Samuel Thé

Éric Thiébaut

Loïc Denis

Ferréol Soulez

Exploiting the scaling indetermination of bi-linear models in inverse problems

Keywords: Inverse problem, Bi-linear models, Scaling indetermination, Blind deconvolution

Many inverse problems in imaging require estimating the parameters of a bi-linear model, e.g., the crisp image and the blur in blind deconvolution. In all these models, there is a scaling indetermination: multiplication of one term by an arbitrary factor can be compensated for by dividing the other by the same factor.

To solve such inverse problems and identify each term of the bilinear model, reconstruction methods rely on prior models that enforce some form of regularity. If these regularization terms verify a homogeneity property, the optimal scaling with respect to the regularization functions can be determined. This has two benefits: hyper-parameter tuning is simplified (a single parameter needs to be chosen) and the computation of the maximum a posteriori estimate is more efficient.

Illustrations on a blind deconvolution problem are given with an unsupervised strategy to tune the hyper-parameter.

I. INTRODUCTION

There are many problems in signal and image processing that can be formulated as inverse problems: the observed signals or images do not directly correspond to the parameters of interest but can be related via a model. Estimating the parameters of interest requires inverting the model. It is often the case that the model also depends on other parameters, e.g., instrumental parameters, and that these parameters need to be jointly estimated from the data. Such inverse problems are often called "blind" reconstruction problems to emphasize that, beyond the reconstruction of the parameters of interest, another set of parameters must be estimated. The family of bi-linear models covers a wide range of such models.

In order to identify each set of parameters in a blind reconstruction problem, regularization terms are introduced. These terms favor solutions in good agreement with our prior knowledge, for example with sufficient temporal or spatial smoothness.

There is a scaling indetermination to all bi-linear models: the model is unchanged when one set of parameters is multiplied by a factor and the other is divided by the same factor. This indetermination is solved by the regularization and the domain constraints imposed on the parameters.

We show in this paper that regularization terms that verify a homogeneity property lead to a closed form expression of the optimal scaling, for a given weighting of each regularization term. This optimal scaling can be beneficial by (i) simplifying the tuning of the relative weights of regularization terms (socalled hyper-parameters of the blind reconstruction problem); (ii) leading to a more efficient reconstruction algorithm; (iii) being applicable to an unsupervised reconstruction strategy based on the minimization of a criterion like Stein's unbiased risk estimate SURE [START_REF] Stein | Estimation of the mean of a multivariate normal distribution[END_REF].

The paper is organized as follows: Section II derives the expression of the optimal scaling and describes the proposed method. Section III illustrates this general methodology on a blind deconvolution problem, i.e., the joint estimation of a crisp image and of the blur. Section IV concludes the paper.

II. PROPOSED METHOD

A. Bi-linear models and scaling invariance

We consider a class of inverse problems for which the direct model of the data writes:

d ≈ x ⊗ y , (1)
where d ∈ D, x ∈ X and y ∈ Y are real-valued vectors: d represents the measurements while x and y represent unknowns which are combined by the bi-linear operator ⊗.

The above symbol ≈ is to account for discrepancies due to the approximations of the model x ⊗ y and to measurement noise. Since the direct model x ⊗ y is bi-linear, it verifies the property:

∀(α, β, x, y) ∈ R 2 ×X×Y, (α x)⊗(β y) = (α β) (x⊗y) (2)
Instances of such problems are numerous in signal and image processing, for example:

• Blind deconvolution [START_REF] Ayers | Iterative blind deconvolution and its applications[END_REF] with x the observed object, y the instrumental point spread function (PSF) and ⊗ the bi-dimensional convolution operator. • Blind source separation [START_REF] Comon | Handbook of Blind Source Separation: Independent component analysis and applications[END_REF] where x and y are matrices 1while ⊗ denotes the usual matrix product.

• Low-rank approximations where a matrix d is approximated as the matrix product of a tall matrix and a wide matrix. To account for the approximate relation in Eq. ( 1), the inverse problem of recovering the unknowns x and y given the data d involves fitting the model x ⊗ y to the data. Formally, this fitting can be expressed by:

min x∈X,y∈Y G(x ⊗ y) (3)
where G: D → R implements a distance between the model x ⊗ y and the data d.

As a consequence of the Property (2), the relation

G((α x) ⊗ (y/α)) = G(x ⊗ y) (4)
holds for any α ∈ R ≡ R\{0}. This means that there is a multiplicative degeneracy in the inverse problem which cannot be disentangled by the data alone whatever the metric implemented by G. In the general case, the Problem (3) is extremely ill-posed due to its numerous degeneracies (such as swap, shift or reducibility degeneracies in the blind deconvolution case [START_REF] Soulez | Superresolution with optically-motivated blind deconvolution[END_REF]). Hence, it is necessary to add some a priori knowledge on both unknowns x and y by the means of regularization penalties and/or by tightening the feasible sets X and/or Y [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. The inverse problem then amounts to solving the following optimization problem:

min x∈X,y∈Y F(x, µ, y, ν) = G(x ⊗ y) + µ J (x) + ν K(y) (5)
where J : X → R + and K: Y → R + are regularization penalties to enforce the prior constraints, µ ∈ R + and ν ∈ R + are hyper-parameters to tune the relative weights of the different terms of the objective function F. Many a priori models have been proposed in the literature, such as quadratic regularization [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF], total variation (TV) [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] or edge-preserving smoothness [START_REF] Charbonnier | Deterministic edge-preserving regularization in computed imaging[END_REF].

B. Exploiting the scaling indetermination

In order to exploit the scaling indetermination of the direct model, we require that the regularization penalties J (x) and K(y) be homogeneous functions of degree q > 0 and r > 0 respectively. That is:

J ∈ Ω(X, q) and K ∈ Ω(Y, r) , (6)
where Ω(U, p) denotes the set of homogeneous functions from the convex cone 2 U to R + of degree p. In other words, we assume that X and Y are convex cones and that:

J (α x) = α q J (x) ≥ 0 , (7a) K(α y) = α r K(y) ≥ 0 , (7b)
hold for any x ∈ X, y ∈ Y and α ∈ R + . Under these assumptions, the identity:

F(α x, µ, y/α, ν) = F x, µ α q , y, ν α -r , ( 8
) 2 A convex cone U is such that: u ∈ U ⇒ α u ∈ U for any α ≥ 0. Examples of convex cones are R n or R n + .
holds ∀α ∈ R + . This identity shows that scaling the unknown parameters in such a way that the model is left unchanged amounts to changing the regularization terms of the objective function.

We now introduce E(α) the value of the objective function in Eq. ( 8) as a function of the scaling factor α:

E(α) = F(α x, µ, y/α, ν) = G(x ⊗ y) + µ α q J (x) + ν α -r K(y) , (9)
for any given x ∈ X, y ∈ Y, µ > 0, ν > 0 and α = 0 such that J (x) > 0 and K(y) > 0. The derivative of E(α) writes:

E (α) = q µ α q-1 J (x) -r ν α -r-1 K(y) .
Since J (x) > 0 and K(y) > 0, E(α) is strictly convex on R + . The optimal scaling factor which minimizes E(α) is α the only positive root of E (α):

α(x, µ, y, ν) = r ν K(y) q µ J (x) 1 q+r . ( 10
)
Replacing the optimal scaling parameter α in the objective function yields:

min α>0 F(α x, µ, y/α, ν) = F(α x, µ, y/α, ν) α= α(x,µ,y,ν) = H x, y, η(µ, ν) (11)
where:

H(x, y, η) = G(x ⊗ y) + η J (x) r q+r K(y) q q+r , (12) and:
η(µ, ν) = (r/q) q q+r + (q/r) r q+r

(µ r ν q ) 1 q+r . ( 13
)
The criterion defined in Eq. ( 12) shows that when considering the shape of the solution (that is x and y up to some scaling factor), the regularity of the two variables is mutually dependent, indeed only the value of µ r ν q matters to set the importance of the regularization. As a consequence a single hyper-parameter, η, instead of two, µ and ν, is sufficient to tune the regularization level. This property can be exploited to simplify the tuning of the regularization. Minimization problems of the form of equation ( 5) generally have several local minima. The solution reached by minimization algorithms, therefore, depends on the optimization path followed during the iterations. Exploiting relation (10) to select, at each iteration, the optimal scaling can also help avoiding some poor local minima.

There is no closed form solution for the general Problem (5), so most practical algorithms to solve this problem are iterative: starting from some initial guess x (0) , y (0) ∈ X × Y, the objective function is progressively minimized to yield improved estimates x (m) , y (m) ∈ X × Y after m iterations.

To take advantage of the scaling invariance there are several possible strategies, the most obvious ones are:

1) Choose a regularization level η > 0 and minimize the objective function H(x, y, η) defined in Eq. ( 12).

Algorithm 1: Rescaled alternating strategy with two hyper-parameters (one is arbitrary).

Input: µ > 0, ν > 0, α 0 > 0 and y (0) ∈ Y s.t. K y (0) > 0. Output: A local optimum (x , y ) ∈ X × Y. for k = 0, 1, 2, . . . , m -1 do while true do x (k+1) ≈ x y (k) , α q k µ see Eq. (14a) α k+1/2 = α x (k+1) , µ, y (k) , ν see Eq. (10) if k > 0 or α k+1/2 ≈ α k then break; α k ← α k+1/2 y (k+1) ≈ y x (k+1) , α -r k+1/2 ν see Eq. (14b) α k+1 = α x (k+1) , µ, y (k+1) , ν see Eq. ( 10
) Result: (x , y ) = α m x (m) , y (m) /α m
2) Choose the regularization levels µ > 0 and ν > 0 (one of which can have an arbitrary fixed value) and minimize the criterion F(x, µ α q , y, ν α -r ) with α the best scaling factor given by Eq. ( 10) and either set before the first iteration, i.e. according to x (0) and y (0) , or at every stages of the optimization, i.e. according to x (k) and y (k) . Optimally scaling the variables at every iterations (or the hyper-parameters) has been shown to speed-up the convergence [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF] so the former variant should be more efficient. Remarks: (i) Whatever the strategy, it is necessary that J x (k) > 0 and K y (k) > 0 for all iterates k. (ii) Owing to the identity in [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF], the scaling of the hyper-parameters in the 2nd method is equivalent to a scaling of the variables.

Directly minimizing the objective function H(x, y, η) defined in Eq. ( 12) is not very practical as it requires to solve a non-linear problem whether the variables x and y are jointly, hierarchically or alternately updated. The 2nd strategy is easier to implement if x and y are updated alternately as it can re-use existing methods to solve the two half-problems of estimating x given y and µ and estimating y given x and ν:

x(y, µ) = arg min x∈X F(x, µ, y, ν) = arg min x∈X {G(x ⊗ y) + µ J (x)} , (14a)
y(x, ν) = arg min y∈Y F(x, µ, y, ν) = arg min y∈Y {G(x ⊗ y) + ν K(y)} . ( 14b
)
As previously discussed, these updates should be performed with rescaled hyper-parameters to apply the optimal scaling factor given in Eq. (10) before each update. These considerations suggest the alternating method implemented by our Algorithm 1 whose output (up to a scaling of the unknowns) depends on a single tuning hyper-parameter, the other being fixed at an arbitrary value.

An advantage of the proposed algorithm is that only y (0) has to be provided. However, since x (0) is unknown, the optimal In blue, the unscaled algorithm, that is alternated minimization of F (x, µ, y, ν) as defined in [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. In red, the proposed algorithm.

initial scaling α 0 cannot be computed by the closed form expression (10). To automatically determine a good initial scaling, our algorithm implements a simple alternated method to jointly estimate x (1) and α 0 . This simple procedure ensures that the other iterations (and hence the output of the algorithm) do not depend on the scaling of y (0) . In other words, taking y (0) = β y inp for a given y inp gives the same results whatever the value of β > 0. As shown by Fig. 1, this feature is important to avoid that the solution and the convergence speed be driven by the scaling of the initial variables.

The updates of the variables x and y according to Eq. ( 14a) and (14b) do not have to be very precise, it is enough to seek for an approximate solution ensuring a sufficient reduction of the objective function. Starting at the previous estimate, a few iterations of the optimization algorithms designed to solve Eq. ( 14a) and (14b) will do the job. Thanks to the symmetry of the problem, the roles of (x, X, µ) and (y, Y, ν) can be exchanged and the algorithm started with an initial x (0) .

III. APPLICATION TO BLIND DECONVOLUTION

Blind deconvolution is typical of the class of problems considered in this paper and we choose it to demonstrate the interest of exploiting the scale invariance property. The objective of image blind deconvolution [START_REF] Ayers | Iterative blind deconvolution and its applications[END_REF], [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF], [START_REF] Thiébaut | Strict a priori constraints for maximum likelihood blind deconvolution[END_REF] is to recover the object of interest, say x, and the instrumental point spread function (PSF), say y, given a single observed image, the data d, of the object, this image being degraded by the blurring due to the PSF and by the noise. Such a problem is challenging but has a great interest in situations where the PSF is unknown or expensive to calibrate (e.g., astronomy, medical imaging, etc.).

A. Statement of the problem

Assuming that the noise statistics is well approximated by a multi-variate Gaussian distribution, an excellent way to measure the distance between the model x ⊗ y and the data d is provided by the co-log-likelihood of the data: where ⊗ denotes discrete convolution product, implemented by means of the fast Fourier transform (FFT), W is the precision matrix of the data and u

G(x ⊗ y) = d -x ⊗ y 2 W , (15)
2 W def = u T • W• u is a Mahalanobis squared norm.
In practice, we considered independent noise so that the precision matrix W is diagonal with diagonal terms given by:

W j,j = 1/Var(d j ) if j th data pixel is valid, 0 otherwise. ( 16
)
This definition provides simple means to consistently account for defective pixels.

As strict constraints such as positivity are beneficial to solve the blind deconvolution problem of images [START_REF] Thiébaut | Strict a priori constraints for maximum likelihood blind deconvolution[END_REF], we impose that both the object and the PSF be nonnegative everywhere (for physical reasons, this is always true for the PSF in optics). This means that X and Y are equal to R n + with n the number of pixels.

For this paper, we consider the following two regularizations:

H 2 (u) = j D j • u 2 , (17)
H 1 (u) = j D j • u 2 + ε u 2 , (18)
where ε ≥ 0 is another tuning parameter and D j : U → R p yields a discrete approximation of the p-dimensional gradient of its argument (p = 2 for images) at j-th position. These regularizations are homogeneous functions of respective degrees 2 and 1, they both favor the smoothness of the variables u. The second regularization, H 1 (u), implements total variation (TV) regularization [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] for ε = 0 and a homogeneous version of edge-preserving smoothness [START_REF] Charbonnier | Deterministic edge-preserving regularization in computed imaging[END_REF] for ε > 0.

B. Results

We generated synthetic data given an image of the galaxy M51 (Fig. 2) and the PSF of the 3.60 m Canada-France Hawaii telescope equipped with one of the first adaptive optics systems (Fig. 3). The resulting data, shown in Fig. 5, is the true object blurred by the PSF and with additional i.i.d. random Gaussian noise of standard deviation σ = 1.5 ADU which corresponds to a PSNR of 80.

For the initial guess of the PSF y (0) , we took the instrumental response of a perfect 3.60 m telescope, that is an Airy function (Fig. 4).

For the regularizations, we first considered the same quadratic regularization, given in Eq. ( 17), for the object x and the PSF y. As the PSF is rather smooth (as expected from the physics) while the object has sharp features (stars), we also considered a homogeneous edge-preserving regularization for the object while keeping a quadratic smoothness for the PSF: J (x) = H 1 (x) and J (y) = H 2 (y).

As the considered objective functions are differentiable (for ε > 0), we used VMLM-B [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF], a limited memory quasi-Newton method implementing bound constraints, to solve the sub-problems (14a) and (14b) on the nonnegative orthant. To ensure convergence, the alternating strategy implemented by Algorithm 1 is applied for m = 200 iterations, while each of the sub-problems is solved by 10 iterations of VMLM-B.

We used the SURE criterion [START_REF] Stein | Estimation of the mean of a multivariate normal distribution[END_REF], an unbiased estimation of the mean squared error, to determine the regularization level. We fixed ν and run Algorithm 1 for a wide range of values of µ to produce the curves shown by Fig. 6. A global minimum of SURE criterion is obtained for µ 0.05 when x and y have the same quadratic regularizations and for µ 0.02 when edge-preserving regularization is applied to x. Figures 9 and7 show the recovered instrumental PSF and object both with quadratic smoothness priors and for the optimal value µ = 0.05. Figures 10 and8 show the instrumental PSF and object recovered when imposing quadratic smoothness to the PSF and edge-preserving smoothness to the object for the optimal value µ = 0.02 in that case. Clearly, the value of the hyperparameter µ found by the proposed procedure is correct in the two considered cases. Besides, imposing more suitable edgepreserving smoothness is beneficial for the object in particular for the recovering of sharp features such as the foreground stars and the cores of the two galaxies.

IV. CONCLUSIONS

We have demonstrated that the scaling indetermination of bi-linear models in inverse problems can, theoretically and for homogeneous regularizations, be exploited to simplify the tuning of the regularization (as one less tuning parameter is necessary) and to speed-up convergence. We have proposed a simple alternating strategy to solve the inverse problem in this context and whatever the scaling of the initialization. We have applied our algorithm to the blind deconvolution of astronomical images under various regularizations whose importance is automatically tuned by minimizing Stein's unbiased estimator of the mean weighted error. On the basis of simulated images, we have shown that the proposed algorithm was able to recover good estimates of the object and the instrumental point spread function (PSF) given a single blurred and noisy image.

1 Fig. 1 .

 11 Fig.1. Convergence curves for two versions of a blind deconvolution algorithm initialized with y (0) = β y inp with different values of β but same input PSF y inp . In blue, the unscaled algorithm, that is alternated minimization of F (x, µ, y, ν) as defined in[START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. In red, the proposed algorithm.

Fig. 2 .

 2 Fig. 2. The true object (the galaxy M51).

Fig. 3 .

 3 Fig. 3.The instrumental PSF (courtesy Franc ¸ois Roddier).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Initial PSF y (0) assumed in Algorithm 1.

Fig. 6 .

 6 Fig.6. The SURE criterion R(µ) as a function of µ only (i.e., for ν optimally set), for quadratic regularization for both the object and the PSF (blue line), and for edge-preserving and quadratic regularizations for the object and the PSF respectively and a given value of ε (orange dashed line).

Fig. 7 .

 7 Fig. 7. The recovered object with a quadratic smoothness regularization.

Fig. 8 .Fig. 9 .

 89 Fig. 8. The recovered object with an homogeneous edge-preserving smoothness regularization.

Fig. 10 .

 10 Fig. 10. The recovered PSF with a quadratic smoothness regularization while imposing edge preserving smoothness for the object.

In the case of blind source separation, x may represent the end-members while y represent the mixture coefficients.


Keywords: in-situ sensor, soil water content, salinity, complex permittivity, self-balanced admittance bridge, electronics drift

The article presents an in-situ sensor to monitor soil moisture and water salinity by measuring soil permittivity and conductivity at a frequency 24 MHz via an admittance approach. It addresses difficulties encountered in other permittivitybased sensors. A self-balanced bridge provides on one measurement a digital output linear to capacitance and conductance of electrodes in soil with low offset, high phase resolution, and taking into account all parasitic impedances. Circuit design and procedures to achieve it are described, using electric circuit theory and component specifications. Probe geometry, two parallel cylinders, permits insertion in soil from surface, low disturbances and a high sample volume around electrodes. It provides an analytical relation to convert admittance to soil permittivity.

Calibration of a sensor with air, alcohols and water fixes its sensitivity and residual phase and inductance for an uncertainty lower than 2% over a range up to 80 and 200 mS•m -1 . Identical sensors can use same parameters after adjusting in one simple operation bridge phases. Thermal drift and its consequences on soil variable measurement are investigated and physical origins identified. It permits to reduce the phase drift within ±150ppm• • C -1 rad to keep accuracy.

Introduction

The article presents an in-situ sensor able to monitor key variables of a soil such as its water content (Θ V ) and water salinity (σ ion ) at a depth or along a profile. Sensor relies on the measurement of permittivity ε r of a soil and its conductivity σ, which vary with Θ V and σ ion . The device produces a rapid response linear to soil quantities ε r and σ with potential deviations intrinsically or automatically compensated. These features and other characteristics facilitate sensor duplication and use at reduced cost, and its integration into networks for field autonomous and continuous operations. Sensors integrate circuits to acquire and process data, and to manage the operations. It uses recent technology of communication to send data once measured in a web-accessed data base where data are retrieved. Networks of in-situ sensors to determine fields of soil variables over large surfaces offer a good alternative to large-volume averaging and more indirect instruments, such as remote sensing from space or air-born radiometers based on surface emissivity [START_REF] Schmugge | Remote sensing in hydrology[END_REF]] and counters of neutrons produced from cosmic rays and backscattered by H atom present in soil [START_REF] Zreda | Cosmos: The cosmic-ray soil moisture observing system[END_REF]].

We limit the scope of present article to the instrument, and do not deal with the control and communication part of the sensor to obtain networks of autonomous and wireless sensors. This is the object of an article recently submitted.

Soil -the about one-meter-thick piece of land in contact with atmosphere and substrate of plant roots -is able to store some water within its pores and to restore it at time scales from few hours to years, depending on its structure. This capability is crucial in soil interaction with other environment components through the water, energy and material cycles, like atmosphere, subsoil, plants... and to control natural phenomena such as landslide and drought from short-term forcing (a strong rain or a dry period), and local climate change in longer terms. Monitoring Θ V is not only necessary in meteorological and climate models, but also in agriculture for effective irrigation, in industrial processes to control material quality (like for concrete) and their aging. Water salinity in soil is important to measure as well to infer nutrient quantity for plants or to control potential contamination from leaching [START_REF] Rhoades | Soil salinity assessment: Methods and interpretation of electrical conductivity measurements[END_REF]].

Whereas air is a gas made homogeneous thanks to convection, soil is an heterogeneous and multi-phase medium where diffusive mechanisms dominate for transport. As a consequence, air relative humidity is easy to measure, not soil moisture.

The standard method-soil sample weighting and drying in oven-provides directly the mass of water content, but is too destructive and human-time consuming for continuous automated monitoring [START_REF] Reynolds | The gravimetric method of soil moisture determination[END_REF]].

One of the most promising methods to meet these objectives is based on soil dielectric permittivity. Permittivity measures the strength of medium electric dipoles to respond to the electric field induced by the sensor. The normalized or relative quantity, ε r (using the vacuum permittivity ε 0 = 8, 854 pF.m -1 for reference), is ε r = 80 for water at 20 • C, due to its strong molecular dipoles, whereas it varies from ε r = 2 to 5 for a dry soil, and is ε r = 1 for air (confounded with that of vacuum). Literature abounds on relations of conversion of ε r into Θ V , from extended Archie's laws [START_REF] Revil | Effective conductivity and permittivity of unsaturated porous materials in the frequency range 1 mhz-1ghz[END_REF]] to Topp's empirical correlation [START_REF] Topp | Electromagnetic determination of soil water content: Measurements in coaxial transmission lines[END_REF]]. Moreover, instruments relying on this principle can provide output easily digitized, and their operations be automated.

Permittivity should not be confused with soil conductivity σ, which results in part from water free charges or ions, and not its molecular dipoles. From a theoretical point of view the two quantities ε r and σ are intricately linked in the equations governing electromagnetic fields in a medium. For this reason ε r and σ are grouped together within the complex expression ε r of the relative permittivity (with j 2 = -1):

ε r = ε r -j σ ε 0 2π f , (1)
where f is the frequency of the electric field applied.

Resolution of the equations in case of porous media like soils and rocks, owing to many simplifications, constitutes the bulk of the literature devoted to the ε r -Θ V conversion (see for instance [START_REF] Chelidze | Electrical spectroscopy of porous rocks: A review i. theoretical models[END_REF]; [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF]; [START_REF] Revil | Effective conductivity and permittivity of unsaturated porous materials in the frequency range 1 mhz-1ghz[END_REF]]). Formally, the expression 1 is valid over a frequency range from mHz to 1 GHz, provided each part in ε r includes terms associated to mechanisms in the medium other than those resulting from water dipoles or its ions. Their importance depends on the frequency range of f . For instance below f = 10 MHz ions close to grain surface act as dipoles of which permittivity exceeds that of water molecules [START_REF] Campbell | Dielectric properties and influence of conductivity in soils at one to fifty megahertz[END_REF]; Chelidze and Gueguen (1999); [START_REF] Descarpentries | Contribution à l"étude de la conductivité et de la polarisation complexe dans les milieux hétérogènes sable-argile-eau[END_REF]; [START_REF] Eller | A capacitive soil moisture sensor[END_REF]; [START_REF] Sen | The frequency dependent dielectric and conductivity response of sedimentary rocks[END_REF]; [START_REF] Kessouri | Mesure simultanée aux fréquences moyennes et cartographie de la permittivité diélectrique et de la conductivité électrique du sol[END_REF]]. Due to thermal diffusivity each type of dipoles presents a relaxation or phase delay to follow the direction of an electric field, when f rises past the characteristic frequency f rlx of the dipoles [START_REF] Debye | Polar molecules[END_REF]]. The effect reduces the contribution of the dipoles to ε r , down to zero as f f rlx . It increases σ around f rlx , independently of ion transport.

Already many sensors have been developed along the permittivity principle, and commercialized. Actually, each manufacturer specializes in a particular measurement technique to determine soil water content. The same technique is used to the family of products proposed by the manufacturer. They only differ by electrode geometry or by adding new features such as converting instrument analog output to a digital one. Although their range is wide, we can classify techniques according to how they exploit electromagnetic fields to sense ε r (see also [START_REF] Chavanne | Autonomous sensors for measuring continuously the moisture and salinity of a porous medium[END_REF]]):

-Those using the phase velocity or travel time of an electromagnetic wave propagating along a guide in a soil. The velocity varies with the real part of the root square of ε r (eq. 1), approximated to ε r at low conductivity. -Others operating with the amplitude of sensor electromagnetic fields to determine the capacitance (or admittance if they are able to measure simultaneously the conductance part in parallel) of electrodes embedded in a soil. Their output can be potentially linear to ε r .

Commonly used sensors of the propagation group are the Time Domain Reflectometry Systems (TDRs). They are composed of a step-pulse generator and a waveform analyzer connected to probes in soil (e.g. the module TDR100 with probes CS659, commercialized by Campbell Scientific Inc., Logan, UT, USA ["TDR100 user guide" (2015)]). Because their output, pulse-travel time along the guide, is readily related to soil permittivity, TDRs gained a sort of standard status as in-situ soil moisture sensors. Guide consists in simple parallel rods inserted in soil, of which the open ends induce a partial reflection (a transmission version with the guide in the form of loop also exists). Topp established in 1980 his correlations between ε r and Θ V thanks to these sensors [START_REF] Topp | Electromagnetic determination of soil water content: Measurements in coaxial transmission lines[END_REF]]. However, high-speed electronics to generate the step excitation, record the return wave and determine a time of travel from the waveform make them costly and energy consuming, which prevent TDRs from continuous monitoring in an autonomous mode. Even the last product of Campbell Scientific, TDR soilvue ["SoilVUETM10 Manual" (2019)], which is more compact and able to measure various soil variables at different depths, remain expensive and difficult to operate autonomously (its price for 6 channels amounts to 1700 $ w/o VAT, not including the external datalogger required to store its digital output and assure their transfer).

A compact and low-cost version of TDRs, called reflectometer, consists in an electronic ring oscillator which triggers an impulse once it receives the return wave [START_REF] Qu | Calibration of a novel low-cost soil water content sensor based on a ring oscillator[END_REF]]. It generates thus a cyclic output of which period is related to the wave travel time. Campbell Scientific proposes some of them, like CS616 and more recently CS650, the digital version ["CS650 and CS655 Water Content Reflectometers Manual" (2016)]. However, the output is not as directly related to travel time as for TDRs, such as Campbell Scientific did not retain the technique for its last product.

Sensors belonging to the capacitance family are cheaper than TDRs due to the use of lower electronic speed (signal-rise time higher than few ns, versus less than 10 ps in the case of TDRs). They operate at a frequency f close to or lower than 100 MHz. At this frequency electromagnetic wavelength in air is 3 m -but only 60 cm at ε r = 25 -, higher than a typical electrode dimension of 10 cm. Sensor circuit and its output can be modeled and interpreted thanks to the electric circuit theory [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF]; [START_REF] Kelleners | Calibration of capacitance probe sensors using electric circuit theory[END_REF]], which assumes negligible propagation effect. Sensor electrodes embedded in a soil form a capacitor which is represented by an admittance composed of electrode capacitance C and conductance G in parallel, Y = G + j C 2π f . Hence Y is readily related to the apparent permittivity of the medium ε r in eq. ( 1):

ε r = C ε 0 g -j G ε 0 g 2π f = -j Y ε 0 g 2π f . ( 2
)
where g is a factor with unit of length depending only on electrode design and dimensions. In the case of the bi-cylindrical geometry adopted for our sensors it varies from 0.1 to 0.15 m (see in section 2.3). Range of techniques to determine Y or, in most cases, only C are wide. They are intended to reduce costs but at the expense of accuracy. EnviroSCAN and recent TriSCAN probes marketed by Sentek Pty Ltd., Stepney, Australia give the frequency of an electronic oscillator consisting of an inductor and a capacitor in parallel [START_REF] Kelleners | Calibration of capacitance probe sensors using electric circuit theory[END_REF]]. The capacitor is partly made of electrodes in soil. Decagon Devices Inc., Pullman, WA, USA (now METER group) commercializes sensors which use the root mean square of partial-charging cycles of the same capacitor (from ECH 2 O to GS3, including EC5, 10HS, 5TE...) [H. [START_REF] Bogena | Evaluation of a low-cost soil water content sensor for wireless network applications[END_REF]]. An hybrid method uses the propagation and reflection of a sinus wave in a coaxial line ended by the electrodes [START_REF] Campbell | Dielectric properties and influence of conductivity in soils at one to fifty megahertz[END_REF]; [START_REF] Gaskin | Measurement of soil water content using a simplified impedance measuring technique[END_REF]]. From the complex coefficient of reflexion at the end of the coaxial line is determined line terminal impedance. It corresponds to the impedance of electrodes considered as an open ended transmission line of which permittivity is that of soil. Sensors of Stevens System (Portland, OR, USA), such as HydraProbe, as well as those of Delta T Devices Ltd. (Cambridge, UK), like PR2 and Theta Probe, are based on this technique, each operating at one frequency (50 and 100 MHz, respectively). In the case of Delta T sensors, due to multiple reflexions at coaxial and probe ends (propagation effect at f = 100 MHz), an empirical approach is necessary and limits instrument output to permittivity [START_REF] Gaskin | Measurement of soil water content using a simplified impedance measuring technique[END_REF]]. HydraProbe provides both ε r and σ, but no information on the precise procedure is available.

Attempts have been made to determine more directly the admittance Y in eq. 2, based on Ohm's law. The difficulty lies in the phase detection of Y . In the 1990s, Hilhorst designed an application-specific integrated circuit to determine Y at f = 20 MHz [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF]]. The circuit contained in particular actuated switches, phase shifters and synchronous detectors to separate the two components of Y . Because of the poor accuracy of component gains -±20% -, of phase errors (up to 15 degrees) and high sensitivity to temperature, measurement of on-board references must be carried out alongside each electrode measurement to offset errors [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF]]. From this work a sensor was developed and commercialized by Delta-T Devices Ltd, WET Sensor, which outputs both ε r and σ. To avoid the phase-resolution difficulty, Rêgo et al. proposed to measure the amplitude of Y at two distinct frequencies, 500 kHz and 8 MHz [START_REF] Rêgo Segundo | A novel low-cost instrumentation system for measuring the water content and apparent electrical conductivity of soils[END_REF]; Rêgo Segundo, Silva Pinto, Almeida Santos, and de Barros [START_REF] Rêgo Segundo | Capacitive impedance measurement: Dual-frequency approach[END_REF]]. C and G are determined by calculating the expression of Y amplitude at both frequencies. However, because of circuit parasitic impedances not accounted for, separation of the two parts of Y is still complex, requiring multi-parameter correlations. Moreover, choice of frequencies may pose a problem due to dipole relaxations in this range.

In summary most of techniques based on ε r to determine soil moisture are simple enough to reduce their cost, but determination of soil permittivity is often too indirect. Manufacturers have to resort to empirical conversions to obtain ε r from instrument output, if they provide it at all. Indeed, they prefer to give a correlation between output and soil water content from calibration with soil samples. These correlations are specific to the instrument and to type of soil, as they include effects from both without discrimination. All literature on soil permittivity becomes useless. The oversimplification of these techniques makes often the physical modeling of their operations difficult and thus prevents some in-depth understanding. Kelleners et al. showed the interest of this type of study in the case of Enviroscan: a simple electric circuit of which components are identified to sensor parts (inductor, access tube...) explains the low dependence of LC-oscillator frequency to medium permittivity above 30, as well as its strong dependence on conductance [START_REF] Kelleners | Calibration of capacitance probe sensors using electric circuit theory[END_REF]]. Above a medium conductivity of 100 mS•m -1 sensor output is dominated by σ not ε r .

To put in evidence sensor artifacts independently of soil related effects, a sensor must be tested with actual standards of complex permittivity like liquids of known permittivity and conductivity σ, as stressed by Jones et al. [START_REF] Jones | Standardizing characterization of electromagnetic water content sensors[END_REF]]. They also assure a better contact with electrodes than a soil. The authors provided guidelines to characterize a sensor and its output: sensitivity to ε r , influences of instrument temperature and of medium conductivity, medium sample volume... They evaluated seven commercial sensors as an illustration [START_REF] Blonquist | Standardizing characterization of electromagnetic water content sensors[END_REF]]. Three of them were based on waveform analysis to retrieve a time travel, like TDR100. Conversion to ε r is based on the same physical dependence of electromagnetic speed to ε r . For accuracy actual value of the guide length is determined with liquid calibration. Each of other sensors -CS616, Hy-draProbe, Theta Probe and ECH 2 O -necessitates an empirical relation deduced from liquid calibration or given by the manufacturer when available. It presents a non linear relation, with low sensitivity beyond ε r 20 to 30 except for the reflectometer CS616. As far as sensitivity to conductivity is concerned, output of time travel sensors presents an increasing dependence, with an error of about δε r = ±1 at σ = 100 mS•m -1 and δε r = ±3 at σ = 200 mS•m -1 . Except HydraProbe of behavior similar to TDRs, other sensors show bigger influence (error of δε r ≈ ±10 at ε r = 40 and σ = 100 mS•m -1 , increasing beyond). Sensor ECH 2 O is even more impacted (negative value at ε r = 40 and σ = 100 mS•m -1 ). Study of temperature influence requires standards which do not vary with temperature or are corrected from the variation, especially for sensors of which guide or electrodes are directly attached to electronics. The layout makes difficult to control electronics temperature independently of electrode one. Authors of the study have to determine variations of liquid permittivity with temperature before sensor characterization [START_REF] Jones | Standardizing characterization of electromagnetic water content sensors[END_REF]], which adds some uncertainty. At the end of the day they report for time travel sensors an absolute drift of δε r ≈ +0.1 to 0.15 • C -1 between 5 to 40 • C and around ε r = 40, or a relative one of +3 500 ppm• • C -1 (ppm: part per million). Sensors CS616, HydraProbe and ECH 2 O present similar drifts, being lower for Thetaprobe. Seyfried et al., using air of which ε r remains constant with temperature, reported for the previous version of CS616, CS615, an absolute drift for sensor output of -0.0005 ms• • C -1 around an average of 0.745 ms, or a relative drift of -700 ppm• • C -1 . Sign and amplitude is not the same as for CS616. Is this because working with a too low permittivity? using a different sensor although with the same technique? or is it resulting from natural dispersion of specifications of sensor electronic parts? An empirical study with limited number of sensors can not answer the questions.

Models EC5 and 5TE, proposed after ECH 2 O by Decagon (5TE provides a multivariable digital output), were studied thoroughly by Rosenbaum et al using the same methodology as for Jones et al [START_REF] Rosenbaum | Correction of temperature and electrical conductivity effects on dielectric permittivity measurements with ech2o sensors[END_REF]]. They intended to establish empirical algorithms to correct for influences of instrument temperature and medium conductivity, taking into account cross-correlation with permittivity (up to ten coefficients for each influence). They provide as well an empirical relation to convert sensor output into ε r . Ten sensors for each product were tested in an attempt to account for sensor variability [START_REF] Rosenbaum | Sensor-tosensor variability of the ech o ec-5, te, and 5te sensors in dielectric liquids[END_REF]]. Mixtures of 2-isopropoxyethanol and water were used for the work. Their permittivity and its dependence with temperature and frequency were determined during the work. Typically, relative drift of liquid ε r due to temperature is about -6 000 ppm• • C -1 . Instrument relative drift varies with permittivity from about -3 000 ppm• • C -1 at ε r = 20 to -5 000 ppm• • C -1 at ε r = 40. Sensitivity to conductivity is lower than for ECH 2 O sensor but still important, depending on ε r : at ε r = 40 absolute error varies from δε r = -9 at σ = 100 mS•m -1 to δε r = -4 at 250 mS•m -1 while at ε r = 26 error δε r varies from -5 at 50 mS•m -1 to +6 at 250 mS•m -1 . In the case of Decagon recent product GS3, it is observed in waterε r = 80 -a decrease of δε r = -21 at σ = 180 mS•m -1 (Figure 6 in [START_REF] Chavanne | Autonomous sensors for measuring continuously the moisture and salinity of a porous medium[END_REF]]).

Along with the digital conversion of measurements, manufacturers have added for its own interest conductivity measurement to their recent products (as well as soil temperature at electrode level): TDR soilvue and reflectometers CS650 of Campbell Scientific, TriSCAN probe for Sentek Pty, 5TE and GS3 for Decagon/Meter..., while keeping the same technique to determine soil water content. Both techniques are thus independent, for instance they do not operate at the same frequency (except for WET Sensor. Information about hydraprobe is insufficient to decide on it). Exists the risk of measuring a conductivity different from the conductivity at f used in eq. ( 1), due to the contribution from dipole relaxations depending on f . Sensors of Campbell Scientific use the amplitude attenuation of trains of 20-µs-wide pulses at 10 kHz frequency [START_REF] Bechtold | Accurate determination of the bulk electrical conductivity with the tdr100 cable tester[END_REF]]. Moreover, it does not seem that this capability is used to correct conductivity influence on permittivity measurement. No real difference on output sensitivity between EC5 and 5TE or GS3 of Decagon. Only Campbell Scientific reports for CS650 the use of an algorithm to correct σ dependence.

The development of our own technique -self-balanced bridge -is intended to address the difficulties encountered in previous sensors. It uses a capacitance approach, actually an admittance one by measuring simultaneously the two quantities C and G in eq. 2. Hence a linear relation is obtained from sensor output to soil apparent complex permittivity at a frequency f (currently 24 MHz), as expressed in eq. 1.

Section 2 exposes instrument characteristics and procedures to obtain an output linear to ε r and σ. Electric circuit theory is used to model the instrument analytically and determine any deviation from linearity, interferences between measured C and G, and other influences on its output. The work links these errors to different parts of the sensor and their known characteristics. The section details modifications on circuit and procedures to offset or reduce the errors, and gives physical relations to account for residues. Physical relations, along with component specifications, allow to assess importance of these errors over the whole range of sensor operation, before any calibration, which is not possible with empirical relations. Choices for sensor designelectronics at surface and bi-cylindrical geometry for parts in soil -are justified to reach simplicity, low disturbance, ease of installation and a large sample volume. Drawbacks -e.g. long lead -are dealt with.

The section 3 presents results from calibration of a sensor with pure liquids and some saline solutions. Electronics drift from thermal variations are studied in detail to identify its physical origin and assess its impact on the final result, water content and salinity during field trials. On-board reference admittance with low drift is used for this work. Means to reduce the drift are indicated.

Design of the instrument has to incorporate constraints of costs and energy consumption by limiting the number of parts, using standard and efficient ones, and by simplifying procedures specific to each sensor. The compromise with measurement quality is detailed. The work is in a research and development phase with successive generations of two or three prototypes. The instrument is made of a sensing part -pair(s) of cylindrical stainless steel electrodes in the soil -, the lead to connect the electrodes to measuring electronics, and the measuring circuit itself inside a separate box at the soil surface (Fig. 1). The housing contains as well units of analog-to-digital conversion, processing and communication assembled in a separate circuit, presently. Battery is enclosed within. Electronics is not compact enough to be inserted in cylinders. The choice is also motivated to have a better radio transmission than being buried and an ease of intervention (for a change of battery). Box dimensions are 14 cm by 14 cm in surface and 6 cm in height, in the case of the last generation sensor (a more compact housing will be used for the next generation). It is fitted with 3 cm high legs at its corners to isolate it from soil surface.The box is laterally 10 cm apart from the point where electrodes are inserted in order to limit perturbations on rain-flow patterns above ground.

Materials and Methods

Sensor design avoids to dig a trench and permits a rapid installation in soil. Two parallel holes are first bored with a large guide from soil surface. The pair of cylinders containing lead and electrodes are inserted, maintained parallel with a spacer, which is removed after installation. A permanent spacer between electrodes, or above, will modify flow of water around them, and therefore will make measurement of moisture not representative of it in soil beyond electrode sampling volume. There may be a risk of a change of electrode spacing due to soil movements. However, we do not observe this as the pair of cylinders in soil still fits the spacer after several months of experiment.

Two types of sensors are developed according to their number of pair of electrodes. Multi-channel sensors measure soil variables at different depths thanks to stacked electrodes on same cylinders [START_REF] Chavanne | Presentation of a complex permittivity-meter with applications for sensing the moisture and salinity of a porous media[END_REF]]. They share the same measuring circuit. Controlled relays connect successively the different pair of electrodes to the circuit. A one-channel sensor has a unique pair of electrodes to determine soil variables at one depth. The electrode geometry is identical for both sensors, with size smaller in the case of one-channel sensor.

Following subsections detail the sensor chain of conversion from raw data -circuit output -to admittance of electrodes in soil, to soil permittivity according to three stages:

-Determination of admittance at the input of measuring circuit taking into account its sensitivity, offset and phase errors.

-Accounting for parasitic impedances of lead.

-Conversion of electrode admittance to soil complex permittivity.

We also indicate the relations presently used to convert permittivity into soil moisture and its water salinity for field measurements.

2.1. Self-Balanced Wheatstone Bridge The core technology of the measuring circuit is a self-balanced Wheatstone bridge, of which a simplified diagram is shown Fig. 2. The entire circuitry is analog. It determines at a frequency f the admittance Y x = G x + j C x 2πf at its input through two direct voltages, V G and V C , each proportional to one admittance part, conductance G x or capacitance C x . The situation corresponds to an on-board admittance Y x with resistance r S = 0 Ω and infinity capacitance C S in series. Bridge contains two branches, similar on their structure, each dedicated to one part of Y x . An excitation voltage v exc , from a quartz-tuned oscillator, is applied between admittance ends. The resulting current i x is balanced owing to a feedback loop made of a current/voltage converter, and a de-modulator and an integrator in each branch. Feedback input is the unbalanced signal at f . Integrator output are the feedback DC voltage Vc and Vg which build up until the bridge is balanced and signal sent to demodulators becomes zero The loop thus provides V G and V C proportional to each part of the current to be balanced, and, as a result, to G x and C x . Voltages are fed to modulators to construct the alternating current i eq which offsets i x . Voltages are limited by component supply and linearity conditions to a range from -3.5 to 3.5 V. They are digitized by an Analogto-Digital Converter (A/DC) in the digital circuit. Data reported in the whole article, except in Fig. 3, correspond to the digitalized output. In the case of last generation sensors voltages are digitized by a bipolar 24-bit A/DC (MCP3903-I/SS of Microchip Technology Inc.), after a voltage division by 7 to accommodate ADC full scale of 1 V. Considering 22 as the effective number of bits the theoretical resolution would be few µV. Actually, due to bridge noise, we observe a resolution of ±0.2 mV from repeatability of measurements at same bridge input. For one measurement the A/D converter performs a sampling to reduce the noise (typ. 50 samples).

v exc G x i x C x G eq C eq i eq V C A i=0 Bridge f 24 MHz feed back loop /2 V G  C/G  G/C relay C S r S
Bridge equilibrium is reached in less than 10 ms (screen shot of an analog output in Fig. 3 when a new admittance is connected), which permits to lower its time of activation. This feature and direct voltages as an output make the bridge suitable for continuous measurement in real time in an autonomous device.

Ideally bridge output should verify:

G x = G eq V G , C x 2πf = C eq 2πf V C , (3)
Parameters G eq and C eq are bridge sensitivity parameters. Their values are fixed by amplifier gains, a voltage divisor and either a resistance or a capacitance according to the branch. These components -mainly resistors -are chosen to present a low dispersion from their nominal value, such as 0.1% for resistors and 1% for capacitors. As a result, bridges of a series of same sensors, i.e. with identical components, will present same sensitivity within specification uncertainty of the critical components. A reference admittance on bridge circuit -see further down -permits as well to control the values of G eq and C eq . Uncertainty on G eq and C eq is assessed as 1%.

Sensitivity parameters are usually varied from G eq = 5 to 10 mS•V -1 , and from C eq = 30 to 50 pF•V -1 . Combining eq. 3 with eq. 2 with a value of 0.14 m for g (assuming also parasitic impedances seen in subsection 2.2 negligible), and taking into account V G and V C range, the capacitive branch covers a ε r range corresponding to values from air to water. The conductive one measures up to around 150 mS•m -1 (or 1 500 µS•cm -1 ). It is possible to raise G eq to extend the range of soil conductivity, at the cost of resolution.

Actually, the linear relationship of eq. 3 corresponds to ideal operations of the bridge in which no offset and phase errors exist. We examine now how to compensate them automatically or by a simple operation.

Correction of offsets A first departure from eq. 3 results from offsets of active components, which gives an output in absence of admittance Y x . Correction is achieved owing to a processor-controlled relay. It disconnects the bridge from electrodes in order to measure open circuit voltages V G0 and V C0 , in addition of V G and V C . Eq 3 is computed with their difference, ∆V G = V G -V G0 and ∆V C = V C -V C0 . Residual offset would be about ±0.3 mV.

As we will see in section 3.3 the operation permits equally to correct part of drifts with temperature of electronics. Furthermore, it avoids determination of V G0 and V C0 for each bridge of a series of same sensors, at the modest cost of a controlled relay. Indeed, as offsets differ due to specification dispersion for a same component, V G0 and V C0 vary from one bridge to another.

Correction of phase errors

The main advantage of the bridge, of which digital electronics are still not capable at this frequency, is its phase resolution, i.e. its ability to discriminate, at the end of the chain of conversion, between soil water content and conductivity [START_REF] Chavanne | Comments to: A novel low-cost instrumentation system for measuring the water content and apparent electrical conductivity of soils[END_REF]]. The tangent of the phase θ of the complex permittivity (eq. 1) is:

tan θ = ε r ε 0 2πf σ = ε r 1.33 σ(mS • m -1 ) C x 2πf G x . (4)
Last equality assumes parasitic contributions of lead to be negligible (see next subsection). In this approximation θ is also the phase of the input admittance Y x = Y 0x e j θ or the inverse of the so-called loss angle.

Let's examine the condition on θ to retrieve accurately a variable when it is smaller than its complex conjugate. For a soil with a large apparent conductivity such as σ = 200 mS•m -1 -or 2.00 mS•cm -1 -(according to Section 4 of [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF]], the bulk conductivity for clayey soils can exceed σ = 100 mS•m -1 . In our trial in Alps σ remains below this value), to obtain the permittivity with a resolution of δε r = ±0.1 at 24 MHz, the system would need a phase resolution of δθ ±0.0007 rad or 0.035 • . It is inherently difficult, especially for a field sensor. The phase tangent itself for a typical permittivity ε r = 25 is tan θ = 0.16. By comparison in the sandy soil of our plot conductivity can be of order of σ = 10 mS•m -1 for a permittivity ε r = 8, which gives tan θ about 1. Both parts are of same importance. A resolution of δε r = ±0.1 requires a less stringent error of δθ ±0.015 rad.

Limitation of the bridge on phase resolution results from phase delays of active components, up to -0.1 rad at 24 MHz. They induce interferences between the two branches of the bridge. Correction to obtain Y x , at first order is:

G x G eq ∆V G -ϕ C/G C eq 2πf ∆V C , C x 2πf C eq 2πf ∆V C + ϕ G/C G eq ∆V G , (5)
where ϕ C/G represents the overall phase error in the capacitive branch, while ϕ G/C is that in the conductive branch.

In order to keep this default negligible, a phase shifter with unity gain is added in each branch (represented by barred circles noted ϕ C/G or ϕ G/C in Fig. 2). Its transfer function T ϕY (symbol Y represents either C/G or G/C) is:

T ϕY = 2 1 -j/(R ϕY C ϕY 2πf ) -1 1 + 2j R ϕY C ϕY 2πf , (6)
where R ϕY and C ϕY are values of the passive constituents of the shifter.

A 100 Ω potentiometer added in series with a 100 Ω resistor constitutes R ϕY . Potentiometer is adjusted to offset phase error ϕ Y of each bridge branch. Capacitance C ϕY is usually 330 pF. As the quantity R ϕY C ϕY 2πf is larger than 5, T ϕY is approximated in first order by the last expression in eq. 6. Its amplitude is close to one while its angle matches to -ϕ Y after adjustment. Its tangent is tan ϕ Y 1/(R ϕY C ϕY πf ). Due again to specification dispersion phase errors differ from a bridge to another. As a consequence, shifters of each bridge must be adjusted against a reference. At the difference of the offset compensation with V G0 and V C0 , there is no automatic procedure to avoid a specific calibration. Nevertheless, the process is facilitated by the addition in bridge circuit of a channel with reference resistor R ref and capacitor C ref in parallel connected to the bridge through a controlled relay. The channel also permits to calibrate the values of bridges sensitivities G eq and C eq . To offset ϕ G/C in the G branch, capacitor is disconnected in the reference channel and the shifter is adjusted to reduce ∆V C to zero at ±0.3 mV in eq. 5. Differentiating the equation gives a potential resolution for ϕ G/C with R ref = 220 Ω and

C eq 2πf = 8.1 mS•V -1 : δϕ G/C R ref C eq 2πf δ∆V C = ±0.0005 rad.
The procedure assumes pure reference resistor and capacitor. In practice the resistor possesses a stray capacitance and the capacitor an equivalent series resistance ESR or a parallel loss. They are a surface-mount (SM) thin metal layer resistor and a SM ceramic multilayer capacitors of COG type, respectively, in a compromise between quality and cost (COG or NPO means temperature compensating with a drift lower than ±30 ppm• • C -1 ). To determine ESR we use a bridge and a high quality capacitor, ATC 100 B Series Porcelain Multilayer Capacitors, with ESR less than 0.02 Ohm for 47 pF below 100 MHz i.e. a pure capacitor at 24 MHz and bridge resolution. Bridge error ϕ C/G is thus offset. We then tested the standard COG capacitor with this bridge. It did not present a detectable phase, i.e. it also behaves like a pure capacitor at 24 MHz. As far as ϕ G/C is concerned an additional difficulty to stray capacitance is the existence of the inductance due to circuit traces, in series with the reference channel. By using a reference resistor of the same series or with same stray capacitance as of R ref but with a higher resistance -4K7Ω -, in order to make negligible the inductance contribution in the output (see below in the case of lead), stray capacitance is determined as 0.2 pF. With the reference resistor of 220 Ω we can assess the parasitic bridge inductance (in the current design its value is 40±3 nH), and even offset it by a capacitor like for the compensation of wire inductance presented below.

Because electronic devices present a drift with temperature change, shifters can introduce small perturbations during field measurement and deteriorate the phase resolution. The point is examined in the subsection 3.3. At this stage bridge offsets and phase errors are corrected and its measurement sensitivity for capacitance and conductance is known within 1%. Bridges of a sensor series with identical component are interchangeable. They gave close output for a same admittance at bridge input.

We now look how to determine electrode admittance Y from the admittance Y x . complex permittivity ε r , by electrode inductance L el and its resistance R el .

Modeling Contributions of Lead and Electrodes to Measured Admittance

Compensation of lead impedance Because the lead from bridge to electrodestwo thin wires -is always surrounded by sheath and air with constant properties, its contribution to Y x is modeled by localized components equivalent to an impedance in series, Z S , and an admittance in parallel, Y p , to Y el (Fig. 4(a)). Relations between Y x and Y el are thus:

Y x = Y el + Y p 1 + Z S (Y el + Y p ) or Y el = -Y p + Y x 1 -Z S (Y x ) . (7)
The admittance Y p = G p +j C p 2πf is dominated by capacitive influence between wires and traces with values C p usually lower than 1 pF. The impedance Z S results mainly from wire self-inductances L S and produces non linear interferences between G el and C el . Development of eq. 7 shows that a conductive soil introduces in the capacitive signal a term -G 2

x L S in F, while the conductive one contains G x C x 2πf L S 2πf in S. The lead between each electrode and the circuit is made of a 250-µm-thin silver-plated Cu wire. Its self-inductance, in absence of magnetic material, is a purely geometrical parameter proportional to wire length, with a coefficient of roughly 1 µH per meter. It is enclosed in a 8-mm-outer-diameter tube made of nonmagnetic stainless steel, like for electrodes. Tube is not electrically grounded as it is observed that a tube grounded from electronics side produces a perturbation on bridge output much larger than the fringing effect, which probably results from an electronic interference between electrodes and leads [START_REF] Chavanne | Study of fringe effects of a tworod capacitor embedded in a medium in order to deduce its permittivity[END_REF]].

In the case of the sensor with the deepest electrodes, with the top at 50 cm from surface, length of each lead is as high as 65 cm and, as a consequence, leads present a large inductance L S . However, the addition in series of a capacitor of capacitance C S permits to offset large part of L S . Hence, the overall impedance Z S is modeled according to:

Z S = r S + j L S 2πf + 1 j C S 2πf = r S + j L S 2πf 1 - 1 L S C S 2πf 2 = r S + j L S 2πf , (8)
where L S is the residual lead inductance. Its precise determination requires a calibration with liquids Wire resistance remains negligible, even accounting for skin effect at 24 MHz. However, to prevent amplifier saturation or distortion in case of short cut or presence of pure capacitance between electrodes, a small resistance r S is added (less than 5 Ω).

Correction of electrode inductance

Y el = tanh(γ) γ Y , (9)
where tanh is hyperbolic tangent, while γ :

γ 2 = -a + j b , or γ = -a/2 + ( a 2 + b 2 )/2 + j a/2 + ( a 2 + b 2 )/2 , with a = L el Cω 2 -R el G , b = L el Gω + R el Cω .
Assuming nonmagnetic medium and currents confined at electrode surface due to skin effect, the quantity L el is mathematically linked to electrode capacitance C according to 4π L el /µ 0 ≡ 4π ε r ε 0 h 2 /C or L el g = µ 0 h 2 (µ 0 being the vacuum magnetic permeability). Therefore, in the case of electrode simple bi-cylindrical geometry inductance value is known and fixed. The same applies to R el , which is found almost negligible (about 0.2 Ω).

The implicit relation between Y el and Y is solved by successive iterations with Y ≡ Y el for the first step.

Conversions to Medium Complex Permittivity

The conductance G and the capacitance C of the admittance Y of electrodes inserted in the medium under study is related to medium permittivity ε r and conductivity σ according to eq. 2, which depends on the geometric factor g.

As the measurement technique is independent of electrode geometry, any choice for it is possible. However, the bi-cylindrical geometry, as shown in Fig. 4(b), allows a large sample volume of soil, and, as a result, deduced soil moisture by the sensor is more representative of average soil moisture close to the sensor than a small volume. Each part of the medium contributes to sensor signal according to a weight proportional to the energy density of electric field at that part [START_REF] Chavanne | Sample volume of a capacitance moisture sensor in function of its geometry[END_REF]. The density is more evenly distributed between two parallel cylinders than for two electrodes along a unique cylinder with same characteristics such as for PR2 of Delta T. In the latter case electric energy remains concentrated too close to cylinder surface where soil has been disturbed. The situation is made worse when electrode surface is protected by a low permittivity layer -tube access like for Enviroscan sensor or a veneer -acting as a low capacitance in series with the medium. Sensor becomes insensitive to medium permittivity when increasing. It explains thus the decrease of sensitivity with soil permittivity for the planar design of the probe presented in [H. R. [START_REF] Bogena | Effective calibration of low-cost soil water content sensors[END_REF]]. In the case of the bi-cylindrical geometry the optimum requires a high electrode diameter Φ such as the ratio to electrode spacing D, Φ/D, is ideally between 0.3 to 0.5. Too thin electrodes like for Campbell's TDRs and reflectometers, including SoilVue (Φ/D = 0.1), concentrate the electric field at electrode surface.

The geometry also provides a simple analytical relation for g. In the case of two parallel cylinders of height h, spacing D and diameter Φ the theoretical relation is (see appendix 2 in [START_REF] Chavanne | Sample volume of a capacitance moisture sensor in function of its geometry[END_REF]]):

g th = π h arccosh(D/Φ) , ( 10
)
where arccosh is the inverse of the hyperbolic cosinus. The value of g th varies from 0.085 m (multi-channel sensors with h = 50 mm, D = 100 mm, Φ = 30 mm) to 0.14 m (one-channel sensor in Fig. 1 with h = 70 mm, D = 20 mm, Φ = 8 mm). The relation is valid in absence of fringing effect, i.e. with the electrode electric field in the medium independent of h, perpendicular to electrode axis and bound by the transverse planes at each electrode end. By using its whole length multi-channel sensors are designed to limit this effect to its end channels, or guards [START_REF] Chavanne | Presentation of a complex permittivity-meter with applications for sensing the moisture and salinity of a porous media[END_REF]]. On the other hand, the one-channel sensor with its sole pair of cylindrical electrodes is affected. It is accounted for in eq. 10 by a dimensionless factor, g/g th [START_REF] Chavanne | Study of fringe effects of a tworod capacitor embedded in a medium in order to deduce its permittivity[END_REF]]. It depends mostly on the geometric ratio h/D, and tends towards 1 as the ratio increases. As sensor ratio in our case is h/D = 3.5, effect including both ends amounts to g/g th = 1.2. It also raises the actual height of the sample volume, all the more as the ratio h/D is lower. However, the total increase represents less than 30% of h for the one-channel sensor (Fig. 9 in [START_REF] Chavanne | Study of fringe effects of a tworod capacitor embedded in a medium in order to deduce its permittivity[END_REF]]).

Sensor Duplication With Same Geometry Impedances in series and in parallel of lead, inductance of electrodes and Factor g are only geometry dependent. Tolerance on compensation capacitors is less than 1%. For sensors equipped with same wires, tubes and electrodes of identical lengths (within 1 mm of tolerance), all these parameters remain unchanged. Change of length is negligible, even taking into account thermal dilations of stainless steel and copper rods or wires (for a length l they are lower than 1/l δl/δT ∼ 20 ppm• • C -1 ).

In subsection 2.1 we have seen how to reduce close to zero bridge off-sets and phase errors, owing in part to an on-board reference admittance.

Bridges with identical components can present same sensitivities with an uncertainty of about 1%. They can be considered identical. Consequently, for a manufacturing series of same sensors the calibration with references to determine all sensor parameters, from bridge sensitivity to g, for one or few of them would be valid for others. This point is important to build a large network of sensors without extensive costs.

Conversions to Medium Water Content and Water Salinity

Among the large number of available relations we have so far limited our choice to the simple Topp's correlation, obtained with TDRs [START_REF] Topp | Electromagnetic determination of soil water content: Measurements in coaxial transmission lines[END_REF]]. For a sandy soil made of more than 80% of quartz (less than 10% of clay), such as in the case of the Paris trial plot, the correlation appears appropriate. It may be less the case for a clayed calcareous soil.

As far as water pore conductivity σ ion is concerned, we use the simple semi-empirical relation suggested by Hilhorst M. after validation with samples of various soils [START_REF] Hilhorst | Dielectric characterisation of soil[END_REF][START_REF] Hilhorst | A pore water conductivity sensor[END_REF]]:

σ ion = σ ε rw ε r -ε r(σ=0) , ( 11
)
where ε rw is the relative permittivity of water at soil temperature.

The function assumes that the only contribution to soil conductivity σ results from ionic current in pore water. Dissipation from dipole relaxations is thus neglected. The ratio (ε r -ε r(σ=0) )/ε rw accounts for the water volume fraction to deduce σ ion from σ. Hilhorst M. verified the relation from ε r measurements on samples at different conductivities σ with a sensor operating at 20 MHz. He determined the constant ε r(σ=0) , which represents the asymptotic value of medium permittivity at low σ. In absence of medium calibration, Hilhorst M. proposes the value ε r(σ=0) = 4.1. However, during a dry spell in which ε r has became low, field measurements in the case of the sandy soil have shown the necessity for a smaller value, 2 or even 1, in order to avoid extreme or negative values of σ ion .

More complex relations are also available [START_REF] Friedman | Soil properties influencing apparent electrical conductivity: a review[END_REF]]. Whatever the method, the quantity σ ion is controlled by the difference between σ and ε r . As a consequence, it is very sensitive to any instrument bias or phase error between permittivity and conductivity. Hilhorst suggests to restrict the use of equation 11 to sensors measuring σ and ε r simultaneously with the same instrument, that is measuring directly the complex permittivity like in our case.

Results, Discussion and Perspective

Calibration with liquids

A calibration with various fluids of known permittivity ε r is necessary to fix precisely some sensor parameters, especially the residual lead inductance L S in eq. 8. Bridge output is converted into the admittance at bridge input thanks to eq. 5. Parasitic admittances of the lead are taken into account with eq. 7, using the expression in eq. 8. The conversion also includes the contribution of electrode inductance L el thanks to eq. 9. Value of L el is known and fixed. It is also the case for bridge sensitivity and phases, if the bridge has been adjusted and the lead is short (lower than 20 cm). Electrode admittance is converted into fluid permittivity and conductivity owing to eq. 2 and factor g given in subsection 2.3.

Fig. 5 shows the result in the case of a one-channel sensor using air, octanol-1, propanol-1, methanol and water at around T = 22 • C. For liquids only the electrodes are covered to eliminate fringing effects, as observed in a previous work [START_REF] Chavanne | Study of fringe effects of a tworod capacitor embedded in a medium in order to deduce its permittivity[END_REF]], and thus to use the factor g th for conversion. Permittivity of pure alcohols are well documented, in particular from laboratory of standards [START_REF] Bottreau | On a multiple reflection time domain method in dielectric spectroscopy: Application to the study of some normal primary alcohols[END_REF]; Weast (1986-87); [START_REF] Gregory | Tables of the complex permittivity of dielectric reference liquids at frequencies up to 5 ghz[END_REF]; [START_REF] Kaatze | Reference liquids for the calibration of dielectric sensors and measurement instruments[END_REF]]. Variations with temperature and with frequency (due to relaxation of molecular dipoles) are given. Uncertainty from the literature is about δε r = 0.15. Data are more accurate than values deduced from measurements performed in a non specialized laboratory (as can be inferred from overestimate by nearly 10% of the static permittivity of glycerol and propanol-1 in [START_REF] Jones | Standardizing characterization of electromagnetic water content sensors[END_REF]). Moreover, working with mixtures of liquids to reach desired permittivity adds uncertainty on reproducibility. Our setup adds some uncertainty estimated at δε r = 0.15. Solutions of methanol at different KCl salinity, and of water, are used. Because of strong molecular dipoles the two liquids present large dissolution, which is not the case of other alcohols. Solution conductivity has been measured with conductimeter 340i of the society WTW (Germany), calibrated against standard aqueous KCl solutions. The absolute uncertainty is about ±2 µS.cm -1 with a relative uncertainty of ±1%.

In Fig. 5 ε r and σ deduced from the conversion are plotted against measured solution conductivity. The graphs show as well the result when not accounting for L S and/or L el . As expected from equations variable ε r is most affected at high salinity. Contribution of L S is the most important, but this depends on the compensation efficiency.

Fig. 5 corresponds to the calibration of the sensor presenting the longest lead among current prototypes, 650 mm from electrode to measuring circuit. It allows to reach a depth of about 50 cm in soil. As a consequence, effects from propagation of electromagnetic waves along the lead wires and reflection can not be completely neglected (prototypes with short lead present a much lower effect). It represents, however, perturbations only dependent on lead characteristics. The different relations and the number of parameters remain unchanged, except that some parameters, bridge sensitivities and phases, differ from values at bridge level (by about +40% for sensitivities and 0.020 rad for phase errors). In total in Fig. 5 five parameters are accurately determined thanks to 9 points of calibration with liquids, each giving a permittivity and a conductivity for each sensor output, or a total of 18 values over a large range in permittivity and conductivity. Overall uncertainty for each variable is assessed at 2%, obtained for the highest conductivity. Air is used to fix precisely the complex admittance in parallel Y p , within δε r = ±0.1 and δσ = ±5 µS.cm -1 .

It is possible to extend the length beyond 65 cm to reach larger depth. Lead inductance must be tightly offset and calibration performed with care. We must assure both electrodes remain parallel mechanically. Fig. 6(a) presents almost one year of continuous measurement of soil variables by a one-channel sensor in Paris. Data have been obtained from sensor output according to procedures and conversion relations detailed in section 2. A series from a rain gauge is also reported. The trial has taken place in a plot on the green roof of laboratory building. Over a year we observe approximately two weather patterns, a wet and cold or cool one during winter and late autumn, while summer and spring experimented drier and hotter one. Paris is submitted to various influences from a continental climate (more present in summer) to an oceanic one from Atlantic (more pronounced during winter, especially winter 2019/20 which, as a result, was one of the mildest over the last 20 years).

Long Term Continuous Measurements

The same sensor has then been installed in an observatory at 800-1000 m elevation on sedimentary mountains, located in Southern Alps close to Digne-les-Bains. Fig. 7 shows series of the same soil variables as in Fig. 6 since sensor installation. Environment is very different from previously, with a clayey calcareous soil and a harsher weather.

The plot experimented frosts and snow falls. At electrode depth no frost was observed as permittivity remained high and, therefore, water was still liquid (or nearly, as towards 22th of January Θ V starts to drop abnormally), but probably frost occurred at soil surface. Frost and snow cover act as a thermal barrier, blocking the diurnal cycle, as well as limiting water flow. After a large rain and snow precipitation in early December, then first snow melting and lastly thaw, water content increased and remained high.

From the end of December until the end of January the plot experimented again snow falls and melting with periods of cover, as reported schematically in Fig. 7.

Water content started to diminish after the end of February and low precipitation. Fine structure of the clay-rich soil permits a good retention to maintain a good water content. As to be expected, σ ion deduced from eq. 11 is controlled by soil temperature and flow of incoming fresh water, decreasing with both (be aware of scale translation and reduction for σ ion ). By contrast, because of the nature of soil in the plot for Paris, a sand mostly made of quartz, water drains more rapidly than in Draix in absence of sufficient rain. Fig. 6(b) shows one of these periods around 15th of April. Salinity quite logically increases as soil becomes drier and warmer. Its diurnal oscillation with soil temperature is becoming more marked. An interesting feature is the apparition of a diurnal cycle as well in the series of water content. In the Alps plot too water content are modulated by diurnal cycles, especially at the end of November before the soil becomes wetter and in early spring 2021. Because of the thermal dependence of ionic conductivity in water via its viscosity, σ ion and σ are naturally influenced by soil temperature T soil . It is more difficult to understand the variation of Θ V or ε r . As mentioned in introduction, before any physical interpretation at soil level, effects from the instrument itself have to be ruled out or shown to be small. Fig. 8(a) shows the variations of soil electrical quantities ε r and σ, along with T soil , over two diurnal cycles in September 2020 when temperatures experimented large changes. Fluctuations of ε r and σ appear too different from each other to deduce any interference between both quantities. Moreover, it would require much larger phase error or parasitic impedance than what is left after all works to offset them, as described in section 2; conductivity presents a diurnal variation of δσ = ±0.3 mS•m -1 . From eq. 4 and 5 variation of δε r = ±0.1 on ε r would require ϕ G/C close to 0.5 rad.

On the other hand, the series in Fig. 8(a) and 8(b) present some accidents and do not follow perfectly cycles of T soil , in particular the series of σ and salinity. A potential cause is bridge electronics influenced by temperature inside the housing at soil surface. The change of weather seen in Fig. 6(b) also corresponds to a smaller sky cover, which leads to stronger variations of air temperature, and therefore of electronics temperature T elc during a diurnal cycle. Sensor setup with electrodes distant from electronics makes variations of T soil and T elc different in time and amplitude, and permits to dissociate their effects. Its drawback is large variations of T elc if no care is taken. Previously, we gave some arguments to rule out an electronic influence on the apparent oscillation of water content [START_REF] Chavanne | Presentation of a complex permittivity-meter with applications for sensing the moisture and salinity of a porous media[END_REF]]. However, with more available data during field trials such as that of T elc and output from a reference channel in bridge circuit, we assess in more depth to which degree influence of T elc is still felt, its physical origin, and hence how to further reduce it, as detailed below.

N.B.: Each sensor has a designation according to their generation and their number in it (like 4G3 or 3G1). Generation concerns the evolution of sensor system in terms of control and energy, not necessarily that of the bridge, as board of the latter is removable.

Temperature Influence Due to Electronics Drift

From a measuring channel

Bridge sensitivity (eq. 3) but also its phases (eq. 5) can vary with electronics drift, and hence modify the output without change at bridge input. Fig. 9 shows variations of bridge outputs corresponding to data in Fig. 8. The time interval experimented one of the highest temperature variations over a day, both in air and soil, during the trial in Paris (Fig. 6(a)). Temperature T elc is equally reported to study its influence. On day 14 th of September 2020, T elc rose from 15 to 45 • C, or by more than δT elc = 30 • C. As a consequence, bridge output, V G and V C when connected to electrodes, and V G0 and V C0 when in open circuit, present following variations, when their estimation is possible:

δV G ∼ -? mV δV C ∼ -15 mV δV G0 ∼ -15 mV δV C0 ∼ -12 mV δ(∆V G ) ∼ ? mV , δ∆V C ∼ -3 mV . ( 12
)
As they are intrinsic to bridge circuit, voltages V G0 and V C0 follow closely T elc . From values in eq. 12 their dependence is about 0.5 mV• • C -1 , which is consistent with the temperature drift of offsets of active components reported in specifications (±0.1 mV• • C -1 for one component). Its sign of variation is not relevant, as confirmed in the case of another bridge for which variations are reversed (see Fig. 11).

Output V G and V C is also influenced by T elc variations. However, another diurnal cycle is superposed with attenuated amplitudes and shifted times. It corresponds to the influence of T soil as evidenced by series in Fig. 10, in which T soil is reported. Time shift between cycles is less than few hours as electrodes remain close to soil surface at about 10 cm. The difference of output, ∆V G and ∆V C , should cancel the influence from T elc . It is not exactly the case, in particular for G branch. Changes of components in this circuit have probably created some fragility. It is not visible in a more recent bridge (see below). Both voltages still follow more closely T soil than in the case of V G and V C . Nevertheless, determination of salinity from eq. 11 is sensitive to small perturbations on conductivity calculated from ∆V G and, as a consequence, σ ion is impacted by T elc cycles as seen in Fig. 8(b). The same data from a more recent bridge are also examined with the use of its reference channel constituted of a resistor of value R ref = 220 Ω. Output for an admittance mounted on surface board is only influenced by T elc , as observed in Fig. 11, which permits a more thorough study of instrument interferences. On 14 th of September, from dawn to mid-day T elc rose from 15 to 55 • C, or by more than δT elc = 40 • C (more than in the case of the one-channel sensor probably due to the type of housing for this sensor not as optimized as for the previous one). Bridge output varies accordingly:

From a reference channel in bridge circuit

δV Gref ∼ +3.5 mV δV Cref ∼ +9.5 mV δV G0 ∼ +6.5 mV δV C0 ∼ +4.5 mV δ(∆V Gref ) ∼ -3 mV , δ(∆V Cref ) ∼ +5 mV . ( 13
)
The correction of V G0 and V C0 is either insufficient, as in the case of ∆V Cref , or too large, as in the case of ∆V Gref . Output V G0 and V C0 permits to cancel drifts from bridge offsets but not from its gains, in particular from bridge sensitivity G eq and C eq (in eq. 3) and from phase shifters of functions T ϕC/G and T ϕG/C (eq. 6). The drifts from the latter ones reintroduce phase errors, δϕ Y , in the relations of conversion. Retaining only first order perturbations in eq. 5, relations between variations due to drifts for the reference channel are:

δ 2 G ref ∼ δ 2 G eq (∆V Gref ) 2 + G 2 eq δ 2 ∆V Gref , δ 2 C ref (2πf ) 2 ∼ (C eq 2πf ) 2 δ 2 ∆V Cref + δ 2 ϕ G/C (G eq ∆V Gref ) 2 . ( 14
)
The output ∆V Cref in absence of capacitance C ref to measure, is still affected by T elc through V G and the drift of the shifter in the G branch.

The bridge sensitivity G eq is deduced from calibration by the reference channel with data from Fig. 11

(a), G eq 1/(R ref ∆V Gref ) = 1/(0.22 • 0.476) = 9.5 mS•V -1 at 20 • C. For the C branch sensitivity is C eq 2πf 47/0.86 • 0.15 = 8.1 mS•V -1 at 20 • C.
Part of the drifts in bridge operations results from the temperature drift of passive components used for these operations. If surface-mount (SM) ceramic multilayer capacitors are of type COG(NPO) with a drift lower than ±30 ppm• • C -1 , no special care was taken in the choice of resistors. Specifications of the SM thick-film resistors in the circuit indicate a drift up to δR/R ∼ ±200 ppm• • C -1 , which amounts to ±0.8% for δT elc ∼ 40 • C.

In the following we assume that the drifts of resistances account for observed variation of ∆V Gref and ∆V Cref in Fig. 11 via G eq , the resistance in the reference channel, R ref , and ϕ G/C . Sensitivity drift δ 2 G eq is assumed mainly controlled by the drift of a resistance (δR/R) eq according to:

δ 2 G eq ∼ (δR/R) 2 eq G 2 eq . (15)
Drifts from reference components, which impair the analysis but not instrument performance, are:

δ 2 G ref ∼ δ 2 R ref /R 4 ref (δR/R) 2 ref 1/R 2 ref , δ 2 C ref (2πf ) 2 ∼ 0 . ( 16
)
Drift δϕ G/C for phase shifter of branch G is inferred from eq. 6:

δ 2 ϕ G/C δ 2 tan ϕ G/C ∼ δ 2 2 R ϕG/C C ϕG/C 2πf ∼ 2 R ϕG/C C ϕG/C 2πf 2 (δR/R) 2 ϕG/C .
Hence, by combining drift assessments and values for G eq and C eq with the uncertainty relations in eq.14, we have:

   (δR/R) 2 ref + (δR/R) 2 eq ∼ (δ∆V Gref /∆V Gref ) 2 = (3/476) 2 or (0.6%) 2 , δϕ G/C ∼ (δR/R) ϕG/C /(R ϕG/C C ϕG/C πf ) ∼ (C eq 2πf /G eq )δ∆V Cref /∆V Gref 1% . ( 17
) In the first relation, variation of ∆V G during the diurnal cycle is consistent with the temperature drift of resistors. Moreover, when subsisting for the standard resistors in G eq and R ref resistors with drift lower than δR/R ∼ ±25 ppm•

• C -1 , drift of G eq has been reduced below ±100 ppm• • C -1 .
Probably, drifts from other components such as the multipliers through their gain (ideally one) has to be included. Moreover, because the sign of resistor drift is random, all gains fixed by a ratio of resistances present as well a thermal drift. All of this should explain that in the second relation of eq. 17 about bridge phase errors, the equivalent drift for (δR/R) ϕG/C is from three to five times as high as for a standard resistor. Nevertheless, substitution with a better resistor for R ϕG/C in the shifter has also lowered the drift on ϕ G/C below ±150 ppm• • C -1 rad, or 0.006 rad during a change of electronics temperature of 40 • C.

In general, what would be the influence of a thermal phase drift on the monitoring of ε r in presence of high soil conductivity? By differentiation of eq. 4 and approximating drift on phase θ with that on ϕ G/C , δϕ G/C , induced error on ε r , δ elc ε r , is:

δ elc ε r ∼ δϕ G/C σ(mS • m -1 ) 1.33 . ( 18
)
For σ = 200 mS•m -1 , phase drift would produce an amplitude variation of δ elc ε r ±1 with δϕ G/C ∼ ±150 ppm• • C -1 rad and δT elc 40 • C. Fluctuation would be in phase with that of T elc , not that of soil temperature T soil . Let's examine the consequence of this relation in the case of the plot of Southern French Alps where soil conductivity is higher than in Paris plot. . Actually ε r varies in phase with T soil , with an amplitude of δε r ∼ 1.5. It is modestly affected by variation of T elc and a phase thermal drift. The observation confirms the result from eq. 18. Conductivity is not large enough to produce an instrumental effect. However, it shows the necessity of a low shift drift.

In this case δε r is a genuine effect at soil level, independent of the instrument. As such study is beyond article scope it does not intend to find the origin of this dependency. We just notice that the change on ε r is in phase with temperature one, which is opposite to water-permittivity dependence with temperature.

Tests in which the channel reference has only a capacitor give similar results for the drifts of sensitivity C eq and phase ϕ C/G of the C branch.

Unitary price of resistors with tighter specification is still low ($0.6 versus $0.07 for standard ones, at 0.25 W w/o VAT for a purchase of 50 units). Their assembly in case of few prototypes adds fees as these components are not mounted by default on assembly machines. However, assembly for series of 50 and 100 boards will make this fixed cost negligible.

The use of these resistors to control bridge sensitivity and phases, in addition of a better thermal insulation, should avoid a correction of instrument thermal drift specific to each bridge. Nevertheless, through the reference channel in bridge circuit it is still possible to derive phase drifts δϕ G/C and δϕ C/G . The high bridge resolution (few tenths of mV as seen in Fig. 11(a)) also permits these corrections and study these delicate effects.

Conclusion

The article has provided a comprehensive description of an in-situ sensor to monitor soil moisture and water salinity using soil permittivity ε r and conductivity σ at a frequency 24 MHz as intermediate via an admittance approach. Soil complex permittivity presents a good sensitivity to soil quantities and numerous conversions exist in the literature. Instrument originality relative to existing sensors lies on the use of a self-balanced Wheatstone bridge. It is able to provide on one measurement a digital output proportional to the capacitance and conductance of electrodes in soil, hence its complex permittivity. The article has detailed from the instrument output to the electrodes all potential deviations from linearity and interferences between the two admittance parts: bridge offset and phase errors, as well as parasitic impedances. They have been modeled using electric circuit theory, and related to circuit components specifications. The work provides the means to compensate the largest part of them intrinsically or automatically (controlled relays, series capacitors and a reference admittance in bridge circuit), and analytical relations to correct residual effects, like electrode inductance.

The whole sensor is made of an electronic housing placed at ground surface and two parallel steel cylinders, with lead to bridge circuit inside and electrodes at the bottom, which are inserted in soil. The design permits easy installation from surface, low disturbances and a high sample volume around electrodes. It provides an analytical relation to convert admittance to soil permittivity and conductivity.

Calibration of a sensor with electrodes at 50-cm depth using air, alcohols and water fixes its sensitivity, as well as its residual phase and inductance, within an uncertainty of 2% over a range up to ε r = 80 and σ = 200 mS•m -1 . Identical sensors can use same parameters after adjusting each phase errors in one simple operation.

Bridge thermal drifts during diurnal cycles and its consequences on soil variable measurement are investigated to identify physical origins. As a result, a judicious choice of resistor types permits to reduce thermal drift of phase within ±150ppm• • C -1 rad in order to keep sensor accuracy even with high conductivity and temperature changes.

Figure 1 .

 1 Figure 1. One-channel sensor of last generation during a field trial. Box containing electronics, and aerial part of lead towards electrodes in the soil.

Figure 2 .

 2 Figure 2. Schematic diagram of the self-balanced Wheatstone bridge to measure conductance Gx and capacitance Cx at frequency f . Bridge excitation at f , vexc, is provided by a quartz-tuned analog oscillator. A feedback loop assures a fast equilibrium and provides direct voltages V G and V C proportional to Gx and Cx with respective coefficients Geq and Ceq. A controlled relay and phase shifters ϕ G/C and ϕ C/G permit to correct deviations from perfect linearity and phase resolution. When connected to electrodes through a lead a resistor r S and a capacitor C S in series ensure protection and compensation of wire inductance (see subsection 2.2).

Figure 3 .

 3 Figure 3. Screen shot of an oscilloscope in recording mode. Bridge analog output of the capacitive branch V C -see Fig. 2 -is recorded when a sensor channel with high capacitance is connected. Bridge reaches its new equilibrium in less than 10 ms (time division of 20 ms. Noise in amplitude due to low resolution of oscilloscope).

Fig. 4 Figure 4 .

 44 Fig. 4(a) breaks up the contributions to the admittance Y x measured at bridge input. Admittance Y el of electrodes at lead ends differs from Y , due specifically to medium

  To determine Y from Y el the pair of electrodes are treated as an open-ended transmission line. It leads to the resolution of a differential equation along electrode length of which parameters include L el and R el , as well as Y . Taking into account the boundary conditions at electrode ends, Y el corresponding to the admittance at the top ends, the solution is:

Figure 5 .

 5 Figure 5. Calibration of a one-channel sensor fitted with a 650-mm-long lead using fluids of known permittivity εr and conductivity σ (air, octanol-1, propanol-1, methanol and water at around T = 22 • C. Methanol and water at different salinity) to adjust some parameters of the electric circuit model depicted in Figure 4(a). The main one is the residual lead inductance L s . Electrode geometry and dimensions impose the value of electrode inductance L el . A series resistor of rs = 4.9 Ω inserted for protection is accounted for. Plots show the influence of both inductances in the raw data. Calibration results for (a) permittivity εr (b) conductivity σ.

Figure 6 .

 6 Figure 6. Monitoring of soil water content Θ V (right hand scale), its salinity σ ion (translated and reduced in left hand scale) and soil temperature T soil (lhs) by a one-channel autonomous sensor in a plot on a roof in Paris (a) over the trial time; (b) over a period around 15th of April 2020. Transition from a wet and cold weather to a drier and hotter one. Precipitations from a close rain gauge are also reported (lhs).

Figure 7 .

 7 Figure 7. Series as in Fig. 6 from the same sensor since November 2020 in the observatory in Southern Alps. Winter period with occurrences of snow cover.

Figure 8 .

 8 Figure 8. Soil variables against soil temperature T soil corresponding to series in Fig. 6(a) over few diurnal cycles in mid-September 2020 (time in x axis as day:hour). (a) Electric variables εr (right hand scale) and σ (left hand scale), and (b) physical variables water content (rhs) and its salinity (lhs).

Figure 9 .Figure 10 .

 910 Figure 9. Bridge output corresponding to time series in Fig. 8, against electronics temperature T elc (time in x axis as day:hour). Voltages V G and V C are the output when bridge is connected to electrodes, while V G0 and V C0 when it is in open circuit. Voltages ∆V G and ∆V C are their differences. (a) Data for the G branch and (b) for the C branch.

Figure 11 .

 11 Figure 11. Bridge output for an on-board reference channel in a multi-channel sensor against electronics temperature T elc during two days in mid-September 2020. (a) data for the G branch and (b) for the C branch.

Figure 12 .

 12 Figure 12. Soil electric variables εr (right hand scale) and σ (translated and reduced in lhs) corresponding to data in Fig. 7 over few diurnal cycles in early spring 2021 (time in x axis as day:hour), (a) against electronics temperature T elc (translated and reduced in lhs) and (b) against soil temperature T soil (lhs).

Fig. 12

 12 Fig. 12 shows variations of ε r and σ corresponding to data in Fig. 7 over few days in early spring 2021. They are reported against temperature T elc and soil temperature T soil . Soldering fragility of sensor bridge has been reduced. No algorithm correction is used. Taking a phase drift of δϕ G/C ∼ ±250 ppm• • C -1 rad for the bridge used, relation (18) gives an error δ elc ε r ∼ ±0.15, with δT elc ∼ 30 • C (from -4 to 26 • C) and a soil conductivity around σ = 40 mS•m -1. Actually ε r varies in phase with T soil , with an amplitude of δε r ∼ 1.5. It is modestly affected by variation of T elc and a phase thermal drift. The observation confirms the result from eq. 18. Conductivity is not large enough to produce an instrumental effect. However, it shows the necessity of a low shift drift.In this case δε r is a genuine effect at soil level, independent of the instrument. As such study is beyond article scope it does not intend to find the origin of this dependency. We just notice that the change on ε r is in phase with temperature one, which is opposite to water-permittivity dependence with temperature.

Acknowledgements

This work is supported in part by the grant ANR11EQPX0011 of the Agence Nationale de la Recherche allocated to the program "Critex"-Challenging equipment for the temporal and spacial exploration of the Critical Zone at the catchment scale-(20 March 2013). It won the calls to submit projects for the "2012 Facilities of Excellence" competition named Equipex. The program Critex is a French national project, which groups together 21 laboratories from more than 20 universities and four research organizations. It will allow the study of surface and underground water relating to the characteristics of the natural environment and surrounding human activities. It will thus contribute to a better management of water resources in varied contexts. IPGP contribution number: 4258

Disclosure statement

The authors declare no conflict of interest.