Time Series Of Measurements Research Articles

Simultaneous measurement of temperature and C2H4 concentration in hydrocarbon flames using interference-immune differential absorption spectroscopy at 3.3 μm

In diagnostic applications based on tunable diode laser absorption spectroscopy, the measurement of target substances can be influenced by factors such as background thermal radiation in the combustion environment, extinction caused by solid or liquid particles, and other interfering absorptions. In this work, we developed a differential absorption diagnostic technique based on wavelength pairs, utilizing an interband cascade laser near 3.3 μm to simultaneously measure temperature and C2H4 concentration in hydrocarbon flames. Based on a detailed study of the C2H4 spectrum in this region and considering the optimal standard for spectral lines, two wavelength pairs were selected. The temperature is determined by the ratio of the absorption cross-sections of two wavelength pairs, and the C2H4 concentration is inferred based on the wavelength pair with higher differential absorption. In the initial stage, the system’s accuracy was verified in high-temperature static conditions (T = 300–800 K, P = 1 atm), and continuous time series measurements demonstrated the system’s stability. The limit of detection achieved by Allan-Werle variance analysis is 2.5 ppm at the optimal average time of 100 s. Subsequently, measurements were taken in a hydrocarbon flame. The obtained results indicate an average deviation of 1.021 % between the measured temperature in the flame and the reference value, with a standard deviation of 1.381 % for concentration measurement. All the measurements show that the system can be potentially applied to combustion diagnosis.

Natural and shipping underwater sound distribution in the Northern Adriatic Sea basin and possible application on target areas

The underwater sound distribution generated by natural sources, shipping and trawling activities has been computed by the Quonops© modelling webservice for the Northern Adriatic Sea (NAS) during 2020, a year characterized by the COVID-19 lockdown restrictions. Modelling has been calibrated by using a year-long time series of field measurements covering the domain of interest. Sound levels (50th percentile) ranged between 75 and 90 dB re 1μPa for all the considered frequencies (63 Hz, 125 Hz, 250 Hz third octave bands). Noisier NAS areas match with the shipping lanes and the distribution of trawling activity. Pressure sound indices based on masking effect were computed for two Ecologically/Biologically Significant Marine Areas (EBSA) located in the NAS. Results indicated a significant contribution of vessel and fishery-generated noise to the local soundscape and provide a basis for addressing NAS underwater noise pollution, with special reference to the Marine Spatial Planning approach.

Firms’ cost structure and stock return volatility

PurposeThis paper proposes a framework to identify a pattern in the relationship between firms’ cost structure (i.e. fixed versus variable) and their volatility in stock returns.Design/methodology/approachOur empirical analysis is based on a panel data regression where we use an extended sample period and a time-series regression-based elasticity measure of operating leverage.FindingsWe document significantly higher systematic risk among firms with large fixed costs, a conclusion which confirms theoretical predictions of earlier studies. In new findings, we document high firm-specific risk and high stock return volatility among firms with a fixed cost structure.Originality/valueThe paper fills a gap in the literature by examining the effect of cost structure using various operating leverage measures and other control measures for firm characteristics on idiosyncratic risk. Studies that seek to explain firms’ systematic risks are numerous; conversely, there are relatively fewer studies on the determinants of firms’ specific risks.

PSInet: a new global water potential network.

Given the pressing challenges posed by climate change, it is crucial to develop a deeper understanding of the impacts of escalating drought and heat stress on terrestrial ecosystems and the vital services they offer. Soil and plant water potential play a pivotal role in governing the dynamics of water within ecosystems and exert direct control over plant function and mortality risk during periods of ecological stress. However, existing observations of water potential suffer from significant limitations, including their sporadic and discontinuous nature, inconsistent representation of relevant spatio-temporal scales and numerous methodological challenges. These limitations hinder the comprehensive and synthetic research needed to enhance our conceptual understanding and predictive models of plant function and survival under limited moisture availability. In this article, we present PSInet (PSI-for the Greek letter Ψ used to denote water potential), a novel collaborative network of researchers and data, designed to bridge the current critical information gap in water potential data. The primary objectives of PSInet are as follows. (i) Establishing the first openly accessible global database for time series of plant and soil water potential measurements, while providing important linkages with other relevant observation networks. (ii) Fostering an inclusive and diverse collaborative environment for all scientists studying water potential in various stages of their careers. (iii) Standardizing methodologies, processing and interpretation of water potential data through the engagement of a global community of scientists, facilitated by the dissemination of standardized protocols, best practices and early career training opportunities. (iv) Facilitating the use of the PSInet database for synthesizing knowledge and addressing prominent gaps in our understanding of plants' physiological responses to various environmental stressors. The PSInet initiative is integral to meeting the fundamental research challenge of discerning which plant species will thrive and which will be vulnerable in a world undergoing rapid warming and increasing aridification.

Controls on Water-Column Respiration Rates in a Coastal Plain Estuary: Insights from Long-Term Time-Series Measurements

Controls on Water-Column Respiration Rates in a Coastal Plain Estuary: Insights from Long-Term Time-Series Measurements

Nitrate trend reversal in Dutch dual-permeability chalk springs, evaluated by tritium-based groundwater travel time distributions

Historical use of fertilizer and manure on farmlands is known to have a lasting impact on ecosystems and water resources, but few studies assess the legacy of nitrate pollution on groundwater and surface water after farming applications were reduced. We studied the response of nitrate in spring water to a reduction of nitrogen fertilizer applications in agriculture realized since the mid-1980s. We assessed the travel time distribution of groundwater based on a time series of tritium measurements for 90 springs and small brooks that drain a dual porosity chalk aquifer. The travel time distributions were constrained using the tritium data in combination with time series of nitrate concentrations, applying a shape-free travel time distribution model. A clear trend reversal of nitrate concentrations was observed and simulated for springs with a large fraction of young water (< 30 years old) whereas the nitrate response in springs with relatively older water was attenuated and delayed. We conclude that obtaining a time series of tritium data helps to constrain age distributions of water that is discharged from dual permeability aquifers. The fraction of water aged <30 years was a meaningful parameter to distinguish between different types of springs. Nitrate trends in springs that drain large fractions of young water (> 0.6) show higher peak concentrations, shorter lag-time between leaching and outflow peaks and steeper declines after trend reversal, relative to trends in springs which are dominantly fed by older groundwater. The study thus shows that the nitrate legacy of groundwater systems is strongly determined by the range of their travel time distributions, and trend reversal in receiving springs and surface waters may appear within 10 to 15 years after measures to reduce nitrate losses from farming.

Early warning signals for bifurcations embedded in high dimensions

Recent work has highlighted the utility of methods for early warning signal detection in dynamic systems approaching critical tipping thresholds. Often these tipping points resemble local bifurcations, whose low dimensional dynamics can play out on a manifold embedded in a much higher dimensional state space. In many cases of practical relevance, the form of this embedding is poorly understood or entirely unknown. This paper explores how measurement of the critical phenomena that generically precede such bifurcations can be used to make inferences about some properties of their embeddings, and, conversely, how prior knowledge about the mechanism of bifurcation can robustify predictions of an oncoming tipping event. These modes of analysis are first demonstrated on a simple fluid flow system undergoing a Hopf bifurcation. The same approach is then applied to data associated with the West African monsoon shift, with results corroborated by existing models of the same system. This example highlights the effectiveness of the methodology even when applied to complex climate data, and demonstrates how a well-resolved spatial structure associated with the onset of atmospheric instability can be inferred purely from time series measurements.

The enigma of Li-rich giants and its relation with temporal variations observed in radial velocity and stellar activity signals

Context. Despite the large number of studies focused on the characterisation of Li-rich stars and understanding the mechanisms leading to such enrichment, their origin remains a mystery. Aims. Magnetic activity, particularly the phenomena usually associated with it (e.g. spots and plages), and the Li abundance (A(Li)) of stars, are in general thought to be connected. As of today, however, just how they are connected is unclear. In this work, we study a sample of young but evolved intermediate-mass red giants that are inhabitants of open clusters where planets have been searched for. Our aim is to use radial velocity (RV) and stellar activity indicator signals to look for relations between Li abundances and stellar activity or variability. Methods. We explored how the standard deviation (STD), peak-to-peak amplitude (PTP), mean, and median of typical stellar activity indicators (BIS, FWHM, Teff, and Hα index) change as a function of the Li content of 82 red giants. Furthermore, we computed weighted Pearson correlation coefficients (ρw) between time series of RV measurements and the stellar activity indicators for the stars in our sample. To aid our results, we also studied generalized Lomb–Scargle periodograms (GLSP) to capture possible significant periodic temporal variations in our data. Results. Our analysis indicates that the STD and PTP of BIS and FWHM, the mean and median of the Hα index, and υ sin(i) increase exponentially with A(Li) in our sample of red giants. Significant temporal variations and correlations between RVs and activity indicators also tend to be found preferentially for stars where high A(Li) is observed. Most of the Li-rich stars in our sample either show strong correlations of RV with at least one of the stellar activity indicators or reveal significant periodic temporal variations in their GLSPs of stellar activity indicators that are consistent with those found for RV.

Variability of saltwater flow in the Hoburg Channel, Baltic Sea: in situ measurements vs NEMO modelling

A half-year long time series of the bottom layer velocity measured in situ in the Hoburg Channel displayed seven-day oscillations of the saltwater flow. The flow was characterized by alterations of surges with the increase of northward velocity to approximately 0.2–0.3 m/s and blockages when the northward velocity vanishes or becomes small negative. The measured time series of the northward velocity component was surprisingly highly correlated with the simulation by NEMO reanalysis at the correlation coefficient of 0.82 and the 95 % confidence limits of 0.76–0.86. The seven-day oscillations were accompanied by almost synchronous oscillations of the southeast component of the wind vector. It can be considered convincing evidence that the seven-day oscillations in the saltwater flow were caused by wind forcing.

Single-cell decoding of drug induced transcriptomic reprogramming in triple negative breast cancers

BackgroundThe encoding of cell intrinsic drug resistance states in breast cancer reflects the contributions of genomic and non-genomic variations and requires accurate estimation of clonal fitness from co-measurement of transcriptomic and genomic data. Somatic copy number (CN) variation is the dominant mutational mechanism leading to transcriptional variation and notably contributes to platinum chemotherapy resistance cell states. Here, we deploy time series measurements of triple negative breast cancer (TNBC) single-cell transcriptomes, along with co-measured single-cell CN fitness, identifying genomic and transcriptomic mechanisms in drug-associated transcriptional cell states.ResultsWe present scRNA-seq data (53,641 filtered cells) from serial passaging TNBC patient-derived xenograft (PDX) experiments spanning 2.5 years, matched with genomic single-cell CN data from the same samples. Our findings reveal distinct clonal responses within TNBC tumors exposed to platinum. Clones with high drug fitness undergo clonal sweeps and show subtle transcriptional reversion, while those with weak fitness exhibit dynamic transcription upon drug withdrawal. Pathway analysis highlights convergence on epithelial-mesenchymal transition and cytokine signaling, associated with resistance. Furthermore, pseudotime analysis demonstrates hysteresis in transcriptional reversion, indicating generation of new intermediate transcriptional states upon platinum exposure.ConclusionsWithin a polyclonal tumor, clones with strong genotype-associated fitness under platinum remained fixed, minimizing transcriptional reversion upon drug withdrawal. Conversely, clones with weaker fitness display non-genomic transcriptional plasticity. This suggests CN-associated and CN-independent transcriptional states could both contribute to platinum resistance. The dominance of genomic or non-genomic mechanisms within polyclonal tumors has implications for drug sensitivity, restoration, and re-treatment strategies.

Short-term time-series observations of phytoplankton light-absorption and productivity in Prydz Bay, coastal Antarctica

The optical characteristics of coastal Antarctic waters exhibit complexity due to the dynamic hydrography influenced by meltwater intrusion, which alters nutrient levels, thermohaline structure, and optically active substances (OAS) regimes. Studies on bio-optical variability and its implications on phytoplankton productivity (PP) are scanty in coastal polar regions. On this backdrop, time-series measurements (72 h at 6 h intervals) of bio-optical properties such as phytoplankton biomass (chlorophyll-a), absorption (aph), and total suspended matter (TSM) concurrently with PP were measured to understand their interplay and variability in relation to the ambient physicochemical settings in the under-sampled Prydz Bay, coastal Antarctica. Our findings revealed thermohaline stratification within the bay, likely attributed to the inflow of less saline meltwater from nearby glaciers and minimal wind activity. The consistent presence of sub-surface chlorophyll maximum (SCM) beneath the stratified layer underscored the light-acclimatization response of shade-adapted phytoplankton. Surface waters exhibited higher TSM compared to deeper layers, indicating glacial melt influence, while the depth of the sunlit layer remained relatively stable, suggesting limited water mass movement and/or variability in OAS at the study site. An inverse relation between chlorophyll-a and chlorophyll-a-specific phytoplankton light absorption (a*ph(λ)) manifested ‘pigment package effect’ within the prevailing phytoplankton community, implying reduced light-absorption efficiency and consequent lower PP. Compared to chlorophyll-a, the phytoplankton light absorption (aph(λ)) emerged as a better proxy for explaining PP variability. Nutrient availability was not limiting, which was conducive to micro (large) phytoplankton growth. Classification of phytoplankton size classes (micro, nano, and pico) based on the B/R ratio (aph at Blue (443 nm)/Red (676 nm) region) confirmed the dominance of larger (micro) phytoplankton that are more susceptible to package effect, thus have implications on reduced PP potential of this polar marine ecosystem.

Red blood cell dynamics during malaria infection challenge the assumptions of mathematical models of infection dynamics.

For decades, mathematical models have been used to understand the course and outcome of malaria infections (i.e., infection dynamics) and the evolutionary dynamics of the parasites that cause them. A key conclusion of these models is that red blood cell (RBC) availability is a fundamental driver of infection dynamics and parasite trait evolution. The extent to which this conclusion holds will in part depend on model assumptions about the host-mediated processes that regulate RBC availability i.e., removal of uninfected RBCs and supply of RBCs. Diverse mathematical functions have been used to describe host-mediated RBC supply and clearance, but it remains unclear whether they adequately capture the dynamics of RBC supply and clearance during infection. Here, we use a unique dataset, comprising time-series measurements of erythrocyte (i.e., mature RBC) and reticulocyte (i.e., newly supplied RBC) densities during Plasmodium chabaudi malaria infection, and a quantitative data-transformation scheme to elucidate whether RBC dynamics conform to common model assumptions. We found that RBC clearance and supply are not well described by mathematical functions commonly used to model these processes. Furthermore, the temporal dynamics of both processes vary with parasite growth rate in a manner again not captured by existing models. Together, these finding suggest that new model formulations are required if we are to explain and ultimately predict the within-host population dynamics and evolution of malaria parasites.

Estimating the uncertainties of satellite derived soil moisture at global scale

This study attempts to derive the uncertainty of the soil moisture estimation from passive microwave satellite mission at global scale. To do so, the approach is based on the sensitivity of the Soil Moisture and Ocean Salinity (SMOS) soil moisture retrieval quality to the land surface characteristics within its footprint (presence of forest, topography, open water bodies, sand, clay, bulk density and soil organic carbon content). First, we performed a global assessment of SMOS using in situ measurements from the International Soil Moisture Network (ISMN) as reference, with more than 1900 ISMN stations and 10 years of SMOS data. This assessment shows that the ubRMSD scores vary greatly between locations (with a mean of 0.074 m3m−3 and an interquartile range of 0.030 m3m−3). Second, the scores are analyzed for different surface conditions within the satellite footprint. The best agreement between the ground measurement and SMOS time series are obtained for low forest cover, low topographic complexity, and marginal presence of open water bodies within the SMOS footprint. Soil parameters also have an impact, with better scores for sandier soils with a high bulk-density and low soil organic carbon content. Finally, we propose to extrapolate the obtained relationships, using a multiple linear regression, in order to derive a global map of SMOS uncertainties based on surface conditions. This map of predicted uncertainties show a diverse range of ubRMSD values across the globe (with a mean of 0.076 m3m−3 and an interquartile range of 0.031 m3m−3) depending on the surface characteristics. At the ISMN site location, the predicted ubRMSD shows similar results than the comparison between SMOS and the in situ measurements. The map of predicted SMOS ubRMSD represents an upper bound estimate of the SMOS uncertainty, as it includes the uncertainties of the in situ sensor measurements and the scale mismatch. Further investigations will focus on the different components of this uncertainty budget to obtain a better assessment of the absolute uncertainties of SMOS soil moisture retrievals across the globe.

Nonlinear spatial evolution of degenerate quartets of water waves

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Volumetric PTV For Fermenting Bread Dough Visualized By Micro X-Ray Computed Tomography

To elucidate the relationship between the formation of internal structures and changes in viscoelastic properties for improving the quality stability of porous and heterogeneous materials, we focused on bread dough that forms an internal bubble structure through yeast metabolism. Specifically, we conducted a time-series measurement of the internal structure and viscoelastic properties during fermentation, particularly in the heterogeneous condition when additives are mixed into the homogeneous dough. We performed a time-lapse imaging using a digital camera to investigate the inflation volume and rate of fermenting bread dough. We used X-ray CT to visualize the internal bubble structure during the fermenting process and evaluated the changes in viscoelastic properties using a dynamic mechanical analysis. The results indicated that the addition of raisins to the bread dough negatively affected its inflation, made the bubble structure of the dough near the raisins larger and coarsened, and altered the viscoelastic properties.

Image Deconvolution and Point-spread Function Reconstruction with STARRED: A Wavelet-based Two-channel Method Optimized for Light-curve Extraction

We present starred, a point-spread function (PSF) reconstruction, two-channel deconvolution, and light-curve extraction method designed for high-precision photometric measurements in imaging time series. An improved resolution of the data is targeted rather than an infinite one, thereby minimizing deconvolution artifacts. In addition, starred performs a joint deconvolution of all available data, accounting for epoch-to-epoch variations of the PSF and decomposing the resulting deconvolved image into a point source and an extended source channel. The output is a high-signal-to-noise-ratio, high-resolution frame combining all data and the photometry of all point sources in the field of view as a function of time. Of note, starred also provides exquisite PSF models for each data frame. We showcase three applications of starred in the context of the imminent LSST survey and of JWST imaging: (i) the extraction of supernovae light curves and the scene representation of their host galaxy; (ii) the extraction of lensed quasar light curves for time-delay cosmography; and (iii) the measurement of the spectral energy distribution of globular clusters in the “Sparkler,” a galaxy at redshift z = 1.378 strongly lensed by the galaxy cluster SMACS J0723.3-7327. starred is implemented in jax, leveraging automatic differentiation and graphics processing unit acceleration. This enables the rapid processing of large time-domain data sets, positioning the method as a powerful tool for extracting light curves from the multitude of lensed or unlensed variable and transient objects in the Rubin-LSST data, even when blended with intervening objects.

Two fractional order cumulative residual time series measures based on Rényi entropy

In the study of time series analysis, Kullback-Leibler divergence can measure the difference between the distribution diverges of time series, and transfer entropy is usually used to quantify the causal influence among time series. In order to analyze the relationship between time series more comprehensively, two novel fractional order times series measures, i.e., Rényi cumulative residual Kullback-Leibler (RCRKL) and Rényi cumulative residual transfer entropy (RCRTE), are proposed in this paper. Some mathematical properties of RCRKL are proved as a distance metric, and the zeros and bounds of RCRTE are studied. Then, some examples on synthetic data are provided to study the effect of parameter changes on metric values. Finally, the proposed two fractional-order time series measures are applied to pavement rutting time series, for which a RCRKL-based clustering is designed and the causal relationships between influence factors and rutting are detected by RCETE. The results demonstrate that the RCRKL-based clustering algorithm performs better than those based on the baseline distance measures. Meanwhile, RCRTE can also detect the true causality between rutting and its influence factors and reject the false causality. In summary, the proposed RCRKL and RCRTE take the advantages of both cumulative residual entropy and Rényi entropy, and provide a more comprehensive insight for time series analysis.

Distribution and sea-to-air fluxes of nitrous oxide and methane from a seasonally hypoxic coastal zone in the southeastern Arabian Sea

The seasonal variability, pathways, and sea-to-air fluxes of nitrous oxide (N2O) and methane (CH4) in the coastal environment, where coastal upwelling and mudbanks co-exist are presented based on the monthly time-series measurements from November 2021 to December 2022. Upwelling-driven hypoxic water's shoreward propagation and persistence were the major factors controlling the N2O concentrations, while the freshwater influx and sedimentary fluxes modulate CH4 concentrations. The N2O concentrations were high during the southwest monsoon (up to 35 nM; 19 ± 8 nM)), followed by spring inter-monsoon (up to 19 nM; 10 ± 5 nM), and lowest during the northeast monsoon (up to 13 nM; 8 ± 2 nM), whereas the CH4 levels were high during the spring inter-monsoon (8.4 to 65 nM), followed by southwest monsoon (6.8 to 53.1 nM) and relatively lower concentrations during the northeast monsoon (3.3 to 32.6 nM). The positive correlations of excess N2O with Apparent Oxygen Utilisation (AOU) and the sum of nitrate and nitrite (NOx) indicate that nitrification is the primary source of N2O in the mudbank regime. The negative correlation of CH4 concentrations with salinity indicates considerable input of CH4 through freshwater influx. CH4 exhibited a highly significant positive correlation with Chlorophyll-a throughout the study period. Furthermore, it displayed a statistically significant positive correlation with phosphate (PO43−) during the northeast monsoon while a strong negative correlation with PO43− during the spring inter-monsoon, pointing towards the role of aerobic CH4 production pathways in the mudbank regime. N2O and CH4 exhibited a contrasting seasonal pattern of sea-to-air fluxes, characterised by the highest N2O fluxes during the southwest monsoon (hypoxia) (13 ± 10 μM m−2 d−1), followed by spring inter-monsoon (12 ± 16 μM m−2 d−1), and the lowest during the northeast monsoon (0.6 ± 3 μM m−2 d−1). Conversely, the highest sea-to-air fluxes of CH4 were noticed during the spring inter-monsoon (74 ± 56 μM m−2 d−1), followed by the southwest monsoon (45 ± 35 μM m−2 d−1), and the lowest values during the northeast monsoon (19 ± 16 μM m−2d−1). Long-term time-series measurements will be invaluable in understanding the longer-term impacts of climate-driven variability on marine biogeochemical cycles in dynamic nearshore systems.

Przewidywanie atlantyckiej południkowej cyrkulacji z wykorzystaniem nieliniowych metod identyfikacji systemu i modelu NARMAX

The Atlantic Meridional Overturning Circulation (AMOC) plays an important role in the coupled ocean-climate system and in global climate change. The analysis of its own behaviour and the understanding its links to other climate dynamics is of paramount importance today as we encounter an increasing pressure to adapt to climate change. Due to the enormous complexity, it is almost impossible to establish accurate models, purely based on first-principle modelling approaches, that can perfectly represent the relationships between the AMOC and other dynamic climate parameters. Data-based or data-driven modelling methods, can therefore provide an attractive alternative solution. Systematic regular and continuous measurement of the AMOC time series began in April 2004. The main objective of the paper is to use the monthly data of the AMOC measured during April 2004-Febuary 2017, together with the North Atlantic Oscillation (NAO) index, and density anomalies of the Gulf of Mexico, Labrador Sea and Norwegian Sea, measured during the same period, to investigate and understand the quantitative relationship between the AMOC and four drivers (NAO and the three density anomaly variables). In doing so, nonlinear system identification methods and the Nonlinear AutoRegressive Moving Average with Exogenous input (NARMAX) method are employed to develop a quantitative model that relates the AMOC to the four drivers. Experimental results show that the derived nonlinear model skillfully captures and represents the dynamics of the AMOC based on the other four variables. One of the findings from this study is that the use of autoregressive variables can help improve the prediction of the AMOC.

General solution for linear 2D anisotropic surface gravity water waves on uniform current

Measuring and modeling ocean waves is a routine task for oceanographers and marine engineers. Accounting for current effects on waves is less common; however, such effects can be non-trivial, particularly for strong currents. Recent research has also found that currents are increasing with global temperatures. Thus, accounting for currents is of potential importance to a growing subset of real-world environments. One significant impact of currents is their modification of dispersion. It is known that, in contrast to dispersion without current, wave–current dispersion is multivalued and anisotropic, meaning it varies with direction and can yield multiple solutions for the same frequency. Strictly speaking, a complete analysis ought to account for all dispersion solutions. This study presents a general wave field solution for linear steady-state two-dimensional surface gravity waves on current. In contrast to existing formulations, it accounts for the complete set of dispersion solutions. The multiple solutions correspond to linearly independent (same frequency) spatial modes with no analogy in models without current. To fully determine the complete temporal and spatial features of a wave field, one must determine the amplitudes of all spatial modes. This can be achieved through the solution of an inverse problem, provided one has sufficient initial conditions. The significance of accounting for all spatial modes is demonstrated through examples. It is shown that a single time-series measurement is insufficient for determining the spatial features of a monochromatic (single frequency) wave. Rather, additional spatial information must be introduced to render the problem well-posed. The theory and methodology presented can be used to improve wave models and measurements.