface is on the order of 980 gal, measuring gravity to an accuracy of 1 microgal requires a resolution of approximately 1 part per billion (10 −9 ). 
     People have been building instruments to measure the acceleration due to gravity, called gravimeters, since 1680. The methods of measuring the acceleration due to gravity have evolved from pendulums that could measure gravity to 1 part in 10,000 (10 −4 ) to today&#39;s instruments that drop an object in a vacuum and measure its position as it free falls over time. Some instruments drop a mass having a retroreflector, an optical device that reflects any incident beam of light directly back at the source, and repeatedly measure the displacement of the mass as it falls using optical interferometry. An ideal curve is then fit to these displacement measurements to calculate the acceleration of the falling mass. This process of dropping a mass may be repeated hundreds of times and the results combined to further reduce the uncertainty of the composite value of the acceleration. 
     Conventional free-fall gravimeters observe a mass free-falling for a distance on the order of tens of centimeters in a vacuum chamber. The position of the falling mass is typically measured using interferometric techniques, bouncing a laser beam off a retroreflector on the falling mass and off a reference retroreflector that is mounted on an active vibration isolation system that has a period of approximately 60 seconds. While the accuracy of a conventional free-fall gravimeter is on the order of 10 −9 , gravimeters of this type may weigh up to 350 kg and have a height of up to 1.5 meters tall. Much of this weight and size is attributable to the complexity of providing a reference retroreflector that approximates a stationary reference in inertial space. This size and weight present limits to the usability of conventional gravimeters. 
     SUMMARY 
     The disclosed gravimeter provides high resolution measurements of the acceleration of a falling mass with a much smaller device than previously available. Such a device enables the precision measurement of the local gravity field in applications not previously possible, such as down-hole measurements in oil wells. 
     In certain embodiments, a gravimeter is disclosed that comprises a base, a reference device coupled to the base wherein the reference device is configured to move along a first axis, a falling device configured to free fall from a first position on a second axis that is parallel to the first axis to a second position on the second axis, a measurement module coupled to the reference device wherein the measurement module is configured to provide a first signal comprising a displacement of the reference device relative to the base and provide a second signal comprising a displacement of the falling device relative to the reference device, and a processing unit configured to accept the first and second signals and compute a displacement of the falling device in an inertial space by processing the first and second signals and subtracting a processed first signal from the processed second signal. 
     In certain embodiments, a method of measuring the relative position of a falling mass in inertial space is disclosed, the method comprising the steps of measuring a first displacement of a reference device relative to a base during an increment of time, measuring a second displacement of a free-falling mass relative to the reference device during the same increment of time, filtering the first displacement, subtracting the filtered first displacement from the second displacement, computing the relative displacement of the falling mass in inertial space and storing the relative displacement and time increment, repeating the steps of measuring an increment of time, measuring the first and second displacements, filtering the first displacement, subtracting the filtered first displacement from the second displacement, and computing the relative displacement and storing the relative displacements and time increments related to a single drop of the mass, calculating a best-fit estimate of the acceleration of the free-falling mass from the stored relative displacements and time increments made during the single drop, repeating the drop and repeating the steps of measuring the first and second displacements, filtering the first displacement, subtracting the filtered first displacement from the second displacement, and computing the relative displacement and storing the relative displacements and time increments, and calculating a best-fit estimate of the acceleration for each drop, and calculating an overall acceleration of the mass by averaging the best-fit estimates of the acceleration from all the drops. 
     In certain embodiments, a computer-readable medium having computer-executable instructions stored thereon for execution by a processor to perform a method of measuring the relative position of a falling mass in inertial space, the method comprising the steps of measuring a first displacement of a reference device relative to a base during an increment of time, measuring a second displacement of a free-falling mass relative to the reference device during the same increment of time, filtering the first displacement, subtracting the filtered first displacement from the second displacement, computing the relative displacement of the falling mass in inertial space and storing the relative displacement and time increment, repeating the steps of measuring an increment of time, measuring the first and second displacements, filtering the first displacement, subtracting the filtered first displacement from the second displacement, and computing the relative displacement and storing the relative displacements and time increments related to a single drop of the mass, calculating a best-fit estimate of the acceleration of the free-falling mass from the stored relative displacements and time increments made during the single drop, repeating the drop and repeating the steps of measuring the first and second displacements, filtering the first displacement, subtracting the filtered first displacement from the second displacement, and computing the relative displacement and storing the relative displacements and time increments, and calculating a best-fit estimate of the acceleration for each drop, and calculating an overall acceleration of the mass by averaging the best-fit estimates of the acceleration from all the drops. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are included to provide further understanding and are incorporated in and constitute a part of this specification, illustrate disclosed embodiments and together with the description serve to explain the principles of the disclosed embodiments. In the drawings: 
         FIG. 1  depicts a schematic concept of a conventional gravimeter. 
         FIG. 2  depicts a transfer function of a spring-mass system. 
         FIGS. 3 and 4  illustrate the concepts of how a low-pass filter may be replicated using a high-pass filter according to certain aspects of the present disclosure. 
         FIG. 5  depicts a schematic of a gravimeter configured to use synthetic vibration isolation according to certain aspects of the present disclosure. 
         FIG. 6  depicts a concept of how the gravimeter synthetic vibration isolation system is implemented according to certain aspects of the present disclosure. 
         FIG. 7  illustrates the frequency response of the physical filtering and the synthetic vibration isolation system according to certain aspects of the present disclosure. 
         FIG. 8  depicts a schematic concept of an optical measurement module according to certain aspects of the present disclosure. 
         FIG. 9  is a block diagram of a gravimeter according to certain aspects of the present disclosure. 
         FIG. 10  is a representation of results of a simulation that compared the error in the raw measurement of acceleration to the residual error in the corrected acceleration according to certain aspects of the present disclosure. 
         FIG. 11  is a flowchart illustrating the method of measuring the acceleration due to gravity using synthetic vibration isolation according to certain aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, numerous specific details are set forth to provide a full understanding of the present disclosure. It will be apparent, however, to one ordinarily skilled in the art that embodiments of the present disclosure may be practiced without some of the specific details. In other instances, well-known structures and techniques have not been shown in detail so as not to obscure the disclosure. 
       FIG. 1  depicts a schematic concept of a conventional gravimeter  10 . The gravimeter comprises a base  18 , which can be considered as the case or structure of the instrument, a reference retroreflector  14  that is coupled to the base  18  through a compliant element  16 , and a falling retroreflector  12 . The inertial reference plane  20  represents a fixed reference in inertial space. The base  18 , reference retroreflector  14 , and falling retroreflector are located at distances X 0 , X 1 , and X 2 , respectively, from inertial reference plane  20 . In the ideal, the reference reflector  14  is stationary in inertial space. In reality, base  18  moves in inertial space due to ground motion and vibration, thermal expansion, and other environmental disturbances. 
     In use, the falling retroreflector  12  starts from a position  12 A, shown as a broken line outline, and free falls toward the reference retroreflector  14 . While falling, the instantaneous position of the falling retroreflector  12  relative to the reference retroreflector  14  is labeled Y 2  in  FIG. 1 . Similarly, the instantaneous position of the falling retroreflector  12  relative to the inertial reference plane  20  is labeled X 2  in  FIG. 1 . While the falling retroreflector  12  is in free fall, a sequence of measurements is taken of the sequential displacements of the falling retroreflector relative to reference retroreflector  14 . If an ideal equation of the motion of a body falling solely under the influence of gravity is fit to the measured displacements, a value of the acceleration due to gravity can be calculated. If the reference reflector  14  were truly fixed in inertial space, the displacement of Y 2  would be equivalent to the displacement of X 2  in inertial space and the accuracy of this measurement would be limited only by the accuracy of the displacement measurement instruments. The motion of the reference reflector  14  caused by motion of the base  18  being transmitted through the compliant member  16 , however, adds error to the measurement of the displacement of Y 2 . One approach to reducing this error is to make the compliant element as soft as possible. 
       FIG. 2  depicts a transfer function  24  of a spring-mass system such as formed by the reference retroreflector  14  and the compliant element  16 . The system will have a natural frequency ω=√{square root over (k/m)} where m is the mass of the suspended element, the reference retroreflector in the example of  FIG. 1 , and k is the spring rate of the compliant element  16 . While most elastic systems are not truly linear over their entire range of motion, most systems can be approximated by a linear spring rate for a limited region of operation, which is what we will do here. The natural frequency  22  is indicated on the horizontal axis by the label ω. At frequencies below the natural frequency  22 , the response is approximately unity, which indicates that a disturbance in the base  18  at these frequencies will pass without reduction to the reference reflector  14 . At frequencies above the natural frequency  22 , the response rolls off at 40 dB/decade, i.e. decreases such that a doubling of the frequency will halve the response. This is commonly referred to as a “low-pass” filter, as low frequency disturbances pass through while high frequency disturbances are attenuated. 
     Returning to  FIG. 1 , the natural frequency of the spring-mass system formed by the reference retroreflector  14  and the compliant element  16  can be reduced by increasing the mass of the reference retroreflector  14  or reducing the spring rate of the compliant element  16 , or both. Some gravimeters add an active control of the attachment point of the compliant element  16  to the base  18  that emulates the behavior of a spring of up to a kilometer in length, such that the most accurate gravimeters have suspension systems with natural frequencies of a few tenths of a hertz. This performance comes at a cost, however, and the gravimeters are large and heavy, up to 350 kg and 1.5 meters tall. 
       FIGS. 3 and 4  illustrate the concepts of how a low-pass filter may be replicated using a high-pass filter according to certain aspects of the present disclosure. In  FIG. 3 , the transfer function of the system of  FIG. 1  is depicted where the disturbance acceleration of the base  18 , whose position is X 0 , is physically filtered in block  30  by the low-pass mass-spring system of reference retroreflector  14  and compliant element  16  to produce the acceleration of the reference retroreflector  14 . This is shown in  FIG. 3  with an input of the acceleration of the base  18 , where X 0  is the position of the base  18  with respect to the inertial reference  20  and the double-dot notation {umlaut over (X)} 0  indicates that this is the second derivative which is acceleration. Similarly, double-dot notation {umlaut over (X)} 1 , indicates the output, which is the acceleration of the reference retroreflector  14  whose position is X 1 . 
       FIG. 4  depicts a computational system that produces a synthetic vibration isolation element  32  using a computed high-pass filter  34 . When the portion of a signal u(t) that remains after passing through the high-pass filter  34  is subtracted from the initial signal in the summation node  36 , the remaining signal v(t) is only the low-frequency components of the signal, similar to the filtering accomplished by the low-pass filter  30  of  FIG. 3 . 
       FIG. 5  depicts a schematic of a gravimeter  40  configured to use synthetic vibration isolation according to certain aspects of the present disclosure. The inertial reference  20 , the base  18 , and the falling retroreflector  12  are the same as presented in  FIG. 1 . In this case, however, the reference retroreflector  44  is fixedly coupled to the base  18 . A measurement device (not shown) is coupled to the reference device  42  that is coupled through compliant element  46  to the base  18 . The measurement device measures the displacement relative to the reference device  42  of the falling retroreflector  12  along optical path  50  and from the reference device  42  to the reference retroreflector  44  along optical path  48 . The relative positions of the base  18 , reference device  42 , and falling retroreflector  12  in inertial space are given by variables X 0 , X 1 , and X 2 . 
     The following equations describe the basic dynamics of the gravimeter of  FIG. 5 . The subscripts of variables such as mass are consistent with the subscripts of the position variables shown in  FIG. 5 . In equation (1), the motion of the falling retroreflector  12  is governed solely by gravity g, wherein m 2  is the mass of the falling retroreflector  12  and x 2  and {umlaut over (x)} 2  are the position and acceleration, respectively, of the falling retroreflector  12  as shown in  FIG. 5 .
 
 m   2   ·{umlaut over (x)}   2   =−m   2   ·g   (1)
 
     The motion of the reference device  42  is governed both by gravity g and the forces applied by the compliant element  46  as shown in equation (2), wherein k 1  is the spring rate of compliant element  46 , m 1  is the mass and x 1  and {umlaut over (x)} 2  are the position and acceleration, respectively, of the reference device  42 , and x 0  is the position of the base  12  and the coupled reference retroreflector  44  as shown in  FIG. 5 .
 
 m   1   ·{umlaut over (x)}   1   =−m   1   ·g−k   1 ( x   1   −x   0 )  (2)
 
     If the static displacement due to gravity 
                 m   1     ·   g       k   1           
is ignored and only the motion of the reference device  42  about its static position is considered, as indicated by equation (3)
 
                     x   1     ⇐       x   1     +         m   1     ·   g       k   1                 (   3   )               
then equation (2) becomes
 
 m   1   ·{umlaut over (x)}   1   =k   1 ( x   1   −x   0 )  (4)
 
Now we can state the equations relating Y 1  and Y 2  to X 0 , X 1 , and X 2 :
 
 y   2   =x   2   −x   1   (5)
 
 y   1   =x   1   −x   0   (6)
 
The natural frequency ω 1  of the spring-mass system of the reference device  42  and the compliant element  46  is defined as
 
                     ω   1   2     ≡       k   1       m   1               (   7   )               
The equations of motion (1) and (4) may now be simplified to
 
 {umlaut over (x)}   2   =−g   (8)
 
 {umlaut over (x)}   1 =ω 1   2 ·( x   1   −x   0 )  (9)
 
Equation (9) can then be rearranged into
 
 {umlaut over (x)}   1 +ω 1   2   ·x   1 =ω 1   2   ·x   0   (10)
 
     At this point, the equations (11) and (8) can be converted to Laplace transforms and rearranged into 
                         X   1     ⁡     (   s   )       ·     (         s   2     +     ω   1   2         ω   1   2       )       =       X   0     ⁡     (   s   )               (   11   )                   X   2     ⁡     (   s   )       =     -     g     s   2                 (   12   )               
wherein the notation X 1 (s), for example, indicates that this is a Laplace transform of the real parameter X 1 (t) where t indicates that this is a function of time. Similarly converting the equations (5) and (6) of the Y 1  and Y 2  measurements into Laplace transforms
 
 Y   1 ( s )= X   1 ( s )− X   0 ( s )  (13)
 
 Y   2 ( s )= X   2 ( s )− X   1 ( s )  (14)
 
     Filtering the Y 1 (s) measurement with a linear filter G(s) and subtracting the filtered results from Y 2 (s) produces the equation 
                         Y   2     ⁡     (   s   )       -         Y   1     ⁡     (   s   )       ·     G   ⁡     (   s   )           =       -     g     s   2         -       X   1     ⁡     (   s   )       +         X   1     ⁡     (   s   )       ·       s   2       ω   1   2       ·     G   ⁡     (   s   )                   (   15   )               
which can be rearranged into
 
                         Y   2     ⁡     (   s   )       -         Y   1     ⁡     (   s   )       ·     G   ⁡     (   s   )           =       -     g     s   2         -       (     1   -         s   2       ω   1   2       ·     G   ⁡     (   s   )           )     ·       X   1     ⁡     (   s   )                   (   16   )               
Substituting equation (11) into equation (16) produces:
 
                         Y   2     ⁡     (   s   )       -         Y   1     ⁡     (   s   )       ·     G   ⁡     (   s   )           =       -     g     s   2         -       (     1   -         s   2       ω   1   2       ·     G   ⁡     (   s   )           )     ·     (       ω   1   2         s   2     +     ω   1   2         )     ·       X   0     ⁡     (   s   )                   (   17   )               
From equation (17), it can be seen that the motion X 0  of the base  18  is filtered by the spring-mass system of the reference device  42  and compliant element  46  in the right parenthetical term and then filtered by the linear filter G(s) in the left parenthetical term.
 
     If G(s) is chosen to implement a low-pass filter, which in effect provides a high-pass of the disturbance following the concept of  FIG. 4 , then one example form of G(s) is: 
                     G   ⁡     (   s   )       ≡       ω   1   2     ·     (     1       s   2     +     2   ·     ξ   h     ·     ω   h       +     ω   h   2         )               (   18   )               
where ω 1  is a natural frequency, and ω h  is a filter natural frequency. According to one example, the filter natural frequency is less than the natural frequency. According to one example, the natural frequency is greater than 10 Hz. According to one example, the filter natural frequency is less than 0.5 Hz.
 
     In other embodiments, other forms of G(s) are used to produce other linear and nonlinear filter functions. Substituting equation (18) into equation (17) produces 
                         Y   2     ⁡     (   s   )       -         Y   1     ⁡     (   s   )       ·     G   ⁡     (   s   )           =       -     g     s   2         -       (         2   ·     ξ   h     ·     ω   h     ·   s     +     ω   h   2           s   2     +     2   ·     ξ   h     ·     ω   h     ·   s     +     ω   h   2         )     ·     (       ω   1   2         s   2     +     ω   1   2         )     ·       X   0     ⁡     (   s   )                   (   19   )               
If the value of the damping coefficient ξ h  is chosen to be much less than 1, e.g. a lightly damped system, then equation (19) becomes:
 
                         Y   2     ⁡     (   s   )       -         Y   1     ⁡     (   s   )       ·     G   ⁡     (   s   )           ≈       -     g     s   2         -       (       ω   h   2         s   2     +     ω   h   2         )     ·     (       ω   1   2         s   2     +     ω   1   2         )     ·       X   0     ⁡     (   s   )                   (   20   )               
It can be seen from equation (20) that the disturbance X 0  will be rejected by a cascade of the spring-mass isolation system formed by reference device  42  and compliant element  46  and an initial 40 dB/decade rejection of the low-pass filter.
 
     It should be noted that while the above discussion of filtering and processing of the Y 1  and Y 2  measurements is conducted in the frequency domain, as embodied in the Laplace transforms, the implementation of the signal processing and filtering may be equally accomplished in the time domain as continuous or discrete processes. When implemented in a system comprising digital computers, a discrete time domain structure may be most convenient. In analog optical or electrical systems, a continuous time domain structure may be most convenient. It will be apparent to those of ordinary skill in the art to implement the disclosed methodology using any of the known time domain and frequency domain signal processing methodologies without departing from the scope of the claims. 
       FIG. 6  depicts a concept of how the gravimeter synthetic vibration isolation system is implemented according to certain aspects of the present disclosure. The real-time measurements of Y 1  signal passes through the filter  60  and is subtracted from the real-time measurement of Y 2  to produce a corrected measurement of the inertial displacement X 2  of the falling retroreflector  12 . To avoid introducing errors through aliasing at the resonant frequency of filter  60 , gravity estimates should be averaged over a period corresponding to an integer multiple of the inverse filter resonant frequency as shown in equation (21). 
     
       
         
           
             
               
                 
                   
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       FIG. 7  illustrates the frequency response of the physical filtering and the synthetic vibration isolation system according to certain aspects of the present disclosure. Curve  62  represents the physical filtering of the spring-mass isolation system formed by reference device  42  and compliant element  46 . As shown in  FIG. 7 , curve  62  may be represented by 
                 ω   1   2       (       s   2     +     2   ⁢     ξω   1     ⁢   s     +     ω   1   2       )       ,         
where ω 1  may represent a natural frequency and ξ may represent a damping coefficient. Curve  64  represents the filtering of the synthetic vibration computation comprising the G(s) filter  60 . To reduce the size of a gravimeter, the physical resonant frequency ω 1  can be made relatively high, relying on the filter  60  to remove the lower frequency noise from the computed value of X 2 .
 
       FIG. 8  depicts a schematic concept of a measurement module  100  according to certain aspects of the present disclosure. The measurement module  100  comprises an optics module  70  and a detector module  90  wherein the optics module  70  is, in this example, mounted on the reference device  42  of  FIG. 5 . The falling retroreflector  12  and reference retroreflector  44  are repeated in  FIG. 8  from  FIG. 5 , as are the measurement beams  48  and  50 . Interferometric displacement measurement systems of this type are disclosed in U.S. Pat. Nos. 6,947,621, 7,003,186, and 7,254,290 which are incorporated herein in their entirety. 
     The detector module  90  comprises a laser  73  that produces a beam of laser light at a first frequency. This light is carried, in this example, by polarization-maintaining single mode fiber optic cables  72 . One of the cables connects to an acousto-optic frequency shifter  74  that is coupled to an offset frequency signal generator  75 . The optical frequency shifter  74  changes the frequency of the laser light such that the frequency of the light in the output cable  72 A is different from that of the light carried by cable  72 . The frequency shift is chosen to create a beat frequency between the frequencies of the light in the cables  72  and  72 A. In this example, the cables  72  and  72 A pass from the detector module  90  to the optics module  70 . In other embodiments, the beam of laser light passes from the detector module  90  to the optics module  70  before the frequency shift and the optical frequency shifter  74  is a part of the optics module  70 . 
     Once inside the optics module  70 , cable  72  feeds the laser light into coupler  76 B while cable  72 A feeds into coupler  76 A. Coupler  76 A splits the beam into two approximately equal beams and feeds them into output cable  78 A and lightwave circuit  80 A. Similarly, coupler  76 B feeds the light from cable  72  into output cable  78 B and lightwave circuit  80 B. Output cable  78 A feeds the light into a collimating lens  79 A which creates measurement beam  48  that travels to reference retroreflector  44  and returns through lens  79 A into output cable  78 A. When this reflected measurement light reaches couple  76 A, it is guided into coupler cable  81 A and into coupler  82 A. At the same time, the light passing through lightwave circuit  80 B also is carried to coupler  82 A. In coupler  82 A, the light that sent out on measurement beam  48  is combined with the light from lightwave  80 B and transferred into a single mode optical cable  83 A and carried to detector  84 A. 
     In a parallel set of devices, the light in output cable  78 B passes through lens  79 B to become measurement beam  94  which bounces off of a fixed retroreflector  92  within the optics module  70  and returns into output cable  78 B, is guided by coupler  76 B into coupler cable  81 B. At the same time, light from lightwave circuit  80 A arrives at coupler  82 B. In coupler  82 B, the light that sent out on reference beam  94  is combined with the light from lightwave  80 A and transferred into a single mode optical cable  83 B and carried to detector  84 B. 
     The balanced detectors  84 A and  84 B convert the optical signals into electrical signals that are passed to phase detector  86  that samples and combines the signals from balanced detectors  84 A and  84 B to determine the displacement of the reference retroreflector  44 . The output  96  of the phase detector  86 , in this example, is a signal that comprises information about the displacement of reference retroreflector  44 . A duplicate of system  100  (not shown), configured to direct beam  48  to falling retroreflector  12  instead of reference retroreflector  44 , provides a signal that comprises information about the displacement of falling retroreflector  12 . 
       FIG. 9  is a block diagram of a gravimeter  40  according to certain aspects of the present disclosure. The gravimeter  40  comprises a base  18  to which is coupled a reflective fiducial  44 , in this example a corner cube retroreflector. A reference device  42  is coupled to the base  12  through a compliant element  46 , in this example a spring. The reference device  42  is configured to move along axis  96 . A measurement module  100 , comprising an optics module  70  and a detector module  90 , is coupled to the reference device  42 . The optics module  70  is coupled to the reference device  42 . In the example configuration of  FIG. 9 , the detector module  90  is coupled to the optics module but not to the reference device  42 . In other embodiments, the detector module  90  is coupled to the reference device  42 . A processing unit  92  is coupled to the detector module  90  of the measurement module  100 . The processing unit  92  comprises storage media  99  in which are stored the executable instructions to operate the gravimeter  40  and perform the method of computing the acceleration of the falling mass. Storage media  99  may be any computer-readable medium, including Random Access Memory (RAM), Read Only Memory (ROM), Programmable ROM (PROM), Erasable PROM (EPROM), rotating magnetic hard disks, solid-state memory (SSD), flash memory data storage devices including “thumb drives” and flash cards such as Secure Digital (SD) cards and Memory Sticks, optical disks including Compact Disks (CDs) and Digital Video Disks (DVDs), magnetic stripes, removable magnetic media such as Zip and Jaz drives, magnetic tape, magnetically or optically scannable images including Magnetic Ink Character Recognition (MICR) characters, barcodes and two-dimensional matrix codes, or any other suitable storage device. The processing unit  92  also comprises a processor  98  that is coupled to the storage media  99  and the detector module  90 , the processor  98  configured to retrieve the executable instructions from the storage media  99  and accept signals from the detector module  90 . Processor  98  may be a single processor or a set of one or more processors, and may be part of a computer system such as a standard personal computer (PC), a laptop or other portable computer, a workstation, or a central computer accessed from a remote terminal. A falling device  12  comprising a reflective fiducial  13 , in this example a corner cube retroreflector, is configured to move along axis  94  that is parallel to axis  96 . In certain embodiments, axis  94  is nominally coincident with axis  96 . 
       FIG. 10  is a representation of results from a simulation that compared the error in the raw measurement of acceleration to the residual error in the corrected acceleration according to certain aspects of the present disclosure. The values presented on this plot are representations of the output of the simulation and should not be interpreted as actual results. The simulation generated a white noise acceleration signal of approximately 5.6 micro-g/sqrt(Hz) and 126 micro-g root-mean-square (RMS) that was used as the motion of the base  18  of the gravimeter  40  of  FIG. 5 . This input resulted in the reference device  42  moving over a distance of several hundred nanometers, with most of the motion occurring at the natural frequency of the spring-mass system formed by the reference retroreflector  14  and the compliant element  16 . The plot of  FIG. 10  shows how repeated measurements were generated of the displacements Y 1  and Y 2  during repeated drops of the falling retroreflector  12 , exactly as the gravimeter would be used, that included the noise in the position of the reference device  42 . These measurements were converted into a raw estimate of the gravitational acceleration of the falling retroreflector without correction, e.g. treating the Y 2  measurement as an X 2  measurement. After subtracting the true acceleration, the errors of these estimates are plotted as points  105 , where point  105 A is the first drop, point  105 B is the second drop,  105 C is the third drop, and so on. It can be seen that the error in the individual measurements ranges up to 200 milligal or more in both the positive and negative directions. Table 1 shows that the standard error of the uncompensated results of this simulation were 89899 microgal (˜90 milligal) and the mean error of approximately 5000 simulated drops is 503 microgal. This is almost three orders of magnitude larger than the microgal-class performance required for geodesy applications. 
     The simulation also implemented the synthetic vibration isolation filtering disclosed herein and estimated the error in each of the measurements by processing the Y 1  signal through the filter G(s). This is converted to an estimate of the acceleration of the reference device  42  and plotted as points  110 , where again  110 A is the first drop and so on. The residual error, e.g. the difference between each pair of data points  105  and  110 , is plotted as data points  115 , where again  115 A is the difference between  105 A and  110 A, and so on. It can be seen that the estimates  110  of the acceleration of the reference device  42  are very close to the errors in the measurements  105  and that the residual errors  115  are much smaller than the raw errors  105 . Table 1 shows that the standard error is reduced by a factor of approximately 40× while the mean error is reduced by over 100×, demonstrating the validity of the disclosed method. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 without 
                 with 
               
               
                   
                 compensation 
                 compensation 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 standard error (microgal) 
                 89899 
                 2137 
               
               
                   
                 mean (microgal) 
                 503 
                 3 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 11  is a flowchart illustrating the method of measuring the acceleration due to gravity using synthetic vibration isolation according to certain aspects of the present disclosure. The process of measuring the acceleration due to gravity will be discussed with reference to the gravimeter of  FIG. 5 . Most experimental measurement of the acceleration due to gravity performs hundreds of repetitions of the measurement process followed by combining the results of the entire set of measurements to reduce the error in the final value. The process starts with the repeated and uninterrupted measurement of the displacement of the reference device  42  with respect to the base  18  over regular increments of time in step  210 . These measurements are filtered using a process such as depicted in  FIG. 6 . After sufficient time is allowed for the filter to settle, repeated drops of the freefalling mass may commence. For each drop, the displacement of the falling retroreflector  12  with respect to reference device  42  is measured in step  215  at regular increments of time coincident with the measurements of the displacement of the reference device  42  with respect to the base  18  in step  210 . The measurements of the displacement of the reference device  42  with respect to the base  18  are filtered in step  220 . The outputs of filter  60  which are coincident in time with the measurements of the displacement of the falling retroreflector  12  are subtracted from the measurements of the displacement of the falling retroreflector  12  in step  225  and the relative displacement of the free-falling mass in inertial space is computed and stored along with the time increment in step  230 . If the drop is not finished, the processes branches at step  235  along the NO path to step  210 . If the drop is complete, the process follows the YES path from step  235  to step  240 . The resulting time series of corrected displacements and time increments are processed to provide a single estimate of acceleration due to gravity during the drop which is stored along with the estimates from previous drops in step  240 . If there are more drops to be performed in this series of measurements, as most experiments include multiple drops of the free-falling mass up to hundreds of drops, the process branches at decision point  245  along the “YES” path to step  210 . After the series of drops is complete, the process branches at decision point  245  on the “NO” path to step  250  wherein a best estimate of the acceleration of the free-falling mass is calculated from the respective estimates of all the drops. In this example, step  250  averages the estimates from each drop. In certain other embodiments, other statistical methods known to those of ordinary skill in the art may be employed to combine the estimates of acceleration due to gravity of one or more of the set of drops. This completes the example method. 
     It can be seen that the disclosed embodiments of a gravimeter comprising a synthetic vibration isolation provide a greatly improved accuracy in the measurement of the acceleration of a falling object under the influence of gravity compared to a similar system that uses only physical vibration isolation. The use of the synthetic vibration isolation system disclosed herein in place of the physical vibration isolation system of the current gravimeters enables the size and weight of the gravimeter to be reduced by orders of magnitude. Small gravimeters of this accuracy may be used in a variety of application where the existing instruments cannot be used, such as down-hole drilling applications. 
     The previous description is provided to enable any person skilled in the art to practice the various aspects described herein. While the foregoing has described what are considered to be the best mode and/or other examples, it is understood that various modifications to these aspects will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein, but is to be accorded the full scope consistent with the language claims, wherein reference to an element in the singular is not intended to mean “one and only one” unless specifically so stated, but rather “one or more.” Unless specifically stated otherwise, the terms “a set” and “some” refer to one or more. Pronouns in the masculine (e.g., his) include the feminine and neuter gender (e.g., her and its) and vice versa. Headings and subheadings, if any, are used for convenience only and do not limit the invention. 
     It is understood that the specific order or hierarchy of steps in the processes disclosed is an illustration of exemplary approaches. Based upon design preferences, it is understood that the specific order or hierarchy of steps in the processes may be rearranged. Some of the steps may be performed simultaneously. The accompanying method claims present elements of the various steps in a sample order, and are not meant to be limited to the specific order or hierarchy presented. 
     Terms such as “top,” “bottom,” “front,” “rear” and the like as used in this disclosure should be understood as referring to an arbitrary frame of reference, rather than to the ordinary gravitational frame of reference. Thus, a top surface, a bottom surface, a front surface, and a rear surface may extend upwardly, downwardly, diagonally, or horizontally in a gravitational frame of reference. 
     “Inertial space” is defined as the ideal frame of reference that is non-accelerating. In particular, an inertial reference does not have vibratory or rotational motion. 
     A phrase such as an “aspect” does not imply that such aspect is essential to the subject technology or that such aspect applies to all configurations of the subject technology. A disclosure relating to an aspect may apply to all configurations, or one or more configurations. A phrase such as an aspect may refer to one or more aspects and vice versa. A phrase such as an “embodiment” does not imply that such embodiment is essential to the subject technology or that such embodiment applies to all configurations of the subject technology. A disclosure relating to an embodiment may apply to all embodiments, or one or more embodiments. A phrase such an embodiment may refer to one or more embodiments and vice versa. 
     The word “exemplary” is used herein to mean “serving as an example or illustration.” Any aspect or design described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects or designs. 
     All structural and functional equivalents to the elements of the various aspects described throughout this disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims. Moreover, nothing disclosed herein is intended to be dedicated to the public regardless of whether such disclosure is explicitly recited in the claims. No claim element is to be construed under the provisions of 35 U.S.C. §112, sixth paragraph, unless the element is expressly recited using the phrase “means for” or, in the case of a method claim, the element is recited using the phrase “step for.” Furthermore, to the extent that the term “include,” “have,” or the like is used in the description or the claims, such term is intended to be inclusive in a manner similar to the term “comprise” as “comprise” is interpreted when employed as a transitional word in a claim.