Directional resistivity imaging using harmonic representations

1. A method for making a downhole directional resistivity measurement, the method comprising:

(a) rotating a downhole directional resistivity tool in a borehole, the tool including at least one transmitting antenna, at least one axially spaced receiving antenna, and at least one azimuth sensor;

(b) causing the tool to acquire a plurality of data pairs, each data pair comprising a directional resistivity measurement and a corresponding azimuth angle;

(c) processing a summation of the data pairs acquired in (b) to calculate a first order harmonic representation at a plurality of selected azimuth angles.

2. The method of claim 1 , wherein the processing in (c) is performed downhole.

3. The method of claim 1 , wherein (b) comprises causing the tool to acquire at least 100 data pairs.

4. The method of claim 1 , wherein the plurality of selected azimuth angles in (c) comprises from 4 to 64 azimuth angles.

5. The method of claim 1 , wherein each of the data pairs acquired in (b) contributes to each of the first order harmonic representations calculated in (c).

6. The method of claim 1 , wherein the processing in (c) comprises computing a sum of a first order harmonic component of each of the data pairs at each of the selected azimuth angles.

7. The method of claim 1 , wherein the first order harmonic representations are calculated in (c) according to the following mathematical equation:

d

⁡

(

ϕ

k

)

=

∑

i

⁢

d

i

⁢

cos

⁡

(

ϕ

k

-

ϕ

i

)

,

k

=

1

,

2

,

…

⁢

,

m

wherein d(φ k ) represent the first order harmonic representations at the selected azimuth angles φ k , d i represents the directional resistivity measurement in the i-th data pair and φ i represents the corresponding azimuth angle in the i-th data pair such that d i and φ i in combination represent the plurality of data pairs, and m is an integer.

8. The method of claim 1 , further comprising:

(d) storing the first order harmonic representations calculated in (c) in downhole memory.

9. The method of claim 1 , further comprising:

(d) transmitting selected ones of the first order harmonic representations calculated in (c) to the surface.

10. The method of claim 1 , further comprising:

(d) repeating (b) and (c) at a plurality of distinct measured depths in the borehole.

11. The method of claim 1 , wherein the directional resistivity tool acquires the plurality of data pairs in (b) during rotation of the tool in (a).

12. A method for making a downhole directional resistivity measurement, the method comprising:

(a) rotating a downhole directional resistivity tool at a first longitudinal position in a borehole, the tool including at least one transmitting antenna, at least one axially spaced receiving antenna, and at least one azimuth sensor;

(b) causing the tool to acquire a plurality of data pairs, each data pair comprising a directional resistivity measurement and a corresponding azimuth angle;

(c) processing a summation of the data pairs acquired in (b) to calculate a first order harmonic representation at a plurality of selected azimuth angles;

(d) repositioning the downhole tool and rotating it at a second longitudinal position in the borehole and repeating (b) and (c).

13. The method of claim 12 , wherein the processing in (c) is performed downhole.

14. The method of claim 12 , further comprising:

(e) assigning a first measured depth value to the first order harmonic representations calculated in (c) and a second measured depth value to the first order harmonic representations calculated in (d).

15. The method of claim 12 , wherein the processing in (c) and (d) comprises computing a sum of the first order harmonic components of each of the data pairs at each of the selected azimuth angles.

16. The method of claim 12 , wherein the first order harmonic representations are calculated in (c) and (d) according to the following mathematical equation:

d

⁡

(

ϕ

k

)

=

∑

i

⁢

d

i

⁢

cos

⁡

(

ϕ

k

-

ϕ

i

)

,

k

=

1

,

2

,

…

⁢

,

m

wherein d(φ k ) represent the first order harmonic representations at the selected azimuth angles φ k , d i represents the directional resistivity measurement in the i-th data pair and φ i , represents the corresponding azimuth angle in the i-th data pair such that d i and φ k in combination represent the plurality of data pairs, and m is an integer.

17. The method of claim 12 , further comprising:

(e) storing the first order harmonic representations calculated in (c) and (d) in downhole memory.

18. The method of claim 12 , further comprising:

(e) transmitting selected ones of the harmonic representations calculated in (c) and (d) to the surface.

19. A downhole measurement tool comprising:

a tool body having a longitudinal axis;

at least one transmitting antenna deployed about the tool body;

at least one receiving antenna deployed about the tool body, the receiving antenna being axially spaced from the transmitting antenna; and

a controller configured to (i) cause the tool to acquire a plurality of data pairs while rotating in a subterranean borehole, each data pair comprising a directional resistivity measurement and a corresponding azimuth angle; and (ii) process downhole a summation of the data pairs acquired in (i) to calculate a first order harmonic representation at a plurality of selected azimuth angles.

20. The measurement tool of claim 19 , wherein the controller is further configured to (iii) repeat (i) and (ii); and (iv) assign a first measured depth value to the first order harmonic representations calculated in (ii) and a second measured depth value to the first order harmonic representations calculated in (iii).

21. The measurement tool of claim 19 , wherein the controller is further configured to perform at least one of the following steps:

(iii) store the first order harmonic representations calculated in (ii) in downhole memory; and

(iv) transmit selected ones of the first order harmonic representations calculated in (ii) to the surface.

22. The downhole tool of claim 19 , wherein the receiving antenna comprises a tilted antenna or a transverse antenna.

23. The downhole tool of claim 19 , wherein the transmitting antenna comprises an axial antenna.

24. The downhole tool of claim 19 , wherein at least one of the transmitting antenna and the receiving antenna comprises a non-planar antenna.