Apparatus and method for multi-sensor estimation of a property of an earth formation

1. An apparatus for estimating a property of an earth formation, the apparatus comprising:

a carrier configured to be deployed according to a selected position relative to the formation, the carrier including at least one sensor array, the at least one sensor array including a plurality of sensors configured to estimate at least one property, each of the plurality of sensors located at a known position relative to one another; and

a processor in operable communication with the plurality of sensors and configured to estimate uncertainties of the location of the plurality of sensors based on a plurality of sequential time-lapse measurements from each of the plurality of sensors, the plurality of sequential time-lapse measurements including a first measurement from at least two of the plurality of sensors at a first time at which the carrier is deployed according to the selected position and a second measurement from the at least two of the plurality of sensors at a later time at which the carrier is deployed according to the same selected position.

2. The apparatus of claim 1 , wherein the processor is configured to apply an inversion algorithm to data generated by the plurality of sensors and construct a model of the formation based on the data.

3. The apparatus of claim 2 , wherein the inversion algorithm includes a model field residual represented by the following equation:

Φ(σ,Δ h )=∥ Aσ+BΔh−G∥ 2 ,

wherein “σ” is a selected property, “Δh” is a vector representing an initial displacement of the apparatus, “A” is a forward problem matrix, “B” is a matrix representing an impact of a displacement on the selected property measurement data and “G” is the vector of a field related to the selected property.

4. The apparatus of claim 3 , wherein the data includes gravity measurement data, the model is a model of formation densities, the selected property is an unknown anomalous density and “G” is a vector of the gravity measurement data.

5. The apparatus of claim 2 , wherein the data is gravity measurement data, and the model is a model of a density distribution of the formation.

6. The apparatus of claim 1 , wherein the at least one sensor array includes:

a first sensor array including a first plurality of sensors configured to estimate at least one property, each of the first plurality of sensors arrayed axially and located at a known position relative to one another;

a second sensor array including a second plurality of sensors configured to estimate the at least one property, each of the second plurality of sensors arrayed axially and located at a known position relative to one another, the second sensor array offset axially from the first sensor array by a fixed distance.

7. The apparatus of claim 6 , wherein the axis is a vertical axis parallel to a direction of gravitational acceleration.

8. The apparatus of claim 1 , wherein the carrier is configured to be conveyed into a borehole, the plurality of sensors being disposed at the carrier.

9. The apparatus of claim 1 , wherein the plurality of sensors are configured to each take the plurality of sequential time-lapse measurements at the selected position, each of the plurality of measurements being temporally separated within a selected time period.

10. A method of estimating a property of an earth formation, the method comprising:

deploying a carrier according to a selected position relative to the formation, the carrier including at least one sensor array including a plurality of sensors, the plurality of sensors located at a known position relative to one another;

estimating at least one property via the plurality of sensors by taking at least two sequential time-lapse measurements from each of the plurality of sensors, the at least two sequential time-lapse measurements including a first measurement from at least two of the plurality of sensors at a first time at which the carrier is deployed according to the selected position and a second measurement from the at least two of the plurality of sensors at a later time at which the carrier is deployed according to the same selected position; and

estimating uncertainties of the location of the plurality of sensors within the selected time period based on the at least two sequential time-lapse measurements from each of the plurality of sensors.

11. The method of claim 10 , further comprising applying an inversion algorithm to data generated by the at least two measurements and constructing a model of the formation based on the data.

12. The method of claim 11 , wherein the inversion algorithm includes a model field residual represented by the following equation:

Φ(σ,Δ h )=∥ Aσ+BΔh−G∥ 2 ,

wherein “σ” is a selected property, “Δh” is a vector representing an initial displacement of the apparatus, “A” is a forward problem matrix, “B” is a matrix representing an impact of a displacement on the selected property measurement data and “G” is the vector of a field related to the selected property.

13. The method of claim 12 , wherein the data includes gravity measurement data, the model is a model of formation densities, the selected property is an unknown anomalous density and “G” is a vector of the gravity measurement data.

14. The method of claim 11 , wherein the data is gravity measurement data, and the model is a model of a density distribution of the formation.

15. The method of claim 10 , wherein disposing includes disposing the carrier at the selected position in a borehole, the plurality of sensors being disposed at the carrier.

16. The method of claim 10 , wherein the at least one sensor array includes:

a first sensor array including a first plurality of sensors configured to estimate at least one property, each of the first plurality of sensors arrayed axially and located at a known position relative to one another;

a second sensor array including a second plurality of sensors configured to estimate the at least one property, each of the second plurality of sensors arrayed axially and located at a known position relative to one another, the second sensor array offset axially from the first sensor array by a fixed distance.

17. The method of claim 10 , wherein the selected position is on a vertical axis parallel to a direction of gravitational acceleration.

18. A non-transitory computer readable medium encoded with a computer program product comprising machine readable instructions for estimating a property of an earth formation by implementing a method comprising:

deploying a carrier according to a selected position relative to the formation, the carrier including at least one sensor array including a plurality of sensors, the plurality of sensors located at a known position relative to one another;

estimating at least one property via the plurality of sensors by taking at least two sequential time-lapse measurements from each of the plurality of sensors, the at least two sequential time-lapse measurements including a first measurement from at least two of the plurality of sensors at a first time at which the carrier is deployed according to the selected position and a second measurement from the at least two of the plurality of sensors at a later time at which the carrier is deployed according to the same selected position; and

estimating uncertainties of the location of the plurality of sensors within the selected time period based on the at least two sequential time-lapse measurements from each of the plurality of sensors.

19. The non-transitory computer readable medium of claim 18 , wherein the method further comprises applying an inversion algorithm to data generated by the at least two measurements and constructing a model of the formation based on the data.

20. The non-transitory computer readable medium of claim 19 , wherein the inversion algorithm includes a model field residual represented by the following equation:

Φ(σ,Δ h )=∥ Aσ+BΔh−G∥ 2 ,

wherein “σ” is a selected property, “Δh” is a vector representing an initial displacement of the apparatus, “A” is a forward problem matrix, “B” is a matrix representing an impact of a displacement on the selected property measurement data and “G” is the vector of a field related to the selected property.

21. The non-transitory computer readable medium of claim 20 , wherein the data includes gravity measurement data, the model is a model of formation densities, the selected property is an unknown anomalous density and “G” is a vector of the gravity measurement data.

22. The non-transitory computer readable medium of claim 19 , wherein the data is gravity measurement data, and the model is a model of a density distribution of the formation.

23. The non-transitory computer readable medium of claim 18 , wherein deploying includes deploying the carrier at the selected position in a borehole, the plurality of sensors being disposed at the carrier.

24. The non-transitory computer readable medium of claim 18 , wherein the at least one sensor array includes:

a first sensor array including a first plurality of sensors configured to estimate at least one property, each of the first plurality of sensors arrayed axially and located at a known position relative to one another;

a second sensor array including a second plurality of sensors configured to estimate the at least one property, each of the second plurality of sensors arrayed axially and located at a known position relative to one another, the second sensor array offset axially from the first sensor array by a fixed distance.

25. The non-transitory computer readable medium of claim 18 , wherein the selected position is on a vertical axis parallel to a direction of gravitational acceleration.