Radar remote sensing of the spring thaw transition across a boreal landscape

The seasonal transition of the boreal forest between frozen and non-frozen conditions affects a number of ecosystem processes that cycle between winter dormant and summer active states. The relatively short K_u-band wavelength (2.14 cm) of the space-borne NASA scatterometer (NSCAT) is sensitive to changes in dielectric properties, associated with large-scale changes in the relative abundance and phase (frozen or thawed) of canopy and surface water. We used a temporal change detection analysis of NSCAT daily radar backscatter measurements to characterize the 1997 seasonal spring thaw transition period across the 10⁶ km² BOREAS study region of central Canada. In the spring, air temperature transitions from frozen to non-frozen conditions and surface observations of seasonal snow cover depletion were generally coincident with decreases in radar backscatter of more than 2.9 dB, regardless of regional landcover characteristics. We used a temporal classification of NSCAT daily differences from 5-day smoothed backscatter values to derive three simple indices describing the initiation, primary event and completion of the spring thaw transition period. Several factors had a negative impact on the relative accuracy of NSCAT-based results, including periodic gaps in NSCAT daily time-series information and a large (i.e., >2 cm day⁻¹) spring rainfall event. However, these results were generally successful in capturing the seasonal transition of the region from frozen to non-frozen conditions, based on comparisons with regional weather station network information. These results illustrate the potential for improved assessment of springtime phenology and associated ecosystem dynamics across high latitude regions, where field based and optical remote-sensing methods are substantially degraded by frequent cloud cover, low solar illumination and sparse surface weather station networks.     

Satellite radar remote sensing of seasonal growing seasons for boreal and subalpine evergreen forests

We evaluated whether satellite radar remote sensing of landscape seasonal freeze-thaw cycles provides an effective measure of active growing season timing and duration for boreal and subalpine evergreen forests. Landscape daily radar backscatter measurements from the SeaWinds scatterometer on-board the QuikSCAT satellite were evaluated across a regional network of North American coniferous forest sites for 2000 and 2001. Radar remote sensing measurements of the initiation and length of the growing season corresponded closely with both site measurements and ecosystem process model (BIOME-BGC) simulations of these parameters because of the sensitivity of the K_u-band scatterometer to snow cover freeze-thaw dynamics and associated linkages between growing season initiation and the timing of seasonal snowmelt. In contrast, remote sensing estimates of the timing of growing season termination were either weakly or not significantly associated with site measurements and model simulation results, due to the relative importance of light availability and other environmental controls on stand phenology in the fall. Regional patterns of estimated annual net primary production (NPP) and component photosynthetic and autotrophic respiration rates for the evergreen forest sites also corresponded favorably with remote sensing estimates of the seasonal timing of spring thaw and associated growing season length, indicating the importance of these parameters in determining spatial and temporal patterns of NPP and the potential utility of satellite radar remote sensing for regional monitoring of the terrestrial biosphere.     

A novel approach for the characterization of tundra wetland regions with C-band SAR satellite data

A circumpolar representative and consistent wetland map is required for a range of applications ranging from upscaling of carbon fluxes and pools to climate modelling and wildlife habitat assessment. Currently available data sets lack sufficient accuracy and/or thematic detail in many regions of the Arctic. Synthetic aperture radar (SAR) data from satellites have already been shown to be suitable for wetland mapping. Envisat Advanced SAR (ASAR) provides global medium-resolution data which are examined with particular focus on spatial wetness patterns in this study. It was found that winter minimum backscatter values as well as their differences to summer minimum values reflect vegetation physiognomy units of certain wetness regimes. Low winter backscatter values are mostly found in areas vegetated by plant communities typically for wet regions in the tundra biome, due to low roughness and low volume scattering caused by the predominant vegetation. Summer to winter difference backscatter values, which in contrast to the winter values depend almost solely on soil moisture content, show expected higher values for wet regions. While the approach using difference values would seem more reasonable in order to delineate wetness patterns considering its direct link to soil moisture, it was found that a classification of winter minimum backscatter values is more applicable in tundra regions due to its better separability into wetness classes. Previous approaches for wetland detection have investigated the impact of liquid water in the soil on backscatter conditions. In this study the absence of liquid water is utilized.

Owing to a lack of comparable regional to circumpolar data with respect to thematic detail, a potential wetland map cannot directly be validated; however, one might claim the validity of such a product by comparison with vegetation maps, which hold some information on the wetness status of certain classes. It was shown that the Envisat ASAR-derived classes are related to wetland classes of conventional vegetation maps, indicating its applicability; 30% of the land area north of the treeline was identified as wetland while conventional maps recorded 1–7%.     

Monitoring snowmelt across the Arctic forest–tundra ecotone using Synthetic Aperture Radar

Detailed snowpack observations, meteorology, topography and landcover classification were integrated with multi‐temporal SAR data to assess its capability for landscape scale snowmelt mapping at the forest–tundra ecotone. At three sites along an approximately 8° latitudinal gradient in the Fennoscandian mountain range, 16 multi‐temporal spaceborne ERS‐2 synthetic aperture radar (SAR) were used for mapping snowmelt.

Comparison of field measurements and backscatter values demonstrates the difficulty of interpreting observed backscatter response because of complex changes in snow properties on diurnal and seasonal temporal scales. Diurnal and seasonal melt–freeze effects in the snowpack, relative to the timing of ERS‐2 SAR image acquisition, effectively reduce the temporal resolution of such data for snow mapping, even at high latitudes.

The integration of diverse data sources did reveal significant associations between vegetation, topography and snowmelt. Several problems with the application of thresholding for the automatic identification of snowmelt were encountered. These largely related to changes in backscattering from vegetation in the late stages of snowmelt. Due to the impact of environmental heterogeneity in vegetation at the forest–tundra ecotone, we suggest that the potential to map snow cover using single polarization C‐band SAR at the forest–tundra ecotone may be limited to tundra areas.     

基于雷达后向散射特性进行湿地植被识别与分类的方法研究

选择东北地区典型内陆沼泽湿地——三江平原洪河自然保护区作为研究区,结合SAR的极化特性,分析了多时相ENVISAT ASAR不同极化下洪河湿地保护区不同地物植被类型的散射特性,利用长波L波段PAL-SAR数据对植被的可穿透性及水分的敏感性,结合与光学影像TM融合后进行神经元网络分类的方法,应用决策树方法进行了多波段、多时相SAR合成湿地植被识别试验。本文将两种方法相结合,分两步完整识别出沼泽、灌丛、岛状林、草甸、开阔水体及少量农田。     

A combined passive/active microwave remote sensing approach for surface variable retrieval using Tropical Rainfall Measuring Mission observations

The backscattering and emission measured simultaneously by radar and radiometer show promise for the estimation of surface variables such as near-surface soil moisture and vegetation characteristics. In this paper, the 10.7 GHz Tropical Rainfall Measuring Mission (TRMM) microwave imager (TMI) channel and 13.8 GHz precipitation radar (PR) observations are simultaneously used for the estimation of the near-surface soil moisture and vegetation properties. The Fresnel model for soil and a simple model for vegetation are used to simulate the passive microwave emission at 10.7 GHz. To determine the PR backscatter signal from a land surface, a theoretical approach is used based on the Geometric Optics Model for simulating bare soil and a semi-empirical water-cloud model for vegetation. The model parameters required in specifying the nature of the soil and vegetation are calibrated on the basis of in situ soil moisture data combined with remotely sensed observations. The calibrated model is subsequently used to retrieve near-surface soil moisture and leaf area index for assumed values of surface roughness and temperature. Algorithm assessment using synthetic passive and active microwave data shows a nonlinearity effect in the system inversion, which results in a varying degree of error statistics in soil wetness and vegetation characteristics retrieval. The technique was applied on TRMM radar/radiometer observations from three consecutive years and evaluated against in situ near-surface (5 cm) soil moisture measurements from the Oklahoma Mesonet showing a consistent performance.     

Investigating the potential for soil moisture and surface roughness monitoring in drylands using ERS SAR data

Multitemporal ERS-1 and ERS-2 SAR data were acquired for northern Jordan between 1995 and 1997 to investigate changes in the backscatter coefficients of a range of typical desert land surfaces. The changes in backscatter found were ascribed to variations in surface soil moisture, and changes in surface roughness caused by a range of natural and anthropogenic factors. Data collected from monitored sites were input into the Integral Equation Model (IEM). The model outputs were strongly correlated with observed backscatter coefficients (r ²=0.84). The results show that the successful monitoring of soil moisture in these environments is strongly dependent on the surface roughness. On surfaces with RMS height 0.5 cm, the sensitivity of the backscatter coefficient to changes in surface microtopography did not allow accurate soil moisture estimation. Microtopographic change on rougher surfaces has less influence on the backscatter coefficient, and the probability of soil moisture estimation from SAR imagery is greater. These results indicate that knowledge of the surface conditions (both in terms of surface roughness and geomorphology) is essential for accurate soil moisture monitoring, whether in a research or operational context. The potential benefits of these findings are discussed in the context of the Jordan Badia Research and Development Project.     

利用MODIS 影像数据获取融雪信息

融雪信息的获取是积雪监测中的一个重要环节, 使用遥感手段获取融雪信息有观测范围广、获取信息量大、速度快、实时性好等优势。讨论了使用MODIS 影像数据获取融雪信息的方法, 根据研究需要提出了融雪指数Snowmelt Index (SMI) 的概念, 从理论上分析了SMI 在融雪过程各阶段的变化规律。最后文章以乌鲁木齐市一次完整的融雪过程为例, 验证了SM I 的下降幅度在融雪过程中随着积雪中的液态水含量的增多而逐渐减少, 并最终停留在一个较低水平上的变化规律。     

积雪、土壤冻融与土壤水分遥感监测研究进展

积雪、土壤冻融与土壤水分是陆表能量与水分以及碳交换过程研究中的重要因子,为了更好地了解积雪覆盖、雪深/雪水当量、土壤冻融状态和土壤水分等参数的遥感监测领域的发展动态,对这些参数遥感监测方法的研究进展进行了梳理,总结了利用光学与微波遥感,以及多源遥感融合的监测方法,并对该研究领域的发展趋势进行了展望。积雪、土壤冻融与土壤水分的遥感监测能力不断提升,监测算法从单一传感器向多传感器、单波段单一模式向多波段多模式集成,以及卫星虚拟星座综合观测概念的提出,均促进了现有卫星观测地表参数能力的提升;长时间序列产品的开发,对于研究和掌握全球变化大背景下对气候的响应提供了很好的数据基础;同时有助于促进遥感在水文、气象、气候、生态等领域的应用。以上的研究综述,有望对陆表水循环遥感参数反演领域,以及水循环遥感关键参数的应用领域有一定的借鉴作用。     

Comparison between a multi-scattering and multi-layer snow scattering model and its parameterized snow backscattering model

Snow Water Equivalent (SWE) is a crucial parameter in the study of climatology and hydrology. Active microwave remote sensing is one of the most promising techniques for estimating the distribution of SWE at high spatial resolutions in large areas. Development of reliable and accurate inversion techniques to recover SWE is one of the most important tasks in current microwave researches. However, a number of snow pack properties, including snow density, particle size, crystal shape, stratification, ground surface roughness and soil moisture, affect the microwave scattering signals and need to be properly modeled and exploited. In this paper, we developed a multi-layer, multi-scattering model for dry snow based on recent theoretical advances in snow and surface modeling. In the proposed multi-layer model, Matrix Doubling method is used to account for scattering from each snow layer; and Advanced Integral Equation Model (AIEM) is incorporated into the model to describe surface scattering. Comparisons were made between the model predictions and field observations from NASA Cold Land Processes Field Experiment (CLPX) during Third Intensive Observation Period (IOP3) and SARALPS-2007 field experiment supported by ESA. The results indicated that model predictions were in good agreement with field observations. With the confirmed confidence, the analyses on multiple scattering, scatterer shape, and snow stratification effects were further made based on the model simulations. Furthermore, a parameterized snow backscattering model with a simple form and high computational efficiency was developed using a database generated by the multiple-scattering model. For a wide range of snow and soil properties, this parameterized model agrees well with the multiple-scattering model, with the root mean square error 0.20 dB, 0.24 dB and 0.43 dB for VV, HH and VH polarizations, respectively. This simplified model can be useful for the development of SWE retrieval algorithm and for fast simulations of radar signals over snow cover in land data assimilation systems.     

Relationships between radar backscatter and accumulation rates on the Greenland ice sheet

C-band radar backscatter generally seems to be inversely related to accumulation rate across Greenland's dry snow zone. A coupled snow metamorphosis-backscatter model was constructed in order to investigate the observation. Accumulation rate and mean annual temperature were the coupled-model inputs. In a local area where accumulation rate is known very accurately the model predicts the backscatter level along several transects within the calibration uncertainties of the ERS-1 SAR data. The analysis was applied to the remainder of the dry snow zone. The model showed that both accumulation rate and temperature can modulate surface backscatter across Greenland's dry snow zone with accumulation rate being the primary influence.     

Using MODIS snow cover maps in modeling snowmelt runoff process in the eastern part of Turkey

Water perhaps is the most valuable natural asset in the Middle East as it was a historical key for settlement and survival in Mesopotamia, “the land between two rivers”. At present, the Euphrates and Tigris are the two largest trans-boundary rivers in Western Asia where Turkey, Syria, Iran, Iraq and Saudi Arabia are the riparian countries. The Euphrates and Tigris basins are largely fed from snow precipitation whereby nearly two-thirds occur in winter and may remain in the form of snow for half of the year. The concentration of discharge mainly from snowmelt during spring and early summer months causes not only extensive flooding, inundating large areas, but also the loss of much needed water required for irrigation and power generation purposes during the summer season. Accordingly, modeling of snow-covered area in the mountainous regions of Eastern Turkey, as being one of the major headwaters of Euphrates-Tigris basin, has significant importance in order to forecast snowmelt discharge especially for energy production, flood control, irrigation and reservoir operation optimization.A pilot basin, located on the upper Euphrates River, is selected where five automated meteorological and snow stations are installed for real time operations. The daily snow cover maps obtained from Moderate Resolution Imaging Spectroradiometer MODIS at 500 m resolution are compared with ground information for the winter of 2002-2003 both during accumulation and ablation and at accumulation stage for the winter of 2003-2004. The snow presence on the ground is determined from the snow courses performed. Such measurements were made at 19 points in and around the upper Euphrates River in Turkey and at 20 points in the upper portion of the pilot basin for the winters of 2002-2003 and 2003-2004, respectively. Comparison of MODIS snow maps with in situ measurements over the snow season show good agreement with overall accuracies ranging between 62% and 82% considering the shift in the days of comparison. The main reasons to have disagreement between MODIS and in situ data are the high cloud cover frequency in the area and the current version of the MODIS cloud-mask that appears to frequently map edges of snow-covered areas and land surfaces. The effect of elevation and land cover types on validation of MODIS snow cover maps is also analyzed. In order to minimize the cloud cover and maximize the snow cover, MODIS-8 daily snow cover products are used in deriving the snow depletion curve, which is one of the input parameters of the snowmelt runoff model (SRM). The initial results of modeling process show that MODIS snow-covered area product can be used for simulation and also for forecasting of snowmelt runoff in basins of Turkey.     

Investigation of C- and X-band backscattering signatures of Baltic Sea ice

Backscattering signatures of various Baltic Sea ice types and open water leads were measured with the helicopter-borne C- and X-band Helsinki University of Technology scatterometer (HUTSCAT) during six ice research campaigns in 1992–1997. The measurements were conducted at incidence angles of 23° and 45°. The HUTSCAT data were assigned by video imagery into various surface type categories. The ground data provided further classification of the HUTSCAT data into different snow wetness categories (dry, moist and wet snow). Various basic statistical parameters of backscattering signature data were used to study discrimination of open water leads and various ice types. The effect of various physical parameters (e.g. polarization, frequency, snow condition) on the surface type discrimination was investigated. The results from the data analysis can be used to help the development of sea ice classification algorithms for space-borne SAR data (e.g. Radarsat and Envisat). According to the results from the maximum likelihood classification it is not possible to reliably distinguish various surface types in the SAR images only by their backscatter intensity. In general, the best ice type discrimination accuracy is achieved with C-band VH-polarization σ° at an incidence angle of 45°.     

Water resources

Abstract

Water, which was taken for granted and assumed to exist in abundance, has become one of our most precious natural resources. Hence hydrological and water management problems represent key areas of applied science in the current worldwide economic and ecological situations. Satellite remote sensing technologies have been used more and more to satisfy urgent requirements for information relating to water resources. Many efforts have been undertaken in Europe to develop appropriate methods and to advance these systems to an operational level. Main emphasis is given to processes in which land surfaces and inland waters are involved, dealing with the estimation of precipitation, the assessment of soil moisture, surface runoff and the forecasting of runoff, be it from rainfall or from snowmelt. Remote sensing methods are of fundamental importance for soil moisture monitoring on a global and on a regional scale. Experimental research has demonstrated that if, in principle, observations in the visible and thermal infrared regions of the electromagnetic spectrum are able to detect the surface moisture of bare soils, microwave sensors are the most promising tools for quantitative estimates of this parameter. A valuable amount of research on this subject has been performed in Europe and is reviewed within the framework of the most advanced international research. Within Europe regional or local aspects and problems dominate. Therefore the efforts and achievements concerned with these areas and applications, and the progress towards operational use, will be focused on primarily. The most advanced and most promising operational applications exist in the field of snow hydrology to improve power production.     

Satellite and ground-based microclimate and hydrologic analyses coupled with a regional urban growth model

Urban development is shown to induce predictable changes in satellite-based measures of radiant surface temperature and evapotranspiration fraction—as long as certain features of the development are known. Specifically, the vegetation changes that accompany the development and the initial climatic state of the land parcel must be noted. Techniques are also developed for quantifying the effects of urbanization on the surface hydrology at a watershed scale. Streamflow and precipitation data are related graphically in order to determine a watershed's general ratio of stormwater runoff to rainfall, along with any changes in the ratio over time. Four distinct runoff responses, separated by season and antecedent moisture conditions, are distinguishable for a particular basin, with the response during the non-summer months under typical antecedent moisture conditions the most representative of and responsive to land-use patterns. This particular runoff response can be estimated from satellite-derived land cover patterns and certain physical attributes of a basin. These satellite-based microclimate and hydrologic analyses are coupled with an existing urban growth model (SLEUTH). The SLEUTH urban growth model simulates future development scenarios for a region of interest. The resulting changes in urban land use lead to the evolution of site-specific climate and hydrology based on the scheme that is presented in this paper. This study, as well as related tools and bodies of knowledge, can be used to broaden the scientific basis behind land-use management decisions.     

多源遥感驱动的SRM模型在缺资料地区的研究及应用

以MODIS雪盖、风云静止卫星降水、GLDAS气温等多源数据,作为传统SRM模型的输入参数,构建多源遥感驱动的SRM融雪径流模型,并在缺资料地区——青藏高原的年楚河流域进行融雪过程的径流模拟。研究表明融雪后期的瞬时降雪很大程度上影响了插值后积雪覆盖率的精度,在插值的时候考虑降水和气温,排除瞬时积雪干扰,改进线性插值获得每天的积雪覆盖率,可以提高模型模拟精度;遥感驱动的SRM模型在缺资料地区年楚河适用性较好,Nash-Sutcliffe系数(NSE)达到0.681,体积差(Dv)为-0.17%,均方根误差(RMSE)为9.678,模型模拟的精度较高。研究结果可为高寒地区生态水文模型研究提供重要参考,同时可为SRM模型在其他流域尤其是缺资料地区融雪径流计算中的应用提供有效支撑。     

Effects of seasonal hydrologic patterns in south Florida wetlands on radar backscatter measured from ERS-2 SAR imagery

A multi-year study was carried out to evaluate ERS synthetic aperture radar (SAR) imagery for monitoring surface hydrologic conditions in wetlands of southern Florida. Surface conditions (water level, aboveground biomass, soil moisture) were measured in 13 study sites (representing three major wetland types) over a 25-month period. ERS SAR imagery was collected over these sites on 22 different occasions and correlated with the surface observations. The results show wide variation in ERS backscatter in individual sites when they were flooded and non-flooded. The range (minimum vs. maximum) in SAR backscatter for the sites when they were flooded was between 2.3 and 8.9 dB, and between 5.0 and 9.0 dB when they were not flooded. Variations in backscatter in the non-flooded sites were consistent with theoretical scattering models for the most part. Backscatter was positively correlated to field measurements of soil moisture. The MIchigan MIcrowave Canopy Scattering (MIMICS) model predicts that backscatter should decrease sharply when a site becomes inundated, but the data show that this drop is only 1-2 dB. This decrease was observed in both non-wooded and wooded sites. The drop in backscatter as water depth increases predicted by MIMICS was observed in the non-wooded wetland sites, and a similar decrease was observed in wooded wetlands as well. Finally, the sensitivity of backscatter and attenuation to variations in aboveground biomass predicted by MIMICS was not observed in the data.The results show that the inter- and intra-annual variations in ERS SAR image intensity in the study region are the result of changes in soil moisture and degree of inundation in the sites. The correlation between changes in SAR backscatter and water depth indicates the potential for using spaceborne SAR systems, such as the ERS for monitoring variations in flooding in south Florida wetlands.     

The relationship between tibetan snow depth,ENSO, river discharge and the monsoons of Bangladesh

Using satellite estimates of snow depth, we examine the interannual variability of the monsoon rains of Bangladesh, an area greatly affected by land surface hydrological processes including Himalayan snowpack size, snowmelt river flooding, and Bay of Bengal storm surge. For the twentieth century, we found Bangladesh monsoon rainfall (BMR) to be uncorrelated with the All‐Indian Monsoon Index. This result is consistent with previous findings for shorter time records. We next used a short 9‐year record of satellite estimates of April snow depth for the Himalayan region and concurrent seasonal El Niño–Southern Oscillation (ENSO) conditions in the equatorial Pacific to develop an empirical model that explains a high percentage of BMR interannual variability. Inclusion of late spring river discharge levels further improves the empirical model representation of BMR for June–September. These results, though with a limited length satellite record, suggest that BMR interannual variability is constrained by concurrent ENSO conditions, spring Himalayan snowpack size and land surface flooding. The same results could not be obtained from analyses using satellite estimates of snow cover. These findings stress the need for development of a quality longer record of satellite estimated snow depth. The twentieth‐century analysis also indicates that BMR should be considered independently of Indian monsoon rainfall.     

基于MODIS数据的玛纳斯河山区雪盖时空分布分析

基于2000~2010年的MODIS/Terra积雪8 d合成数据(MOD10A2)与DEM数据,通过计算和分析积雪频率与积雪覆盖率,研究了新疆玛纳斯河山区雪盖的时空分布特征。结果表明：① 研究区一月份积雪覆盖丰富,积雪频率高值区主要分布在北部中低山地区、南部中海拔地区以及清水河与塔西河的河源地区;四月与十月的雪盖分布规律相似,总体上积雪频率随高程上升而上升;七月份只有少部分高山区域被积雪覆盖;② 积雪频率始终保持较高水平的区域是玛纳斯河、金沟河、清水河以及塔西河的河源高山地区,而玛纳斯河流域中上游的河谷地区则始终保持较低水平;③ 一月份,1 400 m以下地区的积雪覆盖率超过95%,随着高程上升,迅速下降至2 600 m的最低值约41%,此后逐渐上升至5 000 m以上80%左右;④ 一月、四月和十月份积雪覆盖率在大部分高程带上均表现为北坡、东北坡和西北坡最高,东坡和西坡次之,南坡、东南坡和西南坡最低的规律;七月份各高程带的雪盖分布没有明显的坡向差异。     

Analysis of ERS-1 Synthetic Aperture Radar data from Nordaustlandet,Svalbard

Study of the Earth's terrestrial ice masses (glaciers, ice caps and ice sheets), especially the seasonal variation of different surface conditions such as dry snow, wet snow and bare ice, is of particular importance in relation to possible climatic change. Synoptic monitoring techniques using visible and near-infrared satellite imagery are severely limited by the prevalence of cloud cover in the polar regions, and winter observations are impossible as a result of the absence of solar radiation. Consequently, considerable attention is now being focused on the use of imaging radar in the study of large ice masses. In this paper, we present and interpret a time-series of C-band synthetic aperture radar images acquired using the ERS-1 satellite from the Austfonna ice cap in eastern Svalbard. Winter imagery shows little variability, most of the ice cap having a uniform and high (approximately – 3dB) backscatter attributed to ice lenses or to a large effective grain size. Summer imagery shows considerable topographically-related detail, and backscatter values typically 5 to l0 dB less than in winter, which can be explained on the basis of surface scattering from wet snow. However, the marginal areas of the ice cap show a clearly defined zone of high ( –5dB) backscatter in mid- to late-August. It is proposed that this corresponds to the bare ice zone, the high backscatter values being due to scattering from crevasses and meltwater channels, and that the inner boundary of the zone of enhanced backscatter indicates the position of the transient snow line.