Merging of Airborne Gravity and Gravity Derived from Satellite Altimetry: Test Cases Along the Coast of Greenland

The National Survey and Cadastre - Denmark (KMS) has for several years produced gravity anomaly maps over the oceans derived from satellite altimetry. During the last four years, KMS has also conducted airborne gravity surveys along the coast of Greenland dedicated to complement the existing onshore gravity coverage and fill in new data in the very-near coastal area, where altimetry data may contain gross errors. The airborne surveys extend from the coastline to approximately 100 km offshore, along 6000 km of coastline. An adequate merging of these different data sources is important for the use of gravity data especially, when computing geoid models in coastal regions.The presence of reliable marine gravity data for independent control offers an opportunity to study procedures for the merging of airborne and satellite data around Greenland. Two different merging techniques, both based on collocation, are investigated in this paper. Collocation offers a way of combining the individual airborne gravity observation with either the residual geoid observations derived from satellite altimetry or with gravity derived from these data using the inverse Stokes method implemented by Fast Fourier Transform (FFT).     

基于GRACE Follow-On卫星重力梯度法精确反演地球重力场

由于GRACE Follow-On双星系统等效于基线长为星间距离的一维水平重力梯度仪,因此本文基于GRACE Follow-On卫星重力梯度法开展了精确和快速反演下一代地球重力场的可行性论证研究. 研究结果表明：第一,基于GRACE Follow-On卫星重力梯度法（GFO-SGGM）,利用卫星轨道参数（轨道高度250 km、星间距离50 km、轨道倾角89°、轨道离心率0.001）、关键载荷测量精度（星间距离10^-6 m、星间速度10^-7 m·s^-1、星间加速度10^-10 m·s^-2、轨道位置10^-3 m、轨道速度10^-6 m·s^-1、非保守力10^-11 m·s^-2）、观测时间30天和采样间隔10 s反演了120阶地球重力场,在120阶处累计大地水准面精度为9.331×10^-4 m. 第二,在120阶内,利用将来GRACE Follow-On双星反演地球重力场精度较现有GRACE双星平均提高61倍,因此GRACE Follow-On卫星重力梯度法是进一步提高地球重力场反演精度的优选方法. 第三,下一代GRACE Follow-On计划较当前GRACE计划的优点如下：轨道高度更低（200~300 km）、载荷精度更高（10^-7 ~10^-9 m·s^-1）和星间距离更短（50~100 km）.     

Induction studies with satellite data

The natural variations of the Earth's magnetic field of periods spanning from milliseconds to decades can be used to infer the conductivity-depth profile of the Earth's interior. Satellites provide a good spatial coverage of magnetic measurements, and forthcoming missions will probably allow for observations lasting several years, which helps to reduce the statistical error of the estimated response functions.Two methods are used to study the electrical conductivity of the Earth's mantle in the period range from hours to months. In the first, known as the potential method, a spherical harmonic analysis of the geomagnetic field is performed, and the Q-response, which is the transfer function between the internal (induced) and the external (inducing) expansion coefficients is determined for a specific frequency. In the second approach, known as the geomagnetic depth sounding method, the C-response, which is the transfer function between the magnetic vertical component and the horizontal derivative of the horizontal components, is determined. If one of these transfer functions is known for several frequencies, models of the electrical conductivity in the Earth's interior can be constructed.This paper reviews and discusses the possibilities for induction studies using high-precision magnetic measurements from low-altitude satellites. The different methods and various transfer functions are presented, with special emphasis on the differences in analysing data from ground stations and from satellites. The results of several induction studies with scalar satellite data (from the POGO satellites) and with vector data (from the Magsat mission) demonstrate the ability to probe the Earth's conductivity from space. However, compared to the results obtained with ground data the satellite results are much noisier, which presumably is due to the shorter time series of the satellite studies.The results of a new analysis of data from the Magsat satellite indicate higher resistivity in oceanic areas than in continental areas. However, since this holds for the whole range of periods between 2 and 20 days, this difference probably is not caused purely by differences in mantle conductivity (for which one would expect less difference for the longer periods). Further studies with data from recently launched and future satellites are needed.     

影响四轨法D-InSAR形变测量精度误差的相关性分析

卫星成像的基线、视角、基线倾斜角、斜距、卫星距离地面径向距离以及地面分辨率等因素严重影响着合成孔径雷达差分干涉测量监测地面形变的能力和形变监测结果的精度。本文在分析四轨法D-InSAR基本原理和数据处理流程基础上,详细给出了相位测量误差对形变测量精度影响的定量关系式;分析讨论了基线长度误差、基线倾斜角误差、斜距误差、卫星高度误差和地形因素误差对形变测量精度的影响。从而在定量分析方面得出了这些误差对四轨法D-InSAR形变测量精度影响的结论。     

利用解析法有效快速估计将来GRACE Follow-On地球重力场的精度

本文首次利用解析法有效快速估计了将来GRACE（Gravity Recovery and Climate Experiment） Follow-On地球重力场的精度. 第一,基于功率谱原理分别建立了新的GRACE Follow-On卫星激光干涉星间测量系统星间速度、GPS接收机轨道位置和轨道速度以及加速度计非保守力误差影响累计大地水准面的单独和联合解析误差模型. 第二,利用提出的GRACE卫星关键载荷匹配精度指标和美国喷气推进实验室（JPL）公布的GRACE Level 1B实测精度指标的一致性,以及估计的GRACE累计大地水准面精度和德国波兹坦地学研究中心（GFZ）公布的EIGEN-GRACE02S地球重力场模型实测精度的符合性,验证了本文建立的解析误差模型是可靠的. 第三,论证了GRACE Follow-On卫星不同关键载荷匹配精度指标和轨道高度对地球重力场精度的影响. 在360阶处,利用轨道高度250 km、星间距离50 km、星间速度误差1×10^-9m/s、轨道位置误差3×10^-5m、轨道速度误差3×10^-8m/s和非保守力误差3×10^-13m/s²,基于联合解析误差模型估计累计大地水准面的精度为1.231×10^-1 m. 本文的研究不仅为当前GRACE和将来GRACE Follow-On地球重力场精度的有效快速确定提供了理论基础和计算保证,同时对国际将来GRAIL（Gravity Recovery and Interior Laboratory）月球卫星重力测量计划的成功实施具有重要的参考意义.     

Improvement of ERS-1 orbits using along-track accelerations from DORIS data on SPOT2

In long-arc precise orbit determinations of altimetric satellites such as ERS-1, large errors may occur from mismodelling of aerodynamic drag and solar radiation pressure. Such surface forces for non-spherical satellites require accurate modelling of the effective area and particle-surface interactions, but the dominant source of error is neutral air density as derived from thermospheric models for aerodynamic drag. Several techniques can be employed to alleviate air-drag mismodelling but all require the solution of additional parameters from the tracking data. However, for ERS-1 the sparsity of laser range data limits the application of such empirical techniques. To overcome this, use can be made of the dense DORIS Doppler tracking for SPOT2 which is in a similar orbit to ERS-1. A recent investigation by CNES examined the use of drag scale factors from SPOT2 to constrain the ERS-1 orbit. An improvement to that methodology is to consider along-track mismodelling as observed by timing errors in the Doppler data for each pass of SPOT2. The along-track correction to the acceleration as derived from SPOT2 can then be applied to ERS-1 orbits, solving for a scale factor to absorb systematic errors - particularly that arising from the 50 km altitude difference. Results are presented of the associated improvement in ERS-1 orbits as derived from concurrent SPOT2 arcs. It will be seen that the procedure not only improves the laser range fit, but more importantly, leads to more precise radial positioning as evident in the altimeter and crossover residuals.     

Evaluation of in situ and remote sensing sampling methods for SPM concentrations, Belgian continental shelf (southern North Sea)

Large sets of suspended particulate matter (SPM) concentration data from in situ and remote sensing (moderate resolution imaging spectroradiometer, MODIS) samplings in the Belgian nearshore area (southern North Sea) are combined in order to evaluate their heterogeneity and the sampling techniques. In situ SPM concentration measurements are from a vessel (tidal cycle) and from a tripod. During the tidal cycle measurements, vertical profiles of SPM concentration have been collected; these profiles have been used as a link between satellite surface and near-bed tripod SPM concentrations. In situ time series at fixed locations using a tripod are excellent witnesses of SPM concentrations under all weather conditions and may catch SPM concentration variability with a much finer scale. The heterogeneity has been statistically assessed by comparing the SPM concentration frequency distributions. Tidal cycle, tripod and MODIS datasets have different distributions and represent a different subpopulation of the whole SPM concentrations population. The differences between the datasets are related to meteorological conditions during the measurements; to near-bed SPM concentration dynamics, which are partially uncoupled from processes higher up in the water column; to the sampling methods or schemes and to measurement uncertainties. In order to explain the differences between the datasets, the tripod data have been subsampled using wave height conditions and satellite and tidal cycle sampling schemes. It was found that satellites and low-frequent tidal cycle measurements are biased towards good weather condition or spring–summer seasons (satellite). The data show that the mean surface SPM concentration derived from satellite data is slightly lower than from in situ tidal cycle measurements, whereas it is significantly lower than the mean SPM concentration interpolated to the water surface from the tripod measurements. This is explained by the errors arising from the interpolation along the vertical profiles, but also by the fact that satellite-measured signal saturates in the visible band used to retrieve SPM concentration in very turbid waters.     

Inversion, error analysis, and validation of GPS/MET occultation data

The global positioning system meteorology (GPS/MET) experiment was the first practical demonstration of global navigation satellite system (GNSS)-based active limb sounding employing the radio occultation technique. This method measures, as principal observable and with millimetric accuracy, the excess phase path (relative to propagation in vacuum) of GNSS-transmitted radio waves caused by refraction during passage through the Earth’s neutral atmosphere and ionosphere in limb geometry. It shows great potential utility for weather and climate system studies in providing an unique combination of global coverage, high vertical resolution and accuracy, long-term stability, and all-weather capability. We first describe our GPS/MET data processing scheme from excess phases via bending angles to the neutral atmospheric parameters refractivity, density, pressure and temperature. Special emphasis is given to ionospheric correction methodology and the inversion of bending angles to refractivities, where we introduce a matrix inversion technique (instead of the usual integral inversion). The matrix technique is shown to lead to identical results as integral inversion but is more directly extendable to inversion by optimal estimation. The quality of GPS/MET-derived profiles is analyzed with an error estimation analysis employing a Monte Carlo technique. We consider statistical errors together with systematic errors due to upper-boundary initialization of the retrieval by a priori bending angles. Perfect initialization and properly smoothed statistical errors allow for better than 1 K temperature retrieval accuracy up to the stratopause. No initialization and statistical errors yield better than 1 K accuracy up to 30 km but less than 3 K accuracy above 40 km. Given imperfect initialization, biases ≫ 2 K propagate down to below 30 km height in unfavorable realistic cases. Furthermore, results of a statistical validation of GPS/MET profiles through comparison with atmospheric analyses of the European Centre for Medium-range Weather Forecasts (ECMWF) are presented. The comparisons indicate the high utility of the occultation data in that very good agreement of upper troposphere/lower stratosphere temperature (better than 1.5 K rms, ≪ 0.5 K bias) is found for a region (Europe+USA) where the ECMWF analyses are known to be good, but poorer agreement for a region (Southern Pacific) where the analyses are known to be degraded.     

APOD mission status and preliminary results

On September 20 th, 2015, twenty satellites were successfully deployed into a near-polar circular orbit at 520 km altitude by the Chinese CZ-6 test rocket, which was launched from the Tai Yuan Satellite Launch Center. Among these satellites, a set of 4 Cube Sats conform the atmospheric density detection and precise orbit determination(APOD) mission, which is projected for atmospheric density estimation from in-situ detection and precise orbit products. The APOD satellites are manufactured by China Spacesat Co. Ltd. and the payload instruments include an atmospheric density detector(ADD), a dual-frequency dualmode global navigation satellite system(GNSS) receiver(GPS and Beidou), a satellite laser ranging(SLR) reflector, and an S/Xband very long baseline interferometry(VLBI) beacon. In this paper, we compare the GNSS precise orbit products with colocated SLR observations, and the 3 D orbit accuracy shows better than 10 cm RMS. These results reveal the great potential of the onboard micro-electro-mechanical system(MEMS) GNSS receiver. After calibrating ADD density estimates with precise orbit products, the accuracy of our density products can reach about 10% with respect to the background density. Density estimates from APOD are of a great importance for scientific studies on upper atmosphere variations and useful for model data assimilation.     

Bayesian analysis of individual and systematic multiplicative errors for estimating ages with stratigraphic constraints in optically stimulated luminescence dating

Many dating techniques include significant error terms which are not independent between samples to date. This is typically the case in Optically Stimulated Luminescence (OSL) dating where the conversion from characteristic equivalent doses to the corresponding ages using the annual dosimetry data includes error terms that are common to all produced datings. Dealing with these errors is essential to estimate ages from a set of datings whose chronological ordering is known. In this work, we propose and we study a Bayesian model to address this problem. For this purpose, we first consider a multivariate model with multiplicative Gaussian errors in a Bayesian framework. This model relates a set of characteristic equivalent doses to the corresponding ages while taking into account for the systematic and non-systematic errors associated to the dosimetry. It thus offers the opportunity to deal properly with stratigraphic constraints within OSL datings, but also with other datings possessing errors which are independent from systematic errors of OSL (e.g. radiocarbon). Then, we use this model to extend an existing Bayesian model for the assessment of characteristic equivalent doses from Single Aliquot and Regenerative (SAR) dose measurements. The overall Bayesian model leads to the joint estimation of all the variables (which include all the dose–response functions and characteristic equivalent doses) of a sequence of, possibly heterogeneous, datings. We also consider a more generic solution consisting in using directly the age model from a set of characteristic equivalent dose estimates and their associated standard errors. We finally give an example of application on a set of five OSL datings with stratigraphic constraints and observe a good adequacy between the two approaches.     

Moho depth uncertainties in the Vening-Meinesz Moritz inverse problem of isostasy

We formulate an error propagation model based on solving the Vening Meinesz-Moritz (VMM) inverse problem of isostasy. The system of observation equations in the VMM model defines the relation between the isostatic gravity data and the Moho depth by means of a second-order Fredholm integral equation of the first kind. The corresponding error model (derived in a spectral domain) functionally relates the Moho depth errors with the commission errors of used gravity and topographic/bathymetric models. The error model also incorporates the non-isostatic bias which describes the disagreement, mainly of systematic nature, between the isostatic and seismic models. The error analysis is conducted at the study area of the Tibetan Plateau and Himalayas with the world largest crustal thickness. The Moho depth uncertainties due to errors of the currently available global gravity and topographic models are estimated to be typically up to 1–2 km, provided that the GOCE gravity gradient observables improved the medium-wavelength gravity spectra. The errors due to disregarding sedimentary basins can locally exceed ～2 km. The largest errors (which cause a systematic bias between isostatic and seismic models) are attributed to unmodeled mantle heterogeneities (including the core-mantle boundary) and other geophysical processes. These errors are mostly less than 2 km under significant orogens (Himalayas, Ural), but can reach up to ～10 km under the oceanic crust.     

Sampling error of areal average rainfall due to radar partial coverage

Areal average rainfall is important as it is used as an input for most rainfall-runoff analysis in Hydrology and Water Resources. Different from traditional methods of using rain gauge data, the use of radar rainfall for the estimation of areal average rainfall is very straightforward. However, in some cases with severe terrain blockages, the value of the incomplete radar information is of serious concern. This study investigated this problem and derived an equation for estimating the error involved in the areal average rainfall due to partial radar coverage of a basin or sub-basin. When only partial radar information is available, the sampling error decreases with increasing radar coverage and the number of radar bin clusters. As an application example, this study considered the Han River Basin with its rainfall observations using the Ganghwa rain radar. Among a total of 24 mid-sized sub-basins in the Han River Basin evaluated, only five sub-basins were fully covered by the radar and three were totally uncovered. The remaining 16 sub-basins were covered partially by radar leading to incomplete radar information. The results show that the sampling error ranged from several % to tens % of standard deviation of the areal average rainfall depending on the relative areal radar coverage.     

Investigation of the scale-dependent variability of radar-rainfall and rain gauge error covariance

There is a significant spatial sampling mismatch between radar and rain gauge data. The use of rain gauge data to estimate radar-rainfall error variance requires partitioning of the variance of the radar and rain gauge difference to account for the sampling mismatch. A key assumption in the literature pertaining to the error variance separation method used to partition the variance is that the covariance between radar-rainfall error and the error of rain gauges in representing radar sampling domain is negligible. Our study presents the results of an extensive test of this assumption. The test is based on empirical data and covers temporal scales ranging from 0.25 to 24 h and spatial scales ranging from 1 to 32 km. We used a two-year data set from two high quality and high density rain gauge networks in Oklahoma and excluded the winter months. The results obtained using a resampling procedure show that covariance can be considerable at large scales due to the significant variability. As the variability of the covariance rapidly increases with larger spatial and shorter temporal scales, applications of the error variance separation method at those scales require more caution. The variability of the covariance and one of its constituting variables, the variance ratio of radar and gauge errors, shows simple scaling behavior well characterized by a power-law.     

Estimation of radar-rainfall error spatial correlation

The study presents a theoretical framework for estimating the radar-rainfall error spatial correlation (ESC) using data from relatively dense rain gauge networks. The error is defined as the difference between the radar estimate and the corresponding true areal rainfall. The method is analogous to the error variance separation that corrects the error variance of a radar-rainfall product for gauge representativeness errors. The study demonstrates the necessity to consider the area–point uncertainties while estimating the spatial correlation structure in the radar-rainfall errors. To validate the method, the authors conduct a Monte Carlo simulation experiment with synthetic fields with known error spatial correlation structure. These tests reveal that the proposed method, which accounts for the area–point distortions in the estimation of radar-rainfall ESC, performs very effectively. The authors then apply the method to estimate the ESC of the National Weather Service’s standard hourly radar-rainfall products, known as digital precipitation arrays (DPA). Data from the Oklahoma Micronet rain gauge network (with the grid step of about 5 km) are used as the ground reference for the DPAs. This application shows that the radar-rainfall errors are spatially correlated with a correlation distance of about 20 km. The results also demonstrate that the spatial correlations of radar–gauge differences are considerably underestimated, especially at small distances, as the area–point uncertainties are ignored.     

Preliminary assessment of the navigation and positioning performance of BeiDou regional navigation satellite system

BeiDou regional navigation satellite system(BDS)also called BeiDou-2 has been in full operation since December 27,2012.It consists of 14 satellites,including 5 satellites in Geostationary Orbit(GEO),5 satellites in Inclined Geosynchronous Orbit(IGSO),and 4 satellites in Medium Earth Orbit(MEO).In this paper,its basic navigation and positioning performance are evaluated preliminarily by the real data collected in Beijing,including satellite visibility,Position Dilution of Precision(PDOP)value,the precision of code and carrier phase measurements,the accuracy of single point positioning and differential positioning and ambiguity resolution(AR)performance,which are also compared with those of GPS.It is shown that the precision of BDS code and carrier phase measurements are about 33 cm and 2 mm,respectively,which are comparable to those of GPS,and the accuracy of BDS single point positioning has satisfied the design requirement.The real-time kinematic positioning is also feasible by BDS alone in the opening condition,since its fixed rate and reliability of single-epoch dual-frequency AR is comparable to those of GPS.The accuracy of BDS carrier phase differential positioning is better than 1 cm for a very short baseline of 4.2 m and 3 cm for a short baseline of 8.2 km,which is on the same level with that of GPS.For the combined BDS and GPS,the fixed rate and reliability of single-epoch AR and the positioning accuracy are improved significantly.The accuracy of BDS/GPS carrier phase differential positioning is about 35 and 20%better than that of GPS for two short baseline tests in this study.The accuracy of BDS code differential positioning is better than 2.5 m.However it is worse than that of GPS,which may result from large code multipath errors of BDS GEO satellite measurements.     

Estimating Land Surface Evaporation: A Review of Methods Using Remotely Sensed Surface Temperature Data

This paper reviews methods for estimating evaporation from landscapes, regions and larger geographic extents, with remotely sensed surface temperatures, and highlights uncertainties and limitations associated with those estimation methods. Particular attention is given to the validation of such approaches against ground based flux measurements. An assessment of some 30 published validations shows an average root mean squared error value of about 50 W m^?2 and relative errors of 15–30%. The comparison also shows that more complex physical and analytical methods are not necessarily more accurate than empirical and statistical approaches. While some of the methods were developed for specific land covers (e.g. irrigation areas only) we also review methods developed for other disciplines, such as hydrology and meteorology, where continuous estimates in space and in time are needed, thereby focusing on physical and analytical methods as empirical methods are usually limited by in situ training data. This review also provides a discussion of temporal and spatial scaling issues associated with the use of thermal remote sensing for estimating evaporation. Improved temporal scaling procedures are required to extrapolate instantaneous estimates to daily and longer time periods and gap-filling procedures are needed when temporal scaling is affected by intermittent satellite coverage. It is also noted that analysis of multi-resolution data from different satellite/sensor systems (i.e. data fusion) will assist in the development of spatial scaling and aggregation approaches, and that several biological processes need to be better characterized in many current land surface models.     

利用GRACE卫星精密微波测距确定星间平均电子密度

本文介绍如何利用GRACE两颗卫星之间K波段双频微波精密测距和轨道数据,得到星间平均电子密度.发展了一种将连续轨道电子密度极小对齐到零的方法,以消除整周模糊度;借助CHAMP卫星朗缪探针测量得到的轨道电子密度基值以及GPS掩星数据计算的等离子体垂直梯度标高,进一步修正了GRACE星间电子密度所固有的偏差;从而得到大约500 km高度上长达近十年的全球电子密度数据.为了检验消除偏差后GRACE星间电子密度数据的可靠性,对比了GRACE卫星过Millstone Hill雷达上空时,非相干散射雷达观测到的大致同时和相近位置的电子密度数据,结果显示,二者之间的线性相关系数为0.97,平均偏差为-7.26%,GRACE星间电子密度总体稍微偏低,偏差的标准差为18.6%.为进一步验证本文方法所得数据的可用价值,利用消除偏差后的电子密度数据,对GRACE卫星与CHAMP卫星在近乎相同的地方时而高度不同的近圆极轨道上飞行的情况下,两颗卫星观测到的电子密度随经度和纬度的全球分布进行了对比分析.多方面的对比检验证明,本文方法得到的几乎连续10年的GRACE高度上全球电子密度数据基本可靠,为电离层气候学与天气学研究提供了宝贵资料.     

基于CHAMP、GRACE和COSMIC掩星数据的全球电离层h_mF_2建模研究

基于CHAMP、GRACE和COSMIC的电离层h_mF_2掩星数据,采用最小二乘法建立了一个包含地磁和太阳活动、经度、地方时、年积日和纬度信息的非线性多项式h_mF_2模型(Nonlinear Polynomial Peak Height Model—NPPHM).利用GRACE掩星数据对NPPHM与IRI2012进行了独立检验,结果显示这两个模型在2008年与GRACE数据的相关系数分别为0.798和0.532,均方根误差分别为25.97km和44.56km;在2012年,相关系数分别为0.732和0.488,均方根误差分别为31.39km和42.83km.选取全球不同地区14个测高仪站点数据,并引入相似离度对这两个模型的精度进行了评估,结果表明,NPPHM的相似离度远小于IRI2012,更加接近测高仪观测值.使用Athens站2003—2013年数据对模型进行了检验,结果显示IRI2012与Athens站测高仪数据的平均偏差达到8.11%,NPPHM则只有3.53%,并且在太阳活动低年及每年10月NPPHM的精度要明显高于IRI2012.此外,NPPHM也能够较好模拟出h_mF_2的日变化、季节变化及赤道异常等特性.     

Evaluating the utility of the ensemble transform Kalman filter for adaptive sampling when updating a hydrodynamic model

This paper compares two Monte Carlo sequential data assimilation methods based on the Kalman filter, for estimating the effect of measurements on simulations of state error variance made by a one-dimensional hydrodynamic model. The first method used an ensemble Kalman filter (EnKF) to update state estimates, which were then used as initial conditions for further simulations. The second method used an ensemble transform Kalman filter (ETKF) to quickly estimate the effect of measurement error covariance on forecast error covariance without the need to re-run the simulation model. The ETKF gave an unbiased estimate of EnKF analysed error variance, although differences in the treatment of measurement errors meant the results were not identical. Estimates of forecast error variance could also be made, but their accuracy deteriorated as the time from measurements increased due in part to model non-linearity and the decreasing signal variance. The motivation behind the study was to assess the ability of the ETKF to target possible measurements, as part of an adaptive sampling framework, before they are assimilated by an EnKF-based forecasting model on the River Crouch, Essex, UK. The ETKF was found to be a useful tool for quickly estimating the error covariance expected after assimilating measurements into the hydrodynamic model. It, thus, provided a means of quantifying the ‘usefulness’ (in terms of error variance) of possible sampling schemes.     

Multifractal objective analysis: conditioning and interpolation

 We investigate various ways of statistically estimating multifractal fields from sparse data. First, the problem is set in the general framework of conditional expectations, and the effect of (multi) fractal sampling on the statistics of the measured process is investigated, showing how analytical expressions describing the statistical properties of the phenomenon should be modified by the sampling. Then, several techniques are introduced, our goal being to estimate the intensity of a field at resolution λ, given samples of the process collected by networks at higher resolutions Λ>λ. The general strategy underlying all the estimating techniques presented is to approximate the unknown field values at resolution λ by means of most likely estimates conditional to the available information at resolution Λ>λ. Finally, the procedures are tested on simulated lognormal multifractal fields sampled by means of fractal networks, and the propagation of the errors in a scaling framework is also discussed. These techniques are necessary for estimating geophysical processes in regions where no monitoring stations are present, a scenario often encountered in practice, and may also be of great help in studying natural hazards and risk assessment.