## 1 Introduction
The research is based on the results of forest resource surveys used to study the impact of single or several common tree species, such as selective cutting, thinning intensity, tending, and other forest management measures, on the forest stock and carbon cycle per unit area [18; 19]. Alternatively, carbon storage can be calculated based on the biomass carbon storage conversion coefficient [20; 21]. This type of research typically measures and describes the changes before and after the implementation of single or several tree species forest management measures in a small area, ignoring the influence of other interference factors, and it cannot accurately determine whether changes in carbon sinks are related to forest management measures. Furthermore, there is no correspondence between forest management measures and capital investment, and it is impossible to measure the efficiency of FMI and the difference in the investment efficiency of various forest regions. In addition, studies have carried out annual calculations on the stock of different tree species to assess the quality of a forest stand. For example, the enhanced vegetation index (EVI) was used to establish a biomass allometric growth and inversion model to measure the changes in the biomass of the different forest species to estimate the changes in the carbon sink amount [22; 23]. Furthermore, in the measurement of large-scale forest carbon sinks, some studies have comprehensively measured forest biomass based on land use types and combined with MODIS optical data. The study counted long-term carbon sequestration changes to quantify the carbon sequestration effects of forest management measures such as reforestation, thinning, and forest tending, as well as to assess the related socioeconomic carbon sequestration costs [24; 25; 26]. Such studies have comprehensively considered the changes in forest quality and quantity, and there are few research results. Moreover, most studies on the macro scale use the InVEST model to correspond to the forest species coefficient, and use the change of land type to calculate the total amount of carbon sequestration in forest land [27; 28]. However, the accuracy of this calculation method is low. Although the carbon density coefficient is considered, it is mostly empirical and averaged. It can only be used to measure changes in carbon sinks based on changes in land types. In addition, this method does not consider forest quality and its differences, so it is difficult to accurately measure the increase or decrease in the carbon sink amount in the region, and it is impossible to accurately measure the effect of how FMI might increase the carbon sink amount. In view of this, this paper used panel data from 40 key state-owned forest farms in Heilongjiang Province from 2001 to 2019 to explore the increase in carbon sink efficiency and the differences in FMI in various forest regions under the control of natural meteorological factors. First, this paper combined geographic information system (GIS) and remote sensing (RS) technology based on RS data on vegetation net primary productivity (NPP) and accurately calculated the evolution of carbon sinks in the study area from 2001 to 2019. Second, we controlled for other variables that might affect the increase in forest carbon sinks, introduced the lag term of carbon sinks, used the individual fixed-effect model to determine the FMI lag period, and realized the measurement of the causal relationship between the two. Then, we used the panel threshold model to examine the differences in FMI efficiency under different carbon sink levels and identify forest regions with a higher investment efficiency. Finally, we verified that FMI followed the law of diminishing marginal returns in terms of increasing the carbon sink, and discussed how to rationally plan investment levels.