Moua Rohianuu

Lesparre Nolwenn

Girard Jean-François

Belfort Benjamin

François Lehmann

Younes Anis

On the interest of ground penetrating radar data for the estimation of unsaturated soil parameters

Keywords: coupled hydrogeophysical model, time-lapse ground penetrating radar, unsaturated soil parameters, global sensitivity analysis, Bayesian parameter estimation, uncertainty quantification

In this study, the interest of ground penetrating radar (GPR) time-lapse measurements for the estimation of hydrodynamic unsaturated soil parameters is investigated using synthetic infiltration experiments. Water movement and electromagnetic wave propagation in the unsaturated zone are modeled using a one-dimensional hydrogeophysical model. The GPR travel time data are analyzed for different reflectors: a moving reflector (the infiltration wetting front) and three fixed reflectors located at different depths in the soil. Global sensitivity analysis (GSA) is employed to assess the influence of the saturated hydraulic conductivity, the saturated and residual water contents, and the Mualem-van Genuchten shape parameters α and n of the soil on the GPR travel time data of the reflectors. Statistical calibration of the soil parameters is then performed using the Markov chain Monte Carlo (MCMC) method. The impact of the type of reflector (moving or fixed) is then evaluated by analyzing the calibrated model parameters and their confidence intervals for different scenarios. GSA results show that the sensitivities of the moving and fixed reflectors data to the hydrodynamic soil parameters are different whereas the fixed reflectors have similar sensitivities. Results of parameter estimation show that the use of only data from the moving or fixed reflectors does not allow a good identification of all soil parameters. When both data are combined, all soil parameters can be well estimated with narrow confidence intervals.

Introduction

The vadose zone is defined by the region between the ground surface and the groundwater table. Because of its location, it is at the center of the interactive atmospheric-surface-underground water system. Hence, understanding water flow in the vadose zone is crucial for hydrological modeling and forecasting that can be useful for water resources management, agricultural practices optimization, or geotechnical studies. The porous medium in the vadose zone is filled by both water and air phases.

The air phase is considered infinitely mobile and remains at atmospheric pressure. The movement of water has a non-linear behavior and is characterized by two fundamental hydraulic relationships, namely the water retention and the hydraulic con-1 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License. ductivity functions. Various mathematical expressions can describe these functions in terms of dependent variables and fitting parameters. In this work, we use the Mualem-van Genuchten [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils[END_REF] hydraulic conductivity and water retention models. These models include the following unsaturated soil hydraulic parameters: the saturated hydraulic conductivity, the saturated and residual water contents, and the Mualem-van Genuchten shape parameters α and n.

Different approaches can be applied to estimate the unsaturated soil parameters. A typical and prevalent approach relies on laboratory measurements conducted on soil core samples. The typical measurement approaches can be of various types but common practices rely on hydraulic fluxes measurements [START_REF] Vereecken | On the value of soil moisture measurements in vadose zone hydrology: A review[END_REF]. If this approach can be accurate at the column scale, it is prone to certain limitations when the objective is to deduce the soil parameter values at larger scales.

Indeed, sample analysis through laboratory experiments is unlikely to provide parameter estimates at field conditions since the volume of the analyzed samples is often not representative of the field heterogeneity at the mesoscale (Scharnagl et al., 2011). In addition, the method is invasive and can be labor-intensive for deep or large scales investigations [START_REF] Binley | The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales[END_REF]. Furthermore, the conservation of collected samples can be challenging because of issues of compaction and changes in porosity.

A more appropriate approach for large scales is based on the monitoring of underground water content. In fact, water content measurements can show significant variations because of their sensitivity to different hydrological processes. As a consequence, they are convenient for the estimation of soil parameters of the subsurface at the field scale by inverse modeling.

Water content measuring techniques can be classified into two categories, based on whether they provide information on water content directly or not. The first group uses direct measurements conducted at the point scale with water content sensing techniques, for instance using thermal or electromagnetic sensors (e.g., capacitance or time domain reflectometry, [START_REF] Jones | Standardizing Characterization of Electromagnetic Water Content Sensors: Part 1. Methodology[END_REF][START_REF] Belfort | Laboratory Experiments of Drainage, Imbibition and Infiltration under Artificial Rainfall Characterized by Image Analysis Method and Numerical Simulations[END_REF]. These techniques can yield data with great resolution at one location and give information on the dynamics at the field scale [START_REF] Vereecken | On the value of soil moisture measurements in vadose zone hydrology: A review[END_REF]. In addition, measurements taken at various locations can help to describe the distribution of water content, and thus, allow a good characterization of the state of the soil. However, the installation of sensors is often laborious, time-consuming, and destructive (Huisman et al., 2003;[START_REF] Dal Bo | Geophysical imaging of regolith in landscapes along a climate and vegetation gradient in the Chilean coastal cordillera[END_REF]. Furthermore, their reliability requires an accurate calibration [START_REF] Robinson | Soil Moisture Measurement for Ecological and Hydrological Watershed-Scale Observatories: A Review[END_REF].

The second category uses indirect measurements of the water content, like remote sensing and hydrogeophysical methods with non-invasive devices. Remote sensing techniques use devices that are not in direct contact with the ground, such as unmanned aerial vehicles thermal infrared imagery (Zhang et al., 2019) or airborne ground penetrating radar [START_REF] Edemsky | Airborne Ground Penetrating Radar[END_REF]. These methods provide the mapping of water content at a large scale and in locations where direct sensing measurements cannot be conducted. However, remote sensing methods exhibit a penetration depth of only a few centimeters and are often limited by the vegetation density [START_REF] Vereecken | On the value of soil moisture measurements in vadose zone hydrology: A review[END_REF][START_REF] Robinson | Soil Moisture Measurement for Ecological and Hydrological Watershed-Scale Observatories: A Review[END_REF].

Common hydrogeophysical methods include electromagnetic induction [START_REF] Doolittle | The use of electromagnetic induction techniques in soils studies[END_REF], direct current resistivity [START_REF] De Jong | Monitoring Soil Moisture Dynamics Using Electrical Resistivity Tomography under Homogeneous Field Conditions[END_REF], nuclear magnetic resonance [START_REF] Costabel | Noninvasive Estimation of Water Retention Parameters by Observing the Capillary Fringe with Magnetic Resonance Sounding[END_REF], and ground penetrating radar (GPR) (Huisman et al., 2003;[START_REF] Klotzsche | Measuring Soil Water Content with Ground Penetrating Radar: A Decade of Progress[END_REF] methods. These techniques supply indirect information on hydraulic properties or states, at various scales, from estimated geophysical properties. As mentioned by [START_REF] Binley | The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales[END_REF], such conversion from geophysical to hydraulic properties or states requires the use of robust petrophysical relationships to provide reliable estimates of hydraulic parameters.

Nowadays, GPR is highly used in the field of hydrogeophysics. Different techniques have been reviewed and discussed by Huisman et al. (2003) and [START_REF] Klotzsche | Measuring Soil Water Content with Ground Penetrating Radar: A Decade of Progress[END_REF]. GPR is highly sensitive to water content, and, as such, it can close the gap between the spatial scales covered by direct and remote sensing techniques. Furthermore, the temporal variability of the soil water content can be characterized from time-lapse measurements. For instance, GPR data can be collected during artificial experiments (e.g., infiltration, runoff, drainage, imbibition) that can provide interesting information on the flow characteristics (e.g., Saintenoy et al., 2008;[START_REF] Moysey | Hydrologic trajectories in transient ground-penetrating-radar reflection data[END_REF][START_REF] Scholer | Estimating vadose zone hydraulic properties using ground penetrating radar: The impact of prior information[END_REF][START_REF] Busch | Coupled hydrogeophysical inversion of time-lapse surface GPR data to estimate hydraulic properties of a layered subsurface[END_REF][START_REF] Jonard | Estimation of Hydraulic Properties of a Sandy Soil Using Ground-Based Active and Passive Microwave Remote Sensing[END_REF]Jaumann and Roth, 2018;[START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF]2016;2020). Note however that the hydraulic properties estimated from GPR data are subject to an inherent compromise between a deep investigation and a fine spatial resolution. For instance, the lowest frequencies (typically from 1 GHz down to 100 MHz) allow deeper penetrations (until a maximum depth between 1 m and 3 m in most organic media). [START_REF] Saintenoy | Ground Penetrating Radar: Water Table Detection Sensitivity to Soil Water Retention Properties[END_REF] performed a sensitivity analysis to evaluate the importance of the soil parameters on the reflections caused by the water table. They found that the detectability of the GPR reflections is mainly dependent on the slope of the retention curve in the capillary zone. [START_REF] Moysey | Hydrologic trajectories in transient ground-penetrating-radar reflection data[END_REF] showed that the Mualem-van Genuchten parameter n is the most poorly constrained among all unsaturated soil parameters when considering the two-way travel time (TWT) from various sources of reflection for a laboratory infiltration experiment conducted in a sandbox. At the laboratory scale, [START_REF] Léger | Evaluating hydrodynamic parameters accounting for water retention hysteresis in a large sand column using surface GPR[END_REF] have monitored imbibition-drainage experiments using a single-offset surface GPR. At the field scale, infiltration experiments have been monitored with a single-offset surface GPR [START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF]2016) or off-ground GPR [START_REF] Jadoon | Estimation of Soil Hydraulic Parameters in the Field by Integrated Hydrogeophysical Inversion of Time-Lapse Ground-Penetrating Radar Data[END_REF]. [START_REF] Saito | Tracking Infiltration Front Depth Using Time-lapse Multi-offset Gathers Collected with Array Antenna Ground Penetrating Radar[END_REF] used a more complex multi-offset and multi-channel surface GPR to directly monitor the wetting front progression.

Mono-channel multi-offset technique is usually not suited for monitoring experiments with high temporal variability, as the offset must be adjusted between each measurement. The multi-channel technique has the advantage to be multi-offset and is, therefore, able to simultaneously determine the propagation speed and the depths of reflectors. Time-lapse GPR monitoring of artificial infiltrations is one of the cheapest non-destructive methods that can be easily conducted to estimate the unsaturated soil parameters in field conditions. [START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF] have demonstrated the relevance of such a methodology to evaluate the hydraulic parameters of sandy soil. The authors have investigated synthetic and field examples and showed that the inverted parameters were in agreement with the values obtained in the laboratory for soil samples and with disk infiltrometer measurements. However, in their study, [START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF] didn't assess the reliability of the estimated values since the uncertainty associated with the calibrated parameters has not been evaluated. Furthermore, [START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF] employed only the TWT data obtained from the GPR reflection on the wetting front for the calibration of the soil parameters. The present study extends the work of [START_REF] Léger | Hydrodynamic parameters of a sandy soil determined by ground-penetrating radar inside a single ring infiltrometer[END_REF] by:

-Considering different reflectors at different depths: a moving reflector which corresponds to the infiltration dynamic wetting front and two fixed reflectors located at different depths in the soil. contents, and the Mualem-van Genuchten shape parameters α and n) on the GPR TWT data of the three reflectors using Global Sensitivity Analysis (GSA).

-Performing statistical calibration of soil parameters using the Markov Chain Monte Carlo (MCMC) method and evaluating the reliability of the estimated parameters by analyzing not only the calibrated model parameters but also their associated uncertainty.

-Evaluating the impact of the type of reflector (moving or fixed) by analyzing the calibrated model parameters and their confidence intervals for different scenarios.

The plan of the paper is as follows: Section 2 describes the test case as well as the mathematical and numerical hydrogeophysical models. Section 3 reports on the GSA results of the different TWT signals. Then, Section 4 discusses the results of soil parameter estimation with MCMC for different scenarios.

2 Test case description and numerical solution

Test case description

The test case considered in this work is a hypothetical one-dimensional experiment of water infiltration in a homogeneous sandy soil of 150 cm (Fig. 1a). The infiltration is driven by a constant pressure head of 10 cm applied at the surface of the soil (i.e., a 10 cm water ponding condition is maintained at the top). The medium is initially at the hydrostatic equilibrium with a water table maintained at 100 cm below the soil surface (Fig. 1b). The domain is initially formed by an unsaturated zone of 100 cm thick above a saturated zone of 50 cm thick. We assume the experiment to be monitored with a surface GPR.

The propagating time (i.e., the TWT) of the GPR waves reflected by two types of reflectors are considered (Fig. 1c): (i) the moving infiltration wetting front and (ii) two fixed reflectors corresponding to a local heterogeneity at two different depths.

For instance, these can be small objects that are artificially buried (e.g., moisture sensing probes) or naturally embedded (such as small rocks) in the porous medium. The fixed reflectors are supposed to be small enough compared to the section of the infiltrated area, so they do not significantly perturb the vertical flow. The upper fixed reflector, R50, is located in the initially unsaturated zone at 50 cm depth. The reflector R120 is located in the saturated zone, under the water table, at a distance of 120 cm from the soil surface (Fig. 1a). In the following, the time-lapse TWT signal for reflection caused by the infiltration wetting front is noted TWT f and that from the two immovable diffracting points R50 and R120, are respectively noted TWT 50 and TWT 120 . Water infiltration in unsaturated/saturated soils is governed by the one-dimensional Richards' equation [START_REF] Richards | CAPILLARY CONDUCTION OF LIQUIDS THROUGH POROUS MEDIUMS[END_REF]:

∂θ ∂t = ∂ ∂z K(θ) ∂h ∂z -1 (1)
where h (cm) is the pressure head; z is the depth (cm), taken positive in the downward direction; t is the time (s), θ (cm 3 /cm 3 ) is the actual water content, and K(θ) (cm/s) is the hydraulic conductivity which is a function of water content. The initial condition is a hydrostatic pressure distribution corresponding to a water table at 100 cm depth. The boundary condition at the 125 top of the domain is a fixed Dirichlet condition of 10 cm maintained during the experiment. The boundary condition at the bottom is a piezometric head fixed at -100 cm which corresponds to the water table position (Fig. 1). The interdependencies of the pressure head, conductivity, and water content are described using the standard models of [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils[END_REF]:

S e (h) = θ(h) -θ r θ s -θ r =    [1 + (α|h|) n ] -m if h < 0 1 if h ≥ 0 (2) K(h) =    K s × S e (h) 1/2 [1 -(1 -S e (h) 1/m ) m ] 2 if h < 0 K s if h ≥ 0 (3)
where S e (h) (-) is the effective saturation, θ s and θ r (cm 3 /cm 3 ) are the saturated and residual water contents respectively, K s (cm/s) is the saturated conductivity, m = 1 -1/n, α (1/cm) and n (-) are the Mualem-van Genuchten shape parameters.

Petrophysical and Geophysical relationships

In GPR sounding, pulses of radiofrequency (MHz to GHz) electromagnetic waves are emitted from a transmitting antenna through the sounded medium. The electromagnetic response is then acquired with a receiving antenna. With a surface GPR, both antennas are installed at the surface of the soil Fig. 1). To monitor the experiment of water infiltration with time-lapse GPR, the sounding system is set immobile above the infiltration zone in order to capture the time variation of the electromagnetic response due to the change of saturation.

To describe the dependency of the dielectric permittivity on the water content, we use the complex refractive index model [START_REF] Birchak | High dielectric constant microwave probes for sensing soil moisture[END_REF]. This petrophysical relationship relates the dielectric constant ϵ (-) of a three-phase (water-solid-air) medium to its water content by:

ϵ(z, t) = θ(z, t) √ ϵ w + (1 -ϕ) √ ϵ s + [ϕ -θ(z, t)] √ ϵ a (4)
where ϕ (-) is the porosity, ϵ w = 80, ϵ s = 2.5 and ϵ a = 1 are the relative dielectric constants of water, silica (sand) and air, respectively.

In a low electrical conductivity and non-magnetic soil, the electromagnetic waves propagate at a speed V (cm/ns) [START_REF] Annan | Ground penetrating radar: Principles, procedures and applications[END_REF]:

V = c √ ϵ (5)
where c ≈ 30 cm/ns is the speed of electromagnetic waves in air, and ϵ (-) is the dielectric constant of the porous medium.

Equations ( 4) and ( 5) evidence that GPR waves propagate at a much lower speed in wet conditions. Any source of reflection in the sounded soil produces a reflected wave that is recorded at a time corresponding to the duration of its propagation, from the transmitting antenna, down to the source of reflection, then back up to the receiving antenna, i.e., the TWT of the reflected wave.

We consider a one-dimensional scenario (the offset between the antennas is null) and discretize the domain into N cells i, centered at a depth z i , with element boundaries at z i-1/2 and z i+1/2 . The TWT for the reflection occurring at the interface (i -1/2) between the elements i -1 and i can be expressed as the sum of the vertical TWT in each element above i:

TWT(z i-1/2 ) = 2 i-1 j=1 |l j | V j (6)
in which |l j | (cm) is the length of the element j above i and V j (cm/ns) is the GPR propagation speed in the element j.

A reflection occurs at the interface between two successive elements if the reflection coefficient is not zero. The reflection coefficient expresses the contrast of dielectric constant (due to the contrast of water content) at the interface between the two elements i -1 and i. When the offset between transmitting and receiving antennas is null, the reflection coefficient at interface

(i -1/2) is defined by: r(z i-1/2 ) = ϵ(z i ) -ϵ(z i-1 ) ϵ(z i ) + ϵ(z i-1 ) (7)
where ϵ(z i ) is the dielectric constant of the element i.

For an 800 MHz antenna, the wavelength can typically vary from 6 cm in a wet medium to around 18 cm in a dry medium.

The abrupt change in the reflection coefficient makes the wetting front easily detectable whatever the hydraulic parameters, contrarily to the water table which may be hidden due to the capillary fringe [START_REF] Bano | Effects of the transition zone above a water table on the reflection of GPR waves[END_REF][START_REF] Saintenoy | Ground Penetrating Radar: Water Table Detection Sensitivity to Soil Water Retention Properties[END_REF].

The numerical model

The variation of the water content in the soil during the infiltration is computed using the WAMOS-1D code [START_REF] Belfort | Vadose Zone Modeling in a Small Forested Catchment: Impact of Water Pressure Head Sampling Frequency on 1D-Model Calibration[END_REF]. The model describes the water movement in the porous medium using Richards' equation ( 1), and the constitutive relationships between the pressure, the hydraulic conductivity, and the volumetric water content given by Eq. ( 2) and Eq. ( 3).

The domain of 150 cm depth is discretized with uniform elements of 1 cm thick with homogeneous properties. The WAMOS-1D code solves the system of Eqs. ( 1)-( 3) and yields the vertical distribution of water content at each time step. This distribution is then converted into a vertical dielectric permittivity profile ϵ using the petrophysical relationship Eq. ( 4) and into a GPR wave propagation speed profile V using Eq. ( 5). Then, the time-lapse TWT signals for the fixed objects, TWT 50 and TWT 120 , are calculated at each time step using Eq. (6) (dashed and dotted curves in Fig. 2).

The time-lapse signal TWT f , induced by wave reflection on the wetting front because of the sharp water content variation at the front position is calculated in two steps. First, we search the wetting front position z * i-1/2 , which corresponds to the interface position having the maximum reflection coefficient from Eq. ( 7) as illustrated in Fig. 1. Then, the TWT signal of the wetting front is obtained using TWT f = TWT(z * i-1/2 ) from Eq. ( 6) (solid curve Fig. 2). Note that TWT 50 and TWT 120 signals are induced by fixed objects, thus, these signals exist regardless of the position of the infiltration front. On the other hand, TWT f is induced by the infiltration wetting front whose position varies over time.

Besides, contrarily to TWT 50 , and TWT 120 , the TWT f signal disappears when the wetting front reaches the water table. To avoid numerical issues when simulations are performed with different soil parameter sets, the value of TWT f when the water table is reached, is artificially maintained for the remaining time steps until the end of simulation time. The water table is assumed to be reached when the maximum reflection coefficient of Eq. ( 7) is under a threshold of 10 -2 . This reflects a fully saturated domain with an almost uniform water content distribution (solid curve Fig. 2). An explanation of the computation of all TWT signals is summarized in Fig. 3.

3 Global sensitivity analysis of TWT signals

GSA method

The GSA method evaluates how the outputs of a model are influenced by the variation of the input parameters (Mara and Tarantola, 2008). Among the various forms of GSA, a variance-based sensitivity analysis, allowing the calculation of Sobol sensitivity indices (Sobol ′ , 2001) is employed. Such indices depict the contribution of the variation of any input variable x to the total variance of an output variable y. In our case, the input variables are the unsaturated soil parameters (K s , θ s , θ r , α, n) and the output variables are the TWT signals (TWT f , TWT 50 , TWT 120 ).

Given a model with a set of p independent random parameters X = {x 1 , x 2 , ..., x p } that yields a random response y(X), the two variance-based sensitivity measures, also called Sobol indices (Sobol ′ , 2001) are: the first-order sensitivity index:

S i = Var [E [y(X)|x i ]] Var [y(X)] ∈ [0, 1] (8)
the total sensitivity index:

ST i = E [Var [y(X)|x -i ]] Var [y(X)] ∈ [0, 1] (9)
where x -i = X \x i is the set of all parameters except x i , E() and E(.|.) are respectively the expectation and the conditional 205 expectation operators, Var() and Var(.|.) are the variance and the conditional variance, respectively. The first-order index S i quantifies the contribution of the parameter x i alone to the total variance of y(X), while ST i also includes all interactions of

x i with the other parameters x -i .

To perform a variance-based GSA, a practical approach (to save computational time) is to use Polynomial Chaos Expansion (PCE; [START_REF] Wiener | The Homogeneous Chaos[END_REF]. The PCE approach consists in developing any signal y(X) as a set of orthonormal multivariate 210 polynomials of a degree not exceeding D: where β = β 1 , β 2 , ..., β p ∈ R p is a p th -dimensional index, s β are the PC coefficients, Ψ β are the generalized polynomial chaos of degree |β| = p i=1 β i . In this work, Legendre polynomials are used since uniform distributions are assumed for all uncertain parameters. Uniform 215 distributions express the absence of prior information. This makes all parameter values in the given prior intervals equally likely. Large prior distribution intervals are considered for all unsaturated soil parameters (Table 1).

y(X) = |β|≤D s β Ψ β (X) (10
The number of coefficients for a full PCE representation is P = (p + D)!/p!D!. A training dataset of M realizations of the forward coupled hydrogeophysical model is used to build the PCE surrogate model of order D [START_REF] Fajraoui | Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media[END_REF][START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF][START_REF] Younes | Use of Global Sensitivity Analysis to Help Assess Unsaturated Soil Hydraulic Parameters[END_REF]. The coefficients of the PCE are obtained by searching the best fit (in the least square sense) of the PCE surrogate model to the hydrogeophysical model for the M realizations. To work with low-discrepancy sets, the M realizations correspond to sets of input parameters sampled from their prior probability distributions, using quasi-random Sobol sequences [START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF]. We illustrate the principle of the construction of the PCE with our hydrogeophysical model in 2 show that the relative difference between the two variances is very small for all investigated times.

Note that although the relative variance error for the TWT 50 at t = 2000 s is the largest (around 7%), it remains insignificant since the total variance of the signal at this time is negligible (less than 2 ns 2 ). The variance of the forward hydrogeophysical model is therefore well reproduced by the PCE surrogate model which will be employed for the GSA of the TWT signals using the variance decomposition.

GSA results

The temporal distribution of the output variance of the three TWT signals (TWT f , TWT 50 and TWT 120 ) are represented Fig. 4.

For each TWT signal, the variance is represented by the black curve and the relative contributions of the uncertain parameters to the variance are represented by the shaded area. The blank region between the variance curves and the shaded area represents interactions between parameters.

TWT f has a different behavior from the TWT signals of fixed reflectors TWT 50 and TWT 120 (Fig. 4). Although the TWT signals of fixed reflectors have different variance magnitudes, they exhibit similar behavior (Fig. 4b and4c). The variance of the TWT signal is five times more significant for TWT 120 than for TWT 50 . This is in agreement with the physics since the zone of the porous medium affecting the GPR wave is more important for the TWT 120 signal than for the TWT 50 . In addition, the period of influence of the unsaturated parameters (θ r , α, n) is also more important for TWT 120 than for TWT 50 since saturated conditions for the reflector R120 are reached much later than for R50. Since fixed reflectors exhibit similar behavior, in the following, we comment on the results of TWT f and TWT 120 signals.

GSA of the TWT f signal

TWT f variance is zero at the beginning of the infiltration (Fig. 4a) which means that the TWT f signal is not affected by the initial conditions. Indeed, the infiltration wetting front and the TWT f signal start at zero for all parameter sets. Then, the variance of the signal increases until a maximum of 60 ns 2 , reached at around 3 min. After that, the variance decreases, but keeps a significant value of around 25 ns 2 (Fig. 4a). Concerning parameter sensitivities, at the beginning, the TWT f signal is mainly affected by K s . The influence of this parameter decreases over time and reaches zero for long times when steady-state conditions (corresponding to a fully saturated soil) are reached. The parameter θ s has a moderate influence on the TWT f signal. Its influence is not observable at short times since unsaturated conditions occur. Overall, the most influential parameter on the TWT f signal is the van Genuchten parameter α. This parameter seems influential even for saturated conditions. Note that this numerical artifact is observed because the value of TWT f is artificially maintained when the infiltration wetting front reaches the water table, while physically the signal disappears. The effects of the parameters θ r and n are not observable (Fig. 4a). The blank region between the variance curve and the shaded area in this figure is due to the interaction between the parameters. To estimate this interaction, we plot the difference between the total (ST i ) and the first order (S i ) sensitivity indices for all parameters (Fig. 5a). This difference reflects the interaction between the parameters over time. Interactions between parameters are negligible for all parameters (ST i ≈ S i ), except for K s and α (Fig. 5a). Hence, the interaction between these two parameters 260 affects the variance of the TWT f signal as represented by the blank region between the variance curve and the shaded area (Fig. 4a). To evaluate further the effect of the unsaturated soil parameters on the TWT f , we plot the marginal effect of each parameter (Fig. 6). The marginal effect can be easily derived from the PCE coefficients and reflects the effect of one parameter on the output signal. Fig. 6 depicts the marginal effects of each hydraulic parameter, i.e. their influence on the TWT signals as a function of their value when they vary over the range of their prior distribution interval, while the other hydraulic parameters are kept fixed at their center value. This representation allows determining the regions of influence of the hydraulic parameters, given that the stronger the slope of the marginal effect curve, the higher the influence of the parameter. These marginal effects can vary over time, so we represent them at the three time steps (t 1 = 1 min, t 2 = 5 min, and t 3 = 200 min) highlighted with dotted vertical black lines in Fig. 4. The oscillations are caused by numerical artifacts related to the degree of the polynomials used in the PCE model. From Fig. 6a, it can be noticed that:

-K s is highly influential at the beginning of the experiment. At t 1 = 1 min, the TWT f signal varies almost linearly with K s . Indeed, at the beginning of the experiment, when K s increases, the wetting front is more advanced, thus, the GPR wave propagates at a lower speed and the TWT f signal increases. At t 2 = 5 min, the TWT f signal is sensitive only for small K s values. Indeed, for high K s values, the soil is fully saturated and the perturbation of the high value of K s doesn't change the TWT f signal. At t 3 = 200 min, the soil is fully saturated for almost all K s values, thus, the TWT f signal becomes insensitive to K s .

θ s has no influence at the first times (t 1 = 1 min) since unsaturated conditions occur. For long times, the TWT f signal is very sensitive to θ s with an almost linear behavior. Indeed, when the soil is fully saturated, the dielectric permittivity and thus the TWT f signal is almost proportional to θ s .

the sensitivity of TWT f to θ r is moderate and can be observed only at the beginning of the experiment (unsaturated conditions) with an almost linear behavior observable at t = 1 min and 5 min. The positive slope of the curve is consistent with the physics of the process (when θ r increases, the speed of the electromagnetic wave decreases, and the TWT f signal increases).

-The van Genuchten parameter α is highly influential notably for long times (t 3 = 200 min). A small variation of the parameter α can induce a strong variation of the TWT f signal. Notably, the sensitivity of α is very high for α ≤ 0.05

cm -1 .
-The sensitivity of TWT f to the parameter n is almost zero (flat curves) at all times (t = 1 min, 5 min, and 200 min).

The parameter n has therefore a negligible effect on the TWT f signal and, as a consequence, it is expected to be poorly identifiable from the TWT f data.

GSA of the TWT 120 signal

The variance of the TWT 120 signal is nonzero at the beginning of the experiment which means that the TWT 120 signal is affected by the initial conditions (Fig. 4c). Indeed, at the very beginning, the pressure distribution is hydrostatic and the water content distribution in the column is obtained from Eq.2 which depends on all soil parameters except K s . Therefore, the speed of the GPR wave depends on the initial water content distribution which is dependent on the unsaturated soil parameters θ s , 14 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License. θ r , α, and n. The most influential parameter at the beginning of the experiment is the parameter α. Over time, the effect of this parameter reduces, whereas the effect of θ s increases. For long times, θ s becomes the only sensitive parameter. The parameter K s is also very sensitive. Its effect starts at zero, and increases until a maximum is reached at around 3 min, then it slowly decreases and becomes negligible after 100 min. As with the TWT f signal, interactions between parameters are moderate. The difference between the total and first-order Sobol indices is negligible for all parameters except after 1 min for the parameters K s , α and θ s (see Fig. 5b). This interaction corresponds to the blank region, between the variance curve and the shaded area in -As for the TWT f signal, K s is highly sensitive, especially for t = 1 min and 5 min.

-The saturated water content θ s is very influential for all times. The TWT 120 varies almost linearly with θ s even at the beginning (t 1 = 1 min), since the fixed reflector is located in the lower saturated region.

-As for the TWT f signal, θ r is sensitive only at the beginning of the experiment (unsaturated conditions) with an almost linear behavior at t = 1 min and 5 min. When θ r increases, the water content increases, and hence, the TWT 120 increases.

-The van Genuchten parameter α is highly sensitive. However, contrarily to the TWT f signal where α is highly sensitive at long times (t 3 = 200 min), the sensitivity of α for the TWT 120 signal is high at short times (t 1 = 1 min). For long times, the influence of α disappears since the soil becomes fully saturated. The negative slope of the curve of the TWT 120 signal as a function of α observed at the beginning of the experiment is consistent with the physics of the process. Indeed, when α increases, the capillary fringe thickness decreases, hence, the water content in the unsaturated zone decreases, and thus the TWT 120 signal decreases.

-Surprisingly, and contrarily to the TWT f signal which showed a flat curve for the marginal effect of the parameter n for all parameter values and at all investigated times, the TWT 120 signal is sensitive to n at the beginning of the experiment (t 1 = 1 min) with a high sensitivity for n < 3.5 and a moderate sensitivity (the curve has a small slope) for n ≥ 3.5.

Bayesian soil parameter estimation from the TWT signals

In this section, we estimate the unsaturated soil parameters in a Bayesian framework using the Markov Chain Monte Carlo (MCMC) sampler [START_REF] Vrugt | Validity of First-Order Approximations to Describe Parameter Uncertainty in Soil Hydrologic Models[END_REF][START_REF] Vrugt | Inverse Modeling of Subsurface Flow and Transport Properties: A Review with New Developments[END_REF]. The statistical calibration is performed for a GPR monitored infiltration experiment in order to address the following questions:

1. Can we obtain an appropriate estimation of all unsaturated soil parameters from TWT data? 2. What is the impact of the kind of TWT data (moving/fixed reflectors) and of the number of reflectors on the calibrated model parameters and their confidence intervals? 3. What is the optimal set of TWT measurements that yields good reliability of all unsaturated soil parameters? https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

The MCMC method has been successfully employed in various inverse hydrological problems (e.g., [START_REF] Fajraoui | Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media[END_REF][START_REF] Younes | Global sensitivity analysis and Bayesian parameter inference for solute transport in porous media colonized by biofilms[END_REF][START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF]Younes et al., 2018). The method generates random sequences of parameter sets that asymptotically converge toward the target joint posterior distribution by searching the ensemble of possible parameter sets that satisfactorily fit the observations. The converged sets can then be used to assess the quality of the parameter estimation such as the optimal parameter values and the 95% Confidence Intervals (CIs) which allow for evaluating the reliability of the parameters via uncertainty quantification.

In the sequel, the MCMC method is performed with the DREAM (ZS) software [START_REF] Laloy | High-dimensional posterior exploration of hydrologic models using multiple-try DREAM(ZS) and highperformance computing[END_REF], which is an efficient MCMC sampler. The vector of unknowns is formed by the five unsaturated soil parameters (K s , θ s , θ r , α, n). A reference solution is generated by simulating the hydrogeophysical problem formed by the system of equations ( 1)-( 6) using the following reference parameter values K * s = 0.08 cm/s, θ * s = 0.4, θ * r = 0.07, α * = 0.145 cm -1 , n * = 2.68, as shown in Table 3. The TWT f , TWT 50 , and TWT 120 signals used as calibration data are deduced from the results of the simulation using the reference parameter values. These TWT signals are then independently corrupted using a normally distributed noise with a standard deviation σ = 0.5 ns.

The TWT f , TWT 50 and TWT 120 calibration signals, illustrated before noise corruption in Fig. 2, increase almost linearly until reaching a plateau. For the TWT 50 signal, the plateau is reached when the infiltration front attains the R50 reflector and the value of the plateau corresponds to the time needed by the electromagnetic wave to make a round trip from the surface to a 50 cm depth of a full saturated porous medium. For the TWT 120 signal, the plateau signal is reached when the infiltration front attains the water table (the domain becomes fully saturated) and the value of the plateau corresponds to the time needed by the electromagnetic wave to make a round trip from the surface to a 120 cm deep of a fully saturated porous medium. For TWT f , the plateau value is also reached when the infiltration front attains the water table and the value of the plateau corresponds to the time needed by the electromagnetic wave to make a round trip from the surface to the water table at 100 cm deep.

The reliability of the unsaturated soil parameters is assessed for 5 different scenarios of measurement sets. In the first scenario, only data of the wetting-front TWT f signal are used for the calibration. The second and third scenarios use only the TWT 50 and TWT 120 signal, respectively, obtained from reflection on the fixed reflector R50 and R120. The fourth scenario uses both data of TWT f and TWT 120 as fitting data. The last scenario investigates the benefit of adding a fixed reflector by using data of the TWT f , TWT 50 and TWT 120 signals as conditioning information.

In the five scenarios, the MCMC sampler uses three parallel chains and a total number of 50000 runs. The last 25% of the runs that adequately fit the model onto observations are used to estimate the joint posterior distribution.

The MCMC results of the five studied scenarios are given in Table 3 which depicts, for each parameter, the mean estimated value, its posterior CI size, and the ratio of prior to posterior intervals. Note that the CI and the last indicator are calculated from the standard deviation by assuming a Gaussian posterior distribution. A small CI indicates an accurate estimation of the parameter. A significant difference between the prior and posterior intervals is a sign of the high sensitivity of the model to that parameter [START_REF] Dusek | Transport of bromide and pesticides through an undisturbed soil column: A modeling study with global optimization analysis[END_REF].

Results of table 3 for scenario 1 using only data of the TWT f signal for the estimation of the unsaturated soil parameters show that: (bold), size of the posterior confidence intervals (CIs) (between brackets), and ratio of prior to posterior intervals (italic).

-An accurate estimation of K s , the most sensitive parameter (Fig. 4a), is obtained with a CI of 0.037 cm/s and a variation interval reduced by 4.

-A fair estimate of the parameters θ s with a standard deviation of 0.031 (-) and a reduction of the interval of variation by 5. This result is relatively surprising as this parameter did not show a strong influence on TWT f sensitivity (Fig. 4a).

-The parameter θ r is not well estimated. Indeed, although its mean estimated value is very close to its reference value, the associated uncertainty of 0.14 is large and the posterior interval is as large as the prior one, which indicates the low reliability of the estimation.

-A poor estimation of α, while the sensitivity analysis showed it has a strong influence on TWT f (Fig. 4a). Its CI is large, with a value of 0.167 cm -1 and its posterior interval size is half the prior one. -The TWT f signal yields a mean estimated value n = 2.75 ± 0.47 which is close to the reference value n * = 2.68. The parameter n is quite well identified since its posterior interval is 9 times smaller than the prior interval. This is relatively surprising since the sensitivity of n is negligible (Fig. 4a and6a5).

In summary, using only data of the TWT f signal as conditioning information for the hydrogeophysical model calibration yielded well mean estimated parameter values, close to the reference values for all unsaturated soil parameters. However, the examination of the associated uncertainties, showed that only K s , θ s , and n are correctly identified (with narrow posterior intervals with respect to the prior ones). This points out the importance of statistical calibration methods for highly nonlinear problems to investigate not only estimated parameter values but also the associated uncertainties.

The estimation of the unsaturated soil parameters for scenarios 2 and 3, using only data of the TWT 50 or TWT 120 signal for the calibration shows that:

-The parameters K s and θ s , which are the most sensitive parameters during most of the experiment (Fig. 4b and4c), are well identified with small CI size and strong reductions by at least 6 for K s , and 19 for θ s , of their intervals of variation.

We note that the TWT 120 signal allows a much better estimate of both K s and θ s as their CIs are smaller than the ones estimated with TWT 50 . It is especially true for K s where there is almost a factor 2 between the reduction ratios.

-The soil parameters θ r , α, and n, although sensitive (Fig. 4b and4c), cannot be identified from the TWT 50 and TWT 120 signals since their posterior intervals are as large as, or at best two times smaller than their prior ones.

The results of scenario 4 which combines data of TWT f and TWT 120 signals show that:

-Both parameters K s , θ s , and n are very well identified, with very narrow posterior intervals showing a strong reduction by 46, 37, and 17 of their prior intervals, respectively.

-The parameters θ r and α are reasonably well estimated with mean values very close to their reference and intervals of variation reduced by 8 and 9, respectively.

Finally, the results of the last scenario which combines data of TWT f , TWT 50 and TWT 120 signals, show performances very similar to scenario 4. Additional information from TWT 50 helped to reduce slightly the posterior intervals of K s , θ s , and α that in that case show a reduction of 49, 44, and 10 times their prior intervals, respectively.

The results of MCMC for this last scenario are shown in Fig. 7 where diagonal plots depict the inferred posterior parameter distributions and the off-diagonal scatterplots represent the pairwise correlations in the MCMC draws. Almost bell-shaped posterior distributions are obtained for all unsaturated soil parameters. Negligible correlations are observed between the parameters, except moderate correlations observed between K s and θ r (r=-0.78) and between n and θ r (r=0.64) .

Note that the parameter n is relatively well estimated as the target reference value 2.68 is located in the high sensitivity region (n < 3.5) (Fig. 6). In the case of a reference value located in the low sensitivity region (n ≥ 3.5), the calibration of the hydrogeophysical model using TWT f and TWT 120 signals yields a much poorer identification of the parameter n. For instance, The results of GSA showed that the TWT f signal of the wetting front is different from that of the two fixed reflectors which had similar behavior. For the fixed reflectors, the magnitude of the variance (and therefore the sensitivity of the soil parameters) is more pronounced for deeper reflectors. The TWT f signal is highly sensitive to K s and α and moderately sensitive to θ s . A low sensitivity was observed for θ r , whereas the parameter n was insensitive. The TWT 120 signal of the fixed reflector located at 120 cm depth is highly sensitive to K s , θ s and α, and moderately sensitive to θ r . The van Genuchten parameter n has a high sensitivity for n < 3.5 and a poor sensitivity for n ≥ 3.5.

The reliability of the unsaturated soil parameters has been assessed for 5 different scenarios of TWT measurement sets.

When only data of the TWT f signal are used as conditioning information for the model calibration, all estimated parameter values were very close to the reference values. However, analyzing the associated uncertainties showed that only K s , θ s , and n were correctly identified (with narrow posterior intervals). Further, using only data of the TWT 50 or TWT 120 signals for the calibration allows also only a good identification of K s and θ s with a strong reduction of their intervals of variation. The best results, in terms of parameter reliability, are obtained with the combination of TWT f with fixed reflectors. In this case, the four parameters K s , θ s , θ r , and α are very well identified with very narrow posterior interval. The van Genuchten parameter n is estimated with a low uncertainty but its estimation degrades in the low sensitivity region n ≥ 3.5. We note that the deeper reflectors provide more information as the inversion of its signal furnishes parameters with lower uncertainty. Then using two or three reflectors in addition to the wetting front signal doesn't reduce consequently the uncertainty of the parameters.

Figure 1 .

 1 Figure 1. Test case and experimental device illustration at an advanced time step (a). R50 and R120 are fixed reflectors considered in this experiment. TX and RX refer to the transmitter and receiver antennas of the GPR system. Effective saturation Se (b) and reflection coefficient r (c) profiles with depth.

5

  https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

6

  https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Figure 2 .

 2 Figure 2. Hydrogeophysical model responses for Ks = 0.08 cm/s, θs = 0.4 (-), θr = 0.07 (-), α = 0.145 cm -1 , n = 2.68 (-). TWTf corresponds to the TWT signal for the wetting front, while TWT50 and TWT120 are the TWT signals for fixed objects located at 50 and 120 cm below the surface, respectively.

Figure 3 .

 3 Figure 3. Summary of the working process of the forward hydrogeophysical model and how it is used to build the PCE surrogate model.

Fig. 3 .

 3 Fig.3.

  11 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Figure 4 .

 4 Figure 4. Time distribution of the variance of TWTf (a), TWT50 (b) and TWT120 (c). The shaded area under the variance curve represents the partial marginal contributions of the uncertain parameters; the blank region between the shaded area and the variance curves represents the contribution of interactions between the parameters. The marginal effects shown in Fig.6 are represented at three time steps t1 = 1 min, t2 = 5 min, and t3 = 200 min, highlighted here (dotted black lines).

  12 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Figure 5 .

 5 Figure 5. Difference between the total (STi) and the first order (Si) sensitivity indices for all parameters for the TWTf (a) and the TWT120 (b) signals.

  13 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Figure 6 .

 6 Figure 6. Marginal effects of the unsaturated soil parameters Ks, θs, θr, α and n on the TWTf and TWT120 signals at three different times t1 = 1 min, t2 = 5 min, and t3 = 200 min, highlighted in Fig.4.

Fig. 4c .

 4c Fig.4c. The marginal effects of the soil parameters on the TWT 120 signal are plotted in Fig.6b for t = 1 min, 5 min, and 200 min. The curves in this figure show that:

  https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Figure 7 .

 7 Figure 7. MCMC solutions using TWTf , TWT50 and TWT120 signals for the calibration of the hydrogeophysical model. The diagonal plots represent the inferred posterior parameter distributions, showing the estimated mean value (dotted black line) and the target value (hard red line). The off-diagonal represents the pairwise correlations between parameters.

  The results of this study highlight the high benefit of combining TWT signals of fixed and moving (infiltration wetting front) reflectors for very good identification of all the unsaturated soil parameters. It also points out the role of GSA to assess the influence of the parameters on the output signals and the necessity to perform statistical calibration to assess the reliability of model parameters by evaluating not only estimated parameter values but also their associated uncertainties. experiments under the supervision of GJF and LN, and validation of all co-authors. MR prepared the manuscript with reviewing and editing contributions from all co-authors. would like to acknowledge the High-Performance Computing Center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources. Part of the computing resources was funded by the Equipex Equip@Meso project (Programme Investissements d'Avenir) and the CPER Alsacalcul/Big Data. 21 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License.

Table 1 .

 1 Prior intervals of the unsaturated soil parameters for both GSA and Bayesian estimation.

		TWTf	TWT50	TWT120
	t = 50 s
	Var_HYD_model	14	5	15.9
	Var_PCE_model	13.4	5	15.8
	Var_error	4.3%	0.9%	0.5%
	t = 150 s
	Var_hyd_model	54.6	5.5	23.3
	Var_PCE_model	53.9	5.4	23.2
	Var_error	1.3%	1.4%	0.4%
	t = 2000 s
	Var_hyd_model	28.8	1.5	9.9
	Var_PCE_model	27.1	1.4	9.3
	Var_error	5.7%	7.4%	5.2%

Table 2 .

 2 Variance of TWTf, TWT50 and TWT120 signals at t = 50 s, 150 s and 2000 s calculated with the PCE surrogate model and with the hydrogeophysical model.

Table 3 .

 3 16 https://doi.org/10.5194/egusphere-2022-936 Preprint. Discussion started: 29 September 2022 c Author(s) 2022. CC BY 4.0 License. First line: Reference values used to build the synthetic calibration data. Then for the different scenarios: estimated mean values

		Ks (cm/s)	θs (-)	θr (-)	α (1/cm)	n (-)
	X *	0.08	0.40	0.07	0.145	2.68
	Scenario 1	0.081	0.39	0.076	0.211	2.75
	TWTf	(0.037)	(0.031)	(0.14)	(0.167)	(0.93)
		4	5	1	2	9
	Scenario 2	0.074	0.4	0.081	0.173	5.79
	TWT50	(0.023)	(0.008)	(0.061)	(0.269)	(9.99)
		6	19	2	1	1
	Scenario 3	0.078	0.4	0.089	0.167	5.93
	TWT120	(0.011)	(0.007)	(0.053)	(0.195)	(9.36)
		13	24	2	1	1
	Scenario 4	0.08	0.4	0.074	0.151	2.72
	TWTf,	(0.003)	(0.004)	(0.015)	(0.029)	(0.5)
	TWT120	46	37	8	9	17
	Scenario 5	0.079	0.4	0.073	0.149	2.68
	TWTf,	(0.003)	(0.004)	(0.015)	(0.027)	(0.49)
	TWT50,	49	44	8	10	17
	TWT120

Acknowledgements

The doctoral position of the first author is co-funded by the Grand-Est Region and the University of Strasbourg. This research work is partly funded by the French National Research Agency through the Exciting project ANR-17-CE06-0012-01. Finally, computing time was provided by the HPC-UdS. Acknowledgements. The doctoral position of the first author is co-funded by the Grand-Est Region and the University of Strasbourg. This research work is partly funded by the French National Research Agency through the Exciting project ANR-17-CE06-0012-01. The authors

Code availability. The hydrological code WAMOS-1D, the coupled hydrogeophysical model, and the global sensitivity analysis tools can be provided upon request by the authors.

Data availability. All data presented in this paper is available upon request from the authors.

Author contributions. GJF and LN formulated the aims of the research. BB, LF, and YA provided the hydrological code. YA provided the resources for sensitivity analysis and Bayesian estimation studies. MR established the hydrogeophysical coupled model and ran all

Competing interests. The authors declare that they have no conflict of interest.


Ly Dai Hung
email: hunglydai@gmail.com

FAST AND SUSTAINABLE DEVELOPMENT IN DIGITAL REVOLUTION: AN APPLICATION OF KUZNETS CURVE

Keywords: Digital Revolution, Fast and Sustainable Development, Kuznets Curve

Objective:

The paper investigates the fast and sustainable development pattern with the digital revolution by employing the Kuznets curve. The data is a cross-section sample of 180 economies over 1990-2021 time period. And the research objective is to consider whether there exists a fast and sustainable development space in the current digital revolution around the world.

Methodology:

The research methodology combines the qualitative and quantitative analysis. The qualitative analysis examines the application of well-known social and environmental Kuznets curve in the digital revolution. Then, the quantitative analysis explores the empirical evidence on the relationship of economic growth with the social and environmental results, under the impact of the digital revolution.

Findings:

The qualitative analysis shows that the Kuznets curve is well-recorded in the literature of income inequality and environmental reservations. And the digital revolution tends to push forward the achievement of social and environmental results. Moreover, the quantitative analysis records that both the social and environmental Kuznets curves exist. And the digital revolution can change the pattern of Kuznets curve in the direction that enhance the fast and sustainable development.

Implications:

The evidence suggests that the digital revoluation is crucial for the fast and sustainable development cross economies around the world. Thus, one possible policy mechanism can be the stimulation of digital revolution so that an economy can first achieve the social Kuznets curve then the environmental Kuznets curve.

INTRODUCTION

The fast and sustainable development is crucial for an economy to catch up with the world leading economy. The fast development creates the financial resources for the sustainable development, while the sustainable development, in turn, can ensure the fast development. By that way, the concept of fast and sustainable development is mainly employed in the public policy architecture, especially on the long-term strategy development.

The current paper analyzes the fast and sustainable development by referring to the recent digital revolution. There are many contribution of digital revolution toward the economic development, one of which is to the fast and sustainable development. The paper records that the dependence pattern of fast and sustainable development on the digital revolution can follow one inverted-U shaped curve.

Since these curves can be depicted as one Kuznets curve, the impact of digital revolution on the fast and sustainable development can be imaged as one extension of Kuznets idea on the economic growth and economic inequality over the development space.

The paper belongs to the literature on the nexus between economic growth and sustainable development. The economic growth depends on the technology progress since at the long run equilibrium, the output per capita grows with the total factor productivity growth rate [START_REF] Solow | A contribution to the theory of economic growth[END_REF][START_REF] Swan | Economic growth and capital accumulation[END_REF]. And the sustainable development refers to the combination of economic, social and environmental aspects of development stage [START_REF] Schumpeter | The theory of economic development[END_REF].

The paper makes a contribution on the economic growth and sustainable development literature. Its evidence shows that the digital revolution can contribute significantly on the achievement of fast and sustainable development. In particular, our evidence extends the Kuznets curve on the relationship between economic growth and income inequality to the environmental protection, under the impact of digital revolution.

The paper is structured as follows. The first section on Introduction would briefly describe the main results. Then, the second section on Framework analyzes the theory and data, while the third section on Evidence investigates the fast and sustainable development. Finally, the fourth section on Conclusion would summarize the paper and discuss the public policy.

ANALYSIS FRAMEWORK

Theory

The fast and sustainable development is one quite new concept on the economic development literature. The sustainable development is well-recorded a state of development in which the economic growth goes along with the social achievement and environmental protection. And the fast development arises mainly from the strategy architecture along the development process, which is also realized on the recent long-term social economic strategy in the Vietnam economy.

There are two types of sustainable development, including the weak sustainability and strong sustainability. Indeed, the weak sustainability is attained at the intersection between the economic, social and environmental aspects of the development process. And the strong sustainability is attained at each stage of economic development, which also requires the social and environmental achievement at the same time.

The weak sustainability is originated from the seminal paper by [START_REF] Hotelling | The economics of exhaustible resources[END_REF] on the exhaustible resources. Accordingly, the resource has a limited supply, then, tends to reduces over time. The optimal way is to explore the resource so that the growth rate of the resource price is equal to the real interest rate on the financial market. Apply this concept, usually mentioned as Hotelling rule, many authors such as [START_REF] Solow | An almost practical step toward sustainability[END_REF], has extended this line of research and form the weak sustainability. In general, this type of sustainability is based on the idea that the natural capital, which is given by the nature, can be substituted by the physical capital, which is created by the human.

The strong sustainability is developed along with the literature on the ecology [START_REF] Daly | Sustainable development: from concept and theory to operational principles[END_REF]Harwick, 1978). This concept considers that the natural capital can not be perfectly substituted by the physical capital. Therefore, the economic development needs to equipped with the social and environmental achievement at each economic stage. This concept can be thought as the combination of economic -social -environmental results in each development stage.

While the sustainable development is well recorded in the literature, the concept of fast development is mainly referred on the public policy discussion. In particular, an economy achieving the fast development when its economic growth rate is higher than its average growth rate in the past, or higher than the average of world economies. The first case is the time-series fast development while the second case is the cross-section fast development [START_REF] Dai | Fast and Sustainable Development Space in Vietnam by a Multi-Dimension Approach[END_REF]. Recently, the longterm development strategy of Vietnam economy follows the first concept, in which, the fast development can be thought as the combination of economic growth rate being higher than 6.4%, the average growth rate over 1990-2020, with the stability of income distribution.

Data

The data is one cross-section sample of 122 economies over 1990-2021. Each variable is averaged over time period so that the analysis can focus on the long-run role of digital revolution on the fast and sustainable development.

First, the economic growth rate, denoted by (aGDPpcgrowth), is measured by the growth rate of output per capita at constant 2015 USD price. The output, in turn, is the gross domestic product (GDP). This variable captures the growth rate of living standard, which is consistent with the concept of economic development.

Moreover, it also reflects the technological progress since at the long run equilibrium, the output per capita grows with the total factor productivity progress rate. The data covers the 1990-2020 period, and is from the world development indicators, constructed by the World Bank (2022).

Second, the income inequality, denoted by (aGini), is measured by the gini index. This index covers the distribution of income across population, with a higher index reflecting more inequality. The data covers the 1990-2020 period, and is from the world development indicators by World Bank (2022).

Third, the environmental protection is measured by the genuine savings per GDP on percentage, denoted by (aGenSav), that is the gross savings substracting by the environmental damage. A higher genuine savings per GDP means that an economy is more efficient on its economic activities. The data covers the 1990-2020 period, and is from the world development indicators by World Bank (2022).

Fourth, and also the focal variable, is the digital revolution, which is measured by the financial services coverage, denoted by (LnFinance). This variable is the ratio of adults having the financial services over the whole population, on percentage. The data is extracted from the world financial index (World FinDex) constructed by the World Bank (2022) by 3-year survey from 2011 to 2021.

Model

The empirical model is specified as one cross-section ordinary least squares (OLS), with the following form: ). These terms presents an inverted-U shaped curve for the dependence pattern of dependence variable on the technology revolution.

𝑌 ! = 𝛼 " + 𝛽 #$%&%'( " 𝐿𝑛𝐹𝑖𝑛𝑎𝑛𝑐𝑒 ! + 𝛽 #$
The specification of one quadratic function is inspired by the Kuznets curve, introduced in the seminal paper by Kuznets (1955), and is also intensively analyzed by recent authors such as Piketty (1996) on various aspects from the social to environmental achievements. Accordingly, the income inequality follows one quadratic fuction of income growth rate. For low income growth, a higher growth rate is associated with more income inequality. But for high enough income growth, a higher growth rate is associated with less income inequality. The turning point of these two patterns is the also maximum value of the quadratic function.

EMPIRICAL EVIDENCE

Table I presents the regression results of three dependent variables, including the economic growth, income inequality and environmental protection on the technology revolution. In column (1), the economic growth follows a quadratic function of financial coverage. The regression coefficients can be imagined as one inverted-U shaped curve for which an increase of financial coverage is first associated with higher economic growth rate then goes along with less economic growth rate. The decreasing portion of the curve can be explained by the base effect: at a higher technology level, the contribution of technology on the economic growth can goes down. Aghion and Howitt (1998) also establish that the economies with low technology level can have a high economic growth by imitating the advanced technology. But for the economies with high technology level, especially the economy at the world technology frontier, their economic growth rate would be low since they have to invent new technology, which has verly low probability of success. In brief, the environmental protection follows one inverted-U shaped as an Kuznets environmental curve on the financial revolution.

CONCLUSION

The paper analyzes the contribution of digital revolution on the fast and sustainable development by both qualitative and quantitative analysis. The quantitative analysis covers one sample of 120 economies over 1990-2021, with the digital revolution is proxied by the financial coverage ratio. The research method is one cross-section empirical model with an emphasizes on the quadratic regression function, which can be depicted by an inverted-U shaped, a type of Kuznets curve.

The evidence records the existence of Kuznets curve on the dependence pattern of the income inequality and environmental protection on the digital revolution. These curve implies that ony for a high digitial revolution level, proxied by low financial coverage ratio, more financial coverage is associated with less income inequality and lower environmental protection.

The evidence suggests that the digital revoluation is crucial for the fast and sustainable development cross economies around the world. Thus, one possible policy mechanism can be the stimulation of digital revolution so that an economy can first achieve the social Kuznets curve then the environmental Kuznets curve.

Figure 1 :

 1 Figure 1: Kuznets Curve of Income Inequality on Technology Revolution

Figure ( 1

 1 Figure (1) depicts the dependence pattern of income inequality on the technology revolution. There exists clearly one inverted-U-shaped curve, which is similar to the Kuznets curve, on the income inequality pattern. The turning point of the income inequality is at the financial coverage value around 3.8. And another feature of this curve is the concentration of economies around the income inequality value around 40.

Figure 2 :

 2 Figure 2: Kuznets Curve of Environmental Protection on Technology Revolution

Table 1 :

 1 Cross-Section Regression Results of Economic Growth, Income

	Inequality and Environmental Protection on Technology Revolution
		(1)	(2)	(3)
	VARIABLES	Economic Growth	Income	Environmental
		(aGDPpcgrowth)	Inequality	Protection
			(aGini)	(aGenSav)
	Financial Coverage	8.698***	139.0***	4.441**
	(LnFinance)	(2.649)	(25.98)	(1.894)
	Squared Value of	-1.057***	-17.97***	-0.747***
	LnFinance
	(LnFinance2)	(0.354)	(3.248)	(0.243)
	Constant	-15.57***	-226.0***	-4.525
		(4.925)	(51.53)	(3.656)
	Observations	122	110	112
	R-squared	0.116	0.268	0.408

Notes: Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1


Ly Dai Hung
email: hunglydai@gmail.com

TERM-STRUCTURE OF FOREIGN DIRECT INVESTMENT INTO VIETNAM ECONOMY

Keywords: Foreign Direct Investment, Term-Structure, Qualitative Analysis, Quantitative Analysis, Vietnam Economy

Objective:

The paper investigates the term structure of foreign direct investment into Vietnam economy by dividing the short-term, medium-term and long-term contribution on the domestic economic growth. The data is an annual sample of Vietnam economy over 2007-2021.

Methodology:

The research methodology is first based on the recent literature review on the impact of foreign direct investment on the economic growth rate of the host economy. Then, the paper examines the contribution of foreign direct investment into the Vietnam's economic growth by the term structure.

Findings:

The empirical evidence records that the foreign direct investment stimulates the domestic economic growth by promoting the net exports in the medium-term and by the capital accumulation process in the long-term. In the short-term, the foreign direct investment also provides the foreign currency, which affect directly the equilibrium foreign exchange rate. Therefore, the foreign direct investment is a crucial growth engine for the Vietnam economy.

Implications:

The evidence suggests that the foreign direct investment can be a priority for the public policy to enhance the domestic economic growth rate. Beside the domestic sector, including the state-owned firms and private firms, the foreign direct investment needs to be managed so that its contribution to domestic economic growth is maximized.

INTRODUCTION

The foreign direct investment (FDI) is the main driver of cross-border capital flows in the recent financial globalization over last decades. This type of capital flows contributes significantly on the economic growth in the host countries, and also entails the allocation of trading value chain around the world. The current context of high inflation rate and low economic growth prospects (World Bank, 2022) also stresses the importance of FDI capital flows as the connection channel across economies to solve the global stagflation prospect.

The current paper investigates the contribution of foreign direct investment on the economic growth by considering the term-structure of this capital inflows.

The long-term analysis examines the contribution of FDI capital inflows into domestic capital accumulation; The medium-term analysis explores the contribution of FDI capital inflows into exports and imports; And the short-term analysis focuses on the contribution of FDI capital inflows into domestic currency stability.

The empirical evidence records that the foreign direct investment stimulates the domestic economic growth by promoting the net exports in the medium-term and by the capital accumulation process in the long-term. In the shor-term, the FDI also contributes to more stable domestic currency. Therefore, the foreign direct investment is a crucial growth engine for the Vietnam economy.

The paper belongs to the literature on the foreign capital inflows. The capital inflows exert positive impact on the economic growth, but only when the financial development level of the host country is high enough [START_REF] Prasad | Foreign Capital and Economic Growth/Comments and Discussion[END_REF]. The FDI also generates one dynamic sector which accounts for a significant share on the exports and imports of the host country. By that mechanism, the FDI can also affect the trade-offs between the output and inflation in an economy (Hung, 2021). In the short-run, an economy facing the policy trilemma in the international finance would have to deal with the foreign currency supply originated from the FDI inflows. The fluctuation of foreign exchange rate can be an absorpter of the fluctuation on the foreign capital inflows (Obsfeld et al, 2019).

Our result synthesizes the impact of FDI inflows on an economy by three terms, including the short-term, medium-term and long-term. Therefore, the paper provides a multi-aspect impact of FDI inflows on the economic growth of the host economy.

The paper is structured as follows. The first section on Introduction would briefly describe the main results. Then, the second section on Framework analyzes the theory and data, while the third section on Evidence investigates the termstructure of foreign direct investment. Finally, the fourth section on Conclusion would summarize the paper and discuss the public policy.

FRAMEWORK

Theory

In the long-term, the foreign direct investment (FDI) contributes on the economic growth by enhancing the domestic capital accumulation process.

According to the neoclassical growth model (Solow, 1956;Swan, 1956), an economy tends to converge to one long-run equilibrium in which the capital stock per effective unit of labor is constant. Along this convergence path, the economy needs to accumulate the capital, both from domestic and foreign sources, to attain the long-run equilibrium. Thus, the convergence to the steady state economic growth rate is determined by the capital accumulation, then, depending on the foreign capital.

By the long-term aspect of foreign direct investment, the contribution of FDI can be captured by the share of FDI sector over domestic output, measured by the gross domestic product (GDP). Another indicator can be the share of FDI sector over domestic government revenue, which illustrates the significance of FDI sector on the domestic fiscal space. In both theory an practice, a huger fiscal space supports the intervention of government on the domestic economy, such as the stimulus package for aggregate demand, especially on the economic recession period.

In the medium-term, the foreign direct investment contributes on the economic growth by stimulating the foreign trade of goods and services. The foreign trade, in fact, illustrates the foreign demand for domestic goods and services. Thus, according to the idendity equation of aggregate demand, the net exports, which is equal to the exports substracting by the imports, contributes to domestic demand, then, economic growth.

By the medium-term aspect of foreign direct investment, the contribution of FDI works through the establishment of one dynamic sector which contributes on the domestic exports and imports. Thus, the medium-term role of FDI can be illustrated by the share of FDI sector over exports, imports, and the relative comparision of net exports by FDI sector with that by domestic sector.

In the short-term, the foreign direct investment contributes on the economic growth by providing the additional source of foreign currency. In the foreign exchange market, the equilibrium exchange rate is jointly determined by the supply and demand for foreign currency. The supply of foreign currency comes from the capital inflows such as the foreign direct and indirect investment, foreign-currency debts, aid flows and even remittances from domestic workers in foreign economies.

By the short-term aspect of foreign direct investment, the contribution of FDI is captured by the supply of foreign currency, which affects the foreign exchange rate between domestic and foreign currency. Then, one way to examine this channel is to analyze the co-movement of the growth rate of FDI disbursed value with the depreciation rate of domestic currency.

Data

The data is a time-series annual sample of Vietnam economy covering the 1990-2021 period. The starting point is 1990 when the Vietnam economy stimulates the international integration into the world economy with some

EVIDENCE

Long-Term

Figure 1 shows the long-term contribution of FDI on the economic growth.

The share of FDI sector raises from 15,16% in 2005 to 20,02% in 2021, with only a sligh decrease over 2010-2012. The increase of GDP share by the FDI sector proves the significant role of this sector toward the domestic growth rate. As recent theory and evidence, the positive contribution of FDI into domestic GDP can work through the domestic capital accumulation process. An economy needs the foreign capital inflows to build up domestic capital stock in the convergence pathway to its long-run equilibrium, or steady state.

Another aspect of the long-term contribution by FDI is the share of FDI sector over domestic government revenue. In theory, a huger government revenue can support more space for the fiscal policy implementation. In the Vietnam economy, as Figure 1 shows, the share of FDI sector over government revenue raises from 9% in 2005 to 18% in 2021. On other hand, the share by FDI sector doubles over a period of 16 years.

Medium-Term

Figure 2 recors the medium-term contribution of foreign direct investment into the domestic economy. The line figures show that the share of FDI sector over both exports and imports tends to increase over time. The share of FDI sector over exports raises from 28% in 1995 to 73% in 2021, while its share over imports also goes up from 18% in 1995 to 65% in 2021. Overall, the share of FDI sector over total exports and imports increases from 21% in 1995 to 70% in 2021, which means a 3,5-folds increase over a period of 26 years. Another indicator of FDI contribution on the foreign trade is revealed by the comparision of the net exports by the FDI sector with that by the domestic sector.

By the indentity equation of GDP, the net exports are one component of GDP.

According the view point by Vietnam government, the net exports together with private consumption and investment jointly composes the three-horse vehicle which drive the domestic economic growth rate. As Figure 2 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 2 0 1 7 2 0 1 8 2 0 1 9 2 0 2 0 2 0 2 1 Net Exports by Domestic Sector (Million USD,RHS) Net Exports by FDI Sector (Million USD, RHS)

Share over Domestic Exports (%) Share over Domestic Imports (%)

Share over Total Exports and Imports (%)

Short-Term

Figure 3 shows the correlation between the foreign direct investment inflows and the domestic currency. In details, we compute the annual growth rate of disbursed FDI on percentage and the depreciation rate of domestic currency on percentage. The disburse FDI illustrates the absorption capacity of an economy toward the foreign capital. And its growth rate shows the change of absorption capacity over time. Moreover, the value of domestic currency is illustrated by the foreign exchange rate of domestic currency, VND, over the foreign currency, USD.

We also employ the official nominal exchange rate rather than the real rate or effective rate for the reason that the official nominal rate is the common reference for the foreign trade and banking activities, while the real or effective exchange rate is more useful in accessing the strength of one currency. 1 9 9 4 1 9 9 6 1 9 9 8 2 0 0 0 2 0 0 2 2 0 0 4 2 0 0 6 2 0 0 8 2 0 1 0 2 0 1 2 2 0 1 4 2 0 1 6 2 0 1 8 2 0 2 0 Disbursed FDI Growth Rate (%,yoy) VND Depreciation Rate (%,yoy;RHS)

The data records a high correlation between these two variables: when the disbursed FDI growth rate reduces over time, the VND depreciation rate also goes down. The reduction of disbursed FDI growth rate proves the inflows of FDI tends to approach the aborption capacity of an economy. And the decrease of VND depreciation rate proves that the domestic currency tends to enter a stable zone, in which, its value has less standard deviation around the mean. Therefore, the combination of low disbursed FDI growth rate and low VND depreciation rate shows that the FDI inflows are associated with more stable domestic currency.

CONCLUSION

The paper investigates the term structure of foreign direct investment into Vietnam economy by dividing the short-term, medium-term with long-term contribution on the domestic economic growth. The data is an annual sample of Vietnam economy over 2007-2021.

The empirical evidence records that the foreign direct investment stimulates the domestic economic growth by promoting the net exports in the medium-term and by the capital accumulation process in the long-term. In the shor-term, the FDI also contributes to more stable domestic currency. Therefore, the foreign direct investment is a crucial growth engine for the Vietnam economy.

The evidence suggests that the foreign direct investment can be a priority for the public policy to enhance the domestic economic growth rate. Beside the domestic sector, including the state-owned firms and private firms, the foreign direct investment needs to be managed so that its contribution to domestic economic growth is maximized.

REFERENCE

  breakthrough laws such as the Law of Firm, Law of Foreign Investment. And the ending point is 2021 when the Vietnam economy is recovering from the recent Covid-19 pandemic. With a long time period of over 30 years, the data sample is useful to shed a new light into the term-structure of the foreign direct investment into the Vietnam economy. The foreign direct investment (FDI) is measured by the disbursed values in billion USD in current price. And the long-term contribution of FDI into the economic growth is measured by the share of FDI sector over the gross domestic product in percentage. The medium-term contribution of FDI into the foreign trade is measured by the share of FDI sector over the exports, imports and total exports and imports value. The short-term contribution of FDI into the foreign exchange market is measured by the fluctuation of official VND/USD exchange rate over time. The data on the foreign direct investment, economic growth, exports and imports are all from the Vietnam General Statistic Office (GSO, 2022). And the data on the official exchange rate is from the World Bank Macroeconomic Monitoring Database (WB, 2022).

Figure 1 :

 1 Figure 1: Foreign Direct Investment and Economic Growth

  shows, while the domestic sector continuously run the deficit trade balance, the FDI sector have the surplus trade balance. For example, in 2021, the domestic sector has a deficit trade balance of 25 billion USD while the foreign sector has surplus trade balance of 28 billion USD. This results in an overall surplus trade balance at about 3 billion USD in 2021. Therefore, the trade surplus by the FDI sector is strong enough to counteract the trade deficit by the domestic sector.

Figure 2 :

 2 Figure 2: Foreign Direct Investment and Foreign Trade

Figure 3 :

 3 Figure 3: Foreign Direct Investment and Domestic Currency

  Hung, Ly Dai. "Output-inflation trade-off in the presence of foreign capital: Evidence for Vietnam." South Asian Journal of Macroeconomics and Public Finance 10.2 (2021): 179-192.


Ly Dai Hung
email: hunglydai@gmail.com

GLOBAL TRADE CYCLE AND FINANCIAL CYCLE IN VIETNAM ECONOMY

Keywords: Global Trade Cycle, Financial Cycle, Quantitative Analysis, Qualitative Analysis, Vietnam economy

Objective:

The paper investigates the jointly dynamics of international trade cycle and financial cycle in the Vietnam economy. The data is a time-series sample of Vietnam economy over the 1995-2021 time period. And the research objective is to analyze the co-movement of foreign trade and financial cycle, incorporating the recent Covid-19 pandemic.

Methodology:

The research methodology combines both qualitative analysis with quantitative one. In particular, the qualitative analysis reviews the recent literature on the global trade and financial cycle, with a focal point on the developing economies. Then, the quantitative analysis considers the time-series analysis of Vietnam economy, as one typical small open economy.

Findings:

The global trade and financial cycle are considered as a central role within the current international monetary system. The trade cycle can be effective in the short and medium term while the financial cyle exists for a longer term. In the Vietnam economy, the trade and financial cycle co-moves closedly, especially during the time period of structural break such as Covid-19 pandemic.

Implications:

The results uncover that the public policy needs to be adjusted according to the points in the global trade and financial cyle. The policy at the peak of cycle can be tightened while the policy at the trough of cycle needs to be expanded. And the policy at both the peak and trough needs to be accompanied with an excellent communication to the whole economy.

INTRODUCTION

Global trade and financial cycle are at the central point of recent discussion on the international integration, based on a well-established background with both their theoretical and empirical evidence [START_REF] Borio | The financial cycle and macroeconomics: What have we learnt?[END_REF]. Rey (2016) prove that there exist a global financial cyle which affects both the trading and credit supply of economies in the world, and Obstfeld et al (2019) and Obstfeld (2020) show that the exchange rate fluctuation can create a channel for one economy to absorb the external shocks coming from the global cycles.

Recently, Miranda-Agrippino and Rey (2020) detects there exist common factors causing the co-movement of various macroeconomic variables on the global scale, i.e, a global cycle. Their evidence records that the United States economy plays a central role in the global financial cycle, which works through the exchange rate fluctuation and international credit supply as well as cross-border capital flows.

And the China economy determines mostly the global trade cycle through its large impact on the world commodity and exports as well imports of goods and services.

With an in-depth international integration, with the trade openness of about 184% in 2021, the Vietnam economy is affected directly by the external shocks originating from the world. Therefore, understanding the trade and financial cylecycle in the Vietnam economy, then, the comparison if these cycles with the global cycles are important for both the academic and policy discussion.

The current paper investigates the trade and financial macroeconomic variables in the Vietnam economy and compares them with their global counterparts. In particular, the foreign trade is captured by the exports and imports per gross domestic product (GDP), while the finance is illustrated by the domestic credit supply to the private sector per GDP.

In the Vietnam economy, the trade and financial cycle co-moves closely, and both can have lagged effects on the domestic economic growth rate. Compared with the global trade and financial cycle, illustrated by the east Asia & pacific and the world economy, there exists a difference between domestic and global cycle.

The domestic credit supply only has a common tendency with the global one after the 2008 world financial crisis while the domestic exports and imports only have a common tendency with the global one before the 2008 crisis.

The paper is structured as follows. After the first section in the Introduction, the second section on the Analysis framework presents the data and method. Then, the third section on empirical evidence investigates the trade and financial cycles in the Vietnam economy compared with the world. Then, the fourth section concludes.

ANALYSIS FRAMEWORK

We employ a quantitative analysis method that is based on the correlation between the foreign trading value and financial indicators with the real economy.

The correlation, by definition, measures the co-movement of these variables over time, then can capture the existence of the trade and financial cycle in the Vietnam economy. Moreover, we also compute the cyclical component of the trading and financial variables by using the Hodrick -Prescott (HP) filter. The cyclical component is the variation of one variable from its trend value.

In detail, we measure the foreign trading cycle by the value of total exports and imports per gross domestic product (GDP) in percentage. The financial cycle is measured by the value of domestic credit to the private sector per GDP, also on percentage. And the real economy is measured by the growth rate of rthe eal gross domestic product in constant 2010 national price.

For the data, we collect the sample from various sources. In particular, the exports and imports per GDP percentage are from the Vietnam General Statistic Office (GSO, 2022). The domestic credit to the private sector percentage is from the State Bank of Vietnam (SBV, 2022). And the economic growth rate on inrcentage is from the Vietnam GSO (2022).

We also collect the data for the East Asia & Pacific economies and world economies. These variables illustrate the tendency of the rest of the world, then, show the global trade and financial cycle. The data is from world development indicators.

EMPIRICAL EVIDENCE

Table (1) shows the correlation result of trading and financial variables in our data sample. First, the correlation of credit supply with the exports and imports are both positive, with a quite high value, being 0.92 and 0.85 respectively. This high correlation predicts that there can be a co-movement of the trading cycle and financial cycle in the Vietnam economy. Another important feature is that the correlation of economic growth with the credit supply, exports, and imports are all negative. This value can suggest that these trading and financial variables can have a lagged effect on economic growth, thus, resulting in a negative relationship. Next, we examine the dynamic tendency of credit supply and foreign trade over the 1990-2021 period. Another approach to examining the co-movement of trading and financial variables is the use of cyclical components. 1 9 9 0 1 9 9 2 1 9 9 4 1 9 9 6 1 9 9 8 2 0 0 0 2 0 0 2 2 0 0 4 2 0 0 6 2 0 0 8 2 0 1 0 2 0 1 2 2 0 1 4 2 0 1 6 2 0 1 8 2 0 2 0 Domestic credit to private sector (% of GDP)

Exports of goods and services (% of GDP)

Imports of goods and services (% of GDP) The results uncover that public policy needs to be adjusted according to the points in the global trade and financial cycle. The policy at the peak of the cycle can be tightened while the policy at the trough of the cycle needs to be expanded.

And the policy at both the peak and trough needs to be accompanied by excellent communication with the whole economy.

Figure ( 1

 1 Figure (1) shows that the domestic credit to the private sector as a percentage of GDP tends to go up over time. It reached 100% in 2010, then, nearly 140% in 2021. During this period, both the exports and imports raise over time but with a switching regime. Before 2008, these trading variables moves closely with the credit supply, but right after 2008, their increases are significantly lower than the increase in credit supply. The imports after 2008 are even lower than their peak in 2008, while the exports getsget their peak in 2021.

Figure 1 :

 1 Figure 1 : Credit Supply, Exports and Imports in Vietnam Economy

  Figure (2) records that there exists a quite strong related movement of these variables. During the 1995-2009 period, the credit has an increasing variation around its trend, with a peak of cyclical movement at nearly 27% in 2010. Both the exports and imports follow a similar pattern over the same period. And after 2010, these variables reduce substantially their cyclical variations. The overall tendency is that these trade and financial variables tend to move closely with each other during the 1995-2021 period.Next, we detect the existence of the co-movement of domestic and foreign trading and financial variables.

Figure 2 :

 2 Figure 2: Foreign Trading and Financial Cycle in Vietnam's Economy

Figure ( 3

 3 Figure (3) focuses on the domestic credit to the private sector per GDP.There exist different patterns of domestic and foreign credit supply over the 1995-2009 period. The credit supply of VietnamVietnam's economy tends to increase over time while that of east Asia and the pacific as well as the world economy goes down. But over the next period of 2008 -2021, these three variables increase together. And during the 1995-2021 period, the credit supply per GDP of east Asia & pacific, and the world economy is always higher than the corresponding value of Vietnam's economy. In recent years, the gap between these variables tends to be lower over time.Overall, the increase of east Asia and pacific and world credit supply after 2008 can be a recovery to its high level in the 1990s. But the increase of credit supply in the Vietnam economy tends to follow one uniform pattern, not depending on the global financial crisis in 2008.

Figure 3 :

 3 Figure 3 : Domestic Credit to Private Sector per GDP (%)

  Figure 3 : Exports per GDP (%)

Figure 4 :

 4 Figure 4 : Imports per GDP (%)

  the joint dynamics of the international trade cycle and financial cycle in the Vietnam economy. The data is a time-series sample of Vietnam's economy over the 2005-2021 time period. And the research objective is to analyze the co-movement of foreign trade and the financial cycle, incorporating the recent Covid-19 pandemic. The global trade and financial cycle are considered as a central role within the current international monetary system. The trade cycle can be effective in the short and medium term while the financial cycle exists for the longer term. In the Vietnam economy, the trade and financial cycle co-moves closely, and both can have lagged effects on the domestic economic growth rate. In comparison with the global trade and financial cycle, illustrated by the east Asia & pacific and world economy, there exists a difference between domestic and global cycles. The domestic credit supply only has a common tendency with the global one after the 2008 world financial crisis while the domestic exports and imports only have a common tendency with the global one before the 2008 crisis.

Table 1 :

 1 Correlation of Variables

	Variables	Credit	Exports	Imports	Growth
	Credit	1.00
	Exports	0.92	1.00
	Imports	0.85	0.92	1.00
	Growth	-0.57	-0.54	-0.40	1.00
			Source : Author's calculation from data sample.