Accurate high-resolution measurements are essential to improve our understanding of environmental processes. Several chemical sensors relying on membrane separation extraction techniques have slow response times due to a dependence on equilibrium partitioning across the membrane separating the measured medium (i.e., a measuring chamber) and the medium of interest (i.e., a solvent). We present a new technique for deconvolving slow-sensor-response signals using statistical inverse theory; applying a weighted linear least-squares estimator with the growth law as a measurement model.
The solution is regularized using model sparsity, assuming changes in the measured quantity occur with a certain time step, which can be selected based on domain-specific knowledge or L-curve analysis. The advantage of this method is that it (1) models error propagation, providing an explicit uncertainty estimate of the response-time-corrected signal; (2) enables evaluation of the solution self consistency; and (3) only requires instrument accuracy, response time, and data as input parameters. Functionality of the technique is demonstrated using simulated, laboratory, and field measurements. In the field experiment, the coefficient of determination (

High-resolution in situ data are crucial to observe high variability in environmental processes when surrounding environmental parameters are continuously changing.
Many contemporary measurement techniques have a limited response time due to signal convolution inherited from a diffusion process, such as in Vaisala radiosondes

A numerical technique has already been proposed to recover fast fluctuations in

Herein, we establish an alternative method for estimating

We assume that the relationship between observed quantity

To formulate the measurement equation (Eq.

We discretize Eq. (

We assume that sensor measurements

To reliably estimate

Model sparsity regularization (see e.g.,

We can now express the theory, relationship between the measurements and the theory, and the smoothness assumption in matrix form as follows.

The definitions of the matrices are as follows, where

The vector

Equation (

As the uncertainties of measurements are already included in the theory matrix

The quality of the solution relies on an appropriate choice of regularization parameter

To test the numerical validity of our method and develop a regularization parameter selection tool, we used a toy model. This gives us the possibility to prove that the method gives well-behaved, consistent solutions as we know the correct results and control all input variables.

We defined the simulated concentration

The measurement chamber partial pressure

As

In our case, the L-curve criterion involves calculating a norm

We have used the first-order differences of the maximum a posteriori solution

We calculated a set of estimates using a wide range of

Estimated property using simulated data and different regularization parameters (time intervals). The simulated measurements

Using

Choosing the best

Even though the step function is an unrealistic scenario in a practical application, it is likely that the variability of a measured property can change considerably within a single time series.
Since the L-curve criterion provides a

We evaluated our proposed technique in a controlled laboratory experiment using a Contros HydroC CH

Four step changes (two up and two down) were approximated by opening the lid and adding 0.2 L of either methane-enriched or ultrapure water. The RT of the sensor was determined to 23 min at 22

The first 2 h had lower noise due to the reduced sampling rate and hence longer internal averaging period, but noise was otherwise constant and unrelated to the measured concentration. We therefore used two separate

At the first step change, the RT-corrected concentration rapidly increased from around 2.6 to

Estimated uncertainty of the RT-corrected data averaged 0.64

Continued evaluation of our proposed technique was applied under more challenging conditions in a field-based study using simultaneous data from two different methane sensors towed over an intense seabed methane seep site offshore west Spitsbergen

We determined the growth coefficient

The measurement uncertainty (

We produced an L curve from a set of estimates of

The comparison between DRB, EB, and RT-corrected EB sensor data collected during the transect offshore west Spitsbergen is shown in Fig.

The untreated EB sensor data clearly show how the convolution creates a strong hysteresis effect and makes the sensor unable to directly detect rapid changes in methane concentration. This results in a low coefficient of determination (

The high

Using the framework of inverse theory allows us to model error behavior by calculating the covariance matrix

The DRB data have a lower median relative (%) uncertainty estimate, but to compare these relative uncertainty estimates directly can be slightly misleading as the relative uncertainty estimate of the RT-corrected data varies in time (Fig.

We presented and successfully applied a new RT-correction algorithm for membrane-based sensors through a deconvolution of the growth-law equation using the framework of statistical inverse problems. The method requires few and well-defined input parameters, allows the user to identify measurement issues, models error propagation, and uses a regularization parameter which relates directly to the resolution of the response-time-corrected data. Functionality testing was done using both a laboratory and a field experiment. Results from the laboratory experiment uncovered features of the experimental setup which were obscured by convolution in the raw data, and the field experiment demonstrated the robustness of the algorithm under challenging environmental conditions. In both tests, the sensors' ability to describe rapid variability was significantly improved, and better constraints on input uncertainty and response time are areas which can potentially further enhance results.

This method and validation experiments using the Contros/HISEM sensors uncover a new set of applications for these and similar sensors, such as ship-based profiling/towing and monitoring of highly dynamic domains. Conventional EB sensors are also more abundant and affordable compared to more specialized equipment, increasing the availability and possibilities for scientists requiring high-resolution data to solve their research questions. Additionally, we believe this deconvolution method could be applicable to other measurement techniques as well, where diffusion processes hamper response time.

Even though

One issue that arose during development of the automatic

To apply our methodology to the EB field sensor dataset and compare with the DRB data, the sensor growth coefficients

Water temperature and salinity have a direct impact on

Figure

The fluctuations on each side of the diagonal elements (non-diagonal elements) in row 100 (see Fig.

All data and code presented in this paper can be
obtained here:

Writing, original draft preparation, and method and software development were done by KOD and JV. Data were curated by KOD and RG. Investigation and formal analysis were done by KOD, JV, and RG. Visualization was done by KOD. The project was administrated and supervised by KOD, RG, JT, and BF. Resources and funding were acquired by RG, JT, and BF. All authors contributed to reviewing and editing the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research leading to these results has received funding from the European Commission's Seventh Framework Programmes ERC-2011-AdG under grant agreement no. 291062 (ERC ICE&LASERS), the ERC-2015-PoC under grant agreement no. 713619 (ERC OCEAN-IDs), and the Agence National de Recherche (ANR SWIS) under grant agreement no. ANR-18-CE04-0003-01. Additional funding support was provided by SATT Linksium of Grenoble, France (maturation project SubOcean CM2015/07/18). This study is a part of CAGE (Centre for Arctic Gas Hydrate, Environment and Climate), Norwegian Research Council (grant no. 223259). We thank the crew of R/V

This research has been supported by the European Research Council, FP7 Ideas: European Research Council (grant no. ICE&LASERS (291062)), the H2020 European Research Council (grant no. OCEAN-IDs – OCEAN in-situ Isotope and Dissolved gas sensing (713619)), the Agence Nationale de la Recherche (grant no. ANR-18-CE04-0003-01), the Société d'Accélération du Transfert de Technologies (grant no. Maturation project SubOcean CM2015/07/18), and the Norges Forskningsråd (grant no. 223259).

This paper was edited by Takehiko Satoh and reviewed by two anonymous referees.