Multi-Dimensional Gas Metering

• Multi-dimensional gas metering is a technique that recovers several gas parameters simultaneously, addressing the limitations of single scalar measurements in utility, optical, and blockchain contexts.
• It encompasses applications from reading multi-dial utility meters and integrated sensor platforms like CPAS to innovative fee metering in EVM-compatible blockchains.
• Key methodologies involve image-based annotation, advanced signal processing with neural networks, and explicit multi-resource fee accounting to enhance accuracy, speed, and privacy.

Multi-dimensional gas metering denotes metering and read-out problems in which a single scalar gas value is insufficient, and the instrument, inference model, or accounting framework must recover several dimensions simultaneously. In the physical metrology literature, this includes joint interpretation of multiple analogue dials, simultaneous sensing of hydrogen and hydrocarbons from one output trace, measurement of two independent gas flows on one chip, inference of phase-wise superficial velocities from full spatiotemporal void-fraction fields, and rapid optical assessment of gas density and leak or flow parameters from repeated refractometric readouts. A distinct computational literature uses “gas metering” for EVM-compatible blockchains and argues that one scalar gas price is too rigid for workloads with varying mixes of compute, storage reads, storage writes, call overhead, calldata, and persistent state growth (Ebadi et al., 2021, Zhuang et al., 2024, 0805.0892, Dave et al., 2020, Silander et al., 2017, Heimbach et al., 18 Jun 2026).

1. Meanings of dimensionality

The cited literature uses dimensionality in several technically distinct ways. In image-based utility metering, dimensionality arises from multiple analogue dials that rotate at different rates and must be interpreted jointly; NRC-GAMMA explicitly frames the left dial and right dial as a multi-dial, multi-resolution reading problem, and also exposes full-image context versus cropped local context under day/night and artifact-rich capture conditions. In gas analysis, conductance-photoacoustic spectroscopy encodes hydrogen concentration in resonance-frequency shift and propane or hydrocarbon concentration in beat-signal amplitude, both extracted from one output trace. In microflow sensing, a single silicon chip supports two independent gas flows by routing them into distinct sealed channels. In multiphase flow metering, a neural network maps the full spatiotemporal void-fraction field to the pair of superficial velocities $(v_g, v_f)$. In optical refractometry, two or three rapid measurements are used to recover gas density, volumetric expansion factors, and leak or flow parameters. In smart-home metering, electricity, water, and natural gas form a multi-resource environment with cross-channel privacy leakage. In EVM workload analysis, dimensionality refers to separate execution resources and persistent state burdens that are collapsed by current scalar gas accounting (Ebadi et al., 2021, Zhuang et al., 2024, 0805.0892, Dave et al., 2020, Silander et al., 2017, Kement et al., 2020, Heimbach et al., 18 Jun 2026).

A common ambiguity is that the phrase is not confined to chemical composition sensing. The literature also uses related multi-dimensional ideas for segmented gaseous read-out architectures. ACHINOS, a multi-ball read-out for the spherical proportional counter, replaces a single anode ball with several small balls distributed on a virtual sphere, allowing field tuning, detector segmentation, and, with individual ball read-out, a spherical time projection chamber with 3D capabilities. This is not a utility meter, but it shows that dimensionality can also be introduced at the level of read-out geometry and segmentation in gas-filled instrumentation (Giganon et al., 2017).

2. Analogue multi-dial utility meters

The paper “NRC-GAMMA: Introducing a Novel Large Gas Meter Image Dataset” describes a practical but under-automated problem: reading conventional gas meters accurately, at scale, and with less labor than manual inspection. NRC-GAMMA was created from an Itron 400A diaphragm gas meter installed in an NRC test house in Ottawa, Canada. The meter has two proving dials and a 5-digit cyclometer display, but the dataset intentionally focuses on the two dial displays because the left dial rotates at $50 \ \mathrm{dm}^3$ per revolution and the right dial at $10 \ \mathrm{dm}^3$ per revolution, creating a multi-dial, multi-resolution reading task (Ebadi et al., 2021).

The dataset contains 28,883 images of the entire gas meter and 57,766 cropped dial images, split evenly into 28,883 left-dial and 28,883 right-dial crops. The cropped images are $160 \times 160$ pixels. Capture ran continuously from 00:05 to 23:59 on January 20, 2020, with images taken on average every 3 seconds. The imaging system used a Raspberry Pi 3b+ with PoE power, a 5-megapixel infrared NoIR camera, and a weatherproof enclosure. Because acquisition covered day and night, the dataset includes variation in lighting, shadowing, glare, and blur from wind, together with other environmental artifacts. The paper positions this variability as essential for robust automatic gas meter reading.

A notable contribution is the annotation and validation workflow. Amazon Mechanical Turk was used for labeling, with a custom GUI allowing annotators to choose the representative dial region directly from a dial image. Annotators were instructed that if they were uncertain between two regions, they should choose the next anticlockwise region. Each image received at least three annotations, and many received six across two rounds. The first round accepted unanimous agreement among three annotators unless one annotator had poor quality. Non-unanimous cases were sent to a second round with three new annotators. If the second round was unanimous, that label was accepted; otherwise the two rounds were merged and mode-based rules were applied. When two modes were tied but adjacent on the dial, both labels were retained by duplicating the image and assigning one label to each copy. Extremely ambiguous cases were sent to visual inspection by three subject experts using majority vote.

The explicit annotator-quality measure is

$Q_a = \frac{A_u}{A_t},$

where $A_u$ is the number of unacceptable annotations and $A_t$ is the total number of annotations completed by an annotator. High-quality annotators satisfy $Q_a < 0.2$, fair annotators satisfy $0.2 \leq Q_a \leq 0.5$, and low-quality annotators satisfy $Q_a > 0.5$. Among 173,298 first-round annotations, about 6% were deemed unacceptable. The paper also reports 1,659 images with two adjacent labels. This matters because dial-position classes are ordered, and some disagreements are neighboring bins rather than arbitrary misclassifications.

NRC-GAMMA is designed for modular pipelines. Full-meter images support end-to-end systems for detection, cropping, and recognition, while left/right dial crops support direct experimentation on recognition. Crop filenames encode a left/right indicator, a timestamp, and the label, allowing alignment with the original full-meter image. The paper does not prescribe one recognition architecture, but situates the dataset within multi-stage AMR systems involving region detection, dial segmentation, and classification or recognition of dial state. Its principal limitation is that it is a single-meter dataset from one installation, so transfer to other meters or installations may require adaptation.

3. Integrated multi-component sensing from one signal

The paper “Conductance-Photoacoustic Spectroscopy for OneSignal Measurement of Multi-components” presents conductance-photoacoustic spectroscopy (CPAS) as a single integrated platform for multi-dimensional gas metering of hydrogen plus hydrocarbon fuel components. The platform combines platinum-modified conductance spectroscopy for hydrogen with beat-frequency photoacoustic spectroscopy for propane or hydrocarbons, so that one output trace carries information about two different gas species (Zhuang et al., 2024).

The sensor is built on a quartz tuning fork (QTF)-based spectrophone. A Pt microwire of 15 $50 \ \mathrm{dm}^3$0m diameter and 2 mm length is bridged across the QTF prongs. Stainless-steel micro acoustic resonators are placed on both sides of the QTF, and the QTF sits near the maximum acoustic pressure region with about 50 $50 \ \mathrm{dm}^3$1m gap from the resonator ends. A laser source, specifically an interband cascade laser centered at 3368 nm, is used for propane absorption. A function generator provides a sinusoidal modulation signal and a step-ramp signal for creating the beat-frequency condition, and the output is processed through a pre-amplifier, lock-in amplifier, and DAQ.

Hydrogen is inferred from frequency shift of the Pt-modified QTF. The pristine resonance is reported as 32760.2 Hz, and the resonance after Pt bridging is 32667.9 Hz. The paper states that the effective resonance frequency shifts in response to hydrogen concentration, and calibration maps that shift to concentration. Platinum is chosen because of high catalytic activity, chemical stability, minimal cross-sensitivity to other gases, and reversible hydrogen response even at low concentrations.

Propane is measured through beat-frequency photoacoustic spectroscopy. The modulation frequency is set slightly different from the QTF resonance, generating a beat-frequency signal whose amplitude correlates with gas concentration. In the experiment, the QTF resonance after Pt modification is about 32667.9 Hz, the modulation frequency is 32600 Hz, and the beat difference is about 70 Hz. The laser is tuned to a propane absorption line at 3369.74 nm, with laser current swept from 90.1 mA to 92.1 mA. The paper uses the difference between a peak and its subsequent trough as the signal amplitude metric.

The proof-of-concept mixture contained 25% $50 \ \mathrm{dm}^3$2, 2% $50 \ \mathrm{dm}^3$3, and balance $50 \ \mathrm{dm}^3$4. From the shown trace, the horizontal periodicity indicates the QTF resonance and therefore hydrogen concentration, while the vertical amplitude peaks correlate with propane absorption and therefore propane concentration. The reported quantitative results are 25% hydrogen and 2% propane. The authors explicitly state that propane is only a demonstration and that the same approach can be extended to methane, ethane, and other hydrocarbons by selecting an appropriate laser chip.

The paper repeatedly emphasizes “one-signal measurement,” “single detection channel,” “single output trace,” and simultaneous determination of hydrogen and hydrocarbon concentrations. It also describes the BF-PAS part as calibration-free and the overall platform as compact, rapid, and real-time. At the same time, the scope is explicitly proof-of-concept. The hydrogen concentration depends on a calibration relation between resonance shift and $50 \ \mathrm{dm}^3$5 concentration; the “calibration-free” claim applies more clearly to the BF-PAS tracking strategy. The excerpt also does not provide a full interference study or a detailed table of detection limit, response time, linearity, or long-term stability.

4. Flow, density, and distributed gaseous read-out

A distinct line of work addresses dimensionality through multiple flows, multiple phases, or repeated density measurements. The 2008 paper on “Single Chip Sensing of Multiple Gas Flows” describes a 4 mm $50 \ \mathrm{dm}^3$6 4 mm silicon chip fabricated in a BCD6 bipolar-CMOS-DMOS process with post-processing micromachining. The sensing element is a differential thermal anemometer with a p-doped polysilicon heater and two thermopiles placed upstream and downstream. A PMMA adapter, thermally bonded to the chip at about 110 °C for five minutes while applying about 5 N of force, defines two independent U-shaped trenches above distinct sensing regions. This allows two independent gas flows to be measured simultaneously on one chip. The heater is biased at 2 V, corresponding to 4 mW per sensor. The two-channel device shows monotonic flow response, and no measurable cross-talk was detected even when one channel was driven with flow rates up to 50 sccm while monitoring the other. The reported low-flow linear fits are about 0.083 mV/sccm and 0.055 mV/sccm for the PN channel branches, and about 0.065 mV/sccm for the PA channel. A reference single-channel device achieved about 0.64 mV/sccm for the PN structure and 0.41 mV/sccm for the PA structure, showing that the two-channel packaging preserved independence at the cost of reduced sensitivity due to channel geometry near a 90° bend (0805.0892).

The 2020 paper “Inference of Gas-liquid Flowrate using Neural Networks” reformulates multiphase metering as learning a direct mapping from a wire-mesh sensor (WMS) void-fraction field to the superficial gas and liquid velocities,

$50 \ \mathrm{dm}^3$7

The WMS in the TOPFLOW facility is installed in a vertical, upward, 50.8 mm ID pipe of length 8.0 m, at inlet pressure 0.25 MPa, with the sensor located at $50 \ \mathrm{dm}^3$8. The WMS operates at 2.5 kHz for 10 s per sample, yielding an input tensor of shape $50 \ \mathrm{dm}^3$9, further sampled to $10 \ \mathrm{dm}^3$0, corresponding to 204.8 ms of measurements. The training database contains 46 permutations of gas and liquid superficial velocity spanning

$10 \ \mathrm{dm}^3$1

across bubbly, slug, churn-turbulent, and annular regimes. The best-performing network is Model E, combining a 3D convolution head and an LSTM tail; the first Conv3D layer uses 16 kernels of size $10 \ \mathrm{dm}^3$2. With mean absolute percentage error as the preferred metric, the model achieves below 7.5% MAPE across all flow regimes and below 5% MAPE in bubbly flow. For a $10 \ \mathrm{dm}^3$3 input, evaluation takes 3.36 ms, roughly two orders of magnitude shorter than the 204.8 ms input window (Dave et al., 2020).

Fast switching dual Fabry-Perot cavity optical refractometry (FS-DFPC-OR) adds another dimension by using rapid measurement sequences to recover density and flow while suppressing drift. The method relies on completing evacuation and readout before cavity-length drift becomes important; Allan-Werle analysis showed a minimum Allan deviation around an integration time of about 0.25 s, with drift becoming important beyond roughly 0.5 s. For closed compartments, two evacuations in rapid succession determine the gas-expansion factor automatically,

$10 \ \mathrm{dm}^3$4

allowing the original external density to be inferred without separately measuring the volumes. For leak metrology, three cavity evacuations are used: two rapid measurements establish the initial density and expansion factor, and a third later measurement determines density change and hence leak or flow rate. The paper emphasizes exceptionally weak temperature dependence because the leading refractivity–density relation is governed by molecular polarizability and only weakly by higher-order corrections and cavity deformation (Silander et al., 2017).

A related instrumentation development, though outside conventional utility metering, is ACHINOS for the spherical proportional counter. The simulated 11-ball geometry uses 11 metallic balls of 2 mm diameter on a 36 mm virtual sphere around a 20 mm diameter bakelite central sphere. In a 300-mm-diameter detector model, the electric field near the shell is about 8 times higher than for a single 2-mm ball at the same 2000 V bias. The measured rise-time full width improved from about 9.6 $10 \ \mathrm{dm}^3$5s in a single-ball SPC to about 1.1 $10 \ \mathrm{dm}^3$6s in an 11-ball ACHINOS, while energy resolution with a $10 \ \mathrm{dm}^3$7 source was 12.4%. The paper’s importance for the present topic is that dimensionality is introduced through segmentation and individual ball read-out, which yields 3D capabilities in a gas-filled detector (Giganon et al., 2017).

5. High-time-granularity natural gas metering and privacy

Multi-dimensional gas metering also appears in smart-home privacy research, where natural gas is treated as one element of a broader high-time-granularity metering environment that includes electricity and water. The paper “Holistic Privacy for Electricity, Water, and Natural Gas Metering in Next Generation Smart Homes” argues that privacy protection limited to electricity is ineffective for appliances that use multiple metered resources. Natural gas metering is described as increasingly capable of high sampling frequencies and high precision, with technologies including traditional diaphragm and rotary meters, high-resolution encoders, turbine gas meters, orifice meters, and ultrasonic and Coriolis meters; the paper states that gas metering can reach very fine resolution and even up to 1 kHz in some cases (Kement et al., 2020).

The privacy problem is framed by cross-resource appliance signatures. HVAC systems consume electricity and natural gas; washing machines and dishwashers consume electricity and water; combination boilers consume electricity, water, and gas. The paper uses Mutual Information (MI) as the leakage metric and evaluates three cases with the Ampds2 dataset. With no load shaping, the reported MI values are 1.91 bits for HVAC power versus metered power, 1.61 bits for HVAC gas versus metered gas, 0.59 bits for washing-machine water versus metered water, and 0.56 bits for washing-machine power versus metered power. With electricity-only shaping using the Best Effort strategy and a 2 kWh battery, HVAC power MI drops to 0.23 bits and washing-machine power MI drops to 0.03 bits, but HVAC gas MI remains 1.82 bits and washing-machine water MI remains 0.49 bits. With holistic shaping of electricity, water, and gas, using a battery for electricity, a tank for water, and no gas tank because storing gas is considered unsafe, the reported values become 0.49 bits for HVAC electricity, 0.58 bits for HVAC gas, 0.17 bits for washing-machine electricity, and 0.01 bits for washing-machine water.

The central conclusion is that leakage from one resource stream can defeat privacy measures on another. In this literature, dimensionality therefore refers not to a richer physical gas state inside one instrument, but to the joint observability of multiple metered resources whose appliance signatures are correlated. This suggests that next-generation gas metering cannot be analyzed solely as a measurement accuracy problem; under HTG operation it is also an inference and privacy problem.

6. Scalar gas pricing versus multi-dimensional fee accounting

A separate computational literature uses “gas metering” to denote transaction pricing in EVM-compatible blockchains. “EVM Workloads in the Wild: Evidence for Multi-Dimensional Gas Metering, State Growth, Delayed Execution, and Parallelism” argues that current scalar gas accounting assumes a stable resource mix and negligible state drift between simulation and execution, and that both assumptions fail in practice. The study runs archival reth nodes for Ethereum and Base, samples 3,000 blocks per day per chain throughout 2025, decomposes transactions into intrinsic gas, execution gas, refunds, and persistent state deltas, and re-executes September 2025 transactions on nearby historical states to measure state sensitivity (Heimbach et al., 18 Jun 2026).

The paper reconstructs total gas as

$10 \ \mathrm{dm}^3$8

Execution gas is then decomposed by opcode category.

Category	Ethereum	Base
Storage read	22.73%	29.23%
Storage write	34.88%	25.53%
Compute	20.21%	24.30%
Calls	10.18%	12.57%
Contract ops	4.47%	4.78%
Logging	7.19%	3.45%
Transient storage	0.15%	0.09%
Epilogue	0.20%	0.04%

These figures show Base as read- and compute-heavy and Ethereum as write-heavy. Mean gas per transaction is 103,284 on Ethereum and 232,936 on Base; execution gas accounts for 79.82% of total gas on Ethereum and 93.30% on Base. The paper further reports that Base has a higher fraction of cold storage reads, with cold SLOADs at 49.7% on Base versus 39.6% on Ethereum, and mean SLOAD accesses per transaction of 58.1 versus 21.0. Ethereum’s gas limit doubling from 30M to 60M during 2025 shifted its own resource profile toward more compute and fewer writes, illustrating that workload composition can change even without opcode repricing.

Persistent state growth is treated as a separate, mispriced resource. Extrapolated 2025 state growth is 38.07 GB for Ethereum and 456.09 GB for Base. On Base, 58.6% of this is storage slots, 24.2% bytecode, and 17.1% account state; on Ethereum, the corresponding shares are 40.0%, 14.3%, and 45.7%. Average per transaction, Base writes 16.29 slots and 431.2 bytes of bytecode, whereas Ethereum writes 1.08 slots and 24.7 bytes of bytecode. The paper argues that permanent state growth is being priced as if it were transient execution, and therefore advocates explicit pricing of state growth through mechanisms such as storage rent, upfront deposits, or per-byte charges.

State sensitivity is quantified by re-executing transactions across lookbacks and computing the coefficient of variation,

$10 \ \mathrm{dm}^3$9

for gas estimates across successful states. The share of transactions with non-zero gas-estimate variance is 13.9% on Ethereum and 46.0% on Base. Mean CV is 0.57% on Ethereum and 6.88% on Base, with 99th percentile CV of 14.84% and 139.74% respectively. MEV and DeFi are the most state-sensitive categories; on Base, MEV has mean CV 9.76%, 99th percentile CV 153.83%, and non-zero variance in 52.4% of transactions. The paper also reports lower read overlap, write overlap, and value consistency across states on Base than on Ethereum, which limits access-list effectiveness and complicates optimistic parallel execution.

In this computational setting, multi-dimensional gas metering means separate pricing of computation, storage reads, storage writes, call-related operations, calldata, and persistent state growth. The paper’s contribution is empirical rather than prescriptive: it does not present a complete protocol redesign, but it provides measurements intended to justify multi-dimensional fee markets, explicit state pricing, and state-aware execution or scheduling mechanisms.