Regular Latitude-longitude Grid Research Articles

A Long-Time-Step-Permitting Tracer Transport Model on the Regular Latitude–Longitude Grid

A Long-Time-Step-Permitting Tracer Transport Model on the Regular Latitude–Longitude Grid

Physics-Inspired Adaptions to Low-Parameter Neural Network Weather Forecast Systems

Abstract Recently, there has been a surge of research on data-driven weather forecasting systems, especially applications based on convolutional neural networks (CNNs). These are usually trained on atmospheric data represented on regular latitude–longitude grids, neglecting the curvature of Earth. We assess the benefit of replacing the standard convolution operations with an adapted convolution operation that takes into account the geometry of the underlying data (SphereNet convolution), specifically near the poles. Additionally, we assess the effect of including the information that the two hemispheres of Earth have “flipped” properties—for example, cyclones circulating in opposite directions—into the structure of the network. Both approaches are examples of physics-informed machine learning. The methods are tested on the WeatherBench dataset, at a resolution of ∼1.4°, which is higher than many previous studies on CNNs for weather forecasting. For most lead times up to day +10 for 500-hPa geopotential and 850-hPa temperature, we find that using SphereNet convolution or including hemisphere-specific information individually leads to improvement in forecast skill. Combining the two methods typically gives the highest forecast skill. Our version of SphereNet is implemented flexibly and scales well to high-resolution datasets but is still significantly more expensive than a standard convolution operation. Finally, we analyze cases with high forecast error. These occur mainly in winter and are relatively consistent across different training realizations of the networks, pointing to flow-dependent atmospheric predictability.

Evaluation of native Earth system model output with ESMValTool v2.6.0

Abstract. Earth system models (ESMs) are state-of-the-art climate models that allow numerical simulations of the past, present-day, and future climate. To extend our understanding of the Earth system and improve climate change projections, the complexity of ESMs heavily increased over the last decades. As a consequence, the amount and volume of data provided by ESMs has increased considerably. Innovative tools for a comprehensive model evaluation and analysis are required to assess the performance of these increasingly complex ESMs against observations or reanalyses. One of these tools is the Earth System Model Evaluation Tool (ESMValTool), a community diagnostic and performance metrics tool for the evaluation of ESMs. Input data for ESMValTool needs to be formatted according to the CMOR (Climate Model Output Rewriter) standard, a process that is usually referred to as “CMORization”. While this is a quasi-standard for large model intercomparison projects like the Coupled Model Intercomparison Project (CMIP), this complicates the application of ESMValTool to non-CMOR-compliant climate model output. In this paper, we describe an extension of ESMValTool introduced in v2.6.0 that allows seamless reading and processing of “native” climate model output, i.e., operational output produced by running the climate model through the standard workflow of the corresponding modeling institute. This is achieved by an extension of ESMValTool's preprocessing pipeline that performs a CMOR-like reformatting of the native model output during runtime. Thus, the rich collection of diagnostics provided by ESMValTool is now fully available for these models. For models that use unstructured grids, a further preprocessing step required to apply many common diagnostics is regridding to a regular latitude–longitude grid. Extensions to ESMValTool's regridding functions described here allow for more flexible interpolation schemes that can be used on unstructured grids. Currently, ESMValTool supports nearest-neighbor, bilinear, and first-order conservative regridding from unstructured grids to regular grids. Example applications of this new native model support are the evaluation of new model setups against predecessor versions, assessing of the performance of different simulations against observations, CMORization of native model data for contributions to model intercomparison projects, and monitoring of running climate model simulations. For the latter, new general-purpose diagnostics have been added to ESMValTool that are able to plot a wide range of variable types. Currently, five climate models are supported: CESM2 (experimental; at the moment, only surface variables are available), EC-Earth3, EMAC, ICON, and IPSL-CM6. As the framework for the CMOR-like reformatting of native model output described here is implemented in a general way, support for other climate models can be easily added.

The E3SM Diagnostics Package (E3SM Diags v2.7): a Python-based diagnostics package for Earth system model evaluation

Abstract. The E3SM Diagnostics Package (E3SM Diags) is a modern, Python-based Earth system model (ESM) evaluation tool (with Python module name e3sm_diags), developed to support the Department of Energy (DOE) Energy Exascale Earth System Model (E3SM). E3SM Diags provides a wide suite of tools for evaluating native E3SM output, as well as ESM data on regular latitude–longitude grids, including output from Coupled Model Intercomparison Project (CMIP) class models. E3SM Diags is modeled after the National Center for Atmospheric Research (NCAR) Atmosphere Model Working Group (AMWG, 2022) diagnostics package. In its version 1 release, E3SM Diags included a set of core essential diagnostics to evaluate the mean physical climate from model simulations. As of version 2.7, more process-oriented and phenomenon-based evaluation diagnostics have been implemented, such as analysis of the quasi-biennial oscillation (QBO), the El Niño–Southern Oscillation (ENSO), streamflow, the diurnal cycle of precipitation, tropical cyclones, ozone and aerosol properties. An in situ dataset from DOE's Atmospheric Radiation Measurement (ARM) program has been integrated into the package for evaluating the representation of simulated cloud and precipitation processes. This tool is designed with enough flexibility to allow for the addition of new observational datasets and new diagnostic algorithms. Additional features include customizable figures; streamlined installation, configuration and execution; and multiprocessing for fast computation. The package uses an up-to-date observational data repository maintained by its developers, where recent datasets are added to the repository as they become available. Finally, several applications for the E3SM Diags module were introduced to fit a diverse set of use cases from the scientific community.

Deriving Arctic 2 m air temperatures over snow and ice from satellite surface temperature measurements

Abstract. The Arctic region is responding heavily to climate change, and yet, the air temperature of ice-covered areas in the Arctic is heavily under-sampled when it comes to in situ measurements, resulting in large uncertainties in existing weather and reanalysis products. This paper presents a method for estimating daily mean clear-sky 2 m air temperatures (T2m) in the Arctic from satellite observations of skin temperature, using the Arctic and Antarctic ice Surface Temperatures from thermal Infrared (AASTI) satellite dataset, providing spatially detailed observations of the Arctic. The method is based on a linear regression model, which has been tuned against in situ observations to estimate daily mean T2m based on clear-sky satellite ice surface skin temperatures. The daily satellite-derived T2m product includes estimated uncertainties and covers the Arctic sea ice and the Greenland Ice Sheet during clear skies for the period 2000–2009, provided on a 0.25∘ regular latitude–longitude grid. Comparisons with independent in situ measured T2m show average biases of 0.30 and 0.35∘C and average root-mean-square errors of 3.47 and 3.20 ∘C for land ice and sea ice, respectively. The associated uncertainties are verified to be very realistic for both land ice and sea ice, using in situ observations. The reconstruction provides a much better spatial coverage than the sparse in situ observations of T2m in the Arctic and is independent of numerical weather prediction model input. Therefore, it provides an important supplement to simulated air temperatures to be used for assimilation or global surface temperature reconstructions. A comparison of T2m derived from satellite and ERA-Interim/ERA5 estimates shows that the satellite-derived T2m validates similar to or better than ERA-Interim/ERA5 against in situ measurements in the Arctic.

Observed Daily Temperature Variability and Extremes over Southeastern USA (1978–2017)

This study presents an analysis of extreme temperature events over southeastern USA from 1978 to 2017. This region is part of the so-called ‘warming hole’ where long-term surface temperature trends are negative or non-significant, in contrast with the remainder of the country. This study examines whether this distinctive characteristic reflects on the region’s trends in temperature extremes. Daily maximum and minimum temperatures from the US Historical Climatology Network were used to compute extreme indices recommended by the Expert Team on Climate Change Detection and Indices. Temperature extreme indices computed for all stations using the RClimDex package were gridded onto a regular latitude–longitude grid, and a spatiotemporal analysis of associated trends was performed. The results point to a tendency toward warming due to increasing trends in the annual occurrence of the hottest day, the warmest night, warm days, warm nights, summer days, tropical nights, and warm spells, as well as decreases in cool nights, cool days, and frost days. Statistically significant trend changes over large portions of the Southeast were dominated by increases in the frequency of the coldest night, summer days, and warm nights, and decreases in cool nights and frost days. Comparison of our results with other global and regional studies indicate that most of the extreme temperature changes over the Southeast are consistent with findings from other parts of the United States (US) and the world. Overall, this study shows that being part of the ‘warming hole’ does not preclude southeastern US from an intensification of temperature extremes, whether it is an increase in warm extremes or a decrease in cold ones. Further, the results suggest that, should the current trends continue in the long term, the Southeast will not be considered as being part of a warming hole anymore.

Comparison of the Sentinel-3A and B SLSTR Tandem Phase Data Using Metrological Principles

The Sentinel-3 mission is part of the Copernicus programme space segment and has the objective of making global operational observations of ocean, land and atmospheric parameters with its four on-board sensors. Two Sentinel-3 satellites are currently on orbit, providing near-daily global coverage. Sentinel-3A was launched on 16 February 2016 and Sentinel-3B on 25 April 2018. For the early part of its operation, Sentinel-3B flew in tandem with Sentinel-3A, flying 30 s ahead of its twin mission. This provided a unique opportunity to compare the instruments on the two satellites, and to test the per pixel uncertainty values in a metrologically-robust manner. In this work, we consider the tandem-phase data from the infrared channels of one of the on-board instruments: the Sea and Land Surface Temperature Radiometer, SLSTR. A direct comparison was made of both the Level 1 radiance values and the Level 2 sea surface temperature values derived from those radiances. At Level 1, the distribution of differences between the sensor values were compared to the declared uncertainties for data gridded on to a regular latitude-longitude grid with propagated pixel uncertainties. The results showed good overall radiometric agreement between the two sensors, with mean differences of ∼0.06 K, although there was a scene-temperature dependent difference for the oblique view that was consistent with what was expected from a stray light effect observed pre-flight. We propose a means to correct for this effect based on the tandem data. Level 1 uncertainties were found to be representative of the variance of the data, expect in those channels affected by the stray light effect. The sea surface temperature results show a very small difference between the sensors that could be in part due to the fact that the Sentinel-3A retrieval coefficients were also applied to the Sentinel-3B retrieval because the Sentinel-3B coefficients are not currently available. This will lead to small errors between the S3A and S3B retrievals. The comparison also suggests that the retrieval uncertainties may need updating for two of the retrieval processes that there are extra components of uncertainty related the quality level and the probability of cloud that should be included. Finally, a study of the quality flags assigned to sea surface temperature pixel values provided valuable insight into the origin of those quality levels and highlighted possible uncertainties in the defined quality level.

Modeling Tangential Vector Fields on a Sphere

ABSTRACTPhysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or divergence-free. We start with constructing parametric models for curl-free and divergence-free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere. Using the Helmholtz–Hodge decomposition, we then construct a class of simple but flexible parametric models for general tangential vector fields, which are represented as a sum of a curl-free and a divergence-free components. We propose a likelihood-based parameter estimation procedure, and show that fast computation is possible even for large datasets when the observations are on a regular latitude–longitude grid. Characteristics and practical utility of the proposed methodology are illustrated through extensive simulation studies and an application to a dataset of ocean surface wind velocities collected by satellite-based scatterometers. We also compare our model with a bivariate Matérn model and a non-stationary bivariate global model. Supplementary materials for this article are available online.

SA-OBS: A Daily Gridded Surface Temperature and Precipitation Dataset for Southeast Asia

This study introduces a new daily high-resolution land-only observational gridded dataset, called SA-OBS, for precipitation and minimum, mean, and maximum temperature covering Southeast Asia. This dataset improves upon existing observational products in terms of the number of contributing stations, in the use of an interpolation technique appropriate for daily climate observations, and in making estimates of the uncertainty of the gridded data. The dataset is delivered on a 0.25° × 0.25° and a 0.5° × 0.5° regular latitude–longitude grid for the period 1981–2014. The dataset aims to provide best estimates of grid square averages rather than point values to enable direct comparisons with regional climate models. Next to the best estimates, daily uncertainties are quantified. The underlying daily station time series are collected in cooperation between meteorological services in the region: the Southeast Asian Climate Assessment and Dataset (SACA&D). Comparisons are made with station observations and other gridded station or satellite-based datasets (APHRODITE, CMORPH, TRMM). The comparisons show that vast differences exist in the average daily precipitation, the number of rainy days, and the average precipitation on a wet day between these datasets. SA-OBS closely resembles the station observations in terms of dry/wet frequency, the timing of precipitation events, and the reproduction of extreme precipitation. New versions of SA-OBS will be released when the station network in SACA&D has grown further.

NCAR_Topo (v1.0): NCAR global model topography generation software for unstructured grids

Abstract. It is the purpose of this paper to document the NCAR global model topography generation software for unstructured grids (NCAR_Topo (v1.0)). Given a model grid, the software computes the fraction of the grid box covered by land, the grid-box mean elevation (deviation from a geoid that defines nominal sea level surface), and associated sub-grid-scale variances commonly used for gravity wave and turbulent mountain stress parameterizations. The software supports regular latitude–longitude grids as well as unstructured grids, e.g., icosahedral, Voronoi, cubed-sphere and variable-resolution grids.

Operational surface UV radiation product from GOME-2 and AVHRR/3 data

Abstract. The surface ultraviolet (UV) radiation product, version 1.20, generated operationally in the framework of the Satellite Application Facility on Ozone and Atmospheric Chemistry Monitoring (O3M SAF) of the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) is described. The product is based on the total ozone column derived from the measurements of the second Global Ozone Monitoring Experiment (GOME-2) instrument aboard EUMETSAT's polar orbiting meteorological operational (Metop) satellites. Cloud cover is taken into account by retrieving cloud optical depth from the channel 1 reflectance of the third Advanced Very High-Resolution Radiometer (AVHRR/3) instrument aboard both Metop in the morning orbit and Polar Orbiting Environmental Satellites (POES) of the National Oceanic and Atmospheric Administration (NOAA) in the afternoon orbit. In addition, more overpasses are used at high latitudes where the swaths of consecutive orbits overlap. The input satellite data are received from EUMETSAT's Multicast Distribution System (EUMETCast). The surface UV product includes daily maximum dose rates and integrated daily doses with different biological weighting functions, integrated ultraviolet B (UVB) and ultraviolet A (UVA) radiation, solar noon UV index and daily maximum photolysis frequencies of ozone and nitrogen dioxide at the surface level. The quantities are computed in a 0.5°×0.5° regular latitude–longitude grid and stored as daily files in the hierarchical data format (HDF5) within 2 weeks from sensing. The product files are archived in the O3M SAF distributed archive and can be ordered via the EUMETSAT Data Centre.

Assessing the direction of climate interactions by means of complex networks and information theoretic tools.

An estimate of the net direction of climate interactions in different geographical regions is made by constructing a directed climate network from a regular latitude-longitude grid of nodes, using a directionality index (DI) based on conditional mutual information (CMI). Two datasets of surface air temperature anomalies-one monthly averaged and another daily averaged-are analyzed and compared. The network links are interpreted in terms of known atmospheric tropical and extra-tropical variability patterns. Specific and relevant geographical regions are selected, the net direction of propagation of the atmospheric patterns is analyzed, and the direction of the inferred links is validated by recovering some well-known climate variability structures. These patterns are found to be acting at various time-scales, such as atmospheric waves in the extratropics or longer range events in the tropics. This analysis demonstrates the capability of the DI measure to infer the net direction of climate interactions and may contribute to improve the present understanding of climate phenomena and climate predictability. The work presented here also stands out as an application of advanced tools to the analysis of empirical, real-world data.

Pangolin v1.0, a conservative 2-D advection model towards large-scale parallel calculation

Abstract. To exploit the possibilities of parallel computers, we designed a large-scale bidimensional atmospheric advection model named Pangolin. As the basis for a future chemistry-transport model, a finite-volume approach for advection was chosen to ensure mass preservation and to ease parallelization. To overcome the pole restriction on time steps for a regular latitude–longitude grid, Pangolin uses a quasi-area-preserving reduced latitude–longitude grid. The features of the regular grid are exploited to reduce the memory footprint and enable effective parallel performances. In addition, a custom domain decomposition algorithm is presented. To assess the validity of the advection scheme, its results are compared with state-of-the-art models on algebraic test cases. Finally, parallel performances are shown in terms of strong scaling and confirm the efficient scalability up to a few hundred cores.

Localized climate change scenarios of mean temperature and precipitation over Switzerland

There is a growing need of the climate change impact modeling and adaptation community to have more localized climate change scenario information available over complex topography such as in Switzerland. A gridded dataset of expected future climate change signals for seasonal averages of daily mean temperature and precipitation in Switzerland is presented. The basic scenarios are taken from the CH2011 initiative. In CH2011, a Bayesian framework was applied to obtain probabilistic scenarios for three regions within Switzerland. Here, the results for two additional Alpine sub-regions are presented. The regional estimates have then been downscaled onto a regular latitude-longitude grid with a resolution of 0.02° or roughly 2 km. The downscaling procedure is based on the spatial structure of the climate change signals as simulated by the underlying regional climate models and relies on a Kriging with external drift using height as auxiliary predictor. The considered emission scenarios are A1B, A2 and the mitigation scenario RCP3PD. The new dataset shows an expected warming of about 1 to 6 °C until the end of the 21st century, strongly depending on the scenario and the lead time. Owing to a large vertical gradient, the warming is about 1 °C stronger in the Alps than in the Swiss lowlands. In case of precipitation, the projection uncertainty is large and in most seasons precipitation can increase or decrease. In summer a distinct decrease of precipitation can be found, again strongly depending on the emission scenario.

A standard test case suite for two-dimensional linear transport on the sphere: results from a collection of state-of-the-art schemes

Abstract. Recently, a standard test case suite for 2-D linear transport on the sphere was proposed to assess important aspects of accuracy in geophysical fluid dynamics with a "minimal" set of idealized model configurations/runs/diagnostics. Here we present results from 19 state-of-the-art transport scheme formulations based on finite-difference/finite-volume methods as well as emerging (in the context of atmospheric/oceanographic sciences) Galerkin methods. Discretization grids range from traditional regular latitude–longitude grids to more isotropic domain discretizations such as icosahedral and cubed-sphere tessellations of the sphere. The schemes are evaluated using a wide range of diagnostics in idealized flow environments. Accuracy is assessed in single- and two-tracer configurations using conventional error norms as well as novel diagnostics designed for climate and climate–chemistry applications. In addition, algorithmic considerations that may be important for computational efficiency are reported on. The latter is inevitably computing platform dependent. The ensemble of results from a wide variety of schemes presented here helps shed light on the ability of the test case suite diagnostics and flow settings to discriminate between algorithms and provide insights into accuracy in the context of global atmospheric/ocean modeling. A library of benchmark results is provided to facilitate scheme intercomparison and model development. Simple software and data sets are made available to facilitate the process of model evaluation and scheme intercomparison.

Vorticity–divergence mass-conserving semi-Lagrangian shallow-water model using the reduced grid on the sphere

The semi-Lagrangian semi-implicit shallow water model on the sphere using the reduced latitude–longitude grid is presented. The key feature of the model is the vorticity–divergence formulation on the unstaggered grid. The new algorithm for the reconstruction of wind components from vorticity and divergence is described. The mass-conservative version of the model is developed. The conservative cascade scheme (CCS) by Nair et al. is modified to provide a locally-conservative semi-Lagrangian advection algorithm for the reduced grid. Some numerical advection tests are carried out to demonstrate the accuracy of the CCS with the reduced grid. The CCS-based discretization for the continuity equation and finite-volume Helmholtz problem solver are introduced to guarantee the mass-conservation.The results for shallow water tests on the sphere are presented. The results for different versions of the model are compared. They are also compared with the results for the same tests available in literature. The impact of the reduced grid is analyzed. The mass-conservative version of the model using the reduced grid with up to 20% reduction of grid points number has approximately the same accuracy as its non-conservative counterpart implemented on the regular latitude–longitude grid.

A European daily high-resolution observational gridded data set of sea level pressure

[1] In this study we introduce a daily high-resolution land-only observational gridded data set for sea level pressure covering the European region as a new addition to the E-OBS gridded data sets of daily temperatures and precipitation amounts. This data set improves upon existing products in terms of spatial resolution and extent. The data set is delivered on the same four spatial resolutions as the other E-OBS data sets: 0.25° by 0.25° and 0.5° by 0.5° on a regular latitude-longitude grid and 0.22° by 0.22° and 0.44° by 0.44° on a rotated pole grid. We show that there is a good agreement in the long-term mean and standard deviation with popular reanalysis grids. In areas with a relatively high number of stations, the gridded data is closer to the station data than the reanalysis products. There is also a very good agreement with daily weather charts for selected storm events.

Using temporal variability to improve spatial mapping with application to satellite data

AbstractThe National Aeronautics and Space Administration (NASA) has a remote‐sensing program with a large array of satellites whose mission is earth‐system science. To carry out this mission, NASA produces data at various levels; level‐2 data have been calibrated to the satellite's footprint at high temporal resolution, although there is often a lot of missing data. Level‐3 data are produced on a regular latitude—longitude grid over the whole globe at a coarser spatial and temporal resolution (such as a day, a month, or a repeat‐cycle of the satellite), and there are still missing data. This article demonstrates that spatio‐temporal statistical models can be made operational and provide a way to estimate level‐3 values over the whole grid and attach to each value a measure of its uncertainty. Specifically, a hierarchical statistical model is presented that includes a spatio‐temporal random effects (STRE) model as a dynamical component and a temporally independent spatial component for the fine‐scale variation. Optimal spatio‐temporal predictions and their mean squared prediction errors are derived in terms of a fixed‐dimensional Kalman filter. The predictions provide estimates of missing values and filter out unwanted noise. The resulting fixed‐rank filter is scalable, in that it can handle very large data sets. Its functionality relies on estimation of the model's parameters, which is presented in detail. It is demonstrated how both past and current remote‐sensing observations on aerosol optical depth (AOD) can be combined, yielding an optimal statistical predictor of AOD on the log scale along with its prediction standard error. The Canadian Journal of Statistics 38: 271–289; 2010 © 2010 Statistical Society of Canada

Efficient and Accurate Calculation of Ray Theory Seismic Travel Time through Variable Resolution 3D Earth Models

Realistic 3D models of the seismic velocity distribution within the Earth have the potential to significantly improve our ability to accurately and precisely locate seismic events around the world ( e.g. , Flanagan et al. 2007). Development of these models by tomographic inversion of observed seismic travel times and the use of such models in seismic event location calculations, require the ability to accurately and efficiently compute predicted source to receiver travel times through 3D velocity structures. Meeting these requirements on a global scale presents two fundamental challenges: 1) how to represent the 3D distribution of seismic velocity in the Earth in models that can be interpolated efficiently, and 2) how to efficiently and accurately compute predicted travel times through these models. We present a 3D variable-resolution modeling methodology and a travel time prediction algorithm based on ray theory that address these issues and are compatible with computational resources ranging from laptop computers to powerful distributed computing systems. A simple representation of the 3D distribution of Earth properties is a regular latitude-longitude grid over the surface of the Earth with grid nodes distributed in an irregular manner in the radial direction (Figure 1A). Finding the grid cell in which an arbitrary point is located in such a model is both straight-forward and efficient. For models that do not extend to polar regions ( e.g. , Pasyanos et al. 2004), this approach provides satisfactory performance, but for global models such as the Crust 2.0 model (Bassin et al. 2000; http://igppweb.ucsd.edu/~gabi/crust2.html) the approach suffers from the fact that meridians intersect at the poles, leading to much smaller grid cells at high latitudes than at the equator, when cell size is evaluated in km2. Over the past couple of decades gridless Earth model representations and several representations involving irregular meshes have been …

Geometrically Exact Conservative Remapping (GECoRe): Regular Latitude–Longitude and Cubed-Sphere Grids

Abstract Land, ocean, and atmospheric models are often implemented on different spherical grids. As a conseqence coupling these model components requires state variables and fluxes to be regridded. For some variables, such as fluxes, it is paramount that the regridding algorithm is conservative (so that energy and water budget balances are maintained) and monotone (to prevent unphysical values). For global applications the cubed-sphere grids are gaining popularity in the atmospheric community whereas, for example, the land modeling groups are mostly using the regular latitude–longitude grid. Most existing regridding schemes fail to take advantage of geometrical symmetries between these grids and hence accuracy of the calculations can be lost. Hence, a new Geometrically Exact Conservative Remapping (GECoRe) scheme with a monotone option is proposed for remapping between regular latitude–longitude and gnomonic cubed-sphere grids. GECoRe is compared with existing remapping schemes published in the meteorological literature. It is concluded here that the geometrically exact scheme significantly improves the accuracy of the resulting remapping in idealized test cases.