Comment on egusphere-2023-140

Abstract. In sequential estimation methods often used in oceanic and general climate calculations of the state and of forecasts, observations act mathematically and statistically as source or sink terms in conservation equations for heat, salt, mass, and momentum. These artificial terms obscure the inference of the system's variability or secular changes. Furthermore, for the purposes of calculating changes in important functions of state variables such as total mass and energy or volumetric current transports, results of both filter and smoother-based estimates are sensitive to misrepresentation of a large variety of parameters, including initial conditions, prior uncertainty covariances, and systematic and random errors in observations. Here, toy models of a coupled mass–spring oscillator system and of a barotropic Rossby wave system are used to demonstrate many of the issues that arise from such misrepresentations. Results from Kalman filter estimates and those from finite interval smoothing are analyzed. In the filter (and prediction) problem, entry of data leads to violation of conservation and other invariant rules. A finite interval smoothing method restores the conservation rules, but uncertainties in all such estimation results remain. Convincing trend and other time-dependent determinations in “reanalysis-like” estimates require a full understanding of models, observations, and underlying error structures. Application of smoother-type methods that are designed for optimal reconstruction purposes alleviate some of the issues.

In sequential estimation methods often used in general climate or oceanic calculations of the state and of forecasts, observations act mathematically and statistically as forcings as is obvious in the innovation form of the equations. For purposes of calculating changes in important functions of state variables such as total mass and energy, or in volumetric current transports, results are sensitive to mis-representation of a large variety of parameters including initial conditions, prior uncertainty covariances, and systematic and random errors in observations. Errors are both stochastic and systematic, with the latter, as usual, being the most intractable. Here some of the consequences of such errors are first analyzed in the context of a simplified mass-spring oscillator system exhibiting many of the issues of far more complicated realistic problems. The same methods are then applied to a more geophysical barotropic Rossby wave plus western boundary current system. The overall message is that convincing trend and other time-dependent determinations in "reanalyis" like estimates requires a full understanding of both models and observations.

A perfect model Monte Carlo experiment was conducted to explore the characteristics of analysis error in a quasigeostrophic model. An ensemble of cycled analyses was created, with each member of the ensemble receiving different observations and starting from different forecast states. Observations were created by adding random error (consistent with observational error statistics) to vertical profiles extracted from truth run data. Assimilation of new observations was performed every 12 h using a three-dimensional variational analysis scheme. Three observation densities were examined, a low-density network (one observation ; every 202 grid points), a moderate-density network (one observation ; every 102 grid points), and a high-density network (; every 52 grid points). Error characteristics were diagnosed primarily from a subset of 16 analysis times taken every 10 days from a long time series, with the first sample taken after a 50-day spinup. The goal of this paper is to understand the spatial, temporal, and some dynamical characteristics of analysis errors. Results suggest a nonlinear relationship between observational data density and analysis error; there was a much greater reduction in error from the low- to moderate-density networks than from moderate to high density. Errors in the analysis reflected both structured errors created by the chaotic dynamics as well as random observational errors. The correction of the background toward the observations reduced the error but also randomized the prior dynamical structure of the errors, though there was a dependence of error structure on observational data density. Generally, the more observations, the more homogeneous the errors were in time and space and the less the analysis errors projected onto the leading backward Lyapunov vectors. Analyses provided more information at higher wavenumbers as data density increased. Errors were largest in the upper troposphere and smallest in the mid- to lower troposphere. Relatively small ensembles were effective in capturing a large percentage of the analysis-error variance, though more members were needed to capture a specified fraction of the variance as observation density increased.

The readings comprising the measurement of fiber diameter on two tops by the projection micro scope were perturbed by moving one-quarter of the readings in each histogram element to the element next below and one-quarter to the element next above, thus simulating random error of observation. It is shown that the resulting increase in between-fiber variance may be neglected. It is further shown that, in the CSIRO Fiber Fineness Distribution Analyser, which is in effect an automatic projection- microscope method, the variance observed between mean diameters is greater than can be accounted for by the observed between-fiber variance, which in turn is greater than can be accounted for by random error of observation, thus implying a source of systematic error in the measurement of between- fiber variance.

The noise scale factor (NSF) is useful in calculating the random errors of lidar signals. A new method is proposed for estimating the NSF of lidar systems. Instead of measuring the solar background in the traditional method that demands special experiments or additional devices, the new method utilizes the molecular backscattered signal fragments of routinely observed lidar profiles to extract the light intensities and shot noises. Based on a 355 nm lidar system, experiments with the two methods have been carried out. The NSF calculated with the new method is 0.6473, very close to the value resulted from the traditional method, which is 0.6496. The experiments indicate that the new method is feasible and accurate. An example of calculating the random errors and signal to noise ratios (SNR) of lidar signals by using NSF is presented. Compared with the results calculated from multiple lidar profiles, the method using NSF eliminates the influence of aerosols and clouds.

Precise Point Positioning (PPP) is an increasingly recognized precisely the GPS/GNSS positioning technique. In order to improve the accuracy of PPP, the error sources in PPP measurements should be reduced as much as possible and the ambiguities should be correctly resolved. The correct ambiguity resolution requires a careful control of residual errors that are normally categorized into random and systematic errors. To understand effects from two categorized errors on the PPP ambiguity resolution, those two GPS datasets are simulated by generating in locations in South Korea (denoted as SUWN) and Hong Kong (PolyU). Both simulation cases are studied for each dataset; the first case is that all the satellites are affected by systematic and random errors, and the second case is that only a few satellites are affected. In the first case with random errors only, when the magnitude of random errors is increased, L1 ambiguities have a much higher chance to be incorrectly fixed. However, the size of ambiguity error is not exactly proportional to the magnitude of random error. Satellite geometry has more impacts on the L1 ambiguity resolution than the magnitude of random errors. In the first case when all the satellites have both random and systematic errors, the accuracy of fixed ambiguities is considerably affected by the systematic error. A pseudorange systematic error of 5 cm is the much more detrimental to ambiguity resolutions than carrier phase systematic error of 2 mm. In the 2 nd case when only a portion of satellites have systematic and random errors, the L1 ambiguity resolution in PPP can be still corrected. The number of allowable satellites varies from stations to stations, depending on the geometry of satellites. Through extensive simulation tests under different schemes, this paper sheds light on how the PPP ambiguity resolution (more precisely L1 ambiguity resolution) is affected by the characteristics of the residual errors in PPP observations. The numerical examples recall the PPP data analysts that how accurate the error correction models must achieve in order to get all the ambiguities resolved correctly.

Over the past decade, the relation between the Balmer-line luminosity of HII galaxies and the velocity width of the emission lines, the relation, has been painstakingly calibrated as a cosmological distance indicator with seemingly spectacular results: the Hubble constant and the energy density of dark energy obtained using the indicator agree remarkably well with the values from canonical indicators. Since most of the luminosity of these young compact starburst galaxies is emitted by a few narrow emission lines, they can be observed with good precision up to redshifts of $z with JWST, making the indicator a potentially unique cosmological probe. However, the precision of the method remains too low to effectively constrain the relevant cosmological parameters, notably the equation of state of dark energy. The scatter of the relation is significantly larger than the random observational errors, so we do not have a good handle on the systematics of the method. In a previous paper, we posited that since the ionizing radiation of these young galaxies fades rapidly over timescales of only a few million years, age differences could be the main underlying cause of the scatter. In this paper, we explore several different ways of explaining the scatter of the correlation, but without success. We show that the majority of HII galaxies are powered by multiple starbursts of slightly different ages, and therefore that the equivalent widths are not reliable chronometers to correct the luminosities for evolution. Thus, it is not likely that the accuracy of the distance indicator can be improved in the near future. Since we do not fully understand either the systematics or the underlying physics of the relation, using large samples of distant HII galaxies may or may not improve the accuracy of the method.

Users of climate records frequently require data at geographical locations where no direct measurements of climatological variables are collected. Climatological conditions at the areas or points of interest have to be estimated by interpolating observations from neighbouring stations. An objective of assessing spatial representativeness of a network of observing stations is to outline the areas for which the network is capable of providing sufficiently accurate climatological information, i.e., where interpolation errors do not exceed a value acceptable to the user. Two statistical methods: Gandin's point‐to‐point optimal interpolation and Kagan's pointto‐area interpolation were applied to monthly, seasonal and annual total precipitation records gathered by a long‐term, high quality, nationwide network of Canadian climate stations. Due to substantial differences in seasonal climate conditions in the Arctic versus the rest of the country, the national network had to be split into two subsets: north and south of 60°N latitude. Both interpolation techniques use a spatial correlation function to compute interpolation errors. The correlation function of departures or ratios from the “first guess” field of long‐term averages can be considered homogeneous and isotropic over a wide range of distances. Exponential functions are especially suitable to model correlation of precipitation. Alone, they can supply an abundance of information about the nature of precipitation, random observational errors and microclimatic uncertainties. The widely scattered northern stations showed poor spatial correlation and consequently unacceptably large interpolation errors in all cases except summer. The conclusion was that at present the climate network does not provide adequate climatological information north of 60° latitude. The southern stations exhibited fairly good spatial correlation, which allowed computation of the longest acceptable interstation distances that satisfy various interpolation error criteria. The results were tabulated to serve as a reference. In Gandin's case, the areas where the interpolation error does not exceed 65% of a local standard deviation were delineated for all months and seasons using so‐called circles of representativeness. This example revealed vast regions without adequate precipitation gauge coverage, especially in summer and winter. Kagan's method, which is concerned with area averages, produced less demanding results. Intuitively, a less dense network is required for area‐average estimates than for point value estimates. At the same time though, acceptable areal relative errors should be set at lower values. Assuming an arbitrary 10% relative error in the areal estimate, the southern half of the country is adequately represented by the network except for some relatively small sections around Hudson Bay.

The estimation of thermal diffusivity using a flash experiment is studied. The sequential estimation method is presented and compared to the original flash method as well as two other methods that account for heat loss. The potential accuracy of the four methods is tested with a numerical simulation. The thermal diffusivity of a stainless steel sample is estimated from experimental data. The estimates given by the different methods are compared and analyzed. The sequential estimation method is shown to be less sensitive to random measurement errors and to provide more information regarding the accuracy of the estimated parameters.

Abstract. The wind extraction due to assimilation of stratospheric trace gas (tracer) data is examined using a 4D-Var (four-dimensional variational) data assimilation system based on the shallow water equations coupled to the tracer continuity equation. The procedure is outlined as follows. First, a nature run is created, simulating middle stratospheric winter conditions. Second, ozone (O3), nitrous oxide (N2O), and water vapor (H2O) (treated in this study as passive tracers) are initialized using Aura Microwave Limb Sounder (MLS) mixing ratios at 850 K potential temperature and are advected by the nature run winds. Third, the initial dynamical conditions are perturbed by using a 6 h offset. Fourth, simulated hourly tracer observations on the full model grid are assimilated with a 4D-Var system in which tracer and winds are coupled via the adjoint of the tracer continuity equation. Multiple assimilation experiments are performed by varying the amount of random observation error added to the simulated measurements. Finally, the wind extraction potential (WEP) is calculated as the reduction of the vector wind root mean square error (RMSE) relative to the maximum possible reduction. For a single 6 h assimilation cycle with the smallest observation error, WEP values are ~60% for all three tracers, while 10-day multi-cycle simulations result in WEP of ~90%, wind errors of ~0.3 m s−1, and height errors of ~13 m. There is therefore sufficient information in the tracer fields to almost completely constrain the dynamics, even without direct assimilation of dynamical information. When realistic observation error is added (based on MLS precisions at 10 hPa), the WEP after 10 days is 90% for O3, 87% for N2O, and 72% for H2O. O3 and N2O provide more wind information than H2O due to stronger background gradients relative to the MLS precisions. The RMSE for wind reach a minimum level of ~0.3–0.9 m s−1 for the MLS precisions, suggesting a limit to which realistic tracers could constrain the winds, given complete global cover age. With higher observation noise levels, the WEP values decrease, but the impact on the winds is still positive up to noise levels of 100% (relative to the global mean value) when compared to the case of no data assimilation.

Real-time prediction of material removal rate for advanced process control of chemical mechanical polishing

The spatial and temporal variability of sensitivities has a significant impact on parameter estimation and sampling design for studies of solute transport in porous media. Physical insight into the behavior of sensitivities is offered through an analysis of analytically derived sensitivities for the one‐dimensional form of the advection‐dispersion equation. When parameters are estimated in regression models of one‐dimensional transport, the spatial and temporal variability in sensitivities influences variance and covariance of parameter estimates. Several principles account for the observed influence of sensitivities on parameter uncertainty. (1) Information about a physical parameter may be most accurately gained at points in space and time with a high sensitivity to the parameter. (2) As the distance of observation points from the upstream boundary increases, maximum sensitivity to velocity during passage of the solute front increases and the consequent estimate of velocity tends to have lower variance. (3) The frequency of sampling must be “in phase” with the S shape of the dispersion sensitivity curve to yield the most information on dispersion. (4) The sensitivity to the dispersion coefficient is usually at least an order of magnitude less than the sensitivity to velocity. (5) The assumed probability distribution of random error in observations of solute concentration determines the form of the sensitivities. (6) If variance in random error in observations is large, trends in sensitivities of observation points may be obscured by noise and thus have limited value in predicting variance in parameter estimates among designs. (7) Designs that minimize the variance of one parameter may not necessarily minimize the variance of other parameters. (8) The time and space interval over which an observation point is sensitive to a given parameter depends on the actual values of the parameters in the underlying physical system.

Effects of observation errors in linear regression and bin-averaged (BA) validation techniques are investigated using the example of marine wind speeds. It is shown that a conventional linear regression systematically underestimates the slope of the regression line, and systematically overestimates the random model error. A BA analysis systematically understimates extreme wind speeds, incorporates spurious nonlinearity, and overestimates random model errors. Correction techniques are suggested for studies in which the observation error can be estimated. Using synthetic data the potential of the correction techniques in illustrated, and it is shown that the above errors are generally not negligible for wind speed validation studies. Practical examplies consider the random errors of anemometers and wind speed estimates from satellites. These examples highlight the importance of the error corrections, and illustrate the difficulty of estimating observation errors. Finally, it is argued that the well-known symmetric slope regression should not be used for the validation of forecast systems. Although the present study deals with marine wind speeds, its results are expected to be valid for a wide range of validation studies.

The sorption of CO2 on coal is of great interest when investigating the possibility of CO2 Enhanced Coalbed Methane or sequestration of CO2 in coal. The measurement of accurate supercritical CO2 sorption isotherms has proven to be a complicated matter. Therefore, a high accuracy manometric apparatus has been constructed and the reliability of the setup itself is thoroughly investigated. Experimental errors are categorized according to the source of the error, instead of the classical categorization of random and systematic errors. These sources are: (1) Background theory fallacies, (2) effects of minor importance, (3) observational errors and (4) interpretational errors. The reliability of the set-up is investigated by examining the errors of the volume ratio measurements. Volume ratio measurements are sorption measurements, but simplified by not using a sample and without the cumulative effect of errors in sorption measurements. The volume ratio stems from a material balance and the expansion of gas from one cell to another. The pressure and temperature of both cells are measured before and after expansion. The density is calculated from the pressure and temperature with an Equation of State, which also must be of high accuracy. The accuracy of volume ratios measured with helium and carbon dioxide is investigated. Helium measurements are the base case that demonstrates optimal measurement accuracy. The observed errors in the experiments with helium correspond to the a priori error estimation (σ ≤ 0.2%). The difficulty with CO2 measurements is identified as the high sensitivity of the CO2 density for changes in pressure, temperature and impurities in the 80-120 bar region at 318 K. The high accuracy manometric apparatus can measure CO2 volume ratios with an uncertainty of less than 2%, except between 90 and 110 bar where the uncertainty is less than 10%. This larger uncertainty is caused by impurities in the CO2 of unknown origin. The influence of leakage, changes in the cell volumes, water desorption from steel and the influx of contaminants are shown to be of negligible influence. The error assessment in this paper will serve as the basis of uncertainty analysis of future high accuracy CO2 sorption measurements.

Light detection and ranging (LiDAR) has the advantage of simultaneous localization and mapping with high precision, making it one of the important sensors for intelligent robotics navigation, positioning, and perception. It is common knowledge that the random measurement error of global navigation satellite system (GNSS) observations is usually considered to be closely related to the elevation angle factor. However, in the LiDAR point cloud feature matching positioning model, the analysis of factors affecting the random measurement error of observations is unsophisticated, which limits the ability of LiDAR sensors to estimate pose parameters. Therefore, this work draws on the random measurement error analysis method of GNSS observations to study the impact of factors such as distance, angle, and feature accuracy on the random measurement error of LiDAR. The experimental results show that feature accuracy is the main factor affecting the measurement error in the LiDAR point cloud feature matching positioning model, compared with distance and angle factors, even under different sensor specifications, point cloud densities, prior maps, and urban road scenes. Furthermore, a simple random measurement error model based on the feature accuracy factor is used to verify the effect of parameter estimation, and the results show that the random error model can effectively reduce the error of pose parameter estimation, with an improvement effect of about 50%.