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ORIGINAL RESEARCH

Sensitivity of the Dual-Probe Heat-Pulse Method to Spatial Variations in Heat Capacity and Water Content

^a CSIRO Land and Water, 120 Meiers Rd., Indooroopilly, QLD 4068, and Dep. of Environmental Engineering, Griffith Univ., Nathan, QLD 4111, Australia
^b Dep. of Agronomy, Kansas State Univ., Manhattan, KS 66506. Contribution no. 07-97-J from the Kansas Agric. Exp. Stn., Manhattan
^* Corresponding author (gjk{at}ksu.edu).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Received 21 November 2006.

	ABSTRACT

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
Dual-probe heat-pulse (DPHP) sensors are useful for measuring soil heat capacity (C) and water content (), yet little is known about their effective measurement volume. We have adapted previous work on well testing to investigate the spatial sensitivity of the DPHP method for measuring both C and . Spatial sensitivity functions were derived by using a perturbation expansion approach in which C and vary with position, but differ only slightly from their uniform values C₀ and ₀, respectively. Because the dimensionless forms of these spatial sensitivity functions are identical, the spatial sensitivity of the DPHP method is the same for both C and measurements. The spatial sensitivity function is not radially symmetric about the heater probe. Instead, the locations of the temperature and heater probes are of equal importance in defining the spatial sensitivity. The spatial sensitivity is greatest in small areas immediately outside of the heater and temperature probes. Far from the sensor, contours of equal spatial sensitivity approach the shapes of a family of ellipses. For a sensor with a probe spacing of 6 mm, the boundary containing 99% of the total spatial sensitivity is closely approximated by an ellipse with area 168 mm² and a major axis 15.6 mm in length. The spatial sensitivity for C is unaffected by the magnitude of the soil thermal properties near the sensor, as long as the thermal properties are uniform. Likewise, for measurements, the spatial sensitivity is independent of the value of near the sensor, as long as is uniform.

Abbreviations: DPHP, dual-probe heat-pulse

	INTRODUCTION

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
The dual-probe heat-pulse (DPHP) method, introduced by Campbell et al. (1991), involves measuring the temperature rise at a fixed distance from a cylindrical heater probe that contains electrical resistance wire. An impulse of heat is released from the probe by passing electrical current through the wire. The change in temperature is then monitored with a thermistor or thermocouple in a second cylindrical probe oriented parallel to the heater probe. Campbell et al. (1991) showed that volumetric heat capacity (C) can be estimated from the maximum temperature rise that occurs in response to heating. Although modifications of their parameter estimation approach (Bristow et al., 1994, 1995; Welch et al., 1996) have made it possible to estimate thermal conductivity and thermal diffusivity, as well as C, with DPHP sensors, in this study we focus on the measurement of C with the original DPHP method of Campbell et al. (1991).

When the DPHP method is used, it yields a single value of C, with the implicit assumption that the measured value is spatially uniform. That is, the method assumes that heat capacity is spatially uniform within the effective measurement volume of the DPHP sensor. It is natural to ask what the size and shape of this effective measurement volume is. It is also natural to ask what averaging process the method uses to give a single estimate of C in the presence of inhomogeneities. One approach that could be used to answer these questions would be to develop a model of heat transfer in the vicinity of a DPHP sensor and allow for a small region of specified size and shape with properties that differ from the rest of the soil. This type of approach has been used to characterize the spatial sensitivity of a variety of geophysical methods (Vela and McKinley, 1970; Butler and Liu, 1991, 1993; Furman et al., 2002). Another approach would be to develop a model of heat transfer in the vicinity of a DPHP sensor that allows for two regions of different properties, separated by a planar boundary. This approach was used by Philip and Kluitenberg (1999) and Kluitenberg and Philip (1999) to obtain initial estimates of the effective measurement volume of a DPHP sensor. A third, more general approach would be to assume a spatial distribution of C that is only slightly different from a uniform distribution corresponding to a known output, and to use a perturbation expansion to find the resultant small change in the measured output. The small change in output is calculated by using spatial convolution integrals of "spatial weighting functions" with the actual variations of the transport parameters (Knight, 1992; Oliver, 1993; Kabala, 2001). The procedure for calculating these weighting functions and, hence, sensitivity coefficients, can be given a rigorous mathematical formulation in terms of Fréchet derivatives, with the weighting functions appearing in the integrals as Fréchet kernels (Parker, 1977, 1994).

Oliver (1990, 1993) used a perturbation technique and Laplace transforms to find solutions for the spatial weighting functions, or Fréchet kernels, for small variations from uniform in storativity and transmissivity for well tests. For cases in which the pumping well and measurement well are co-located and the kernels are radially symmetric, he was able to invert the Laplace transforms and give explicit formulas for the kernels. In the case where the wells are not co-located, he showed that the weighting functions are not radially symmetric, and gave solutions for the Fréchet kernels as expressions involving convolution integrals in the time domain, which must be evaluated numerically. Romeu et al. (1999) used similar techniques and gave a general method to calculate the well-testing spatial weighting function, or Fréchet kernel, for variations in permeability, and gave three examples for three different geometries. Knight and Kluitenberg (2005) showed how to explicitly invert the Laplace transforms of Oliver (1993) in terms of products of Bessel and exponential functions, and derived explicit formulas for the Fréchet kernels for small variations from uniformity in storativity and transmissivity for slug and pumping tests. They used the explicit forms to study the properties of the spatial weighting functions, and showed that, for storativity, most of the measurement sensitivity at any time is contained in an expanding region whose boundary is an ellipse with one focus at each of the pumping and measurement wells.

The heat equation and initial and boundary conditions for measuring soil heat capacity C in the DPHP method are mathematically similar to the Boussinesq equation and initial and boundary conditions for measuring aquifer storativity in a slug test. In this study, we adapted the previous work on well testing to derive the spatial sensitivity of the DPHP method for measuring soil water content to small variations in heat capacity and, therefore, in water content. We used the results of Knight and Kluitenberg (2005) for the special case of the Campbell et al. (1991) method, which uses the maximum temperature rise, in which case the spatial sensitivity does not have singularities around the sensor probes. We determined how the spatial sensitivity to variations in water content is linearly related to the spatial sensitivity for heat capacity, and that the sensitivity to water content does not depend on the actual value of the water content around the probe, as long as the water content is uniform. Inasmuch as the DPHP method uses a heat pulse that is of finite duration rather than instantaneous, we examined the mildly singular behavior of the spatial sensitivity near the probes close to the time of maximum temperature rise.

	The Dual-Probe Heat-Pulse Method

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
The temperature distribution due to an instantaneous release of heat from an infinite line source at (x,y) = (0,0) in a homogeneous medium at zero initial temperature is (Carslaw and Jaeger, 1959, p. 258)

[1]

in which T(r,t) is the temperature (K) at time t (s) and radial distance r = (x² + y²) from the heater (m), q is the heat input per unit length (J m^–1), is the thermal conductivity (W m^–1 K^–1), and is thermal diffusivity (m s^–2). Upon differentiating Eq. [1] with respect to time, and setting the result to zero, we find that Eq. [1] has a maximum at time t = r²/4. Substituting this result into Eq. [1] yields

[2]

where C is the volumetric heat capacity (J m^–3 K^–1) and T_m is the maximum temperature. In the DPHP method of Campbell et al. (1991), Eq. [2] is used to estimate C from measurements of r, q, and T_m. The fact that Eq. [1] satisfies the initial condition T(r,0) = 0 does not result in a loss of generality. The principle of superposition permits application of Eq. [1] to problems with an arbitrary initial uniform temperature T_i. For such problems the solution T(r,t) becomes T(r,t) – T_i, the temperature rise above the initial temperature T_i, and T_m becomes T_m – T_i, the maximum temperature rise above the initial temperature.

Instantaneous heating of the line source cannot be achieved in practice, so a heat pulse of finite duration (typically 8 s) is used. The finite heating duration causes the maximum temperature rise to occur later than t = r²/4, but it has minimal effect on the magnitude of T_m (Bristow et al., 1994). Kluitenberg et al. (1993) showed that the use of Eq. [2] causes <1% error in C estimates in most cases. Although other approaches for estimating C are available (Bristow et al., 1994, 1995; Welch et al., 1996; Bilskie et al., 1998; Mori et al., 2003; Knight and Kluitenberg, 2004), the simplicity of Eq. [2] has made it the method of choice for estimating C.

It is also important to recognize that, in practice, heat is released from a cylindrical heater probe of finite length. At the location of the thermocouple or thermistor in the temperature probe (equidistant from the ends of the probe), the finite dimensions of the heater probe have relatively little effect on the temperature, and the heat source is well approximated by a line source of infinite length (Kluitenberg et al., 1993). It is for this reason that Eq. [1] provides an excellent approximation of the temperature at the location where it is measured in the DPHP method. To perform a complete evaluation of the spatial sensitivity of the DPHP method in three dimensions, it would be necessary to characterize the temperature history at this location, as well as in the volume of soil surrounding both this location and the heater probe. Unfortunately, at positions increasingly farther from the location of the temperature measurement, in any direction, the finite dimensions of the heater probe have an increasingly greater effect on the temperature field, and the use of Eq. [1] becomes questionable. Thus, a complete volumetric characterization of the spatial sensitivity of the DPHP methods would require that we replace Eq. [1] with a corresponding solution that accounts for the finite dimensions of the heater probe. Such a solution is available (Kluitenberg et al., 1995), but its use greatly increases the complexity of the spatial sensitivity analysis. Instead, we restrict our analysis to characterizing the spatial sensitivity of the DPHP method in a plane normal to the line heat source that passes through the location at which temperature is measured in the DPHP method (Fig. 1 ). Within this plane, Eq. [1] offers a reasonable approximation of the temperature history, with the exception of positions near the heater probe and positions far from the heater probe.

View larger version (13K): [in this window] [in a new window]   FIG. 1. Diagram of a dual-probe heat-pulse (DPHP) sensor with heater probe and temperature probe. The plane passing through the heater and temperature probes corresponds to the x–y plane in Fig. 2 and defines the area in which the spatial sensitivity of the DPHP method is characterized. The plane is oriented normal to the probes and passes through the temperature probe as the location where temperature is measured with a thermistor or thermocouple. See Heitman et al. (2003) for a more detailed schematic diagram that includes cutaway views of the probes and sensor body.

View larger version (9K): [in this window] [in a new window]   FIG. 2. Geometrical configuration of the heater probe and temperature probe of a dual-probe heat-pulse sensor. The arbitrary point (x,y) represents the point at which the Fréchet kernel F(x,y,t) is calculated. The heater and temperature probes are separated by distance a, with the heater probe at (x,y) = (–a/2,0) and the temperature probe at (x,y) = (a/2,0). The scalar quantities r1 = [(x + a/2)2 + y2] and r2 = [(x – a/2)2 + y2] give the distance from the point (x,y) to the heater probe and temperature probe, respectively.

	Theory

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

[3]

To examine the effect of heterogeneity in heat capacity, we assume that the heat capacity varies with position, but differs only slightly from its uniform value C₀. Thus, C(x) can be expressed as

[4]

where the constant is small. We also assume that the solution of Eq. [3] can be written as the sum of a zero-order term and a perturbation term of order , so that

[5]

In the context of a DPHP sensor, the form of Eq. [4] and [5] assumes that the line heat source (heater probe) sends out a temperature increase signal that spreads out radially from the line source with decreasing velocity and amplitude. A small proportion of the signal is "scattered" by the small heterogeneities in the heat capacity field, and a new signal spreads out from each scatterer with a nonuniform velocity and decreasing amplitude. At any time t, some of the combined scattered signals from all of the inhomogeneities reach an observation point (temperature probe) as the perturbation term T₁(x,t). The form of Eq. [5] assumes that we can ignore second-order scattering of the previously scattered signal.

The Spatial Sensitivity Problem
In the event that the heat capacity has uniform value C₀, Eq. [3] becomes

[6]

and the definition of the thermal diffusivity becomes = /C₀. Aside from slight differences in notation (C and T used instead of C₀ and T₀, respectively), Eq. [1] is the solution of Eq. [6] for zero initial temperature and instantaneous heat input q from an infinite line source positioned at (x,y) = (0,0). Here we write the solution of Eq. [6] as a function of x and t for the line source at arbitrary position (x'',y'') and obtain

[7]

where r₁ is the radial distance from the line source, defined as

[8]

Like Eq. [1], Eq. [7] can be applied to problems with an arbitrary initial uniform temperature by using the principle of superposition. Examination of Eq. [7] shows that the temperature at a point a distance r₁ from the line source increases to a maximum value at time t = r₁²/4, and then decreases. The maximum value of Eq. [7] at this time is

[9]

We are interested in calculating the sensitivity of the solution of Eq. [3] to small variations in heat capacity C₁, and we have particular interest in the effect of these variations at time t = r₁²/4, when the solution of Eq. [3] exhibits a maximum. In the special case that the heat capacity has uniform value C₀, and changes in heat capacity are spatially uniform, we have C₁ = C₀ and T₁ = T₀, and the effect of the variations C₀ on the temperature T₀ can be obtained from Eq. [7]. Using traditional sensitivity analysis, we write

[10]

where the partial derivative T₀/C₀ is the usual sensitivity coefficient evaluated at (x,t). Evaluating the partial derivative of Eq. [7] gives

[11]

Examination of Eq. [11] shows that the sensitivity coefficient depends on the radial distance from the line source, and varies with time. Upon generalizing this result to consider spatially nonuniform heterogeneities in heat capacity, the sensitivity coefficient becomes a Fréchet derivative.

The Fréchet Derivative
The quantity T₁(x,t) in Eq. [5] is called the Fréchet derivative of T(x,t) with respect to the function C(x), evaluated at C(x) = C₀. The Fréchet derivative T₁(x,t) generalizes the sensitivity coefficient for the dependence of T(x,t) on a parameter C₀ with one degree of freedom, to a spatial weighting function that describes the dependence of T(x,t) on a function C(x) with an infinite number of degrees of freedom. We wish to express the Fréchet derivative as a spatial convolution integral, in terms of the known function C₁(x), as

[12]

where the area of integration is the x–y plane, and the unknown function F(x,t) is called the Fréchet kernel for the volumetric heat capacity. The Fréchet kernel is the spatial weighting function (function of space and time) that determines the weighting given to inhomogeneities in heat capacity.

The connection between the Fréchet kernel and the traditional sensitivity coefficient can be shown by defining the total sensitivity M(x,t) as the spatial integral throughout the entire x–y domain of the Fréchet kernel F(x,t). Thus, the total sensitivity is

[13]

Considering again the special case with spatially uniform change in heat capacity, Eq. [12] becomes

[14]

Comparing Eq. [14] with Eq. [10] shows that

[15]

Upon comparing Eq. [15] with Eq. [13], we see that the traditional sensitivity coefficient is equivalent to the spatial integral of the Fréchet kernel.

Our goal is to find an explicit functional form for the Fréchet kernel that is relevant to the DPHP method. To accomplish this, we will work first in the Laplace domain, and then give an explicit inversion of the transform of the Fréchet kernel. If f (t) is a function of time, the Laplace transform(p) of f (t) is defined by

[16]

where p is the Laplace transform variable.

The Laplace transform of Eq. [3] for a zero initial condition is

[17]

When heat capacity has the uniform value C₀, Eq. [17] becomes the transformed version of Eq. [6]:

[18]

which is the modified Helmholtz equation. The transformed version of Eq. [5] is

[19]

Substituting Eq. [4] and [19] into Eq. [17], and collecting zero-order terms, gives Eq. [18]. Collecting first-order terms in gives the equation satisfied by ₁(x,p) :

[20]

which is the modified Helmholtz equation in ₁(x,p), plus a "source term" depending on C₁(x) and the zero-order solution ₀(x,p).

Because we assume that the boundary conditions for T₀ are the same as those for T, Eq. [20] has zero initial and boundary conditions for ₁. The solution of Eq. [20] in terms of a transformed Green's function, (x',x,p), is

[21]

or

[22]

By definition, the Green's function, (x',x,p), is the solution of Eq. [18] at position x' corresponding to a unit source at x. Taking the Laplace transform of Eq. [12] gives

[23]

and comparing this result with Eq. [22] shows that the Laplace transform of the kernel is

[24]

In the transform domain, the kernel is the product of known transforms of relatively simple functions. In the time domain, it corresponds to a convolution integral that takes on a more complicated form. In the time domain, the heat capacity kernel F is the negative of the convolution of the time derivative of the temperature with the Green's function, or equivalently, the negative of the convolution of the temperature with the time derivative of the Green's function. This means that the spatial sensitivity for heat capacity depends on the rate of change in temperature due to the imposed heat input.

Explicit Form for Fréchet Kernel
To obtain an explicit function for F, we take the position of the line heat source to be at (x,y) = (–a/2,0) and the observation point (temperature sensor) to be a distance a away at the position (x,y) = (a/2,0) as shown in Fig. 2 . The Laplace transform of the temperature at any point (x,y) due to the line source at (–a/2,0) is obtained by taking the Laplace transform of Eq. [7] to give

[25]

where K₀(z) denotes the modified Bessel function of the second kind of order zero and argument z, and r₁ is the radial distance from the line source, defined as

[26]

In the Laplace domain, the Green's function for the temperature at the observation point (a/2,0) due to an instantaneous source at any point (x,y) is

[27]

where r₂ is the radial distance from the observation point, defined as

[28]

In the time domain, the Green's function is the standard solution for the two-dimensional heat equation

[29]

corresponding to the effect at position (a/2,0) of a unit instantaneous release of heat at time t' at position (x,y) (Carslaw and Jaeger, 1959, p. 258).

Substituting Eq. [25] and [27] into Eq. [24] gives the expression

[30]

which is the Laplace transform of the Fréchet kernel for a line heat source at (x,y) = (–a/2,0) and an observation point at (x,y) = (a/2,0). Henceforth, we will understand that the Fréchet kernel is for this geometric configuration, without the dependence on a being explicitly indicated in the notation. If the positions of the line source at (–a/2,0) and the observation point (a/2,0) are interchanged, then r₁ and r₂ must be interchanged in Eq. [30]. It is a consequence of the Reciprocity Theorem (Bruggeman, 1999) that the value of the kernel is unaffected by this interchange.

Apart from differences in the definitions of the coefficients q, , and , Eq. [30] is identical to Eq. [69] of Knight and Kluitenberg (2005). They gave the explicit inversion of their Eq. [69] as their Eq. [75]. Thus, we can use Eq. [75] of Knight and Kluitenberg (2005) to write the explicit inversion of Eq. [30] as

[31]

which has units m K² J^–1. In this expression, K₁(z) denotes the modified Bessel function of the second kind of order one and argument z, and r = (x² + y²). The distance a between the heater probe and the temperature probe is a natural length scale for this problem, and suggests the dimensionless length and time variables X = x/a, Y = y/a, R = r/a, R₁ = r₁/a, R₂ = r₂/a, and = t/a² for a 0. Using these relationships results in the dimensionless form of the Fréchet kernel:

[32]

where F*(X,Y,) = a⁴C₀²F(x,y,t)/q. In terms of dimensionless variables, the heater probe and temperature probe are located at (X,Y) = (–,0) and (X,Y) = (,0), respectively, and are separated by a dimensionless distance of unity.

Because the DPHP method of Campbell et al. (1991) involves measuring the maximum value observed at the temperature probe, we have particular interest in the Fréchet kernel F(x,y,t) at time t = a²/4 when Eq. [7] exhibits a maximum. At this time, Eq. [31] becomes

[33]

The dimensionless equivalent of t = a²/4 is = . Upon substituting = into Eq. [32], we obtain

[34]

which is the special case of Eq. [32] at the time of the temperature maximum. To evaluate Eq. [33] at (x,y) = (–a/2,0) and (x,y) = (a/2,0), we make use of approximations

[35]

for small z, given as Eq. [9.6.8] and [9.6.9] in Olver (1965). Upon setting r₁ = a in Eq. [33] and making use of Eq. [35], we find that Eq. [33] has the limit F(a/2,0,a²/4) = –2q/²eC₀²a⁴ as r₂ 0. By the Reciprocity Theorem of Bruggeman (1999), Eq. [33] has the same limit as r₁ 0. Similarly, Eq. [34] has the finite value –2/²e at (X,Y) = (–,0) and (X,Y) = (,0).

Now that we have a functional form for F(x,y,t), we are able to determine the functional form of the total sensitivity as defined by Eq. [13]. The total sensitivity is the spatial integral of the kernel F(x,y,t) throughout the entire x–y domain. Rather than attempting to formally integrate F(x,y,t), we obtain the form for the total sensitivity by taking advantage of the connection between the Fréchet kernel and the traditional sensitivity coefficient for the case of spatially uniform heat capacity. Upon substituting Eq. [11] into Eq. [15] and replacing r₁ with a, we obtain

[36]

where the notation M(a/2,0,t) indicates the total sensitivity (m³ K² J^–1) as observed by the temperature probe at (x,y) = (a/2,0). As time increases, the total sensitivity decreases from zero to a minimum value of –4q/e²C₀²a² at t = a²/8, and then increases to a limiting value of zero at large time. For a > 0, we can also define a dimensionless form of the total sensitivity at the temperature sensor:

[37]

where M*(,0,t) = a²C₀²M(a/2,0,)/q. As increases, the dimensionless total sensitivity decreases from zero to a minimum value of –4/e² at = 1/8, and then increases to a limiting value of zero for large . In the analysis that follows, we will also need the value of the total sensitivity at the time the temperature maximum occurs at the temperature probe. At time t = a²/4, the total sensitivity is

[38]

which has the dimensionless equivalent

[39]

at dimensionless time = .

	Results and Discussion

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
The Fréchet kernels F(x,y,t) and F(x,y,a²/4) given by Eq. [31] and [33], respectively, are spatial weighting functions for a DPHP sensor with a heater probe at (x,y) = (–a/2,0) and a temperature probe at (x,y) = (a/2,0). Both of these kernels determine the weighting given to inhomogeneities in volumetric heat capacity when a measurement of the temperature at the temperature probe is used to estimate heat capacity. Whereas the kernel F(x,y,t) gives the spatial weighting function for the determination of heat capacity by using a temperature measurement at any time t, the kernel F(x,y,a²/4) gives the weighting function for the determination of heat capacity by using the maximum temperature T_0,m measured at the particular time t = a²/4. Our primary interest lies in the behavior of the kernel F(x,y,a²/4) because heat capacity is estimated by using a measurement of T_0,m in the method of Campbell et al. (1991). Recall, however, that the theory underlying the method of Campbell et al. (1991) assumes an instantaneous input of heat from the line heat source. The fact that a heat input of finite duration is used in practice means that the maximum temperature T_0,m actually occurs at a time slightly greater than t = a²/4. As a result, the kernel F(x,y,a²/4), which is F(x,y,t) evaluated at time t = a²/4, must be considered an approximation of the true spatial weighting function associated with the measurement of T_0,m. This is an issue of potential concern because the behavior of the Fréchet kernel F(x,y,t) changes at time t = a²/4. Earlier, it was noted that the spatial sensitivity for heat capacity depends on the rate of change in temperature due to the imposed heat input. Inasmuch as a temperature maximum occurs at time t = a²/4 at the temperature probe, the rate of change in temperature changes at this time. Thus, we expect the properties of F(x,y,t) to change at time t = a²/4 at the temperature probe. Furthermore, it turns out that the properties of F(x,y,t) also change at the heater probe at time t = a²/4. To justify our use of F(x,y,a²/4) as the approximate spatial weighting function for the DPHP method of Campbell et al. (1991), we first explore the behavior of F(x,y,t) for times slightly before and slightly after t = a²/4. To generalize our results, this analysis is performed using the dimensionless kernel F*(X,Y,) for dimensionless times near = .

Behavior of the Fréchet Kernel near =
Figure 3 shows the dimensionless Fréchet kernel F*(X,Y,) for a DPHP sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). Values of F*(X,Y,) were obtained by evaluating Eq. [32] as a function of X with Y = 0 for dimensionless times = 0.21, 0.23, 0.25, 0.27, and 0.29. It is evident from Fig. 3 that the behavior of the kernel F*(X,Y,) changes markedly at the location (X,Y) = (,0) at dimensionless times near = 0.25. The kernel has no singularity at = 0.25, but it has a negative singularity at (X,Y) = (,0) for < 0.25 and a positive singularity at this point for > 0.25. Because the heat capacity kernel is symmetric about the Y axis, there is a corresponding singularity at (X,Y) = (–,0), the location of the heater probe. The singularities at the heater and temperature probes mean that the spatial sensitivity is very high at, and near, these locations for 0.25. It appears that any heterogeneity in heat capacity at, or near, these locations will therefore have great effect on the temperature recorded at the temperature probe at times other than = 0.25.

View larger version (17K): [in this window] [in a new window]   FIG. 3. Dimensionless Fréchet kernel F*(X,0,) for a dual-probe heat-pulse sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). Results are from Eq. [32] with Y = 0 for dimensionless times = 0.21, 0.23, 0.25 0.27, and 0.29. The curve for = 0.25 is the special case of Eq. [34] for which there are no singularities at the heater probe or temperature sensor.

[40]

An explicit form for I*() can be obtained by using small-argument approximations of K₀ and K₁ near (X,Y) = (,0) (see Appendix). The graph of I*() for a region of dimensionless radius b = 1/12 (Fig. 4 ) shows that the integral of F*(X,Y,) in a small area surrounding the temperature probe varies smoothly as a function of . In particular, note that I*() does not change abruptly near = 0.25. We conclude, therefore, that the kernel F(x,y,a²/4) is a reasonable approximation of the spatial weighting function for heat capacity as determined with the DPHP method of Campbell et al. (1991). Next, we explore the behavior of F(x,y,a²/4) in detail. To generalize our results, this analysis is performed using the dimensionless kernel F*(X,Y,) for dimensionless time = .

View larger version (8K): [in this window] [in a new window]   FIG. 4. Dimensionless integral I*() as defined by Eq. [40]. It is the spatial integral of the Fréchet kernel F*(X,Y,) across a small region of radius R2 = b centered at (X,Y) = (,0), the location of the temperature probe. Results are from Eq. [A5] with b = 1/12. Note that I*() varies smoothly near = 0.25, despite the fact that the kernel F*(X,Y,) has a negative singularity at (X,Y) = (,0) for < 0.25 and a positive singularity at (X,Y) = (,0) for > 0.25.

Behavior of the Fréchet Kernel at = 1/4

View larger version (25K): [in this window] [in a new window]   FIG. 5. Surface representing the Fréchet kernel F*(X,Y,) as a function of X and Y for X 0 at dimensionless time = 0.25. The kernel F*(X,Y,) is for a dual-probe-heat-pulse sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). The vertical line shows the location of the temperature probe at (X,Y) = (,0). Results are from Eq. [34]. The dimensionless kernel is negative everywhere, and there is no singularity at (X,Y) = (,0).

View larger version (34K): [in this window] [in a new window]   FIG. 6. Contour lines representing selected values (numbers on curves) of the dimensionless heat capacity kernel F*(X,Y,) for a dual-probe heat-pulse sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). The dots show the location of the probes. The two contours for F*(X,Y,) = –0.054051 (unlabeled) intersect to form two saddle points. The kernel has a local maximum of F*(X,Y,) = –0.044986 at (X,Y) = (0,0) and global minima of F*(X,Y,) = –0.082842 near (X,Y) = (–,0) and (X,Y) = (,0) (cross symbols). Results are from Eq. [34]. The dimensionless heat capacity kernel is negative everywhere, and has no singularities at (X,Y) = (–,0) and (X,Y) = (,0).

[41]

given as Eq. [9.7.2] in Olver (1965). Upon using Eq. [41] in Eq. [34], we find that the kernel F*(X,Y,) can be approximated by

[42]

for large distances R, which corresponds to large values of R₁ and R₂. The exponential factor in Eq. [42] causes the kernel to decay rapidly with distance from the DPHP sensor, and shows that that F*(X,Y,) approaches zero in the limit as R . This is consistent with the behavior of the dimensionless heat capacity kernel in Fig. 5 and 6. It is noteworthy that the argument of the exponential term in Eq. [42] depends only on the sum R₁ + R₂. A curve on which R₁ + R₂ is constant is an ellipse with a focus at each probe. The form of Eq. [42] therefore indicates that, far from the two probes of the DPHP sensor, the contours of equal spatial sensitivity approach the shapes of a family of ellipses with a focus at each probe. This is illustrated in Fig. 7 , where ellipses (dashed lines) have been superimposed on selected contours from Fig. 6.

View larger version (28K): [in this window] [in a new window]   FIG. 7. Contour lines representing selected values (numbers on curves) of the dimensionless heat capacity kernel F*(X,Y,) for a dual-probe heat-pulse sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). The dots show the location of the probes. The exact values (solid lines) are from Eq. [34] and are identical to the contours for F*(X,Y,) = –0.0001, –0.001, –0.005, –0.02, and –0.04 in Fig. 6. The dashed lines are ellipses, which are approximations of the exact contours.

It is important to recognize the generality of the results presented in Fig. 5 and 6. Upon examining Eq. [34], we see that the dimensionless spatial sensitivity F*(X,Y,) is independent of the parameters q, C₀, and when written in terms of the scaled distances X and Y. This means that the size and shape of the spatial sensitivity of the DPHP sensor to variations in C are independent of the values of C₀ and , as long as C₀ and are uniform in the soil around the sensor. It also means that the size and shape of the spatial sensitivity are unaffected by the magnitude of the heat input q. This result contradicts the hypothesis offered by Ren et al. (2005), who suggested that the sampling size of the sensor would increase as power input to the heater probe is increased. When written in terms of the scaled distances X and Y, the size and shape of the dimensionless spatial sensitivity F*(X,Y,) are also independent of the parameter a. That is, the distance between the heater and temperature probes has no effect on the size and shape of the spatial sensitivity when results are plotted as a function of X and Y, as in Fig. 5 and 6. Inasmuch as X and Y are equivalent to x/a and y/a, respectively, it is clear that only the parameter a influences the size and shape of the spatial sensitivity when considered as a function of the physical coordinates x and y. Thus, changing the distance between the heater and temperature probes offers the only opportunity to manipulate the size of the region in which C₀ is measured by using the DPHP method of Campbell et al. (1991).

Contours of Equal Fractional Spatial Sensitivity
Given that we have a function F*(X,Y,) that describes the spatial sensitivity of the DPHP method in the X–Y plane, it is natural to ask if we can calculate the fractional spatial sensitivity (fraction of the total spatial sensitivity) for a region contained within a particular contour of the kernel. This can be accomplished by integrating the kernel F*(X,Y,) across region and then dividing the result by the total sensitivity for the entire X–Y plane. We know from Eq. [39] that the total spatial sensitivity for the X–Y plane at time = is M*(,0,) = –1/e. Thus, the fractional spatial sensitivity for region can be expressed as

[43]

where is the region contained within a particular contour of F*(X,Y,). We used Monte Carlo integration (Press et al., 1992, p. 304) to estimate the value of the dimensionless spatial integral in Eq. [43] because this method is well suited for problems involving regions of complex geometry. Inasmuch as Monte Carlo integration is a stochastic method, evaluation of Eq. [43] yields an estimate of P*(,0,,), rather than the exact value. Fortunately, the error of the estimate—characterized by a one-standard-deviation error estimate (Press et al., 1992)—can be kept small by using a large number of points to sample the region over which integration is performed. Monte Carlo integration can also be used to estimate the dimensionless area of the region over which integration is performed. This is accomplished by integrating a function that has the value unity within region and the value zero outside region . Thus, after a region is defined by selecting a particular F*(X,Y,) contour of interest, Monte Carlo integration can be used to estimate the value of P*(,0,,) for region as well as the dimensionless area of region .

Figure 8 shows the contours of the Fréchet kernel F*(X,Y,) for which Eq. [43] was evaluated. The numbers on the contour lines are values of P*(,0,,), which indicate the fraction of the total spatial sensitivity contained within a particular contour. The values of F*(X,Y,) for each contour are provided in Table 1. An iterative procedure was used to identify the contours of F*(X,Y,) that yielded the particular values of fractional spatial sensitivity (0.1, 0.2, 0.3, etc.) shown in Fig. 8 and Table 1. Estimates of P*(,0,,) are given in Table 1, along with their corresponding one-standard-deviation error estimates. Also shown in Table 1 are estimates (and corresponding error estimates) of the dimensionless area contained within each contour of F*(X,Y,). Contours in Fig. 8 with values of P*(,0,) = 0.7 or greater each define a region that is approximately elliptical in shape. The region containing a fractional spatial sensitivity of 0.5 is the doughnut-shaped region (dimensionless area 1.018) contained between the two contours labeled P*(,0,) = 0.5. The region containing a fractional spatial sensitivity of 0.3 consists of two parts, with each part contributing exactly half of the total area (Table 1) for that region. The regions containing fractional spatial sensitivities of 0.1 and 0.2 also consist of two parts of equal area. The results in Fig. 8 and Table 1 are also general in that they are independent of the parameters a, q, C₀, and .

View larger version (32K): [in this window] [in a new window]   FIG. 8. Contour lines representing selected values of the dimensionless heat capacity kernel F*(X,Y,) for a dual-probe heat-pulse sensor with heater probe at (X,Y) = (–,0) and temperature probe at (X,Y) = (,0). The dots show the location of the probes. The numbers on the contour lines are values of P*(,0,), which indicate the fraction of the total spatial sensitivity for the region contained within a particular contour. The fractional spatial sensitivity for each contour of F*(X,Y,) was obtained from Eq. [43]. Table 1 shows the value of F*(X,Y,) for each contour, as well as the area of the region bounded by each contour.

View this table: [in this window] [in a new window]   TABLE 1. Values of the dimensionless Fréchet kernel F*(X,Y,) and the fractional spatial sensitivity P*(,0,) for the contours in Fig. 8. Also given are dimensionless areas of the regions defined by the contour lines in Fig. 8. The errors for P*(,0,) and dimensionless area are one-standard-deviation error estimates for the Monte Carlo integration method.

Extension to the Measurement of Water Content
An important extension of the DPHP method is the measurement of soil volumetric water content (Bristow et al., 1993; Tarara and Ham, 1997; Song et al., 1998; Campbell et al., 2002; Basinger et al., 2003; Heitman et al., 2003; Mori et al., 2003; Ochsner et al., 2003; Ren et al., 2003). First suggested by Bristow et al. (1993), indirect estimation of water content via the DPHP method exploits the linear relationship between heat capacity and water content. If the heat capacity of the soil gas phase is assumed negligible, C can be considered a weighted sum of the heat capacities of the soil water and soil solid constituents (Kluitenberg, 2002):

[44]

where C_w is the volumetric heat capacity of water, is the volumetric water content (m³ m^–3), _b is the bulk density (kg m^–3), and c_s is the specific heat of the soil solid (mineral and organic) constituents (J kg^–1 K^–1). This relationship shows that can be determined from the DPHP measurement of C₀ if _b and c_s are known. It also shows that changes in water content can be determined from measured changes in C₀ without knowing _b and c_s.

The linear relationship between heat capacity and water content suggests that the sensitivity of the solution of Eq. [3] to small variations in water content is linearly related to the sensitivity to small variations in heat capacity. For the special case of uniform heat capacity C₀ and uniform water content ₀, this can be seen from the following application of the chain rule:

[45]

where the result C₀/₀ = C_w is obtained by differentiation of Eq. [44]. To examine the effect of heterogeneity in water content, we assume that the water content varies with position, but differs only slightly from a uniform value ₀. Thus, (x) can be expressed as

[46]

where the constant is small. Upon substituting Eq. [46] into Eq. [44] and comparing the result with Eq. [4], we find that C₀ and C₁(x) take on the form

[47]

We now return to the definition of the Fréchet kernel for the volumetric heat capacity, as given in Eq. [12]. Substituting C₁(x) = C_w1(x) from Eq. [47] into Eq. [12] yields

[48]

where F(x,y,t) = C_wF(x,y,t). Thus, the kernels for heat capacity and water content differ only by the constant factor C_w. That is, at any location in the x–y plane, the spatial sensitivity for the measurement of volumetric water content with the DPHP method and the spatial sensitivity of the DPHP heat capacity measurement differ only by the factor C_w. An important consequence of this linear scaling is that the dimensionless forms for F(x,y,t) and F(x,y,t) are identical. That is, F*(X,Y,) = F*(X,Y,), where F*(X,Y,) = a⁴C₀² F(x,y,t)/qC_w. This is important because it means that the results presented in Fig. 3 through 8 are valid for the measurement of volumetric water content with the DPHP method.

Knight (1992) showed that the spatial sensitivity for the time domain reflectometry (TDR) method for measuring soil water content is independent of the water content around the TDR instrument, as long as the water content is uniform. This is in contrast to the neutron moisture meter, which has a volume of influence whose size depends on the soil water content around the sensor. When the DPHP dimensionless spatial sensitivity F*(X,Y,) for soil water content at the dimensionless time = is written in terms of the scaled distances X and Y, it is independent of the other parameters. This means that the size and shape of the spatial sensitivity of the DPHP sensor to water content variations are independent of the value of water content around the sensor, as long as the water content is uniform.

	Conclusions

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
We have adapted previous work on well testing to obtain an explicit expression for the Fréchet kernel F(x,y,t), or spatial sensitivity function, for a DPHP sensor. The expression for F(x,y,t), given by Eq. [31], characterizes the spatial weighting given to small inhomogeneities in soil volumetric heat capacity when a measured temperature is used to estimate the heat capacity. For the particular time t = a²/4, the Fréchet kernel F(x,y,a²/4) gives the spatial sensitivity of C₀ for the method of Campbell et al. (1991), in which C₀ is determined using only the maximum temperature T_0,m recorded at the temperature probe. In practice, the maximum temperature T_0,m occurs at a time slightly greater than t = a²/4 because instantaneous heating is approximated by heat releasing within a short time interval. Although the Fréchet kernel F(x,y,t) has singularities at the location of the heater and temperature probes for times t a²/4, we have shown that the singularities are integrable. Thus, the expression for F(x,y,a²/4), given by Eq. [33], provides a reasonable approximation of the spatial sensitivity for the time at which T_0,m is actually recorded.

To generalize the results of our analysis, the behavior of F(x,y,a²/4) was explored by using its dimensionless equivalent, F*(X,Y,). Important properties of F*(X,Y,) are that it is negative throughout the x–y domain and that interchanging the location of the heater and temperature probes has no effect on the spatial sensitivity function. Examination of F*(X,Y,) revealed that the spatial sensitivity function is not radially symmetric about the heater probe, as suggested by Campbell et al. (1991). Instead, the location of the temperature probe and the heater probe are of equal importance in defining the spatial sensitivity of the DPHP method for measuring C₀. The magnitude of F*(X,Y,) is greatest in small areas immediately to the outside of the heater and temperature probes. Thus, the spatial sensitivity to small variations in heat capacity is greatest in these small areas. Far from the probes of the sensor, contours of equal spatial sensitivity approach the shapes of a family of ellipses. The effective outer boundary of the method therefore can be approximated by an ellipse with a focus at each probe.

We have shown that the total spatial sensitivity within the X–Y plane for time t = a²/4, or dimensionless time = , is a finite quantity. This result allowed us to calculate the fraction of the total spatial sensitivity (fractional spatial sensitivity) for a region contained within any given contour of F*(X,Y,). The contours of equal fractional spatial sensitivity are particularly useful for obtaining simplified descriptions of the effective outer boundary of the DPHP method. For example, for a sensor with probe spacing a = 6 mm, we showed that the boundary containing 99% of the total spatial sensitivity is closely approximated by an ellipse with area 168 mm² and a major axis 15.6 mm in length.

Analysis of the DPHP method for measuring volumetric water content revealed that the spatial sensitivity to small variations in water content is identical to the spatial sensitivity for the measurement of C₀ with the DPHP method. Because the dimensionless Fréchet kernel for ₀ at time = is identical to F*(X,Y,), the graphical results presented in Fig. 3 through 8 are equally valid for the measurement of water content with the DPHP method.

An important finding regarding the measurement of C₀ with the DPHP method is that the spatial sensitivity is unaffected by the values of the soil thermal properties, as long as the thermal properties are uniform in the soil around the sensor. That is, the size and shape of the spatial sensitivity of the DPHP sensor to variations in C₀ are independent of the values of C₀ and , as long as C₀ and are uniform in the soil around the sensor. Similarly, for the measurement of ₀ with the DPHP method, the size and shape of the spatial sensitivity of the DPHP sensor to water content variations are independent of the value of water content around the sensor, as long as the water content is uniform.

Finally, we recall that our investigation into the spatial sensitivity of the DPHP method was based on a perturbation expansion approach that treated heterogeneity in heat capacity as small, first-order variations about its uniform value of C₀. In principle, this approach is identical to traditional, first-order sensitivity analysis. Although the approach is useful and valid, it is important to recognize its limitations. It does not account for the higher order variations in heat capacity that might occur if the volume of soil sampled by a DPHP sensor contains regions with distinctly different thermal properties. Thus, our results are not applicable for situations in which the sensor is in close proximity to the soil surface, the wall of a sample container, a wetting front, a large soil void, or any other interface separating regions of distinctly different thermal properties. That said, the analysis presented here still has considerable utility in such situations insofar as it can be used to estimate the distance of separation from the interface required for the interface to have minimal influence. The more challenging scenario—one that may occur frequently in field applications—is that in which the heterogeneity in heat capacity within the volume of soil sampled by the sensor deviates slightly from the assumed distribution (i.e., small, first-order variations about a uniform value) in some unknown and undetectable manner. Although our results cannot provide an exact characterization of the spatial sensitivity of the DPHP method for such a scenario, they still have much value in providing a first-order approximation of the spatial sensitivity. Lastly, we note that our analysis does not consider the possibility of air gaps between the soil and the probes of the sensor. Explicitly accounting for the presence of air gaps would greatly increase the complexity of the spatial sensitivity analysis.

	Appendix

TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 
The integral I*() defined by Eq. [40] is the spatial integral of the Fréchet kernel F*(X,Y,) across a small region of radius R₂ = b centered at (X,Y) = (,0). An explicit form for I*() can be obtained by using an approximation of F*(X,Y,) valid for the small region surrounding the point (X,Y) = (,0). Upon making use of Eq. [35] and the identity R₁² + R₂² = 2R² + , Eq. [32] becomes

[A1]

Setting R₁ = 1 in Eq. [A1] and substituting the result into Eq. [40] yields

[A2]

which becomes

[A3]

upon making use of the substitution z = R₂²/4. We now rearrange Eq. [A3] to obtain the form

[A4]

Upon evaluating the integrals in Eq. [A4], we obtain the final result

[A5]

where E₁(z) is the exponential integral function with argument z (Gautschi and Cahill, 1965) and = 0.57722 is Euler's constant.

	ACKNOWLEDGMENTS

 
Most of this work was conducted while J.H. Knight was a Visiting Professor in the Department of Agronomy at Kansas State University, supported by a grant from the Kansas State University President's Faculty Development Awards Program. This work was also supported by funding from the National Aeronautics and Space Administration (Grant NAG 9-1399) and the National Science Foundation (Grant ECS-0410055).

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TOP ABSTRACT INTRODUCTION The Dual-Probe Heat-Pulse Method Theory Results and Discussion Conclusions Appendix REFERENCES

 

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