U.S. Department of the Interior

U.S. Geological Survey

U.S. Geological Survey Bulletin 2190

Joint Inversion of Acoustic and Resistivity Data for the

Estimation of Gas Hydrate Concentration
Joint Inversion of Acoustic and Resistivity Data for the 

Estimation of Gas Hydrate Concentration

By Myung W. Lee

U.S. Geological Survey Bulletin 2190

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

This publication is only available online at:

http://geology.cr.usgs.gov/pub/bulletins/b2190/

Any use of trade, product, or firm names in this publication

is for descriptive purposes only and

does not imply endorsement by the U.S. Government

Version 1.0 2002

Published in the Central Region, Denver, Colorado

Manuscript approved for publication January 9, 2002

Graphics by author and Gayle M. Dumonceaux

Photocomposition by Gayle M. Dumonceaux

Edited by L.M. Carter
III

Contents

Abstract ......................................................................................................................................... 1

Introduction ................................................................................................................................... 1

Acknowledgments........................................................................................................................ 1

Theory ............................................................................................................................................ 1

Inversion Equation ............................................................................................................. 1

Derivatives for the Acoustics........................................................................................... 2

Derivatives for the Resistivity .......................................................................................... 2

Modified Joint Inversion (MJI) ........................................................................................ 3

Inversion Examples ...................................................................................................................... 3

Original Joint Inversion (OJI) ........................................................................................... 3

Modified Joint Inversion Scheme (MJI) ........................................................................ 4

Discussion ..................................................................................................................................... 7

Weighted Error ................................................................................................................... 7

Simultaneous Estimation of c and Rw ............................................................................ 8

Behavior of the Joint Inversion Method ........................................................................ 8

Understanding the Coupling Constant ........................................................................... 8

Comparison of Various Inversion Results...................................................................... 9

Conclusion ..................................................................................................................................... 10

References Cited .......................................................................................................................... 10

Figures

1–7. Graphs showing:

1. Result of original joint inversion of acoustics and resistivity............................... 3

2. Estimated resistivity on connate water using original joint inversion and

observed formation resistivity with respect to depth............................................ 5

3. Result of joint inversion of acoustics and resistivity using a constant

resistivity of connate water Rw=0.4 ohm-meter ..................................................... 6

4. Result of modified joint inversion of acoustics and resistivity with a

coupling constant γ=1 ................................................................................................. 6

5. Estimated resistivity of connate water using modified joint inversion with a

coupling constant γ=1 and observed formation resistivity with

respect to depth........................................................................................................... 7

6. Result of modified joint inversion of acoustics and resistivity with a

coupling constant γ=5 ................................................................................................ 9

7. Result of averaging the individual gas hydrate estimates from P-wave,

S-wave, and resistivity data ...................................................................................... 10

1. Values used in the inversion ................................................................................................. 4

2. Modeled velocities and resistivity from the joint inversion .............................................. 5

3. Results of estimated gas hydrate concentrations using various methods.................... 5

Tables
1

Joint Inversion of Acoustic and Resistivity Data for the 

Estimation of Gas Hydrate Concentration

By Myung W. Lee 

Abstract

Downhole log measurements, such as acoustic or electrical 

resistivity logs, are frequently used to estimate in situ gas 

hydrate concentrations in the pore space of sedimentary rocks. 

Usually the gas hydrate concentration is estimated separately 

based on each log measurement. However, measurements are 

related to each other through the gas hydrate concentration, so 

the gas hydrate concentrations can be estimated by jointly 

inverting available logs. Because the magnitude of slowness of 

acoustic and resistivity values differs by more than an order of 

magnitude, a least-squares method, weighted by the inverse of 

the observed values, is attempted. Estimating the resistivity of 

connate water and gas hydrate concentration simultaneously is 

problematic, because the resistivity of connate water is independent of acoustics. In order to overcome this problem, a coupling 

constant is introduced in the Jacobian matrix. In the use of different logs to estimate gas hydrate concentration, a joint inversion of different measurements is preferred to the averaging of 

each inversion result.

Introduction

Gas hydrate has become an important research topic, 

because of its significance as a potential resource and because it 

is a controlling element in global warming, as well as a factor 

relevant to sea floor stability (Sloan and others, 1999). In this 

context, the estimation of in situ gas hydrate amounts present in 

sediments is important, and downhole well-log measurements 

are frequently used to quantify accumulations.

Gas hydrate in the pore space increases the elastic velocities and electrical resistivities of the host sediments. Therefore, 

downhole acoustic and electrical resistivity logs are most commonly used to identify and quantify natural gas hydrate 

resources.

Collett (1983, 1995) used downhole measurements extensively not only to identify the presence of gas hydrate in sediments and but also to quantify the amount of gas hydrate. 

Mathews (1986) used electrical resistivity logs to estimate gas 

hydrate saturation at the NW-Eileen #2 well, Alaska, and the 

deep sea drilling project Site 570 at Blake Ridge. Recently, 

Collett and Ladd (2000) also used resistivity logs to estimate gas 

hydrate amounts at Blake Ridge for Ocean Drill Program leg 

164, Sites 994, 995, and 997, as did Miyairi and others (1999) at 

the Mallik 2L-38 gas hydrate research well, northeastern 

Canada. Acoustic data have likewise been used to estimate 

hydrate concentrations. (For example, see Collett, 1983; 1995; 

Guerin and others, 1999; Lee and Collett, 1999; Lee, 2000.)

These estimates of gas hydrate were all derived based on 

measurements from only one log type, except that Miyairi and 

others (1999) used a statistical inversion of the logs. Because 

log measurements—P-wave, S-wave, and resistivity—are 

related to each other through the gas hydrate in the pore space, 

gas hydrate concentrations can be estimated by jointly inverting 

acoustic and resistivity logs.

This paper presents a method of estimating gas hydrate 

concentration by jointly inverting acoustic slowness (P-wave 

and S-wave slowness) and resistivity. Theoretical predictions 

of slownesses are computed from a weighted equation (Lee 

and others, 1996), and resistivities of sediments are computed 

from Archie’s equation (1942). This inversion method was 

applied to the log data acquired at the Mallik 2L-38 gas 

hydrate research well, Mackenzie Delta, Canada (Dallimore 

and others, 1999). 

Acknowledgments

Well logs were acquired by Schlumberger Ltd. at the 

Mallik 2L-38 gas hydrate research well, which was drilled to 

investigate gas hydrate in a collaborative research project 

among Japan National Oil Company, Japan Petroleum Exploration Company, Geological Survey of Canada, and U.S. Geological Survey. I thank J.J. Miller and Y.K. Jin for many 

helpful comments.

Theory

Inversion Equation

Gas hydrate concentration can be estimated by a joint 

inversion of the acoustic (P- and S-wave) and resistivity data. 

Theoretical development of geophysical inversion methods has 

been extensive (Aki and Richards, 1980; Lines and Treitel, 

1984; Tarantola, 1987). The derivation of the inversion method 

in this article closely follows Lee (1990). The method can be 

formulated by the Taylor series expansion of acoustic and resistivity equations, assuming that the unknown variables are the 

gas hydrate concentration (c) and the resistivity of the connate 

water (Rw).
2 Joint Inversion of Acoustic and Resistivity Data —Gas Hydrates

Column vectors O and T are defined as follows:

O = column vector of the slownesses of P-wave (Sp), and 

S-wave (Ss), and the resistivity (Rt). 

T = column vector of the computed or theoretical slowness 

of P-wave, slowness of S-wave, and resistivity. 

Acoustics and resistivity can be approximated by the 

following equation:

where m is a parameter vector consisting of c and Rw, G is a 3×2 

Jacobian matrix, 

which is a column vector of ( ). 

The least-squares solution of in Equation (1) for a 

given observation O is attained by minimizing the mean-squared 

error . Because the magnitude of resistivity 

is different from the values of slowness, a weighted least-squares 

method is attempted; the error function is defined as

 

where i = 1, 2, and 3, which are values that represent indices for 

the P-wave slowness, S-wave slowness, and resistivity respectively; Ei is the component of the error function; Ti is the component of the computed value; and Oi

 is the component of the 

observed value. Then the solution can be written as

 

where is the generalized inverse of the 3×2 weighted Jacobian matrix (element of Gwe is the element of G in Equation (1) 

divided by the corresponding observed value), and I3 is a 3×1 

column vector consisting of 1’s. The generalized inverse of 

can be obtained using a singular value decomposition 

(SVD), and the solution can be written as

 

where V is a 2×2 matrix of orthonormal eigenvectors, U is a 3×3 

matrix of orthonormal eigenvectors, is a diagonal matrix of 

two singular values, and the superscript T denotes a matrix 

transpose.

In this analysis, the derivatives for the Jacobian G are 

obtained from a weighted equation for the acoustic data (Lee and 

others, 1996) and the Archie equation (Archie, 1942) for the 

resistivity measurement. 

Λ

Derivatives for the Acoustics

A weighted equation (WE) can be written as follows, using 

slowness of the constituents (based on Lee and others, 1996):

 

where

Sp = slowness by the WE

Sp1 = slowness by the Wood equation

Sp2 = slowness by the time average equation

W = a weighting factor

= porosity

c = gas hydrate concentration.

The slowness by Wood equation (Sp1) and slowness by 

time average equation (Sp2) are given by

 

 

where are densities of gas-hydrate-bearing 

sediment, water, gas hydrate, and modified matrix, respectively; 

Sw, Sh, and Sm are the slowness of water, gas hydrate, and modified matrix, respectively. 

Shear-wave velocity is given by the following formula (Lee 

and others, 1996):

 

where Ss is the slowness of the S-wave, and are the ratios 

of S-wave velocity to P-wave velocity for the matrix of non-gashydrate-bearing sediments and for gas hydrate, respectively.

Using Equations (4), (5) and (6), the derivatives can be 

obtained as

where 

 

 

 

Derivatives for the Resistivity

Water saturation (Cw) from the Archie equation is given by

where a, m, and n are constants, Rt is the resistivity of the 

formation, Rw is the resistivity of the connate water, and c, 

Sp mn + 1 ( )

Ss mn + 1 ( )

Rt mn + 1 ( )

Sp mn ( )

Ss mn ( )

Rt mn ( )

∂Sp mn ( )l∂c

∂Ss mn ( )l∂c

∂Rt mn ( )l∂c

∆c

0

0

∂Rt mn ( )l∂Rw

= + + ∆Rw or,

/

/

/ /

T mn + 1 ( ) T mn ( ) ∂T

∂c

------    

n

∆c ∂T

∂Rw

-----------    

n

∆Rw = ++ + ..., or

Sp mn + 1 ( )

Ss mn + 1 ( )

Rt mn + 1 ( )

Sp mn ( )

Ss mn ( )

Rt mn ( )

∂Sp mn ( )l∂c

∂Ss mn ( )l∂c

∂Rt mn ( )l∂c

0

0

∂Rt mn ( )l∂Rw

= + ∆Rw or,

/

/

/

/

,

,

,

∆c

∆Rw

T mn + 1 ( ) T mn = ( ) + G∆m (1)

∆c c = n ++ 1 – cn ∆Rw = Rw

n+1

–Rw

n ; ; and ∆m m = n+1 mn – ,

∆c, ∆Rw

∆m

E OT ( ) – E = 2 ( ):

E Oi Ti ( ) – i = ( / Oi)

2

(2)

(3)

Gwe

-1

Gwe

-1

∆m Gwe I 3 TiOi = – ( ) – -1 I 3 Ti ( ) – O -1E i) = Gwe /

mn + 1 = mn V + Λ – 1UT I( ) 3 – Ti n+1 n -1U T I 3 Ti ( ) – /Oi)

Sp = Wφ( ) 1 – c Sp1 + [ ] 1 – Wφ( ) 1 – c Sp2 (4)

φ

Sp1 ρφ( ) 1 – c S 2 = w

ρ w ρ h ρ m

ρφcSh

2

ρ( ) 1 – φ Sm

2

Sp1 = + +

Sp2 φ 1 – c (5) = ( )Sw + + φcSh ( ) 1 – φ Sm

ρ m ρ h ρ w ρ, , , and

(6) α( ) 1 – φ + βc Ss = Sp

α β

∂Sp

∂c

--------- – WφSp1 WφSp2 Wφ( ) 1 – c ∂Sp1

∂c ------------ [ ] 1 – Wφ( ) 1 – c ∂Sp2∂c = ++ + ------------ (7)

(8)

, and ∂Sp1

∂c ------------ 0.5 ρφ Sw

2 = [ ] – /ρw ρφ ρh ] Sp1 Sh

2 + / / ∂c

∂Sp2

∂c ------------ = – φ Sw + φ Sh ∂c

( ) ∂Sp ∂c )[ ] α ( ) 1 – φ + cβφ – βφ Sp

[ ] α ( ) 1 – φ + cβφ 2

/

∂c = ∂Ss

Cw (9)

aRw

φmRt     = 1/n

w
3

P-wave

S-wave

4

2

0

100

10

1

1.0

0.8

0.6

0.4

0.2

0

0 2 4

A

B

C

1 10 100

0 0.2 0.4 0.6 0.8 1.0

HYDRATE CONCENTRATION FROM P-WAVE

AND S-WAVE DATA (H2)

OBSERVED RESISTIVITY, IN OHM-METERS

OBSERVED VELOCITY, IN KILOMETERS PER SECOND

MODELED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM RESISTIVITY,

P-WAVE, AND 

S-WAVE DATA (H3)

MODELED VELOCITY,

IN KILOMETERS PER SECOND

Joint inversion

Variable Rw and

no coupling

Joint inversion

Variable Rw and

no coupling

Joint inversion

Variable Rw and

no coupling

H3 = H2

which is given by , is the gas hydrate concentration. Using Equation (9),

 

Modified Joint Inversion (MJI)

The solution in Equations (2) or (3) can be written in the 

following way using the element of Jacobian matrix (Gij).

 

 

The solution based on Equation (11) indicates that the 

resistivity measurement does not contribute directly to the estimation of gas hydrate concentration using the inversion scheme 

shown in Equation (1); there is no E3 component in the estimation of c. The only contribution of resistivity measurement to 

the joint inversion is through Rw , not through c; accordingly, the 

estimated c from the P- and S-wave data is affected indirectly 

from the computed resistivity value, which provides minimum 

total error. In other words, the preceding joint inversion scheme 

computes the gas hydrate concentration using P-wave and Swave data and attempts to fit the observed resistivity as closely 

as possible by changing Rw. 

The reason for decoupling resistivity in Equation (11) for 

the estimation of c comes from the fact that the element of G12

and G22 , which are elements of the Jacobian matrix G shown 

in Equation (1), are zeros. In order to avoid this unwanted 

behavior, the elements of G12 and G22 are modified by adding 

an arbitrary constant (γ). As can be seen from the numerical 

example given later, the magnitude of the constants controls the 

coupling between acoustics and resistivity in the joint inversion. 

The joint inversion with and ,where is a nonzero constant, is called a modified joint inversion (MJI), and is 

called a coupling constant in this report. The inversion scheme 

using the original formulation, shown in Equation (1), is called 

the original joint inversion (OJI) to differentiate from MJI.

Inversion Examples

Original Joint Inversion (OJI)

In an investigation of the performance of the joint inversion using more than one type of measurements, parameters 

shown in table 1 were used. The initial values for c and Rw for 

the inversion were 0 and 0.4 ohm-meters (ohm-m), and both c

and Rw were simultaneously estimated. Figure 1 shows the 

result of the joint inversion using P-wave, S-wave, and resistivity (PSR inversion) from the original inversion scheme shown 

in Equation (1). Figure 1A shows the comparison of the 

observed velocity versus the modeled (computed) velocity. 

Most of the scattering is about a 45° line and the average 

computed P-wave velocity is about 1 percent higher than the 

c = ( ) 1 – Cw

(10)

∂Rt

∂c -------- = nRt

1 – c -----------

Rt

Rw

------- ∂Rt

∂Rw

---------- = w w

G11E1 + G21E2

G11 + G21

c cn = c n = n+1 + 2 2

R w

–G31G11E1 – G31G21E2 G11 G21 + ( ) + E3 +

G32( ) G11 + G21

n+1= Rw

n 2 2

2 2 (11)

G12=γ G22=γ γ

γ

Figure 1. Graph showing the result of the original joint inversion 

of acoustics and resistivity. Both gas hydrate concentration (c) 

and the resistivity of connate water (Rw) were simultaneously 

estimated. A, Cross plot of modeled velocities versus observed 

velocities. B, Cross plot of modeled resistivity versus observed 

resistivity. C, Cross plot of gas hydrate concentration estimated 

from acoustics versus that estimated from the joint inversion of 

acoustics and resistivity. 

Inversion Examples
4 Joint Inversion of Acoustic and Resistivity Data —Gas Hydrates

observed velocity; the average computed S-wave velocity is 3 

percent higher than the observed velocity (table 1). When only 

P-wave and S-wave velocities are used in the joint inversion (PS 

inversion), the modeled velocities are 2.72 ±0.40 kilometers per 

second (km/s) and 1.21±0.27 km/s for the P-wave and S-wave 

velocities, respectively. The modeled P-wave velocity from PS 

inversion more closely matches the observed velocity, but there 

is little difference in S-wave velocities. 

Figure 1B shows the modeled and observed resistivity 

(from PSR inversion). Unlike the acoustic case, the computed 

resistivities are almost identical to those observed when Rt is 

greater than about 9 ohm-m. Modeled resistivities are higher 

than the observed resistivities when Rt is less than 9 ohm-m. 

The comparison between estimated gas hydrate concentrations using only acoustic data and estimates based on combined 

acoustic and resistivity data is shown in figure 1C. As the theory 

predicts, the estimates are almost identical. As indicated in 

Equation (11), the gas hydrate concentration is estimated mostly 

from P- and S-wave data, and Rw is estimated by matching the 

observed Rt with the theoretical prediction. 

Figure 2, showing the estimated Rw for the preceding 

example and the observed resistivity (Rt ) with respect to depth, 

indicates that the majority of estimated Rw values falls between 

0.2 ohm-m and 1 ohm-m. A cross plot between the Rw and the 

observed resistivity (Rt) indicates that estimated Rw is roughly 

proportional to the observed resistivity. Figure 2 also indicates 

that the estimated Rw appears to decrease with respect to the 

increasing depth. A least-squares fitting curve for the estimated 

Rw is given by Rw = 2.72-0.00214D, where D is depth in meters 

(m). At D = 1,000 m, the estimated Rw is 0.58 ohm-m. The 

accuracy of the estimated Rw is discussed later.

Figure 3 shows the PSR inversion result when Rw is 

assumed to be a constant of 0.4 ohm-m. When Rw is a constant, 

Equation (11) cannot be used for the inversion solution, and 

only the gas hydrate concentration, c, is estimated. Therefore, 

the Jacobian matrix shown in Equation (1) is a 3×1 column 

matrix. In this case, the solution is given by:

 

 

If the original solution form is used, the results of the PSR 

inversion using a constant Rw approach the results of the resistivity inversion (R inversion) as shown later. 

The modeled resistivity is similar to the observed 

resistivity when resistivity is greater than 8 ohm-m (fig. 3B), and 

modeled acoustics show a large scattering around a 45° line (fig. 

3A). Table 2 indicates that the average of the modeled P-wave 

velocity is 2.84 ±0.46 km/s, which is about 4 percent higher 

compared to the observed value, and 1.35 ±0.33 km/s for the Swave velocity, which is about 15 percent higher. The estimated 

gas hydrate concentrations using acoustic and resistivity data 

are much higher than those based on acoustic data only (PS 

inversion), as shown in figure 3C and table 3. 

Modified Joint Inversion Scheme (MJI)

Result of the MJI, estimating c and Rw simultaneously 

with , is shown in figure 4. Figure 4A shows the comparison 

of the observed velocity versus the modeled (computed) velocity. Like results shown in figure 1A, the scattering between the 

modeled and observed velocities is about the 45° line, and the 

average computed P-wave velocity is about 2 percent higher 

than the observed velocity; the average computed S-wave velocity is about 5 percent higher than the observed velocity (table 2). 

Table 1. Values used in the inversion. 

Meaning Symbol Value Remarks

Slowness of hydrate (s/m) Sh 0.303 Type 1 gas hydrate

Slowness of water (s/m) Sw 0.667

Slowness of modified matrix (s/m) Sm 0.203 30% volume clay content

Ratio of Vs/Vp 0.56 Non-hydrate-bearing sediment

Ratio of Vs/Vp 0.51 Type 1 gas hydrate

Density of gas hydrate (g/cm3) 0.91 Type 1 gas hydrate

Density of water (g/cm3) 1.0

Density of modified matrix (g/cm3) 2.65

Weighting factor W 1.44

Cementation factor m 1.95 Archie’s Equation

Archie’s parameter a 1.05 Archie’s Equation

Exponent n 1.9386 Archie’s Equation 

Resistivity of water (ohm-m) Rw 0.4

α

β

ρh

ρw

ρm

G11E1 + G21E2 G31

G11 + + G21 G31

c cn = c n = n+1 + 2 2

+ E3

2 (12)

γ=1
5

1

0.1

0.01

900 950 1000 1050 1100 1150

DEPTH, IN METERS

ESTIMATED RESISTIVITY OF CONNATE WATER,

IN OHM-METERS

Least-squares fitting curve

Rw=2.72-0.00214D

Coupling constant = 0

Formation resistivity

Rw

1

10

100

OBSERVED FORMATION RESISTIVITY,

IN OHM-METERS

Figure 2. Graph showing estimated 

resistivity of connate water (Rw ) using 

original joint inversion and observed formation resistivity with respect to depth. D, 

depth in meters.

Table 3. Results of estimated gas hydrate concentrations using various methods.

Measurement Estimation1 Estimation2 Remarks

P-wave 0.29±0.25 0.39±0.23

S-wave 0.28±0.24 0.36±0.21

Resistivity 0.33±0.30 0.49±0.24 Constant Rw= 0.4 ohm-m

P- and S-wave 0.28±0.24 0.37±0.22 Joint inversion

P-, S-wave, and resistivity 0.28±0.24 0.37±0.22 OJI with variable Rw 

P-, S-wave, and resistivity 0.30±0.26 0.39±0.23 MJI with γ=1 and variable Rw 

P-, S-wave, and resistivity 0.32±0.30 0.47±0.24 MJI with γ=5 and Rw = 0.4 ohm-m 

P-, S-wave, and resistivity 0.32±0.30 0.47±0.24 OJI with Rw = 0.4 ohm-m

P- and S- wave 0.28±0.25 0.36±0.22 Average 

P-, S-wave and resistivity 0.30±0.26 0.36±0.24 Average

 1 Estimate of gas hydrate concentration averaged from depth range 897–1,109 meters including zero gas hydrate concentration values.

 2 Estimate of gas hydrate concentration averaged from depth range 897–1,109 meters excluding zero gas hydrate concentration values. 

[Percentage (%) in parentheses is the fractional error between observed and modeled values. Model 1, result from 

the joint inversion estimating c and Rw simultaneously using the original equation; Model 2, result from joint inversion estimating c with a constant Rw = 0.4 ohm-m; Model 3, result from the modified joint inversion with γ = 1 and 

estimating c and Rw simultaneously] 

Observed Model 1

(OJI)

Model 2

(Constant Rw)

Model 3

(MJI)

Vp (km/s) 2.72±0.39 2.75±0.38 (1%) 2.84±0.46(4%) 2.78±0.40(2%)

Vs (km/s) 1.17±0.26 1.21±0.27(3%) 1.35±0.33(15%) 1.23±0.28 (5%)

Rt(ohm-m) 17.97±24.99 18.44±23.93 (3%) 17.99±23.32 (0%) 18.26±23.89 (2%)

Table 2. Modeled velocities and resistivity from the joint inversion.

Inversion Examples
6 Joint Inversion of Acoustic and Resistivity Data —Gas Hydrates

4

2

0 

0 2 4

100

10

1

1 10 100

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

C

B

A

OBSERVED VELOCITY, IN KILOMETERS PER SECOND

OBSERVED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM P-WAVE

AND S-WAVE DATA (H2)

HYDRATE CONCENTRATION FROM RESISTIVITY,

P-WAVE, AND 

S-WAVE DATA (H3) MODELED RESISTIVITY, IN OHM-METERS

MODELED VELOCITY,

IN KILOMETERS PER SECOND

S-wave 

P-wave 

Joint inversion

Constant Rw and

no coupling

Joint inversion

Constant Rw and

no coupling

Joint inversion

Constant Rw and

no coupling

H3=0.99H2+0.04

4

2

0

0 2 4

1 10 100

0 0.2 0.4 0.6 0.8 1.0

100

10

1

1.0

0.8

0.6

0.4

0.2

0

OBSERVED VELOCITY, IN KILOMETERS PER SECOND

OBSERVED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM P-WAVE

AND S-WAVE DATA (H2)

C

B

A

MODELED VELOCITY,

IN KILOMETERS PER SECOND MODELED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM RESISTIVITY,

P-WAVE, AND 

S-WAVE DATA (H3)

S-wave

P-wave

Joint inversion

Variable Rw and

coupling constant = 1

Joint inversion

Variable Rw and

coupling constant = 1

Joint inversion

Variable Rw and

coupling constant = 1

H3=1.044H2+0.003

Figure 3. Graph showing the result of joint inversion of acoustics 

and resistivity. Only gas hydrate concentration (c) with a constant 

resistivity of connate water Rw =0.4 ohm-meter (ohm-m) was 

estimated. A, Cross plot of modeled velocities versus observed 

velocities. B, Cross plot of modeled resistivity versus observed 

resistivity. C, Cross plot of gas hydrate concentration estimated 

from acoustics versus that estimated from the joint inversion of 

acoustics and resistivity.

Figure 4. Graph showing the result of the modified joint inversion 

of acoustics and resistivity with a coupling constant γ=1. Both gas 

hydrate concentration (c) and the resistivity of connate water ( Rw ) 

were simultaneously estimated. A, Cross plot of modeled velocities 

versus observed velocities. B, Cross plot of modeled resistivity 

versus observed resistivity. C, Cross plot of gas hydrate concentration estimated from acoustics versus that estimated from the joint 

inversion of acoustics and resistivity.
7

100

10

1

OBSERVED FORMATION RESISTIVITY,

IN OHM-METERS

DEPTH, IN METERS

900 950 1000 1050 1100 1150

1

0.1

0.01

ESTIMATED RESISTIVITY OF CONNATE WATER,

IN OHM-METERS

Rw

Least-squares fitting curve

Rw=1.2 -7.1x10-4D

Formation resistivity Coupling constant = 1

The accuracy of the predicted velocity from MJI is between 

those of OJI with variable Rw and OJI with a constant Rw .

Figure 4B shows the modeled and observed resistivity 

(from PSR inversion). Like the results for the velocities, the 

accuracy of predicted resistivity is between that from Model 1 

(OJI with a variable Rw ) and that from Model 2 (OJI with a 

constant Rw ). 

A comparison of estimated gas hydrate concentration using 

only acoustic data (PS inversion) and acoustic and resistivity 

data (PSR inversion) is shown in figure 4C. A least-squares fitting curve indicates that the gas hydrate concentration from PSR 

inversion is about 4 percent higher than that from the PS inversion.

Figure 5 shows the estimated Rw for figure 4’s example, 

and the observed resistivity (Rt ) with respect to depth. The general trend of the estimated Rw with respect to the depth is similar 

to the result of OJI, but the estimated Rw is somewhat less than 

that from OJI. A least-squares fitting curve for the estimated Rw

is given by Rw =1.2–0.00071D, where D is depth in meters. At 

D = 1,000 m, the estimated Rw is 0.49 ohm-m. 

Table 2 and figure 4 show that the result of MJI is between 

the results from Model 1 and Model 2, which indicates that the 

resistivity measurement contributed to the estimation of gas 

hydrate concentration using MJI with γ=1 when c and Rw are 

simultaneously estimated. The degree of contribution of the 

resistivity to the inversion result is somewhat controlled by the 

magnitude of the coupling constant (γ), which will be addressed 

later.

In summary, the preceding results imply that the joint 

inversion estimating c and Rw simultaneously is feasible. The 

result of OJI estimating c and Rw simultaneously from acoustic 

and resistivity data demonstrates that OJI is closely similar to a 

sequential estimation of gas hydrate concentration based on using 

acoustic and Rw from resistivity data using the estimated gas 

hydrate concentration as input. However, the results of MJI indicate the possibility of a way to control the contribution of acoustic 

and resistivity measurement to the inversion, even though the 

coupling between the acoustic and resistivity is arbitrary.

Discussion

Weighted Error

The solution of the modified joint inversion using arbitrary 

non-zero constants for the G12 and G22 can be written as

 

 

where Qij represents the element of the generalized inverse of 

Gw. Based on the data set from the Mallik 2L-38 gas hydrate 

research well, the average slowness of the observed P-wave is 

0.375±0.052 seconds per kilometer (s/km), the average slowness 

of S-wave is 0.894±0.184 s/km, and the average resistivity is 

18.60±26.43 ohm-m. Let be 

fractional errors in P-wave slowness, S-wave slowness, and 

resistivity respectively. If the fractional errors are of the same 

magnitude or similar, then , because the magnitude of the resistivity is much higher than the magnitude of the 

slowness and S-wave slowness is higher than the P-wave 

Figure 5. Graph showing estimated resistivity of connate water (Rw) using modified joint inversion 

with a coupling constant γ=1 and observed formation resistivity with respect to depth. D, depth in 

meters. 

(13)

c c n = n+1 + Q11E1+ + Q12E2 Q13E3

R R n = n+1

w w +Q21E1+ + Q22E2 Q23 E3

∆Sp /Sp , , ∆Ss /Ss and ∆Rt /Rt

∆Rt >>∆Ss >∆Sp

Discussion
8 Joint Inversion of Acoustic and Resistivity Data —Gas Hydrates

–

slowness. If a least-squares error is not weighted, the solutions 

shown in Equation (13) are affected mostly by E3, which is . 

So the solution can be approximated by

 

 

Therefore, when an unweighted error is used for the joint inversion, Equation (14) implies that P-wave and S-wave do not contribute much to the joint inversion, and the estimation of hydrate 

concentration is dominated by the resistivity measurement. At 

the same time, Equation (14) indicates that Rw and c are not well 

constrained, because the error in Rt controls both the c and Rw

simultaneously. 

When the weighted error is used, the magnitudes of Ei’s are 

on the same order of magnitude. Therefore, each measurement 

contributes to the estimation of gas hydrate concentration and 

resistivity of connate water.

Simultaneous Estimation of c and Rw

Estimating c and Rw by simultaneously using the joint 

inversion method is problematic as mentioned previously, 

mainly because only the resistivity, and not the acoustics, 

depends on both c and Rw. In consequence, some of the elements of the Jacobian matrix are zero. This leads to the decoupling of the resistivity from the estimation of gas hydrate 

concentration as shown in Equation (11). In order to overcome 

this problem, a modified joint inversion formula was introduced. 

However, because the coupling constant (γ) is not based on a 

physical principle and c depends on Rw, the result of Rw estimation should be checked against other estimates. 

Miyairi and others (1999) estimated the Rwa, which is 

defined by from the resistivity logs. The Rwa

should be equal to the Rw in clean water-bearing formations. 

According to Miyairi and others (1999), the resistivity of connate water for the non-gas-hydrate-bearing sediment at the 

Mallik 2L-38 varies between 0.2 and 0.5 ohm-m and decreases 

as depth increases. As shown in figure 2, the majority of estimated Rw from OJI varies between 0.2 and 1.0 ohm-m and 

decreases with depth. Miyairi and others (1999) used Rw=0.48, 

0.38, and 0.29 ohm-m for the depth intervals of 814–933 m, 

933–1,076 m, and 1,076–1,150 m, respectively. The estimated 

Rw from OJI is 0.85, 0.57, and 0.34 ohm-m for the average depth 

interval of 873 m, 1,004 m, and 1,113 m respectively. Comparing the result of OJI inversion and Miyairi and others (1999), the 

rate of resistivity decrease with depth is similar, but Rw values 

differ by 0.05–0.4 ohm-m. The result from MJI indicates that 

the estimated Rw values vary slightly from the OJI values, being 

0.58, 0.49, and 0.41 ohm-m for the depths of 873 m, 1,004 m, 

and 1,113 m respectively, which are about 30 percent higher 

than those used by Miyairi and others (1999). 

Behavior of the Joint Inversion Method

Inversion results indicate that the coupling constant (γ) in 

the Jacobian matrix controls the amount of coupling between the 

acoustics and resistivity in the estimation of gas hydrate concentration. When γ=0, the modified joint inversion reverts to the 

original joint inversion method, and the gas hydrate estimation 

is controlled mostly by the acoustical properties, receiving 

minimal contributions from the resistivity measurements, as 

indicated in figure 1.

The result of modified joint inversion using γ=5, shown in 

figure 6, is almost identical to that from the R inversion (table 

3). Numerical tests indicate that as γ increases, the estimated 

gas hydrate concentration from a joint inversion of acoustic and 

resistivity approaches the result of inversion using only resistivity, and the estimated Rw approaches a constant value, which is 

the initial value. 

Conversely, as γ decreases, the result of MJI approaches 

that of OJI. These observations indicate that the coupling constant controls the degree of resistivity contribution to the estimation of gas hydrate concentration using the joint inversion. 

However, it is difficult to predict the general behavior of inversion results with respect to the coupling constant, except for low 

and high constant values.

Understanding the Coupling Constant

The modified joint inversion formula can be better understood if we assume that erroneous P-wave and S-wave values 

are used in the following way when computing the Jacobian 

matrix:

 

where the superscript t indicates the true computed values from 

the theory and the superscript e indicates the altered values used 

in the inversion. Then, G12 = γ and G22 = γ. Note that only 

elements of the Jacobian matrix, not the theoretical values, are 

altered.

If γ becomes small, the erroneous theoretical values used in 

the modified joint inversion approach the true values. In this 

case, the solution of the modified inversion approaches the solution of the original inversion shown in Equation (11). 

If γ is large, the acoustic model fails to fit the observation 

accurately and errors become large. When the error of computed slowness is large, Equation (1) indicates that the relative 

contribution of G11 and G21 compared to G12 and G22 to the 

overall inversion is insignificant. Therefore, G11≈0 and G21 

≈0 as γ becomes large. In this case, the solution can be 

written as

 

 

 

 

Equation (16) indicates that as the coupling constant 

becomes larger, E3 or the resistivity measurement dominates the 

estimation of c, and the change of Rw is insignificant or Rw

approaches the initial value.

∆Rt

(14)

Q13∆Rt cn+1~~ cn+

R n+1 n +Q23∆Rt ~~ w Rw

Rwa = Rtφ m/a,

(15)

Se

p = S t

p + γ Rw

Se

s = S t

s + γ Rw

(16)

c cn = c n =

+ E3 c n =

n+1~~ + 2γ 2 G31E3 γ G31G32( ) E1 + E2

2γ 2G2

31

~~ G31

as γ→∞ and

R n+1 n ~~ w Rw + E1 + E2

2γ ~~ n Rw as γ→∞
9

P-wave

S-wave

4

2

0

100

10

1

1.0

0.8

0.6

0.4

0.2

0

0 2 4

A

B

C

1 10 100

0 0.2 0.4 0.6 0.8 1.0

HYDRATE CONCENTRATION FROM P-WAVE

AND S-WAVE DATA (H2)

OBSERVED RESISTIVITY, IN OHM-METERS

OBSERVED VELOCITY, IN KILOMETERS PER SECOND

MODELED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM RESISTIVITY,

P-WAVE, AND 

S-WAVE DATA (H3)

MODELED VELOCITY,

IN KILOMETERS PER SECOND

Joint inversion

Variable Rw and

coupling constant = 5

Joint inversion

Variable Rw and

coupling constant = 5

Joint inversion

Variable Rw and

coupling constant = 5

H3=1.138H2-0.0009

Note that the solution of MJI with a large γ approaches the 

result of OJI with a constant Rw. Equation (11) indicates that 

when there is no update of Rw in the inversion, the following 

relation should be valid:

 

Substituting Equation (17) into Equation (11) for c, the result is

 

 

This is identical to the inversion solution of the resistivity and 

identical to Equation (16).

Comparison of Various Inversion Results

Each measurement—P-wave, S-wave, and resistivity—can 

be used separately in the inversion to estimate gas hydrate concentrations. When only Rt is used in the inversion, Rw should 

be an input, because of the presence of two unknowns with one 

observation. Table 3 shows results of the estimation of gas 

hydrate at the Mallik 2L-38 well from various combinations of 

measurements. Among all the estimates, those using only 

resistivity measurements yield the highest average hydrate 

concentrations. 

The average hydrate concentrations estimated from P-wave 

and S-wave slownesses are similar, but the estimated amount is 

less than that from resistivity by about 5 percent when all zero 

concentration values are included and by about 12 percent when 

zero concentration values are excluded. The average gas 

hydrate concentration estimated from the PS inversion is smaller 

than that from the PSR inversion with a constant Rw , or a variable Rw using MJI with γ=1, but the estimates are similar to 

those of PSR inversion with a variable Rw using OJI. 

The results of the simultaneous estimations of c and Rw

from MJI with γ=1 indicate that the estimates of gas hydrate 

concentration are slightly higher, on the order of 2 percent, than 

that from estimates based on acoustics. Estimated gas hydrate 

concentrations indicate that MJI with γ=1 still favors acoustics, 

but the contribution of resistivity to the estimates is apparent, 

and the amount may be controlled by the value of γ. However, it 

is difficult to predict the behavior of the inversion result with 

respect to the coupling constant.

Gas hydrate concentrations by averaging each individual 

estimation are also shown in table 3. The gas hydrate concentrations estimated from the average of P- and S- wave estimates are 

similar to that of the PS inversion. Estimates based on the PSR 

inversion using MJI with γ=1 are similar to the averages of three 

individual estimates. But the cross plots for the modeled and 

observed quantities are substantially different. Figure 7 shows 

the result of averaging gas hydrate concentrations from separate 

estimations from P-wave, S-wave, and resistivity data. The modeled velocities are similar to those from MJI with γ=1; in fact, 

average velocities of modeled P- and S-waves are 2.78±0.38 km/

s and 1.23± 0.26 km/s, respectively. The only difference in the 

velocity estimates is the amount of scattering of the modeled 

velocities. However, the modeled resistivity differs markedly 

from the observed value (fig. 7B), being about 50 percent lower.

Figure 6. Graph showing the result of the modified joint inversion 

of acoustics and resistivity with a coupling constant γ=5. Both gas 

hydrate concentration (c) and the resistivity of connate water (Rw)

were simultaneously estimated. A, Cross plot of modeled velocities 

versus observed velocities. B, Cross plot of modeled resistivity versus observed resistivity. C, Cross plot of gas hydrate concentration 

estimated from acoustics versus that estimated from the joint inversion of acoustics and resistivity.

(17)

(18)

G31 G11E1 ( + G21E2 ) G21 = (G11

2 + 2 ) E3

c cn = c n = n+1 + E3

G31

Discussion
10 Joint Inversion of Acoustic and Resistivity Data —Gas Hydrates

P-wave

S-wave

4

2

0

100

10

1

1.0

0.8

0.6

0.4

0.2

0

0 2 4

A

B

C

1 10 100

0 0.2 0.4 0.6 0.8 1.0

HYDRATE CONCENTRATION FROM P-WAVE

AND S-WAVE DATA (H2)

OBSERVED RESISTIVITY, IN OHM-METERS

OBSERVED VELOCITY, IN KILOMETERS PER SECOND

MODELED RESISTIVITY, IN OHM-METERS

HYDRATE CONCENTRATION FROM RESISTIVITY,

P-WAVE, AND 

S-WAVE DATA (H3)

MODELED VELOCITY,

IN KILOMETERS PER SECOND

Average

Average

Average

H3=1.05H2-0.0007

Various inversion methods provide different estimations of 

gas hydrate concentrations. Judging which estimate is the most 

accurate one is difficult, because gas hydrate concentrations in 

sediments at the Mallik 2L-38 well were not accurately measured. The difference of gas hydrate estimation between resistivity and acoustics may be real, because the lateral resolution 

(or depth of investigation) of each logging tool is different, or 

the difference may be an artifact owing to the inaccuracy of 

acoustic models based on the weighted equation or erroneous 

parameters of the Archie’s constants. 

Conclusion

The following conclusions can be drawn from a joint inversion of acoustics and resistivity measurements for gas-hydratebearing sediments.

1. Simultaneously estimating the gas hydrate concentration and the resistivity of connate water (Rw) is problematic, 

because the original inversion equation for the estimation of gas 

hydrate concentration uses only acoustics decoupled from the 

resistivity measurement. 

2. A modified joint inversion method, implemented by 

inserting a coupling constant where the element of Jacobian 

matrix is zero, partly compensates the adverse effect of decoupling. However, because the coupling constant is not based on 

the physical principles, predicting the effect of the coupling 

constant is difficult except at the large values.

3. The contribution of resistivity to gas hydrate estimates 

can be controlled in part by the magnitude of the coupling constant. As the value of the coupling constant (γ) increases, the 

modified joint inversion favors the resistivity measurement; 

when γ decreases, MJI favors acoustics. 

4. Even though Rw is constrained by P- and S-wave data 

through c, the estimated Rw from the modified joint inversion 

should be checked against Rw values estimated by other 

methods, because Rw is not constrained physically by acoustics.

5. When a constant Rw is used in the gas hydrate estimation, the joint inversion method has no clear advantage over the 

individual inversion. However, when c and Rw are estimated 

simultaneously, the modified joint inversion method may be 

preferred. 

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11

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